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abstract: 'Preheating and other particle production phenomena in the early Universe can give rise to high-energy out-of-equilibrium fermions with an anisotropic stress. We develop a formalism to calculate the spectrum of gravitational waves due to fermions, and apply it to a variety of scenarios after inflation. We pay particular attention to regularization issues. We show that fermion production sources a stochastic background of gravitational waves with a significant amplitude, but we find that typical frequencies of this new background are not within the presently accessible direct detection range. However, small-coupling scenarios might still produce a signal observable by planned detectors, and thus open a new window into the physics of the very early Universe.'
author:
- 'Kari Enqvist, Daniel G. Figueroa and Tuukka Meriniemi'
date: 'March 22, 2012'
title: Stochastic Background of Gravitational Waves from Fermions
---
[**Introduction.**]{} The existence of gravitational waves (GW) is arguably one of the most important predictions of the general theory of relativity that still remains unverified. The measured decay of the orbital period of compact binaries [@HulseTaylor] has nevertheless provided a strong, though indirect, evidence for GW. On theoretical grounds, we expect that the universe today should be permeated by a variety of GW backgrounds of diverse origin. For instance, GW are expected from astrophysical sources, like the collapse of supernovas or the coalescence of compact binaries. They are also expected from high-energy phenomena in the early Universe, see [@Maggiore] for a review. Many plans for direct detection experiments exist, such as the VIRGO interferometer, the Laser Interferometer Gravitational Wave Observatory (LIGO), the European Laser Interferometer Space Antenna (eLISA), the Big Bang Observer (BBO), or the Decihertz Interferometer Gravitational Wave Observatory (DECIGO); all these observatories operate at some typical frequencies ranging from $10^{-4}$ Hz to $10^4$ Hz, and it is highly expected in the community that GW should be detected within this decade.
A number of constraints on GW have been derived from a variety of considerations related to things such as millisecond pulsars [@pulsar] and Big Bang nucleosynthesis [@BBN]. Indeed, there are hopes that cosmology may in near future offer more insight on the existence and nature of GW. From the observations of the Cosmic Microwave Background temperature and polarization [@WMAP7] anisotropies, one can already infer upper bounds on the amplitudes of GW (see also [@Elena]).
During inflation metric perturbations, including tensor modes, i.e. GW, are generated. If the scale of inflation is sufficiently high, the GW background from inflation could be detected directly [@Kamionkowski] with satellite missions such as the Planck Surveyor or the proposed CMBpol satellite. Moreover, post-inflationary dynamics can also be a source of stochastic backgrounds of GW, generated by causal mechanisms very different from the quantum nature of the inflationary GW. After inflation the energy stored in the scalar inflaton field responsible for the superluminal expansion is converted into (almost) all the matter and radiation of the universe. This stage is called reheating, by the end of which the inflaton decay products have thermalized among themselves and the standard hot Big Bang evolution can commence. The process of reheating is not well understood. However, it is often assumed to be initially driven by non-perturbative effects, consisting in violent bursts of particle production known as preheating [@preheating].
In this letter we will focus on the generation of GW right after inflation during preheating or other stages of non-perturbative particle production. Most previous studies of post-inflationary phenomena generating GW have focused only on bosonic sources [@TkachevGW; @Dani; @PhTs], usually scalar fields. Here we want to complete the picture by considering fermionic fields as the source of GW. The phenomena in which fermions are created after inflation correspond to out-of-equilibrium periods in the evolution of the universe. Thus the created fermions have typically a non-thermal spectra and, as a consequence, they have a non-trivial anisotropic stress which will source GW. We will consider the possibility that the inflaton couples to fermions, and discuss the production of GW during fermionic preheating [@Greene; @Baacke]. Our considerations also apply to situations where fermions are produced by a scalar field other than the inflaton. The fermions we observe in Nature must have been generated sometime between the end of inflation and Big Bang nucleosynthesis. Fermionic preheating, or production of fermions in general, is thus no less natural than the usual bosonic production, but it is technically much more difficult to treat.
Since sources of GW produce spectra with distinctive shapes and amplitudes, it is important to characterize all the potential sources. The question then is: how big is the amplitude, and what is the frequency, of the stochastic GW produced by fermions in the early universe? The aim of the present letter is to answer these questions. We first develop a general formalism for computing the GW spectrum generated by an ensemble of fermions, and then apply it to two distinct scenarios: 1) fermions generated in preheating, and 2) thermal Universe into which the fermions are injected. From now on we will work in units $\hbar = c = 1$, with $M_p \approx 2.4\times10^{18}\,{\rm GeV}$ the reduced Planck mass. Summation will be assumed over repeated indices.\
[**Gravitational waves from fermions: the formalism.**]{} Let us consider a flat Friedman-Robertson-Walker background with $a(t)$ the scale factor and $t$ conformal time. GW are the transverse-traceless (TT) part of metric perturbations $h_{ij}$, $$ds^2 = a^2(t)(-dt^2+(\delta_{ij}+h_{ij})dx^idx^j),$$ subject to $\partial_i h_{ij} = h_{ii} = 0$. The spectrum of energy density of a stochastic GW background in comoving momentum $k$ is given as $$\begin{aligned}
\label{eq:GWspectrum}
\frac{d\rho_{_{\rm GW}}}{d\log k}(k,t) &=& \frac{M_p^2k^3|{\dot h}_{k}(t)|^2}{8\pi^2a^2(t)}\,,$$ where $|{\dot h}_k(t)|^2$ is the power spectrum of $\dot h_{ij} = \frac{dh_{ij}}{dt}$. For initial conditions $h_{ij}(t_i) = \dot h_{ij}(t_i) = 0$, the GW spectrum at sub-horizon scales becomes [@Maggiore] $$\begin{aligned}
\label{eq:GWfromUETC}
\frac{d\rho_{GW}}{d\log k}(k,t) &=& \frac{k^3}{4\pi^2M_p^2}{1\over a^4(t)}\int_{t_i}^t dt' \int _{t_i}^t dt''a^3(t') a^3(t'') {\nonumber}\\
&& \times\,\, \cos(k(t'-t''))\,\Pi^2(k,t',t''),\end{aligned}$$ where ${\Pi}^2$ is the unequal time correlator (UTC) of the TT-part of the anisotropic-stress $\Pi_{ij}^{{{\rm TT}}}$, $$\langle {\Pi}_{ij}^{{\rm TT}}\hspace*{-0.5mm}({{\bf k}},t)\,{{\Pi}_{ij}^{{{\rm TT}}}}^{\hspace*{-0.2mm}*}\hspace*{-1mm}({{\bf k}}',t')\rangle = (2\pi)^3{\Pi}^2(k,t,t')\delta_D({{\bf k}}-{{\bf k}}')$$
Let us now consider spin-${1\over2}$ fermions as the source of GW, which can be represented as $$\psi({{\bf x}},t) = \hspace*{-1mm}\int\hspace*{-2mm}\frac{d{{\bf k}}}{(2\pi)^3}e^{-i{{\bf k}}{{\bf x}}}\left\lbrace a_{{{\bf k}},r}{\tt u}_{{{\bf k}},r}(t) + b^\dag_{-{{\bf k}},r}{\tt v}_{{{\bf k}},r}(t) \right\rbrace,\vspace*{-0.5cm}$$ $${\tt u}_{{{\bf k}},r} = (u_{{{\bf k}},+}S_r~~u_{{{\bf k}},-}S_r)^{\rm T}, {\tt v}_{{{\bf k}},r} = (v_{{{\bf k}},+}S_{-r}~~v_{{{\bf k}},-}S_{-r})^{\rm T},{\nonumber}$$ with $r = 1,2$, $S_{1} = -S_{-2} = (1~~0)^{\rm T}$, $S_{2} = S_{-1} = (0~~1)^{\rm T}$, and $a_{r}, b_{r}$ the usual creation/annihilation operators obeying the relations ${\left\lbrace}a_{r}({{\bf k}}), a_{s}^\dag({{\bf q}}){\right\rbrace}$ = ${\left\lbrace}b_{r}({{\bf k}}), b_{s}^\dag({{\bf q}}){\right\rbrace}$ = $ (2\pi)^3\delta_{rs}\delta_D({{\bf k}}-{{\bf q}})$, ${\left\lbrace}a_{r}({{\bf k}}), b_{s}^\dag({{\bf q}}){\right\rbrace}$ = 0. The fermion energy-momentum tensor is given by [@BD] $$\begin{aligned}
T_{ij}({{\bf x}},t) = {1\over2a(t)}\left(\bar\psi\gamma_{(i}\overrightarrow{D}_{j)}\psi - \bar\psi\overleftarrow{D}_{(i}\gamma_{j)}\psi \right),\end{aligned}$$ with $D_i \equiv \partial_i + {1\over4}[\gamma_\alpha,\gamma_\beta]\omega^{{{\alpha\beta}}}_j$ the covariant derivative, $\gamma_i$ the standard (flat-space) Dirac matrices, and $\omega^{{{\alpha\beta}}}$ the spin connection. The TT part of the anisotropic-stress in Fourier space, can be simply obtained by means of the orthogonal projector ${\mathcal P}_{ij} \equiv \delta_{ij} - \hat k_i \hat k_j$, as $\Pi_{ij}^{{\rm TT}}({{\bf k}},t) = {\mathcal P}_{il}T_{lm}{\mathcal P}_{mj} - {1\over2}{\mathcal P}_{ij}{\mathcal P}_{lm}T_{lm}$. We then have all the ingredients to calculate $\Pi^2(k,t,t')$ and obtain the GW spectrum. Using the fact that $v_{{{\bf p}},\pm} = \pm u_{{{\bf p}},\mp}^*$, we find $$\begin{aligned}
\label{eq:GWspectrumII}
\frac{d\rho_{{_{\rm GW}}}}{d\log k}(k,t) &=& {(k^3/M_p^2)\over 8\pi^4a^4(t)}\int \hspace*{-1mm}dp\,d\theta\,p^4{\rm sin}^3\theta\,F(k,p,\theta),{\nonumber}\\
F(k,p,\theta) &=& |I_{+}^{(c)}-I_{-}^{(c)}|^2 + |I_{+}^{(s)}-I_{-}^{(s)}|^2\\
I_{\pm}^{(c)}(k,p,\theta) &\equiv& \int\hspace*{-1mm}{dt'\over a(t')}\,{\rm cos}(kt')\, u_{{{\bf k}}-{{\bf p}},\pm}(t')u_{{{\bf p}},\pm}(t'),{\nonumber}\end{aligned}$$ with $I^{(s)}_{\pm}$ defined analogously by replacing $\cos(kt)$ by $\sin(kt)$. Eq. (\[eq:GWspectrumII\]) is the master set of formulae that describe the GW spectrum at subhorizon scales, as generated by some fermionic field $\psi$ with eigenfunctions $u_{{{\bf k}},\pm}(t)$. The $eom$ for $u_{{{\bf k}},\pm}(t)$ will follow from the Dirac equation. For any process in the early Universe where fermions are excited, one just needs to plug in the solutions $u_{{{\bf k}},\pm}(t)$ into the master equation (\[eq:GWspectrumII\]) to find the spectrum of GW.
The structure of the formulae in Eq. (\[eq:GWspectrumII\]) resembles that of scalar fields sourcing GW. However, in the bosonic case, apart from multiplicative factors, there appears a power $p^6$ instead of $p^4$ in the integrand, there are no polarization indices $+,-$, and of course the fermionic mode functions $u_{{{\bf k}},\pm}(t)$ are replaced by the Klein-Gordon scalar modes $\phi_{{{\bf k}}}(t)$.
Both bosonic and fermionic vacuum expectation values (VEVs), like $\Pi^2$, require regularization. In the case of bosons, this has not been an issue in the literature, since the bosonic UTC are either introduced as a theoretical regularized Ansatz, or in the case of lattice simulations, the ultraviolet (UV) modes causing the divergence are simply not captured. In the fermionic case one cannot skip regularization. To regularize $\Pi^2$, note first that the VEV of the source itself, $\Pi_{ij}^{{\rm TT}}$, needs also to be regularized. Similarly to the flat-space case, regularization of the source’s VEV amounts to a substraction of the zero-point fluctuations or, equivalently, to a time-dependent normal-ordering ($tNO$) procedure, i.e. $\langle \Pi_{ij}^{TT} \rangle_{\rm reg} \equiv \langle 0| \Pi_{ij}^{TT} | 0 \rangle - \langle 0_t| \Pi_{ij}^{TT} | 0_t \rangle$, with $|0\rangle$ the initial vacuum and $|0_t\rangle$ the vacuum at time $t$. This removes the unphysical divergence in the VEV of $\Pi_{ij}^{TT}$ at every time $t$. In practice, $tNO$ amounts to the replacement $(u_{{{\bf p}},\pm}u_{{{\bf p}}',\pm})\to$ $(u_{{{\bf p}},\pm}u_{{{\bf p}}',\pm})_{_{\rm reg}} = |\beta_{{\bf p}}||\beta_{{{\bf p}}'}|u_{{{\bf p}},\pm}u_{{{\bf p}}',\pm}-(\beta_{{\bf p}}\beta_{{{\bf p}}'}u_{{{\bf p}},\mp}u_{{{\bf p}}',\mp})^*$ with $|\beta_{{\bf p}}| = \sqrt{1-|\alpha_{{\bf p}}|^2}$, where $\alpha$, $\beta$ are the canonical Bogoliubov coefficients [@BD] connecting the initial creation and annhilation operators with those at time $t$. To render $\Pi^2$ finite we just need to replace in Eq. (\[eq:GWspectrumII\]) the functions $I^{(c)}_{\pm}$ by $$\begin{aligned}
{\mathcal I}^{(c)}_{\pm} \equiv \int\hspace*{-1mm}{dt'\over a(t')}\,{\rm cos}(kt')\, \left(u_{{{\bf k}}-{{\bf p}},\pm}(t')u_{{{\bf p}},\pm}^*(t')\right)_{\rm reg},\end{aligned}$$ and similarly for ${\mathcal I}^{(s)}_{\pm}$. In this way the convergence of the integration over $p$ is ensured by the suppression of the UV divergent modes.\
[**Stochastic background of gravitational waves from fermions produced in the early Universe.**]{} Several scenarios of the early Universe may create high-energy out-of-equilibrium fermions by non-perturbative effects, for instance a homogeneous scalar field oscillating around the minimum of a potential. In such situation, if fermions are coupled to that field, a non-perturbative population of their modes (respecting Pauli-blocking) takes place [@Greene; @Baacke]. This generates a non-trivial anisotropic-stress that sources a stochastic background of GW. In this letter we consider two different scenarios:
I\) [*(p)reheating after inflation*]{}: Let us assume an inflaton oscillating around the minimum of its potential after the end of inflation. This is the case of chaotic inflation models with polynomial potentials like $V \propto \phi^2$ or $V \propto \phi^4$. In general, the shape of the inflaton potential during inflation is however irrelevant for our purposes. We will just assume that inflation took place at some energy scale $E_I$, and that afterwards the inflaton oscillates with a certain frequency and decreasing amplitude. The expansion of the Universe will be dictated during (p)Reheating by the inflaton energy density. For a polynomial potential $V(\phi)$ it is well known that the scale factor behaves as a power law in time. In the simplest case $V(\phi) = {1\over2}\omega_0^2\phi^2$, $a(t) \propto t^2$, so the expansion of the Universe is matter-dominated (MD).
II\) [*Oscillating scalar field in a thermal era*]{}: It is also possible to consider a scalar field, other than the inflaton, which oscillates coherently around a potential minimum when the Universe has already entered into a thermal era. That is the case, for example, of the curvaton scenario [@curvatons], which is an alternative to single-field inflationary models. After the Universe has reheated, the curvaton oscillates with decreasing amplitude and fixed frequency. The expansion of the Universe is radiation-dominated (RD), driven by the thermal relativistic bath of particles with energy density $\rho_{\rm th} \propto 1/a^4$, and $a(t) \propto t$. We assume oscillations begin at some energy scale $E_I$ way above the electroweak scale.
In both scenarios the oscillatory field $\phi$ behaves as a damped oscillator, $\phi(t)=\Phi(t)F(t)$, with decreasing amplitude $\Phi(t)$ and periodic behaviour $F(t+2\pi/\omega_0) = F(t) \leq 1$, as determined by the choosen potential. We will assume that $\phi$ is coupled to some fermion species $\psi$ via a Yukawa interaction $h\phi\bar\psi\psi$, with $h$ the interaction strength. The Dirac equation yields the equation for the mode functions $u_{{{\bf k}},\pm}$ as $$\begin{aligned}
\label{diracmodes}
\ddot{u}_{{{\bf k}},\pm} + \left({{\bf k}}^2+h^2a^2\phi^2 \pm ih(\dot a\phi + a\dot\phi)\right)u_{{{\bf k}},\pm} = 0~.\end{aligned}$$ We have solved numerically the mode equations (\[diracmodes\]) for the scenarios I and II, scanning for the parameters $h$, $\omega_0$ and $E_I$ and assuming $V(\phi) = {1\over2}\omega_0^2\phi^2$ and initial conditions corresponding to a vanishing particle density. In both scenarios, each time $\phi$ passes through the minimum of the potential, fermions are created out-of-equilibrium by non-perturbative parametric effects [@preheating].
Pauli blocking prevents the fermion occupation numbers to grow arbitrarily. Fermion excitations are forced to fill up a “Fermi-sphere” in momentum space of comoving radius $k_F \sim a(t)q^{1/4}\omega_0$, with $q \equiv h^2(\Phi/\omega_0)^2$ the resonant parameter [@Greene]. The occupation numbers with momenta smaller than $k_F$ oscillates continuously between 0 and 1, whilst the radius $k_F$ grows as $\propto a^{1/4}$. When the energy density of the created fermions has grown to a significant fraction of the energy of the oscillating field $\phi$ – usually within tens of oscillations –, the creation of fermions finally ceases. The excitations of the fermionic modes within the Fermi-sphere source the generation of GW. The non-excited modes, i.e. those outside the sphere, are on the contrary responsible for the UV divergence discussed before. The regularization scheme we impose filters the infrared (IR) modes within the sphere $k \lesssim k_F$ and removes the contribution from the UV ones. It is useful to define the times $t_I$, $t_*$ and $t_{\rm RD}$, as the initial time, the end of GW production, and the first moment when the Universe becomes RD, respectively. In the scenario I, $t_I < t_{*} < t_{\rm RD}$, so in this case the effective equation of state $p/\rho = w$ between $t_I$ and $t_{\rm RD}$ is typically different from that of RD, $w_{\rm RD} = 1/3$ (unless $V(\phi) \propto \phi^4$). It is then convenient to introduce the factor $\epsilon \equiv (a_*/a_{\rm RD})^{(1-3w)}$, see next. In the scenario II, $t_{\rm RD} \leq t_I < t_*$, so that $\epsilon = 1$. Below the Planck scale, GW decouple immediately after production, so we can evaluate the GW energy density spectrum today from the spectrum computed at the time of production. We just have to redshift the amplitude and wavenumbers, $$\begin{aligned}
\label{eq:freqAndAmplitude}
f &=& \epsilon^{1/4}\left({a_I\over a_*}\right)\left(\omega_0\over \rho_*^{1/4}\right)\left({k\over\omega_0}\right)\times5\cdot10^{10}~{\rm Hz}{\nonumber}\\
h^2\Omega_{{_{\rm GW}}}&\equiv& {h^2\over\rho_c}\hspace*{-1mm}\left({d\rho_{{_{\rm GW}}}\over d\log k}\right)_{\hspace*{-1mm}0} = h^2\Omega_{\rm rad}\left(g_0\over g_*\right)^{\hspace*{-1mm}{1\over3}}{\epsilon\over \rho_*}\hspace*{-1mm}\left({d\rho_{{_{\rm GW}}}\over d\log k}\right)_{\hspace*{-1mm}*},{\nonumber}\end{aligned}$$ with $h^2\Omega_{\rm rad} \approx 4\times10^{-5}$, $\rho_* = E_I^4(a_I/a_*)^{3(1+\omega)}$ the energy density at $t_*$, and $(g_0/g_*)^{1/3} \sim \mathcal{O}(0.1)$ the ratio of the number of relativistic $dof$ today to those active at $t_*$.
In Fig. \[fig1\] we show two examples of GW energy density spectra, obtained with the machinery presented in the previous section. The spectra grows in the IR region, reaches a maximum at $k \sim q^{1/4}\omega_0$, and finally falls down in the UV region. The spectra are peaked, as expected on dimensional grounds, around the scale $\sim k_F$, characteristic of the fermions’ dynamics. From the $k\rightarrow0$ limit in Eq. (\[eq:GWspectrumII\]), we would expect the spectra to grow as $\propto k^3$ in the very IR region. Instead we observe a behaviour $\sim k^2$, probably signaling that we are exploring $k$’s too close to the maximum, and that one should look for much smaller $k$’s to probe the $k^3$ behaviour. Unfortunately, due to computer limitations, both the limits $k \ll k_F$ and $k \gg k_F$ are indeed very challenging to probe, so we leave a complete characterization of the IR and UV tails of the GW spectra for a future publication.
![Spectra of GW at the time of production, in the scenario I with $E_I \sim 10^{16}$ GeV and $\omega_0 \sim 10^{14}$ GeV, for $q = 10^2$ (dashed line, peaked at $k = 8q^{1/4}\omega_0$) and $q = 10^{6}$ (solid line, peaked at $k = 6q^{1/4}\omega_0$). The spectra in the scenario II for the same parameters look qualitatively the same.[]{data-label="fig1"}](GWspectra.pdf){width="8cm" height="5cm"}
With our formalism we can obtain nevertheless the most important aspects characterizing any GW spectrum: the spectral shape around the peak, and the amplitude and position of such peak. For instance in the scenario I with $E_I \sim 10^{16}$ GeV and $\omega_0 \sim 10^{14}$ GeV, we find for $q = 10^2$, today’s peak frequency and amplitude as $f \sim 10^{9}$ Hz, $h^2\Omega_{GW} \sim 10^{-20}$, and for $q = 10^6$, as $f \sim 10^{10}$ Hz, $h^2\Omega_{GW} \sim 10^{-14}$. In the scenario II, for the same $E_I$ and $\omega_0$, we find $f \sim 10^9$ Hz, $h^2\Omega_{GW} \sim 10^{-18}$ for $q = 10^2$, and $f \sim 10^{10}$, $h^2\Omega_{GW} \sim 10^{-12}$ for $q = 10^6$. Thus the amplitude of the GW background explored here can indeed be very significant. However the typical frequencies are too large as compared to the range probed by GW observatories ($\sim 10^{-4}-10^4$ Hz). From Eq. (\[eq:freqAndAmplitude\]) we learn that if $\omega_0/\rho_I^{1/4} \ll 1$, we could decrease significantly the frequency towards the observable window. However $\Omega_{\rm GW}$ is suppressed by $(\omega_0^4/\rho_I)$, so the amplitude would be far too small, $\Omega_{GW} \ll 10^{-20}$. We hope to return to a systematic exploration of the parameters in a future publication.
There are some indications for scenarios where a GW background at sufficiently low frequencies and high enough amplitudes could be found. For instance, in hybrid inflation, the frequency $\omega_0$ should be replaced by $\sqrt{\lambda}v$, and the initial energy scale $E_I$ by $\sim \lambda^{1/4}v$, where $\lambda$ and $v$ are the self-coupling and VEV of an auxiliary field coupled to the inflaton. The frequencies would then scale as $f \propto \lambda^{1/4}$ [@Dani] so that for sufficiently small $\lambda$, one could obtain a peak frequency within the observable range. This possibility can be explored by using the machinery developed in this Letter.
Summarizing, we have shown that fermions in the early Universe may be very efficient generators of GW. These waves remain decoupled since the moment of their production, and thus the amplitude and shape of their spectrum probes the physics responsible for their generation. The characteristic spectrum is different from other backgrounds of GW, like those arising from binaries coalescing [@Maggiore], which are decreasing with frequency, or those arising from inflation or self-ordering fields [@GWfromSOSF], which are flat. A comparison of the exact shape of the spectra predicted here versus those generated by scalar fields in phase transitions and (p)reheating requires further study. Here we just want to emphasize that, contrary to naive expectations based on the Pauli principle, we found that fermions are capable of producing a stochastic background of GW with a very large amplitude. Although it may be manifest only at very high frequencies much beyond present/planned GW detectors’ sensitivities, there is however some hope that small-coupling models could give rise to a signal in observable frequency ranges.
[*Acknowledgments*]{} We thank R. Durrer, J. García-Bellido and A. Riotto for useful comments. TM is supported by the Magnus Ehrnrooth foundation, while KE and DGF are respectively supported by the Academy of Finland grants 131454 and 218322.
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abstract: |
We calculate the $B_{u,d,s}\to T$ form factors within the framework of the perturbative QCD approach, where $T$ denotes a light tensor meson with $J^P=2^+$. Due to the similarities between the wave functions of a vector and a tensor meson, the factorization formulas of $B\to T$ form factors can be obtained from the $B\to V$ transition through a replacement rule. As a consequence, we find that these two sets of form factors have the same signs and correlated $q^2$-dependence behaviors. At $q^2=0$ point, the $B\to
T$ form factors are smaller than the $B\to V$ ones, in accordance with the experimental data of radiative $B$ decays. In addition, we use our results for the form factors to explore semilteptonic $B\to
Tl\bar \nu_l$ decays and the branching fractions can reach the order $10^{-4}$.
author:
- Wei Wang
title: $B$ to tensor meson form factors in the perturbative QCD approach
---
Introduction {#section:introduction}
============
In the quark model, a meson is composed of one quark pair and the spin-parity quantum numbers $J^{P}$ of a meson state is consequently fixed by this constituent quark pair, for instance $J^P=0^+$ for pseudoscalar mesons. For p-wave tensor mesons with $J^P=2^+$, both orbital angular momentum $L$ and the total spin $S$ of the quark pair are equal to 1. By making use of the flavor SU(3) symmetry, the nine mesons, isovector mesons $a_2(1320)$, isodoulet states $K_2^*(1430)$ and two isosinglet mesons $f_2(1270),f_2'(1525)$, form the first $^3P_2$ nonet [@Amsler:2008zz]. These mesons have been well established in various processes.
$B$ meson decays into tensor mesons are of prime interest in several aspects. The main experimental observables in hadronic $B$ decays, branching ratios and CP asymmetries, are helpful to inspect different theoretical computations. One exploration concerns the isospin symmetry. For instance the $B\to K_2^*(1430)\eta$ channel has already been observed in 2006 with the branching ratio (BR) $(9.1\pm3.0)\times 10^{-6}$ for the charged channel and a similar one $(9.6\pm2.1)\times 10^{-6}$ for the neutral channel [@Aubert:2006fj]. But the $B\to\omega K_2^*(1430)$ mode possesses a large isospin violation: the BR for the neutral mode $(10.1\pm2.3)\times 10^{-6}$ is about one half of that for the charged mode $ (21.5\pm4.3)\times 10^{-6}$ [@Aubert:2009sx]. Moreover, polarizations of the final mesons in $B$ decays can shed light on the helicity structure of the electroweak interactions. In the standard model, they are expected to obey a specific hierarchy when factorization is adopted to handle the decay amplitudes and the heavy quark symmetry is exploited to derive relations among the involved form factors. In particular the longitudinal polarization fraction is expected to be close to unity. Deviations from this rule have already been experimentally detected in several $B$ decays to two light vector mesons, with the implication of something beyond the naive expectation. Towards this direction $B$ meson decays into a tensor meson can play a complementary role. For example the decay mode $B\to\phi K_2^*(1430)$ is mainly dominated by the longitudinal polarization [@:2008zzd; @Aubert:2008zza], in contrast with the $B\to \phi K^*$ where the transverse polarization is comparable with the longitudinal one [@HFAG].
Despite a number of interesting decay modes have been detected on the experimental side, currently there exist few theoretical investigations on $B$ to tensor transitions. Since a tensor meson can not be produced by a local vector or axial-vector current, the $B\to MT$ decay amplitude is reduced in terms of the $B\to T$ transition and the emission of a light meson $M$. The motif of this work is to handle the first sector with the computation of the $B\to
T$ form factors, and in particular we will use the perturbative QCD (PQCD) approach [@Keum:2000ph] which is based on the $k_T$ factorization. If the recoiling meson in the final state moves very fast, a hard gluon is required to kick the soft light quark in $B$ meson into an energetic one and then the process is perturbatively calculable. Keeping quarks’ intrinsic transverse momentum, the PQCD approach is free of endpoint divergence and the Sudakov formalism makes it more self-consistent. As a direct consequence, we can do the form factor calculation and the quantitative annihilation type diagram calculation in this approach. Our results for these form factors in this work will serve as necessary inputs in the future analysis of the semileptonic and nonleptonic $B$ decays into a tensor meson.
This paper is organized as follows. In Sec. II, we collect the input quantities, including the $B$-meson wave function, light-cone distribution amplitudes (LCDAs) of light tensor mesons. In Sec. \[section:formfactor\], we discuss the factorization property of the $B\to T$ form factors in the PQCD approach. Subsequently we present our numerical results and a comparison with other model predictions is also given. Branching ratios, polarizations and angular asymmetries of the semileptonic $B\to Tl\bar\nu_l$ decays are predicted in Sec. \[sec:semileptonic\]. Our summary is given in the last section.
Wave functions {#section:LCDAs}
==============
We will work in the $B$ meson rest frame and employ the light-cone coordinates for momentum variables. In the heavy quark limit the light tensor meson in the final state moves very fast in the large-recoil region, we choose its momentum mainly on the plus direction in the light-cone coordinates. The momentum of $B$ meson and the light meson can be written as $$\begin{aligned}
P_{B}=\frac{m_{B}}{\sqrt{2}}(1,1,0_{\perp})\;,\;
P_2=\frac{m_{B}}{\sqrt{2}}(\eta,\frac{r_2^2}{\eta},0_{\perp})\;,\label{eq:momentum}
\end{aligned}$$ where $r_2\equiv \frac{m_{T}}{m_{B}}$, with $m_{T},m_B$ as the mass of the tensor meson and the $B$ meson, respectively. The approximate relation $\eta\approx1-q^2/m_{B}^2$ holds for the momentum transfer $q=P_{B}-P_2$. The momentum of the light antiquark in $B$ meson and the quark in light mesons are denoted as $k_1$ and $k_2$ respectively $$\begin{aligned}
k_1=(0,\frac{m_{B}}{\sqrt{2}}x_1,\textbf{k}_{1\perp})\;,\;k_2=(\frac{m_{B}}{\sqrt{2}}
x_2\eta,0,\textbf{k}_{2\perp})\;,\label{eq:fmomentum}
\end{aligned}$$ with $x_i$ being the momentum fraction.
The spin-2 polarization tensor, which satisfies $\epsilon_{\mu\nu}
P^{\nu}_2=0$, is symmetric and traceless. It can be constructed via the spin-1 polarization vector $\epsilon$: $$\begin{aligned}
&&\epsilon_{\mu\nu}(\pm2)=
\epsilon_\mu(\pm)\epsilon_\nu(\pm),\;\;\;\;
\epsilon_{\mu\nu}(\pm1)=\frac{1}{\sqrt2}
[\epsilon_{\mu}(\pm)\epsilon_\nu(0)+\epsilon_{\nu}(\pm)\epsilon_\mu(0)],\nonumber\\
&&\epsilon_{\mu\nu}(0)=\frac{1}{\sqrt6}
[\epsilon_{\mu}(+)\epsilon_\nu(-)+\epsilon_{\nu}(+)\epsilon_\mu(-)]
+\sqrt{\frac{2}{3}}\epsilon_{\mu}(0)\epsilon_\nu(0).\end{aligned}$$ In the case of the tensor meson moving on the plus direction of the $z$ axis, the explicit structures of $\epsilon$ in the ordinary coordinate frame are chosen as $$\begin{aligned}
\epsilon_\mu(0)&=&\frac{1}{m_T}(|\vec p_T|,0,0,E_T),\;\;\;
\epsilon_\mu(\pm)=\frac{1}{\sqrt{2}}(0,\mp1,-i,0),\end{aligned}$$ where $E_T$ and $\vec{p}_T$ is the energy and the magnitude of the tensor meson momentum in the $B$ rest frame, respectively. In the following calculation, it is convenient to introduce a new polarization vector $\epsilon_T$ for the involved tensor meson $$\begin{aligned}
&&\epsilon_{T\mu}(h) =\frac{1}{m_B}
\epsilon_{\mu\nu}(h)P_{B}^\nu,\end{aligned}$$ which satisfies $$\begin{aligned}
&& \epsilon_{T\mu}(\pm2)=0,\;\;\;
\epsilon_{T\mu}(\pm1)=\frac{1}{m_B}\frac{1}{\sqrt2}\epsilon(0)\cdot
P_{B}\epsilon_\mu(\pm),\;\;\;
\epsilon_{T\mu}(0)=\frac{1}{m_B}\sqrt{\frac{2}{3}}\epsilon(0)\cdot
P_{B}\epsilon_\mu(0).\end{aligned}$$ The contraction is evaluated as $\epsilon(0)\cdot P_{B}/m_B=|\vec
p_T|/m_T$ and thus we can see that the new vector $\epsilon_T$ plays a similar role with the ordinary polarization vector $\epsilon$, regardless of the dimensionless constants $\frac{1}{\sqrt2}|\vec
p_T|/m_T$ or $\sqrt{\frac{2}{3}}|\vec p_T|/m_T$.
Tensor meson decay constants are defined through matrix elements of local current operators between the vacuum and a meson state [@Cheng:2010hn] $$\langle T|j_{\mu\nu}(0)|0\rangle=f_Tm_T^2\epsilon_{\mu\nu}^*,\;\;\;
\langle T|j_{\mu\nu\rho}|0\rangle =-if^T_T
m_T(\epsilon_{\mu\delta}^* P_{2\nu}-\epsilon_{\nu\delta}^*
P_{2\mu}).$$ The interpolating current for $f_T$ is chosen as $j_{\mu\nu}=\frac{1}{2} [\bar q_1(0)\gamma_\mu i
\buildrel\leftrightarrow\over D_\nu q_2(0)+\bar q_1(0)\gamma_\nu i
\buildrel\leftrightarrow\over D_\mu q_2(0)]$ with the covariant derivative $ \buildrel\leftrightarrow\over D_\nu =
\buildrel\rightarrow\over D_\nu - \buildrel\leftarrow\over D_\nu $: $\buildrel\rightarrow\over D_\nu=\buildrel\rightarrow\over
\partial_\nu+ig_sA^a_\nu\lambda^a/2$ and $\buildrel\leftarrow\over D_\nu=\buildrel\leftarrow\over
\partial_\nu-ig_sA^a_\nu\lambda^a/2$; the one for $f_T^\perp$ is selected as $
j_{\mu\nu\rho}^\dagger= \bar q_2(0)\sigma_{\mu\nu} i\buildrel\leftrightarrow\over
D_\delta(0) q_1(0).$ These quantities have been partly calculated in the QCD sum rules in Refs. [@Aliev:1981ju; @Aliev:1982ab; @Aliev:2009nn] and we quote the recently updated results from Ref. [@Cheng:2010hn] in Tab. \[Table:Tdecayconstant\]. One interesting feature in these values is that the two decay constants of $a_2(1320)$ are almost equal but large differences are found for $K_2^*(1430)$ and $f_2'(1525)$. In the case of a light vector meson, taking $\rho$ as an example, the transverse decay constant is typically about (20%–30%) smaller than the longitudinal one: $f_\rho^T/f_\rho=(0.687\pm0.027)$ [@Allton:2008pn].
-------------- --------------- --------------- ----------------- ------------------ ------------------- ------------------- -------------------- -- --
$f_{a_2} $ $ f_{a_2}^T $ $ f_{K_2^*} $ $ f_{K_2^*}^T $ $f_{f_2(1270)} $ $f_{f_2(1270)}^T$ $f_{f_2'(1525)} $ $f_{f_2'(1525)}^T$
$107\pm6$ $105\pm 21$ $ 118\pm 5$ $77\pm 14$ $102\pm 6$ $117\pm25$ $126\pm 4$ $65\pm 12$
-------------- --------------- --------------- ----------------- ------------------ ------------------- ------------------- -------------------- -- --
: Decay constants of tensor mesons from Ref. [@Cheng:2010hn] (in units of MeV)
\[Table:Tdecayconstant\]
In the PQCD approach, the necessary inputs contain the LCDAs which are constructed by matrix elements of the non-local operators at the light-like separations $z_\mu$ with $z^2=0$, and sandwiched between the vacuum and the meson state. For tensor mesons, their distribution amplitudes are recently analyzed in Ref. [@Cheng:2010hn] which will provide a solid foundation in our study of $B\to T$ form factors. The LCDAs up to twist-3 for a generic tensor meson are defined by: $$\begin{aligned}
\langle T(P_2,\epsilon)|\bar q_{2\beta}(z) q_{1\alpha} (0)|0\rangle
&=&\frac{1}{\sqrt{2N_c}}\int_0^1 dx e^{ixP_2\cdot z} \left[m_T\not\!
\epsilon^*_{\bullet L} \phi_T(x) +\not\! \epsilon^*_{\bullet
L}\not\! P_2 \phi_{T}^{t}(x) +m_T^2\frac{\epsilon_{\bullet} \cdot
n}{P_2\cdot n} \phi_T^s(x)\right]_{\alpha\beta},
\nonumber\\
\langle T(P_2,\epsilon)|\bar q_{2\beta}(z) q_{1\alpha}
(0)|0\rangle &=&\frac{1}{\sqrt{2N_c}}\int_0^1 dx e^{ixP_2\cdot z}
\left[ m_T\not\! \epsilon^*_{\bullet T}\phi_T^v(x)+
\not\!\epsilon^*_{\bullet T}\not\! P_2\phi_T^T(x)+m_T
i\epsilon_{\mu\nu\rho\sigma}\gamma_5\gamma^\mu\epsilon_{\bullet
T}^{*\nu} n^\rho v^\sigma \phi_T^a(x)\right ]_{\alpha\beta}\;,
\label{spf}\end{aligned}$$ for the longitudinal polarization ($h=0$) and transverse polarizations ($h=\pm1$), respectively. Here $x$ is the momentum fraction associated with the $q_2$ quark. $n$ is the moving direction of the vector meson and $v$ is the opposite direction. $N_c=3$ is the color factor and the convention $\epsilon^{0123}=1$ has been adopted. The new vector $\epsilon_\bullet$ in Eq. is related to the polarization tensor by $\epsilon_{\bullet\mu}\equiv\frac{\epsilon_{\mu\nu} v^\nu}{P_2\cdot
v}m_T=\frac{2\epsilon_{\mu\nu} P_B^\nu}{m_B^2-q^2}m_T$ and moreover it plays the same role with the polarization vector $\epsilon$ in the definition of the vector meson LCDAs. The above distribution amplitudes can be related to the ones given in Ref. [@Cheng:2010hn] by [^1] $$\begin{aligned}
&&\phi_{T}(x)=\frac{f_{T}}{2\sqrt{2N_c}}\phi_{||}(x),\;\;\;
\phi_{T}^t(x)=\frac{f_{T}^T}{2\sqrt{2N_c}}h_{||}^{(t)}(x),\nonumber\\
&&\phi_{T}^s(x)=\frac{f_{T}^T}{4\sqrt{2N_c}}
\frac{d}{dx}h_{||}^{(s)}(x),\hspace{3mm}
\phi_{T}^T(x)=\frac{f_{T}^T}{2\sqrt{2N_c}}\phi_{\perp}(x)
,\nonumber\\
&&\phi_{T}^v(x)=\frac{f_{T}}{2\sqrt{2N_c}}g_{\perp}^{(v)}(x),
\hspace{3mm}\phi_{T}^a(x)=\frac{f_{T}}{8\sqrt{2N_c}}
\frac{d}{dx}g_{\perp}^{(a)}(x).\end{aligned}$$
The twist-2 LCDA can be expanded in terms of Gegenbauer polynomials $C_n^{3/2}(2x-1)$ weighted by the Gegenbauer moments. Particularly its asymptotic form is $$\begin{aligned}
\phi_{||,\perp} (x)&=& 30x(1-x)(2x-1),\label{phiV}\end{aligned}$$ with the normalization conditions $$\begin{aligned}
\int_0^1 dx (2x-1) \phi_{||,\perp} (x)=1.\end{aligned}$$
Using equation of motion in QCD, two-particle twist-3 distribution amplitudes are expressed as functions of the twist-2 LCDAs and the three-particle twist-3 LCDAs. In the Wandzura-Wilczek limit, i.e. with the neglect of the three-particle terms, the asymptotic forms of twist-3 LCDAs are derived as [@Cheng:2010hn] $$\begin{aligned}
h_\parallel^{(t)}(x) & = & \frac{15}{2}(2x-1)(1-6x+6x^2) ,\;\;\;
h_{||}^{(s)}(x) = 15x(1-x)(2x-1),\\
g_\perp^{(a)}(x) & = & 20x(1-x)(2x-1) ,\;\;\; g_\perp^{(v)}(x)
=5(2x-1)^3.
$$
Since the $B$ meson is a pseudoscalar heavy meson, the two structure $(\gamma^\mu\gamma_5)$ and $\gamma_{5}$ components remain as leading contributions. Then, $\Phi_{B}$ is written by $$\Phi_{B} = \frac{i}{\sqrt{6}}
\left\{ (\not \! P_B \gamma_5) \phi_B^A + \gamma_{5} \phi_B^P
\right\},$$ where $\phi_B^{A,P}$ are Lorentz scalar distribution amplitudes. As shown in Ref. [@Lu:2002ny], $B$ meson’s wave function can be simplified into $$\Phi_{B}(x,b) = \frac{i}{\sqrt{6}}
\left[ (\not \! P_B \gamma_5) + m_B \gamma_5 \right] \phi_B(x,b),$$ where the numerically-suppressed terms in the PQCD approach have been neglected. For the distribution amplitude, we adopt the model: $$\begin{aligned}
\phi_{B}(x,b)=N_{B}x^{2}(1-x)^{2}\exp \left[ -\frac{1}{2} \left(
\frac{xm_{B}}{\omega _{B}}\right) ^{2} -\frac{\omega
_{B}^{2}b^{2}}{2}\right] \label{bw} \;,\end{aligned}$$ with $\omega_{B}$ being the shape parameter and $N_B$ as the normalization constant. In the above parametrization form, $\phi_B$ will have a sharp peak at $x\sim 0.1$, in accordance with the most probable momentum fraction of the light quark: $\Lambda_{\rm
QCD}/m_B$. Here $\Lambda_{\rm QCD}$ denotes the typical hadronization scale. In recent years, a number of studies of $B^{\pm}$ and $B_d^0$ decays have been performed in the PQCD approach, from which the $\omega_b$ is found around $0.40\mbox{GeV}$ [@Keum:2000ph; @Lu:2002ny]. In our calculation, we will adopt $\omega_b=(0.40\pm0.05)\mbox{GeV}$ and $f_B=(0.19\pm0.02)\rm{GeV}$ for $B$ mesons. For the $B_s$ meson, taking the SU(3) breaking effects into consideration, we employ $\omega_{b}=(0.50\pm 0.05)\mbox{GeV}$ [@Ali:2007ff] and $f_{B_s}=(0.23\pm0.02)\rm{GeV}$. These values for decay constants are consistent with the recent Lattice QCD simulations [@Gamiz:2009ku] $$\begin{aligned}
f_{B}=(0.190\pm0.01) {\rm GeV},\;\;\;
f_{B_s}=(0.231\pm0.015) {\rm GeV}.\end{aligned}$$
$B\to T$ form factors in the PQCD approach {#section:formfactor}
===========================================
PQCD approach
-------------
The most important feature of the PQCD approach is that it takes into account the intrinsic transverse momentum of valence quarks. The tree-level transition amplitude, taking the first diagram in Fig. \[fig:transition\] as an example, can be directly expressed as a convolution of wave functions $\phi_B$, $\phi_2$ and hard scattering kernel $T_H$ with both longitudinal momenta and transverse space coordinates $$\begin{aligned}
{\cal M}=\int^1_0dx_1dx_2\int
{d^2{\vec{b}}_{1}}{d^2{\vec{b}}_{2}}{\phi}_B(x_1,{\vec{b}}_{1},P_B,t)
T_H(x_1,x_2,{\vec{b}}_{1},{\vec{b}}_{2},t){\phi}_2(x_2,{\vec{b}}_{2},P_2,t).\end{aligned}$$ Individual higher order diagrams may suffer from two generic types of infrared divergences: soft and collinear. In both cases, the loop integration generates logarithmic divergences. These divergences can be separated from the hard kernel and reabsorbed into meson wave functions using eikonal approximation [@Li:1994iu]. When soft and collinear momentum overlap, double logarithm divergences will be generated and they can be grouped into the Sudakov factor using the technique of resummation. In the threshold region, loop corrections to the weak decay vertex will also produce double logarithms which can be factored out from the hard part and grouped into the quark jet function. Resummation of the double logarithms results in the threshold factor $S_t$ [@Li:2001ay]. This factor decreases faster than any other power of ${x}$ as ${x\rightarrow 0}$, which modifies the behavior in the endpoint region to make PQCD approach more self-consistent. For a review of this approach, please see Ref. [@Li:2003yj].
With the inclusion of the Sudakov factors, we can get the generic factorization formula in the PQCD approach: $$\begin{aligned}
{\cal M}&=&\int^1_0dx_1dx_2\int
{d^2{\vec{b}}_{1}}{d^2{\vec{b}}_{2}} {\phi}_B(x_1,{\vec{b}}_{1},P_B,t)
T_H(x_1,x_2,{\vec{b}}_{1},{\vec{b}}_{2},t){\phi}_2(x_2,{\vec{b}}_{2},P_2,t)
S_t(x_2)\exp[-S_B(t)-S_2(t)].\label{eq:PQCD-factorization}\end{aligned}$$ This factorization framework has been successfully generalized to a number of transition form factors, including different final states such as light pseudoscalar and vector meson [@Lu:2002ny; @Chen:2002bq], scalar mesons [@Wang:2006ria; @Li:2008tk], axial-vector mesons [@Wang:2007an; @Li:2009tx] and the $D$ meson case in large recoil region [@Kurimoto:2002sb; @Li:2008ts].
$B\to T$ form factors
---------------------
In analogy with $B\to V$ form factors, we parameterize the $B\to T$ form factors as $$\begin{aligned}
\langle T(P_2,\epsilon)|\bar q\gamma^{\mu}b|\overline B(P_B)\rangle
&=&-\frac{2V(q^2)}{m_B+m_T}\epsilon^{\mu\nu\rho\sigma} \epsilon^*_{T\nu} P_{B\rho}P_{2\sigma}, \nonumber\\
\langle T(P_2,\epsilon)|\bar q\gamma^{\mu}\gamma_5 b|\overline
B(P_B)\rangle
&=&2im_T A_0(q^2)\frac{\epsilon^*_{T } \cdot q }{ q^2}q^{\mu}
+i(m_B+m_T)A_1(q^2)\left[ \epsilon^*_{T\mu }
-\frac{\epsilon^*_{T } \cdot q }{q^2}q^{\mu} \right] \nonumber\\
&&-iA_2(q^2)\frac{\epsilon^*_{T } \cdot q }{ m_B+m_T }
\left[ P^{\mu}-\frac{m_B^2-m_T^2}{q^2}q^{\mu} \right],\nonumber\\
\langle T(P_2,\epsilon)|\bar q\sigma^{\mu\nu}q_{\nu}b|\overline
B(P_B)\rangle
&=&-2iT_1(q^2)\epsilon^{\mu\nu\rho\sigma} \epsilon^*_{T\nu} P_{B\rho}P_{2\sigma}, \nonumber\\
\langle T(P_2,\epsilon)|\bar q\sigma^{\mu\nu}\gamma_5q_{\nu}b|\overline
B(P_B)\rangle
&=&T_2(q^2)\left[(m_B^2-m_T^2) \epsilon^*_{T\mu }
- {\epsilon^*_{T } \cdot q } P^{\mu} \right] +T_3(q^2) {\epsilon^*_{T } \cdot q }\left[
q^{\mu}-\frac{q^2}{m_B^2-m_T^2}P^{\mu}\right],\label{eq:BtoTformfactors-definition}
\end{aligned}$$ where $q=P_B-P_2, P=P_B+P_2$. Similar with the $B\to V$ form factors, we also have the relation $2m_TA_0(0)=(m_B+m_T)A_1(0)-(m_B-m_T)A_2(0)$ for tensor mesons in order to smear the pole at $q^2=0$. In the above definitions the flavor factor, for instance $1/\sqrt 2$ for the isosinglet meson with the component $\frac{1}{\sqrt 2}(\bar uu+\bar dd)$, has not been explicitly specified but will be taken into account in the following numerical analysis. The parametrization of $B\to T$ form factors is analogous to the $B\to V$ case except that the $\epsilon$ is replaced by $\epsilon_T$. In the literature, the $B\to T$ form factors have been previously defined in an alternative form [@Isgur:1988gb] $$\begin{aligned}
\langle T(P_2,\vp)|V_\mu|\overline B (P_B)\rangle
&=&-h(q^2)\epsilon_{\mu\nu\alpha\beta}\vp^{\pp*\nu\lambda}P_\lambda
P^\alpha q^\beta,\nonumber\\
\langle T(P_2,\vp)|A_\mu|\overline B(P_B)\rangle
&=&-i\left\{k(q^2)\vp^{\pp*}_{\mu\nu}P^\nu +\vp^{\pp*}_{\alpha\beta}P^\alpha P^\beta[P_\mu b_+(q^2) +q_\mu b_-(q^2)]
\right\},\label{eq:B TO T-OLD}\end{aligned}$$ where the two sets of form factors are related via $$\begin{aligned}
V&=&-m_B(m_{B}+m_T)h(q^2),\;\;\;\; A_1=-\frac{m_Bk(q^2)}{m_{B}+m_T},\;\;\;
A_2= m_B(m_{B}+m_T)b_+(q^2),\nonumber\\
A_0(q^2)&=&\frac{m_{B}+m_T}{2m_T} A_1(q^2)-\frac{m_{B}-m_T}{2m_T}
A_2(q^2)-\frac{m_Bq^2}{2m_T}b_-(q^2).\label{eq:B TO T1-relation}\end{aligned}$$
In the PQCD approach, the factorization formulae of $B\to T$ form factors can be obtained through a straightforward evaluation of the hard kernels shown in Eq. . But the correspondence between a vector meson and a tensor meson allows us to get these formulas in a comparative way. As we have shown in the above, both LCDAs of a tensor meson and the $B\to T$ form factors are in conjunction with the quantities involving a vector meson and explicitly we have $$\begin{aligned}
\phi_V^{(i)}\leftrightarrow \phi_T^{(i)},\;\;\;
F^{B\to T}&\leftrightarrow& F^{B\to
V},\end{aligned}$$ where $\phi_{V,T}^{(i)}$ and $F$ denotes any generic LCDA and $B\to
(T,V)$ form factor, respectively. The only difference is that the polarization vector $\epsilon$ is replaced by $\epsilon_\bullet$ in the LCDAs but by $\epsilon_T$ in the transition form factors. As a consequence the factorization formulas for the $B\to T$ form factors are derived as $$\begin{aligned}
F^{B\to T}(\phi_T^{(i)})&=& \frac{\epsilon_\bullet}{\epsilon_T} F^{B\to
V}(\phi_V^{(i)}) =\frac{2 m_B m_T}{m_B^2-q^2} F^{B\to
V}(\phi_V^{(i)}).\label{eq:comparison}\end{aligned}$$ As for the expressions of the $B\to V$ form factors, please see Refs. [@Lu:2002ny; @Chen:2002bq] and also our recent update in Refs. [@Wang:2007an; @Li:2009tx].
Form factors in the large recoiling region can be directly calculated since the exchanged gluon is hard enough so that the perturbation theory works well. In order to extrapolate the form factors to the whole kinematic region, we usually use the results obtained in the region $0< q^2<10\rm{GeV}^2$ and recast the form factors by adopting certain parametrization of the $q^2$-distribution. Unlike the other nonperturbative approaches like the QCD sum rules where the analytic properties can be used to constrain the pole structure of the form factors, the PQCD approach is mainly established on the perturbative property of the form factors (i.e. factorization) and in this approach one has to assume the parametrization form in a phenomenological way. In the literature, the popular forms for $B\to P$ and $B\to V$ form factors (P, V denotes a light pseudoscalar meson and a vector meson respectively) include pole form, dipole form and exponential form, and the BK parametrization [@Becirevic:1999kt]. In the small $q^2$ region, these forms do not differ too much as all of them have similar forms by making use of the expansion of $q^2/m_B^2$. Unfortunately the differences increase with the increase of $q^2$. The limited knowledge of the form factors in the large $q^2$ region will inevitably introduce sizable uncertainties. However as a first step to proceed, it is helpful to investigate these form factors by employing one commonly-adopted form. The dipole form has been adopted in the previous PQCD studies [@Li:2008tk; @Wang:2007an; @Li:2009tx] $$\begin{aligned}
F(q^2)&=&\frac{F(0)}{1-a(q^2/m_B^2)+b(q^2/m_B^2)^2}\end{aligned}$$ and this parametrization works well. In contrast, the $B\to T$ form factors receive additional $q^2$-dependence as can be seen from the factorization formulas in Eq. . In this case the following modified form is more appropriate for the $q^2$-distribution of $B\to T$ form factors $$\begin{aligned}
F(q^2)&=&\frac{F(0)}{(1-q^2/m_B^2)(1-a(q^2/m_B^2)+b(q^2/m_B^2)^2)},\label{eq:fit-B-T}\end{aligned}$$ and we shall use this form in our fitting procedure.
Numerical results for the form factors at maximally recoil point and the two fitted parameters $a,b$ are collected in table \[Tab:formfactorsBtoTbeforemixing\]. The first type of errors comes from decay constants and shape parameter $\omega_b$ of $B$ meson; while the second one is from factorization scales (from $0.75t$ to $1.25t$, not changing the transverse part $1/b_i$), the threshold resummation parameter $c=0.4\pm0.1$ and $\Lambda_{\rm
QCD}=(0.25\pm0.05)\rm{GeV}$. The hadron masses are taken from particle data group [@Amsler:2008zz] $$\begin{aligned}
m_{a_2}=1.3183{\rm GeV},\;\; m_{K_2^*}=1.43 {\rm GeV},\;\; m_{f_2(1270)}=1.2751{\rm
GeV},\;\; m_{f_2'(1525)}=1.525 {\rm GeV}.\end{aligned}$$
$F$ $F(0)$ $a$ $b$ $F$ $F(0)$ $a$ $b$
------------------- ---------------------------------- ---------------------------------- ----------------------------------- ------------------------ ---------------------------------- ---------------------------------- -----------------------------------
$V^{B a_2}$ $0.18_{-0.03-0.03}^{+0.04+0.04}$ $1.70_{-0.01-0.05}^{+0.01+0.06}$ $0.63_{-0.01-0.04}^{+0.03+0.09}$ $V^{B f_2(1270)}$ $0.12_{-0.02-0.02}^{+0.02+0.02}$ $1.68_{-0.00-0.05}^{+0.02+0.06}$ $0.62_{-0.00-0.07}^{+0.05+0.10}$
$A_0^{B a_2}$ $0.18_{-0.03-0.03}^{+0.04+0.04}$ $1.74_{-0.05-0.07}^{+0.00+0.06}$ $0.71_{-0.13-0.13}^{+0.00+0.07}$ $A_0^{Bf_2(1270)}$ $0.13_{-0.02-0.02}^{+0.03+0.03}$ $1.74_{-0.02-0.06}^{+0.01+0.05}$ $0.69_{-0.05-0.10}^{+0.04+0.06}$
$A_1^{B a_2}$ $0.11_{-0.02-0.02}^{+0.02+0.02}$ $0.74_{-0.01-0.03}^{+0.02+0.04}$ $-0.11_{-0.03-0.02}^{+0.04+0.03}$ $A_1^{B f_2(1270)}$ $0.08_{-0.01-0.01}^{+0.02+0.01}$ $0.73_{-0.03-0.04}^{+0.01+0.05}$ $-0.12_{-0.09-0.00}^{+0.03+0.04}$
$A_2^{B a_2}$ $0.06_{-0.01-0.01}^{+0.01+0.01}$ $--$ $--$ $A_2^{B f_2(1270)}$ $0.04_{-0.01-0.00}^{+0.01+0.01}$ $--$ $--$
$T_1^{B a_2}$ $0.15_{-0.03-0.02}^{+0.03+0.03}$ $1.69_{-0.01-0.05}^{+0.00+0.05}$ $0.64_{-0.04-0.06}^{+0.00+0.05}$ $T_1^{B f_2(1270)}$ $0.10_{-0.02-0.01}^{+0.02+0.02}$ $1.67_{-0.01-0.08}^{+0.00+0.05}$ $0.62_{-0.03-0.15}^{+0.00+0.05}$
$T_2^{B a_2}$ $0.15_{-0.03-0.02}^{+0.03+0.03}$ $0.74_{-0.01-0.07}^{+0.01+0.01}$ $-0.11_{-0.01-0.09}^{+0.02+0.00}$ $T_2^{B f_2(1270)}$ $0.10_{-0.02-0.01}^{+0.02+0.02}$ $0.72_{-0.04-0.08}^{+0.00+0.03}$ $-0.09_{-0.10-0.11}^{+0.00+0.00}$
$T_3^{B a_2}$ $0.13_{-0.02-0.02}^{+0.03+0.03}$ $1.58_{-0.01-0.05}^{+0.01+0.06}$ $0.52_{-0.04-0.04}^{+0.02+0.05}$ $T_3^{B f_2(1270)}$ $0.09_{-0.02-0.01}^{+0.02+0.02}$ $1.56_{-0.00-0.05}^{+0.03+0.08}$ $0.48_{-0.00-0.04}^{+0.08+0.12}$
$V^{B K_{2}^*}$ $0.21_{-0.04-0.03}^{+0.04+0.05}$ $1.73_{-0.02-0.03}^{+0.02+0.05}$ $0.66_{-0.05-0.01}^{+0.04+0.07}$
$A_0^{B K_{2}^*}$ $0.18_{-0.03-0.03}^{+0.04+0.04}$ $1.70_{-0.02-0.07}^{+0.00+0.05}$ $0.64_{-0.06-0.10}^{+0.00+0.04}$
$A_1^{B K_{2}^*}$ $0.13_{-0.02-0.02}^{+0.03+0.03}$ $0.78_{-0.01-0.04}^{+0.01+0.05}$ $-0.11_{-0.03-0.02}^{+0.02+0.04}$
$A_2^{B K_{2}^*}$ $0.08_{-0.02-0.01}^{+0.02+0.02}$ $--$ $--$
$T_1^{B K_{2}^*}$ $0.17_{-0.03-0.03}^{+0.04+0.04}$ $1.73_{-0.03-0.07}^{+0.00+0.05}$ $0.69_{-0.08-0.11}^{+0.00+0.05}$
$T_2^{B K_{2}^*}$ $0.17_{-0.03-0.03}^{+0.03+0.04}$ $0.79_{-0.04-0.09}^{+0.00+0.02}$ $-0.06_{-0.10-0.16}^{+0.00+0.00}$
$T_3^{B K_{2}^*}$ $0.14_{-0.03-0.02}^{+0.03+0.03}$ $1.61_{-0.00-0.04}^{+0.01+0.09}$ $0.52_{-0.01-0.01}^{+0.05+0.15}$
$V^{B_s K_2^*}$ $0.18_{-0.03-0.03}^{+0.03+0.04}$ $1.73_{-0.00-0.05}^{+0.02+0.05}$ $0.67_{-0.00-0.05}^{+0.05+0.06}$ $V^{B_s f_2'(1525)}$ $0.20_{-0.03-0.03}^{+0.04+0.05}$ $1.75_{-0.00-0.03}^{+0.02+0.05}$ $0.69_{-0.01-0.01}^{+0.05+0.08}$
$A_0^{B_s K_2^*}$ $0.15_{-0.02-0.02}^{+0.03+0.03}$ $1.70_{-0.01-0.05}^{+0.00+0.03}$ $0.65_{-0.03-0.04}^{+0.01+0.00}$ $A_0^{B_s f_2'(1525)}$ $0.16_{-0.02-0.02}^{+0.03+0.03}$ $1.69_{-0.01-0.03}^{+0.00+0.04}$ $0.64_{-0.04-0.02}^{+0.00+0.01}$
$A_1^{B_s K_2^*}$ $0.11_{-0.02-0.02}^{+0.02+0.02}$ $0.79_{-0.01-0.03}^{+0.02+0.03}$ $-0.10_{-0.03-0.02}^{+0.07+0.06}$ $A_1^{B_s f_2'(1525)}$ $0.12_{-0.02-0.02}^{+0.02+0.03}$ $0.80_{-0.00-0.03}^{+0.02+0.07}$ $-0.11_{-0.00-0.00}^{+0.05+0.09}$
$A_2^{B_s K_2^*}$ $0.07_{-0.01-0.01}^{+0.01+0.02}$ $--$ $--$ $A_2^{B_s f_2'(1525)}$ $0.09_{-0.01-0.01}^{+0.02+0.02}$ $--$ $--$
$T_1^{B_s K_2^*}$ $0.15_{-0.02-0.02}^{+0.03+0.03}$ $1.73_{-0.01-0.06}^{+0.00+0.04}$ $0.69_{-0.03-0.11}^{+0.00+0.04}$ $T_1^{B_s f_2'(1525)}$ $0.16_{-0.03-0.02}^{+0.03+0.04}$ $1.75_{-0.00-0.05}^{+0.01+0.05}$ $0.71_{-0.01-0.08}^{+0.03+0.06}$
$T_2^{B_s K_2^*}$ $0.15_{-0.02-0.02}^{+0.03+0.03}$ $0.80_{-0.03-0.08}^{+0.00+0.02}$ $-0.06_{-0.09-0.13}^{+0.00+0.00}$ $T_2^{B_s f_2'(1525)}$ $0.16_{-0.03-0.02}^{+0.03+0.04}$ $0.82_{-0.04-0.06}^{+0.00+0.04}$ $-0.08_{-0.09-0.08}^{+0.00+0.03}$
$T_3^{B_s K_2^*}$ $0.12_{-0.02-0.02}^{+0.02+0.03}$ $1.61_{-0.00-0.04}^{+0.03+0.08}$ $0.52_{-0.01-0.00}^{+0.08+0.14}$ $T_3^{B_s f_2'(1525)}$ $0.13_{-0.02-0.02}^{+0.03+0.03}$ $1.64_{-0.00-0.06}^{+0.02+0.06}$ $0.57_{-0.01-0.09}^{+0.04+0.05}$
: $B\to T$ form factors. $a,b$ are the parameters of the form factors in the parametrization shown in Eq. . The two kinds of errors are from: decay constants of $B$ meson and shape parameter $\omega_b$; $\Lambda_{\rm{QCD}}$, the scales $t$s and the threshold resummation parameter $c$. []{data-label="Tab:formfactorsBtoTbeforemixing"}
A number of remarks on these results are given in order.
1. With terms suppressed by $r_2^2$ neglected, $A_2(q^2)$ can be expressed as a linear combination of $A_0$ and $A_1$ [@Li:2009tx] $$\begin{aligned}
A_2(q^2)=\frac{1+r_2}{1-q^2/m_B^2}\left[(1+r_2)A_1(q^2)-2r_2
A_0(q^2)\right].\end{aligned}$$ We will use this relation for $A_2(q^2)$ in the whole kinematic region instead of a direct fitting.
2. The $B\to f_2(1270)$ form factors are smaller than the other channels due to the factor $1/\sqrt2$ in the flavor wave function of $f_2(1270)$. The smaller transverse decay constants of $K_2^*$ and $f_2'(1525)$ have a tendency to suppress the transition amplitudes. But their larger masses give an enhancement, since both contributions from the twist-3 LCDAs and the correspondence relation in Eq. (\[eq:comparison\]) are proportional to the hadron mass.
3. The parameters $a$ in most transition form factors are roughly $1.7$, but they are around $0.7$ for $A_1(q^2)$ and $T_2(q^2)$. Analogously the parameter $b$ is close to $0.6$ with the exception for $A_1(q^2)$ and $T_2(q^2)$ as it is approaching $0$. The vanishing $b$ implies that the dipole behavior in these two form factors is reduced into the monopole form.
4. In our computation, the asymptotic forms for the LCDAs have been adopted. The twist-2 LCDAs $\phi_{||,\perp}$ can be expanded into Gegenbauer polynomials $C_{n}^{3/2}(2x_2-1)$ (with $x_2$ being the momentum fraction of the quark in the meson) and the twist-3 LCDAs will be expressed in terms of twist-2 ones through the use of equation of motion [@Cheng:2010hn]: $$\begin{aligned}
g_\perp^{(v)}(x_2)&=& \int_0^{x_2} dv \frac{\phi_{||}(v)}{1-v}+ \int_{x_2}^1 dv \frac{\phi_{||}(v)}{v},\nonumber\\
g_{\perp}^{(a)}(x_2)&=& 4(1-x_2) \int_0^{x_2} dv \frac{\phi_{||}(v)}{1-v}
+4x_2\int_{x_2}^1 dv \frac{\phi_{||}(v)}{v},\nonumber\\
h_{||}^{(t)}(x_2)&=& \frac{3}{2} (2x_2-1) \left[\int_0^{x_2} dv\frac{\phi_{\perp}(v)}{1-v}-\int_{x_2}^1 dv\frac{\phi_{\perp}(v)}{v}\right],\nonumber\\
h_{||}^{(s)}(x_2)&=&3(1-x_2) \int_0^{x_2} dv \frac{\phi_{\perp}(v)}{1-v}
+3x_2\int_{x_2}^1 dv \frac{\phi_{\perp}(v)}{v}.\end{aligned}$$ Taking into account the contributions from the next non-zero Gegenbauer moment besides the asymptotic form, i.e. $a_3$, we find $$\begin{aligned}
A_0^{Ba_2}(0)=0.18\pm0.07 a_3,\;\;\; T_1^{Ba_2}(0)= 0.15\pm 0.057 a_3. \end{aligned}$$ In the case of $\pi$ and $\rho$ meson, the first non-zero Gegenbauer moment is around (0.2-0.3) [@Ball:2006wn]. If it were the similar for the tensor meson, we can see that the form factors will be changed by roughly $10\% -20\%$.
5. Since the $B\to T$ form factors are obtained from the $B\to V$ ones, it is meaningful to analyze these two sets of form factors in a comparative way. It is worth comparing their distribution amplitudes. The six LCDAs are functions of $x_2$, with $x_2$ being the momentum fraction of the quark in the light meson. Taking $\rho$ and $a_2$ mesons as an example, these LCDAs are depicted in Fig. \[fig:LCDA-comparison\], where the solid (dashed) lines denote the LCDAs for $\rho$ ($a_2$) meson. For $\rho$ meson LCDAs, the asymptotic form has been used. From this figure, we can see that although the two sets of LCDAs are different in the small-momentum-fraction region $x_2<0.5$, they have similar shapes when $x_2>0.6$. The large-momentum-fraction region, $0.6<x_2<1$ [^2] dominates in the PQCD approach. As one important consequence, the $B\to T$ and $B\to V$ form factors will have several similar properties. For instance the two kinds of form factors will have the same signs and their $q^2$-dependence parameters will also be close.
6. As functions of $q^2$, the $B\to T$ form factors are expected to be sharper than the $B\to V$ form factors, since the former ones contain one more pole structure in the $q^2$-distribution. To illustrate this situation, in Fig. \[fig:q2-dependence\] we show the $B\to \rho$ (dashed lines) and $B\to a_2$ form factors (solid lines) in the region of $0<q^2<10
{\rm GeV}^2$, where the PQCD results for the $B\to \rho$ form factors are taken from our recent update in Ref. [@Li:2009tx]. We also quote them in table \[Tab:Btorhoformfactor\], but only the central values are shown for the $q^2$-dependence parameters $a,b$. The ratio of the $B\to \rho$ and $B\to a_2$ form factors is 0.73 for $A_0$ and 0.77 for $T_1$, respectively.
$F$ $F(0)$ $a$ $b$
-- ------- ---------------------------------- -------- --------- -- --
$V$ $0.21_{-0.04-0.02}^{+0.05+0.03}$ 1.75 0.69
$A_0$ $0.25_{-0.05-0.03}^{+0.06+0.04}$ 1.69 0.57
$A_1$ $0.16_{-0.03-0.02}^{+0.04+0.02}$ 0.77 $-0.13$
$A_2$ $0.13_{-0.03-0.01}^{+0.03+0.02}$ — —
$T_1$ $0.19_{-0.04-0.02}^{+0.04+0.03}$ 1.69 0.61
$T_2$ $0.19_{-0.04-0.02}^{+0.04+0.03}$ 0.73 $-0.12$
$T_3$ $0.17_{-0.03-0.02}^{+0.04+0.02}$ $1.58$ 0.50
: $B\to \rho$ form factors in the PQCD approach [@Li:2009tx]
\[Tab:Btorhoformfactor\]
7. At the maximally recoiling point with $q^2=0$, the $B\to\rho$ and $B\to a_2(1320)$ form factors have different magnitudes. Taking $A_0$ and $T_1$ as an example, in table \[Tab:formfactorcomparison\] we enumerate distinct contributions from the three LCDAs. The matching coefficient $2m_Tm_B/(m_B^2-q^2)$ between the two sets of form factors is roughly $1/2$ at $q^2=0$ and in this case the $B\to T$ transition is expected to be smaller. It is also confirmed by the numerical results in table \[Tab:formfactorcomparison\], where its twist-2 contribution is only one half of the $B\to V$ case. On the contrary this does not occur for the twist-3 LCDAs, as the larger tensor meson mass has compensated the suppression: $m_{a_2}\sim 2m_{\rho}$.
$A_0$ $B\to\rho$ $B\to a_2(1320)$
---------- ------------ ------------------ -- -- --
$\phi$ $0.108$ $0.050$
$\phi^s$ $0.103$ $0.088$
$\phi^t$ $0.040$ $0.046$
total $0.251$ $0.184$
$T_1$ $B\to\rho$ $B\to a_2(1320)$
$\phi^T$ $0.085$ $0.049$
$\phi^a$ $0.047$ $0.046$
$\phi^v$ $0.063$ $0.054$
total $0.194$ $0.150$
: Different contributions to form factors $A_0$ and $T_1$ for $B\to \rho$ and $B\to a_2(1320)$.[]{data-label="Tab:formfactorcomparison"}
![LCDAs of the vector meson $\rho$ (solid lines) and its tensor counterpart $a_2$ (dashed lines). The asymptotic forms are adopted for $\rho,a_2$ meson LCDAs. []{data-label="fig:LCDA-comparison"}](LCDA-phix2.eps "fig:") ![LCDAs of the vector meson $\rho$ (solid lines) and its tensor counterpart $a_2$ (dashed lines). The asymptotic forms are adopted for $\rho,a_2$ meson LCDAs. []{data-label="fig:LCDA-comparison"}](LCDA-phitx2.eps "fig:") ![LCDAs of the vector meson $\rho$ (solid lines) and its tensor counterpart $a_2$ (dashed lines). The asymptotic forms are adopted for $\rho,a_2$ meson LCDAs. []{data-label="fig:LCDA-comparison"}](LCDA-phisx2.eps "fig:") ![LCDAs of the vector meson $\rho$ (solid lines) and its tensor counterpart $a_2$ (dashed lines). The asymptotic forms are adopted for $\rho,a_2$ meson LCDAs. []{data-label="fig:LCDA-comparison"}](LCDA-phiLTx2.eps "fig:") ![LCDAs of the vector meson $\rho$ (solid lines) and its tensor counterpart $a_2$ (dashed lines). The asymptotic forms are adopted for $\rho,a_2$ meson LCDAs. []{data-label="fig:LCDA-comparison"}](LCDA-phivx2.eps "fig:") ![LCDAs of the vector meson $\rho$ (solid lines) and its tensor counterpart $a_2$ (dashed lines). The asymptotic forms are adopted for $\rho,a_2$ meson LCDAs. []{data-label="fig:LCDA-comparison"}](LCDA-phiax2.eps "fig:")
![Transition form factors as functions of $q^2$. Solid (black) and dashed (red) lines correspond to our results of the $B\to\rho$ and $B\to a_2$ channel, respectively. Dotted (blue) lines denote the results in the covariant LFQM, with $V,A_0,A_1,A_2$ for the $B\to a_2$ process and $T_{1,2,3}$ for the $B\to K_2^*$ transition. A minus sign has been added to the LFQM results for $V,A_1,A_2,T_3$ so that they have the same sign with our results. []{data-label="fig:q2-dependence"}](q2-V.eps "fig:") ![Transition form factors as functions of $q^2$. Solid (black) and dashed (red) lines correspond to our results of the $B\to\rho$ and $B\to a_2$ channel, respectively. Dotted (blue) lines denote the results in the covariant LFQM, with $V,A_0,A_1,A_2$ for the $B\to a_2$ process and $T_{1,2,3}$ for the $B\to K_2^*$ transition. A minus sign has been added to the LFQM results for $V,A_1,A_2,T_3$ so that they have the same sign with our results. []{data-label="fig:q2-dependence"}](q2-A0.eps "fig:") ![Transition form factors as functions of $q^2$. Solid (black) and dashed (red) lines correspond to our results of the $B\to\rho$ and $B\to a_2$ channel, respectively. Dotted (blue) lines denote the results in the covariant LFQM, with $V,A_0,A_1,A_2$ for the $B\to a_2$ process and $T_{1,2,3}$ for the $B\to K_2^*$ transition. A minus sign has been added to the LFQM results for $V,A_1,A_2,T_3$ so that they have the same sign with our results. []{data-label="fig:q2-dependence"}](q2-A1.eps "fig:") ![Transition form factors as functions of $q^2$. Solid (black) and dashed (red) lines correspond to our results of the $B\to\rho$ and $B\to a_2$ channel, respectively. Dotted (blue) lines denote the results in the covariant LFQM, with $V,A_0,A_1,A_2$ for the $B\to a_2$ process and $T_{1,2,3}$ for the $B\to K_2^*$ transition. A minus sign has been added to the LFQM results for $V,A_1,A_2,T_3$ so that they have the same sign with our results. []{data-label="fig:q2-dependence"}](q2-A2.eps "fig:") ![Transition form factors as functions of $q^2$. Solid (black) and dashed (red) lines correspond to our results of the $B\to\rho$ and $B\to a_2$ channel, respectively. Dotted (blue) lines denote the results in the covariant LFQM, with $V,A_0,A_1,A_2$ for the $B\to a_2$ process and $T_{1,2,3}$ for the $B\to K_2^*$ transition. A minus sign has been added to the LFQM results for $V,A_1,A_2,T_3$ so that they have the same sign with our results. []{data-label="fig:q2-dependence"}](q2-T1.eps "fig:") ![Transition form factors as functions of $q^2$. Solid (black) and dashed (red) lines correspond to our results of the $B\to\rho$ and $B\to a_2$ channel, respectively. Dotted (blue) lines denote the results in the covariant LFQM, with $V,A_0,A_1,A_2$ for the $B\to a_2$ process and $T_{1,2,3}$ for the $B\to K_2^*$ transition. A minus sign has been added to the LFQM results for $V,A_1,A_2,T_3$ so that they have the same sign with our results. []{data-label="fig:q2-dependence"}](q2-T2.eps "fig:") ![Transition form factors as functions of $q^2$. Solid (black) and dashed (red) lines correspond to our results of the $B\to\rho$ and $B\to a_2$ channel, respectively. Dotted (blue) lines denote the results in the covariant LFQM, with $V,A_0,A_1,A_2$ for the $B\to a_2$ process and $T_{1,2,3}$ for the $B\to K_2^*$ transition. A minus sign has been added to the LFQM results for $V,A_1,A_2,T_3$ so that they have the same sign with our results. []{data-label="fig:q2-dependence"}](q2-T3.eps "fig:")
In the literature, the $B\to T$ form factors have been explored in the ISGW model [@Isgur:1988gb], its improved form ISGW II model [@Scora:1995ty; @Kim:2002rx; @Sharma:2010yx; @Cheng:2010sn] and other relativistic quark models for instance the covariant light-front quark model (LFQM) [@Cheng:2003sm; @Cheng:2004yj; @Cheng:2009ms]. The form factor $T_1$ for $B\to K_2^*$ is also estimated in the technique of QCD sum rules (QCDSR) [@Safir:2001cd], relativistic quark model [@Ebert:2001en] and heavy quark symmetry [@Veseli:1995bt]. We collect the results using these approaches [@Sharma:2010yx; @Cheng:2010sn; @Cheng:2003sm; @Cheng:2004yj; @Cheng:2009ms; @Safir:2001cd] in table \[Tab:BtoVformfactor\] for the convenience of a comparison, where their results have been converted to the new form factors defined in Eq. through the relations in Eq. . Our PQCD results, all uncertainties added in quadrature, are also shown in table \[Tab:BtoVformfactor\]. From this table, we can find many differences among these theoretical predictions. Results for all form factors from the ISGW II model possess a different sign with our results and the magnitudes are typically larger. The two calculations in the same ISGW II model are also different, for instance the prediction in Ref. [@Sharma:2010yx] of $A_2$ for $B\to K_2^*$ is about twice as large as the one in Ref. [@Cheng:2010sn]. The estimate in the QCDSR [@Safir:2001cd] is consistent with our result.
Results in the covariant LFQM are different with ours in several aspects. Firstly, for $A_0$ and $T_{1,2}$ [^3], the LFQM predicts the same sign with our results but the remanent results have negative signs. Secondly, their predictions, except for $A_1$, are much larger than ours in magnitude. Moreover the $q^2$-distribution is also different. In Fig. \[fig:q2-dependence\], we show the LFQM results (dotted lines) in the region of $0<q^2<10 {\rm GeV}^2$, with $V,A_{0,1,2}$ for the $B\to a_2$ process [@Cheng:2003sm] but $T_{1,2,3}$ for the $B\to K_2^*$ transition [@Cheng:2009ms]. A minus sign has been added to $V,A_1,A_2,T_3$ so that they have the same sign with our results. From this figure, we can find that the differences for $A_1,A_2,T_2$ between their results and ours get larger as $q^2$ grows. In particular, the $T_2$ grows faster than $T_1$ with the increase of $q^2$ in the LFQM but it is reverse in our results. In the covariant LFQM the meson-quark-antiquark coupling vertex for a tensor meson contains $\epsilon_{\mu\nu}\frac{p_1^{\prime\nu}-p_2^\nu}{2}
\sqrt{\frac{2}{\beta^{\prime2}}}$, which corresponds to $\epsilon_\mu$ in the case of a vector meson. $p_1'(p_2)$ denotes the momentum of the quark and antiquark in the final meson. The $\beta'$, of the order $\Lambda_{\rm QCD}$, is the shape parameter which characterizes the momentum distribution inside the tensor meson. It is hard to deduce the relative signs from this structure since (1) apart from the longitudinal momentum in $p_1',p_2$, the transverse part might also contribute; (2) it involves the zero-mode terms which are essential for the maintenance of the Lorentz covariance. In this sense the relation between a vector meson and its tensor counterpart is not as simple as the one in the PQCD approach, where $\epsilon$ is replaced by $\epsilon_\bullet$. These different results can be discriminated in the future when enough data is available.
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
$B\to a_2$ $B\to K_2^*$ $B\to f_2$ $B_s\to K_2^*$ $B_s\to f_2'$
------------------------------------------- ----------- --------------------------------- --------------------------------- --------------------------------- --------------------------------- --------------------------------- --
ISGW II [@Sharma:2010yx]([@Cheng:2010sn]) $V$ —$(-0.57)$
$A_0$ $-0.18$ $-0.17(-0.25)$ $-0.08$ $-0.27$ $-0.26$
$A_1$ $-0.35$ $-0.38(-0.23)$ $-0.24$ $-0.39$ $-0.45$
$A_2$ $-0.45$ $-0.53(-0.21)$ $-0.34$ $-0.47$ $-0.59$
LFQM [@Cheng:2003sm; @Cheng:2009ms] $V$ $-0.28$ $-0.28$
$A_0$ $0.20$ $0.26$
$A_1$ $-0.025$ $-0.012$
$A_2$ $-0.17$ $-0.21$
$T_1=T_2$ $0.28$ $0.28$
$T_3$ $-0.25$ $-0.18$
QCDSR [@Safir:2001cd] $T_1$ $0.19\pm0.04$
This work $V$ $0.18_{-0.04}^{+0.05}$ $0.21_{-0.05}^{+0.06}$ $ 0.12_{-0.03}^{+0.03}$ $0.18_{-0.04}^{+0.05}$ $0.20_{-0.04}^{+0.06}$
$A_0$ $ 0.18 $ 0.18 $ 0.13 $ 0.15 $ 0.16
_{ -0.04 } _{ -0.04 } _{ -0.03 } _{ -0.03 } _{ -0.03 }
^{+ 0.06 } ^{+ 0.05 } ^{+ 0.04 } ^{+ 0.04 } ^{+ 0.04 }
$ $ $ $ $
$A_1$ $ 0.11 $ 0.13 $ 0.08 $ 0.11 $ 0.12
_{ -0.03 } _{ -0.03 } _{ -0.02 } _{ -0.02 } _{ -0.03 }
^{+ 0.03 } ^{+ 0.04 } ^{+ 0.02 } ^{+ 0.03 } ^{+ 0.03 }
$ $ $ $ $
$A_2$ $ 0.06 $ 0.08 $ 0.04 $ 0.07 $ 0.09
_{ -0.01 } _{ -0.02 } _{ -0.01 } _{ -0.02 } _{ -0.02 }
^{+ 0.02 } ^{+ 0.03 } ^{+ 0.01 } ^{+ 0.02 } ^{+ 0.03 }
$ $ $ $ $
$T_1=T_2$ $ 0.15 $ 0.17 $ 0.10 $ 0.15 $ 0.16
_{ -0.03 } _{ -0.04 } _{ -0.02 } _{ -0.03 } _{ -0.04 }
^{+ 0.04 } ^{+ 0.05 } ^{+ 0.03 } ^{+ 0.04 } ^{+ 0.05 }
$ $ $ $ $
$T_3$ $ 0.13 $ 0.14 $ 0.09 $ 0.12 $ 0.13
_{ -0.03 } _{ -0.03 } _{ -0.02 } _{ -0.03 } _{ -0.03 }
^{+ 0.04 } ^{+ 0.05 } ^{+ 0.03 } ^{+ 0.04 } ^{+ 0.04 }
$ $ $ $ $
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
: $B\to T$ form factors at maximally recoil, i.e. $q^2=0$. Theoretical results in the ISGW II model [@Sharma:2010yx], the covariant light-front quark model [@Cheng:2003sm; @Cheng:2009ms] and the QCD sum rules [@Safir:2001cd] are also collected for a comparison. Results in the parentheses are from Ref. [@Cheng:2010sn].
\[Tab:BtoVformfactor\]
In the large energy limit, the seven $B\to T$ form factors are expected to satisfy several nontrivial relations [@Datta:2007yk; @Hatanaka:2009gb] and all form factors can be parameterized into two independent functions $\zeta_\perp(q^2)$ and $\zeta_{||}(q^2)$. In the large recoil region, we have checked that our results respect these relations. Moreover, the relative size of these two functions is also of prime interest but it can not be deduced from the large energy limit itself. Our results for the $B\to K_2^*$ transition [^4], $$\begin{aligned}
\zeta_\perp(0)=\frac{|\vec p_{K_2^*}|}{m_{K_2^*}}T_1^{BK_2^*}(0)= (0.29\pm0.09),\;\;\;
\zeta_{||}(0)=\frac{1}{1-\frac{m_{K_2^*}^2}{m_BE_{K_2^*}}}\left(\frac{|\vec
p_{K_2^*}|}{m_{K_2^*}}A_0^{BK_2^*}(0)-\frac{m_{K_2^*}}{m_B}\zeta_\perp(0)\right)=
(0.26\pm0.10),\nonumber\end{aligned}$$ show that they are of similar size, the same conclusion with the $B\to V$ cases. This is not accidental but instead is an outcome of the similar shapes between the vector and tensor meson LCDAs in the dominant region of the PQCD approach. Our result is also accordance with the theoretical estimate in Ref. [@Hatanaka:2009gb] $$\begin{aligned}
\zeta_\perp(0)=0.27\pm0.03^{+0.00}_{-0.01}.\end{aligned}$$
On the experimental side the branching ratio of the color-allowed tree-dominated processes $B^0\to a_2^\pm \pi^\mp$ has been set with an upper limit $$\begin{aligned}
{\cal B}(B^0\to a_2^\pm \pi^\mp)<3.0\times 10^{-4}.\end{aligned}$$ When factorization is adopted this mode can be used to extract the $B\to a_2$ form factor $$\begin{aligned}
|A_0^{B\to a_2}(q^2=0)|&<&7.6 F_+^{B\to \pi}(q^2=0)\simeq1.9,\end{aligned}$$ where penguin contributions have been neglected as a result of their small Wilson coefficients. Unfortunately the above constraint is too loose to provide any useful information on the characters of the tensor mesons. We expect more news on this front from the $B$ factories and other experiment facilities, including the Large Hadron Collider.
At the leading order of $\alpha_s$, both $B\to K^*\gamma$ and $B\to
K_2^*\gamma$ only receive contributions from the chromo-magnetic operator $O_{7\gamma}$, which leads to $$\begin{aligned}
{\cal B}(B\to K^*\gamma)&=& \tau_{B}\frac{G_F^2 \alpha_{\rm
em}m_B^3m_b^2}{32\pi^4}\left(1-\frac{m_{K^*}^2}{m_B^2}\right)^3|V_{tb}V_{ts}^*C_7T_1^{BK^*}(0)|^2,\nonumber\\
{\cal B}(B\to K_2^*\gamma)&=& \tau_{B}\frac{G_F^2 \alpha_{\rm
em}m_B^5m_b^2}{256\pi^4
m_{K_2^*}^2}\left(1-\frac{m_{K^*_2}^2}{m_B^2}\right)^5|V_{tb}V_{ts}^*C_7T_1^{BK^*_2}(0)|^2,\end{aligned}$$ with $C_{7}$ being the Wilson coefficient for $O_{7\gamma}$ and $V_{tb},V_{ts}$ being the CKM matrix element. Assuming that $C_7$ is the same for the above two channels, we obtain the form factors relation $$\begin{aligned}
\frac{ T_1^{BK_2^*}(0)}{ T_1^{BK^*}(0)}&=& (0.52\pm0.08),\end{aligned}$$ from the experimental data [@HFAG] $$\begin{aligned}
{\cal B}(B^-\to K^{*-}\gamma)&=& (42.1\pm1.8)\times 10^{-6},\;\;\;
{\cal B}(B^-\to K^{*-}_2\gamma)=(14.5\pm4.3)\times 10^{-6}.\end{aligned}$$ Our result for this ratio, roughly 0.7, is larger than this value but is consistent with it when hadronic uncertainties from the final mesons are taken into account. It also confirms our results that the $B\to K_2^*$ form factors are smaller than the $B\to K^*$ ones at $q^2=0$ point, in contrast to the LFQM results $T_1^{BK_2^*}(0)\simeq T_1^{BK^*}(0)$ [@Cheng:2009ms].
Semilteptonic $B\to Tl\bar\nu$ decays {#sec:semileptonic}
=====================================
Integrating out the off shell W boson, one obtains the effective Hamiltonian responsible for $b\to ul\bar \nu_l$ transition $$\begin{aligned}
{\cal H}_{\rm eff}(b\to ul\bar \nu_l)=\frac{G_F}{\sqrt{2}}V_{ub}\bar
u\gamma_{\mu}(1-\gamma_5)b \bar l\gamma^{\mu}(1-\gamma_5)\nu_l,
\end{aligned}$$ where $V_{ub}$ is the CKM matrix element. In semileptonic $B\to
Tl\bar\nu_l$ decays, the helicity of the tensor meson can be $h=0,\pm1$ but the $h=2$ configuration is not allowed physically. Using the form factors obtained in the previous section, we can investigate the semileptonic $B\to Tl\bar\nu$ decays with the partial decay width $$\begin{aligned}
\frac{d\Gamma}{dq^2}&=&\sum_{i=L,\pm}
\frac{d\Gamma_{i}}{dq^2},\nonumber\\
\frac{d\Gamma_{L, \pm}}{dq^2}&=&
\frac{ |G_F V_{ub}|^2 \sqrt {\lambda_T} }{256m_{B_s}^3\pi^3q^2}
\left(1- {\frac{m_\ell^2}{q^2}}\right)^2
(X_{L}, X_{\pm})\end{aligned}$$ where $\lambda_T=\lambda(m^2_{B},m^2_T, q^2)$, and $\lambda(a^2,b^2,c^2)=(a^2-b^2-c^2)^2-4b^2c^2$. The subscript $(L,\pm)$ denotes the three polarizations of the tensor meson along its momentum direction: $(0,\pm1)$. $m_l$ represents the mass of the charged lepton, and $q^2$ is the momentum square of the lepton pair. In terms of the angular distributions, we can study the forward-backward asymmetries (FBAs) of lepton which are defined as $$\begin{aligned}
\frac{dA_{FB}}{dq^2} &=& \frac{\int^{1}_{0} dz (d\Gamma/dq^2dz) -
\int^{0}_{-1} dz (d\Gamma/dq^2dz)}{\int^{1}_{0} dz (d\Gamma/dq^2dz)
+ \int^{0}_{-1} dz (d\Gamma/dq^2dz)}\nonumber
\end{aligned}$$ where $z=\cos\theta$ and the angle $\theta$ is the polar angle of lepton with respect to the moving direction of the tensor meson in the lepton pair rest frame. Explicitly, we have $$\begin{aligned}
\frac{dA_{\rm FB}}{dq^2}&=&
\frac{1}{X_L + X_{+} + X_{-}} \left(\frac{\lambda_T}{6m_T^2m_B^2} 2 m_\ell^2
\sqrt{\lambda_{T}}h_0(q^2) A_0(q^2) -\frac{\lambda_T}{8m_T^2m_B^2} 4q^4\sqrt {\lambda_{T}} A_1(q^2) V(q^2)
\right),\label{eq:AAS-tensor}\end{aligned}$$ where $$\begin{aligned}
X_{L} &=& \frac{2}{3}\frac{\lambda_T}{6m_T^2m_B^2} \left[
(2q^2+m_\ell^2) h_0^2(q^2) + 3
{\lambda_{T}} m_\ell^2 A_0^2 (q^2)\right]\,, \nonumber \\
X_{\pm} &=& \frac{2q^2}{3} (2 q^2 + m_\ell^2
)\frac{\lambda_T}{8m_T^2m_B^2} \left[(m_{B}+m_{T})A_1(q^2) \mp
\frac{\sqrt{\lambda_{T}}
}{m_{B}+m_{T}}V(q^2) \right]^2,\nonumber\\
h_0(q^2)&=& \frac{ 1}{2 m_{T}
}\left[(m_{B}^2-m_{T}^2-q^2)(m_{B}+m_{T})A_1(q^2)-
\frac{{\lambda_{T}
}}{m_{B}+m_{T}}A_2(q^2)\right].\label{eq:longitudinal-Ds1}\end{aligned}$$ Integrating over the $q^2$, we obtain the partial decay width and integrated angular asymmetry for this decay mode $$\begin{aligned}
\Gamma=\Gamma_L+\Gamma_++\Gamma_-,\;\;\;
A_{FB} &=& \frac{1}{\Gamma}\int dq^2 \int^{1}_{-1}sign(z) dz (d\Gamma/dq^2dz) \nonumber\end{aligned}$$ with $
\Gamma_{L,\pm}=\int_{m_l^2}^{(m_{B}-m_{T})^2} dq^2
\frac{d\Gamma_{L,\pm}}{dq^2}.$ Physical quantities ${\cal B}_{\rm{L}}$, ${\cal B}_{\rm{+}}$, ${\cal
B}_{\rm{-}}$, and ${\cal B}_{\rm{total}}$ can be obtained through different experimental measurements, where ${\cal B}_{\rm{T}}={\cal
B}_{\rm{+}}+{\cal B}_{\rm{-}}$ and ${\cal B}_{\rm{total}}={\cal
B}_{\rm{L}}+{\cal B}_{\rm{T}}$ with ${\cal B}_{\rm{L}}$, ${\cal
B}_{\rm{+}}$ and ${\cal B}_{\rm{-}}$ corresponding to contributions of different polarization configurations to branching ratios. Since there are three different polarizations, it is also meaningful to define the polarization fraction $$\begin{aligned}
f_L=\frac{\Gamma_L}{\Gamma_L+\Gamma_++\Gamma_-}.\end{aligned}$$
Our theoretical results for the $B\to T l\bar\nu_l$ ($l=e,\mu$) and $B\to T \tau\bar\nu_{\tau}$ decays are listed in Table \[tab:branchratios1\], with masses of the electron and muon neglected in the case of $l=e,\mu$. The $B$ meson lifetime is taken from the particle data group and the CKM matrix element $V_{ub}$ is employed as $|V_{ub}|=(3.89\pm0.44)\times
10^{-3}$ [@Amsler:2008zz].
Some remarks are given in order.
- Most of the total branching ratios are of the order $10^{-4}$, implying a promising prospect to measure these channels at the Super B factories and the LHCb. A tensor meson can be reconstructed in the final state of two or three pseudoscalar mesons.
- The heavy $\tau$ lepton will bring a smaller phase space than the lighter electron, thus the branching ratios of $B\to T
\tau\bar\nu_{\tau}$ decays are smaller than those of the corresponding $B\to T e\bar\nu_e$ decay modes by a factor of 3.
- The positively polarized branching ratio ${\rm Br}_+$ is tiny since the $A_1$ term cancels with the contribution from $V$. The longitudinal contributions are about twice as large as the transverse polarizations and accordingly the polarization fraction $f_L$ is around $(60\%-70\%)$
- For $l=(e,\mu)$ the angular asymmetries are negative since only the second terms in Eq. contribute. In the case of $l=\tau$, the two terms give destructive contributions, resulting in tiny angular asymmetries in magnitude.
- In the polarization fractions and angular asymmetries, the uncertainties from the form factors and the CKM matrix element will mostly cancel and thus they are stable against hadronic uncertainties.
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
${\cal ${\cal B}_+$ ${\cal B}_-$ ${\cal $f_L$ $A_{\rm FB}$
B}_{\rm{L}}$ B}_{\rm{total}}$
--------------------------------------------------------- ---------------------------------- -------------- -------------------------------------------------------- -------------------------- --------------------------------- ----------------------------
$\overline B^0\to a_2^+l\bar $ 0.85 $\sim0.007$ $ 0.30 $1.16 ^{+0.81}_{-0.57} $ $ $-0.186^{+0.003}_{-0.002}$
\nu_l$ _{ -0.42 } _{ -0.15 } 73.3
^{+ 0.60 } ^{+ 0.21 } _{ -0.5 }
$ $ ^{+ 0.4 }
$
$B^-\to f_2^0 (1270)l\bar $ 0.52 $\sim 0.004$ $ 0.17 $0.69 _{-0.34}^{+0.48} $ $ $-0.175\pm0.004$
\nu_l$ _{ -0.26 } _{ -0.08 } 74.9
^{+ 0.36 } ^{+ 0.11 } _{ -0.7 }
$ $ ^{+ 0.6 }
$
$\overline B_s\to K_2^{*+}(1430)l\bar $ 0.50 $\sim0.006$ $ 0.23 $0.73 ^{+0.48}_{-0.33} $ $ $-0.221\pm0.005$
\nu_l$ _{ -0.23 } _{ -0.11 } 68.3
^{+ 0.32 } ^{+ 0.15 } _{ -0.5 }
$ $ ^{+ 0.5 }
$
$\overline B^0\to a_2^+\tau\bar \nu_{\tau}$ $ 0.29 $\sim 0.004$ $ 0.12 $0.41 ^{+0.29}_{-0.20} $ $ 69.9 $0.031\pm0.005$
_{ -0.14 } _{ -0.06 } _{ -0.6 }
^{+ 0.20 } ^{+ 0.08 } ^{+ 0.6 }
$ $ $
$B^-\to f_2^0(1270)\tau\bar \nu_{\tau}$ $ 0.18 $\sim 0.002$ $ 0.07 $0.25 ^{+0.18}_{-0.13} $ $ $0.048\pm0.007$
_{ -0.09 } _{ -0.03 } ^{+ 0.05 } 71.9
^{+ 0.13 } $ _{ -0.8 }
$ ^{+ 0.8 }
$
$\overline B_s\to K_2^{*+}(1430)\tau\bar \nu_{\tau}$ $ 0.16 $\sim 0.003$ $ 0.09 $0.25 ^{+0.17}_{-0.12})$ $ 64.1 $-0.024\pm0.005$
_{ -0.07 } _{ -0.04 } _{ -0.3 }
^{+ 0.10 } ^{+ 0.06 } ^{+ 0.4 }
$ $ $
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
: The branching ratios, polarizations and angular asymmetries for the $b\to ul\bar \nu_l$ ($l=e,\mu$) and $b\to u\tau\bar
\nu_\tau$ decay channels (in units of $10^{-4}$). ${\cal
B}_{\rm{L}}$ and ${\cal B}_{\pm}$ are the longitudinally and transversely polarized contributions to the branching ratios. []{data-label="tab:branchratios1"}
Summary
=======
Inspired by the success of the PQCD approach in the application to $B$ decays into s-wave mesons, we give a comprehensive study on the $B\to T$ transition form factors. Our results will become necessary inputs in the analysis of the nonleptonic $B$ decays into a tensor meson.
The similarities in the Lorentz structures of the wave functions and $B$ decay form factors involving a vector and a tensor meson allow us to obtain the factorization formulas of $B\to T$ form factors from the $B\to V$ ones. Furthermore, the light-cone distribution amplitudes of tensor mesons and vector mesons have similar shapes in the dominant region of the perturbative QCD approach, and thus these two sets of form factors are found to have the same signs and related $q^2$-dependence behaviors. In the large recoil region, we find that our results for the form factors satisfy the relations derived from the large energy limit. The two independent functions $\zeta_\perp$ and $\zeta_{||}$ are found to have similar size at $q^2=0$ point. We also find that the $B\to T$ form factors are smaller than the $B\to V$ ones, which is supported by the experimental data of radiative $B$ decays.
At last, we also use these results to explore semilteptonic $B\to
Tl\bar \nu_l$ decays and we find that the branching fractions can reach the order $10^{-4}$, implying a promising prospect to observe these channels.
Acknowledgements {#acknowledgements .unnumbered}
================
I would like to acknowledge Hai-Yang Cheng and Ying Li for useful discussions. I am very grateful to Pietro Colangelo and Fulvia De Fazio for their warm hospitality during my stay in Bari. This work is supported by the INFN through the program of INFN fellowship for foreigners and also in part by the National Natural Science Foundation of China under the Grant No. 10805037 and 10947007.
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[^1]: The distribution amplitudes $h_{||}^{(s)},h_{||}^{(t)}$ and $g_{\perp}^{(v)}$ correspond to $h_s,h_t$ and $g_v$ in Ref. [@Cheng:2010hn], but our $g^{(a)}_\perp$ differs from their $g_a$ by a factor of 2: $g_\perp^{(a)}=2g_a$. This definition is more convenient in the following analysis of form factors.
[^2]: The dominant region in the PQCD approach can be obtained by the power counting in this approach which has been established in Ref. [@Chen:2001pr]. The typical momentum of the spectator (with the momentum fraction $1-x_2$) is of the order $\Lambda/m_B<0.3$ with the vary of the hadronization scale $\Lambda$. This means that the dominant contribution lies in the region of $0.7<x_2$. This conclusion can also be drawn in a simple way. PQCD is based on the hard scattering picture, in which the endpoint region $x_2\sim 0,1$ is suppressed by the Sudakov factor. In the Feynman diagrams given in Fig.1, if the momentum of the spectator (with momentum fraction $1-x_2$) is getting larger, the gluon and the quark propagators will have larger virtualities. For instance, they are $p_b^2-m_b^2= x_2\eta m_B^2- k_{1\perp}^2$ and $p_g^2=x_1x_2\eta m_B^2 -(k_{1\perp}-k_{2\perp})^2$ with the transverse component $k_{1\perp,2\perp}$ of the order $\Lambda$. Therefore the region $x_2>0.5$ is more important compared with the region of $x_2<0.5$ and thus in our analysis the region of $x_2>0.6$ is chosen.
[^3]: The form factors $T_{1,2,3}$ in this work correspond to the $U_{1,2,3}$ in Ref. [@Cheng:2009ms].
[^4]: We use the definitions of $\zeta_{\perp}$ and $\zeta_{||}$ in Ref. [@Hatanaka:2009gb], but our form factors correspond to theirs with a tilde.
|
---
abstract: |
Let $S$ be a smooth cubic surface over a field $K$. It is well-known that new $K$-rational points may be obtained from old ones by secant and tangent constructions. A Mordell-Weil generating set is a subset $B \subset S(K)$ of minimal cardinality which generates $S(K)$ via successive secant and tangent constructions. Let $r(S,K)=\#B$. Manin posed what is known as the Mordell-Weil problem for cubic surfaces: if $K$ is finitely generated over its prime subfield then $r(S,K)< \infty$. In this paper, we prove a special case of this conjecture. Namely, if $S$ contains a pair of skew lines, both defined over $K$, then $r(S,K)=1$.
One of the difficulties in studying the secant and tangent process on cubic surfaces is that it does not lead to an associative binary operation as in the case of elliptic curves. As a partial remedy we introduce an abelian group $H_S(K)$ associated to a cubic surface $S/K$, naturally generated by the $K$-rational points, which retains much information about the secant and tangent process. In particular, $r(S,K)$ is large as soon as the minimal number of generators of $H_S(K)$ is large. In situations where $K$ is a global field, and weak approximation holds for $K$-rational points on $S$, the group $H_S^0(K)$ surjects onto $\prod_{\upsilon \in \Delta} H_S^0(K_\upsilon)$ for any finite set $\Delta$ of places, where $H_S^0$ is the degree $0$ part of $H_S$. We use this to construct a family of smooth cubic surfaces over ${\mathbb{Q}}$ such that $r(S,{\mathbb{Q}})$ is unbounded in this family. This is the cubic surface analogue of the unboundedness of ranks conjecture for elliptic curves over ${\mathbb{Q}}$.
address: |
Mathematics Institute\
University of Warwick\
Coventry\
CV4 7AL\
United Kingdom
author:
- Samir Siksek
title: |
On the Number of Mordell-Weil Generators\
for Cubic Surfaces
---
\[section\] \[lem\][Proposition]{} \[lem\][Algorithm]{} \[lem\][Corollary]{} \[lem\][Conjecture]{}
[^1]
Introduction
============
Let $C$ be a smooth plane cubic curve over ${\mathbb{Q}}$. The Mordell-Weil Theorem can be restated as follows: there is a finite subset $B$ of $C({\mathbb{Q}})$ such that the whole of $C({\mathbb{Q}})$ can be obtained from this subset by drawing secants and tangents through pairs of previously constructed points and consecutively adding their new intersection points with $C$. It is conjectured that a minimal such $B$ can be arbitrarily large; this is indeed the well-known conjecture that there are elliptic curves with arbitrarily large ranks. This paper is concerned with the cubic surface analogues of the Mordell-Weil Theorem and the unboundedness of ranks.
Let $K$ be a field and let $S$ be a smooth cubic surface over $K$ in ${\mathbb{P}}^3$. By a $K$-line we mean a line $\ell \subset {\mathbb{P}}^3$ that is defined over $K$. If $\ell \not\subset S$ then $\ell \cdot S=P+Q+R$ where $P$, $Q$, $R\in S$. If any two of $P$, $Q$, $R$ are $K$-points then so is the third. The line $\ell$ is tangent at $P$ if and only if $P$ appears more than once in the sum $P+Q+R$. If $B \subseteq S(K)$, we shall write $\operatorname{Span}(B)$ for the subset of $S(K)$ generated from $B$ by successive secant and tangent constructions. More formally, we define a sequence $$B=B_0 \subseteq B_1 \subseteq B_2 \subseteq \cdots \subseteq S(K)$$ as follows. We let $B_{n+1}$ be the set of points $R\in S(K)$ such that either $R \in B_n$, or for some $K$-line $\ell \not \subset S$ we have $\ell \cdot S=P+Q+R$ where $P$, $Q \in B_n$. Then $\operatorname{Span}(B)= \cup B_n$. In view of the Mordell-Weil Theorem for cubic curves it is natural to ask, for $K={\mathbb{Q}}$ say, if there is some finite subset $B \subset S(K)$ such that $\operatorname{Span}(B)=S(K)$. As far as we are aware, the possible existence of such an analogue of the Mordell-Weil Theorem was first mentioned by Segre [@Segre43 page 26] in 1943. Manin [@Ma1 page 3] asks the same question for fields $K$ finitely generated over their prime subfields. He calls this [@Ma2] the Mordell-Weil problem for cubic surfaces. The results of numerical experiments by Zagier (described by Manin in [@Ma2]) and Vioreanu [@V] lead different experts to different opinions about the validity of this Mordell-Weil conjecture. However, Manin [@Ma2] and Kanevsky and Manin [@KM] prove the existence of finite generating sets for rational points on certain Zariski open subsets of rationally trivial cubic surfaces with [**modified**]{} composition operations induced by birational maps $S \dashrightarrow {\mathbb{P}}^2$. However, we are not aware of even a single example in the literature where the existence of a finite set $B$ that generates $S(K)$ via the secant and tangent process is proven. In this paper we give a positive answer to a special case of the Mordell-Weil problem.
\[thm:mw\] Let $K$ be a field with at least $13$ elements. Let $S$ be a smooth cubic surface over $K$. Suppose $S$ contains a pair of skew lines both defined over $K$. Let $P\in S(K)$ be a point on either line that is not an Eckardt point. Then $\operatorname{Span}(P)=S(K)$.
An [*Eckardt point*]{} is a point where three of the lines contained in $S$ meet. Note that Theorem \[thm:mw\] does not make the assumption that the field is finitely generated over its prime subfield! It is certainly true that every smooth cubic surface over an algebraically closed field is generated by a single element. The theorem probably holds for most fields with fewer that $13$ elements, but the proof is likely to involve a tedious examination of special cases and we have not attempted it. It is well-known that a cubic surface with a pair of skew $K$-lines $\ell$, $\ell^\prime$ is birational to $\ell \times \ell^\prime$. The proof of Theorem \[thm:mw\] (given in Section \[sec:mw\]) is an extension of the proof that $S$ is birational to $\ell \times \ell^\prime$.
Now let us write $$r(S,K):=\min \{\#B : \text{$B \subseteq S(K)$ and $\operatorname{Span}(B)=S(K)$} \}.$$ We are unable to show that $r(S,K)$ is finite for cubic surfaces without a skew pair of $K$-lines. However, in some cases we can bound $r(S,K)$ from below. We use this to show that $r(S,{\mathbb{Q}})$ is arbitrarily large as $S$ varies among smooth cubic surfaces over ${\mathbb{Q}}$. To do this we introduce and study a simple analogue of the Picard group of an elliptic curve. Let $$G_S(K)= \bigoplus_{P \in S(K)} {\mathbb{Z}}\cdot P$$ be the free abelian group generated by the $K$-rational points of $S$. Let $G^\prime_S(K)$ be the subgroup generated by all three point sums $P+Q+R$ with $P$, $Q$, $R \in S(K)$ such that
1. there is $K$-line $\ell$ not contained in $S$ with $\ell \cdot S=P+Q+R$, or
2. there is a $K$-line $\ell$ contained in $S$ such that $P$, $Q$, $R \in \ell$.
The [*degree map*]{} $\deg : G_S(K) \rightarrow {\mathbb{Z}}$ is given by $\deg(\sum a_i P_i)=\sum a_i$. Let $$G^{\prime\prime}_S(K)=\{D \in G^\prime_S(K) : \deg(D)=0\}.$$ Let $H_S(K):=G_S(K)/G^{\prime\prime}_S(K)$. If $P \in S(K)$ we denote the image of $P$ in $H_S(K)$ by $[P]$. The degree map remains well-defined on $H_S(K)$: we let $\deg : H_S(K) \rightarrow {\mathbb{Z}}$ be given by $\deg(\sum a_i [P_i ])=\sum a_i$. We shall write $$H^0_S(K)=\{ D \in H_S(K) : \deg(D)=0\}.$$ If $S(K) \neq \emptyset$ then the degree homomorphism clearly induces an isomorphism $$H_S(K)/H^0_S(K) \cong {\mathbb{Z}}.$$ The group $H_S(K)$ will allow us to study $r(S,K)$.
\[thm:two\] Let $p_1,\dots,p_s$ ($s\geq 1$) be distinct primes such that
1. $p_i \equiv 1 \pmod{3}$,
2. $2$ is a cube modulo $p_i$.
Let $M=\prod p_i$ and let $S=S_M/{\mathbb{Q}}$ be the smooth cubic surface given by $$\label{eqn:one}
S_M : x^3+y^3+z(z^2+M w^2)=0.$$ Write $H_S({\mathbb{Q}})[2]$ for the $2$-torsion subgroup of $H_S({\mathbb{Q}})$. Then $H^0_S({\mathbb{Q}}) = H_S({\mathbb{Q}})[2]$ and $$r(S,{\mathbb{Q}}) \geq \dim_{{\mathbb{F}}_2} H_S({\mathbb{Q}})[2] \geq 2s.$$
A prime $p$ satisfies conditions (a) and (b) of the theorem if and only if the polynomial $t^3-2$ has three roots modulo $p$. By the Chebotarëv Density Theorem such primes form a set with Dirichlet density $1/6$ (see for example [@Heilbronn pages 227–229]). We thus see that $r(S_M,{\mathbb{Q}})$ becomes arbitrarily large as $M$ varies. The cubic surface $S_M$ has precisely one ${\mathbb{Q}}$-rational line, which is given by $x+y=z=0$.
Our second theorem concerns a family of diagonal cubic surfaces. To obtain a similar result for diagonal cubic surfaces we need to assume the following conjecture.
\[conj\] (Colliot-Thélène) Let $X$ be smooth and proper geometrically rational surface over a number field $K$. Then the Brauer-Manin obstruction is the only one to weak approximation on $X$.
This conjecture is stated as a question by Colliot-Thélène and Sansuc in [@angers Section V], but since then has been stated as a conjecture by Colliot-Thélène [@CT page 319]. Indeed for the proof of Theorem \[thm:two\] we need the fact that $S_M$ satisfies weak approximation, but fortunately this follows from a theorem of Salberger and Skorobogatov on degree $4$ del Pezzo surfaces.
\[thm:three\] Let $p_1,\dots,p_s$ ($s \geq 1$) be distinct primes $\equiv 1 \pmod{3}$. Let $M=3\prod p_i$ and let $S=S^\prime_M/{\mathbb{Q}}$ be the smooth cubic surface given by $$\label{eqn:two}
S^\prime_M : x^3+y^3+z^3+M w^3=0.$$ Assume that the Brauer-Manin obstruction is the only one to weak approximation for $S$. Then $$r(S,{\mathbb{Q}}) \geq \dim_{{\mathbb{F}}_3} H^0_S({\mathbb{Q}})/3 H^0_S({\mathbb{Q}}) \geq 2s.$$
Theorem \[thm:three\] is less satisfactory than Theorem \[thm:two\] in two obvious ways. The first is that it is conditional on the yet unproven (and probably very difficult) conjecture concerning weak approximation. The second is that we do not know if $H_S^0({\mathbb{Q}})$ contains any elements of infinite order or is merely a torsion group.
The paper is organized as follows. In Section \[sec:class\] we briefly review what we need from the geometry of cubic surfaces. The proof of Theorem \[thm:mw\] is given in Section \[sec:mw\]. In Section \[sec:pre\] we prove some useful results about $H_S(K)$ that follow from the definition and the elementary geometry of cubic surfaces. In particular, we show that $H^0_S(K) = H_S(K)[2]$ if $S$ contains a $K$-line, and that $H^0_S(K) =0$ if $S$ contains a pair of skew $K$-lines. In Section \[sec:local\] we study $H_S(K)$ for local fields $K$. In particular we show that $H^0_S(K)$ is finite and that the map $S(K) \rightarrow H_S(K)$ given by $P \mapsto [P]$ is locally constant. In Section \[sec:real\] we show that $H^0_S({\mathbb{R}})=0$ or ${\mathbb{Z}}/2{\mathbb{Z}}$ depending on whether $S({\mathbb{R}})$ has one or two connected components. In Section \[sec:weak\], for $K$ a number field and $\Delta$ a finite set of places of $K$, we study the diagonal map $\mu_\Delta : H_S(K) \rightarrow \prod_{\upsilon \in \Delta} H_S(K_\upsilon)$; for example if $S/K$ satisfies weak approximation then we show that the map is surjective. The proofs of Theorems \[thm:two\] and \[thm:three\] essentially boil down to proving (enough of) weak approximation and then estimating the size of the target space of the diagonal map $\mu_\Delta$ for $\Delta=\{p_1,\dotsc,p_s\}$. To this end we briefly introduce the Brauer-Manin obstruction (Section \[sec:BM\]) and apply it in Sections \[sec:BMSM\] and \[sec:BMSMd\] to prove (enough of) weak approximation for the surfaces $S_M$ and $S_M^\prime$, where in the latter case we are forced to assume Colliot-Thélène’s Conjecture \[conj\]. Finally we must study $H_S({\mathbb{Q}}_p)$ for the surfaces $S=S_M$, $S=S_M^\prime$. Let $C$ be the plane genus $1$ curve given by the equation $$\label{eqn:C}
C : x^3+y^3+z^3=0.$$ For a prime $p \neq 3$ we shall denote $C_p=C \times {\mathbb{F}}_p$. For $p =p_1,\dotsc,p_s$ it is easily seen that $S_M$ and $S_M^\prime$ both reduce to a cone over $C_p$. This fact is crucial to the proofs of both Theorems \[thm:two\] and \[thm:three\]. In Section \[sec:C\] we shall briefly study $\operatorname{Pic}^0(C_p)/2\operatorname{Pic}^0(C_p)$ and $\operatorname{Pic}^0(C_p)/3\operatorname{Pic}^0(C_p)$. Section \[sec:red\] quickly reviews good choices of parametrizations of lines in ${\mathbb{P}}^3$: a good choice is one that still parametrizes the lines after reduction modulo $p$. In Section \[sec:SM\] we shall construct, for the surface $S=S_M$, a surjective homomorphism $H^0_S({\mathbb{Q}}_p) \rightarrow \operatorname{Pic}^0(C_p)/2\operatorname{Pic}^0(C_p)$. In Section \[sec:proofone\] we use this and the surjectivity of the diagonal map $\mu_\Delta$ for $\Delta=\{p_1,\dotsc,p_s\}$ to deduce Theorem \[thm:two\]. We then turn our attention to the surface $S=S_M^\prime$. In Section \[sec:SMd\] we construct a surjective homomorphism $H^0_S({\mathbb{Q}}_p) \rightarrow \operatorname{Pic}^0(C_p)/3\operatorname{Pic}^0(C_p)$, and conclude the proof of Theorem \[thm:three\] in Section \[sec:prooftwo\].
One reason why $H_S(K)$ may be of interest is that it seems to be intimately related to the Chow group $\operatorname{CH}_0(S)$ of zero-cycles on $S/K$. It is straightforward to see that elements of $G_S^{\prime\prime}(K)$ are zero-cycles that are rationally equivalent to $0$. Thus we have a natural homomorphism $$\epsilon : H_S(K) \rightarrow \operatorname{CH}_0(S).$$ In subsequent papers we plan to address the relationship of $H_S(K)$ with $\operatorname{CH}_0(S)$, as well as various constructions found in [@Ma1] such as Moufang loops, $R$-equivalence and universal equivalence. We will also make a more extensive study of $H_S(K)$ for $K$ a finite or local field, and the natural pairing $\operatorname{Br}(S) \times \prod H_S(K_\upsilon) \rightarrow {\mathbb{Q}}/{\mathbb{Z}}$ induced by the corresponding pairing for the Chow group of zero cycles.
I thank Felipe Voloch for drawing my attention to [@V] (via the website [mathoverflow.net]{}). I am grateful to Martin Bright, David Holmes, Miles Reid and Damiano Testa for helpful discussions, and the referee for many corrections and comments. I would like to thank Jean-Louis Colliot-Thélène for useful correspondence regarding the above conjecture and for drawing my attention to [@SS]. In particular, Professor Colliot-Thélène points out that it should be possible to deduce unboundedness results for cubic surfaces similar to our Theorems \[thm:two\] and \[thm:three\] from unboundedness results for the number of $R$-equivalence classes of Châtelet surfaces [@css Theorem 8.13].
Some Geometry {#sec:class}
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We shall need some basic material on the geometry of cubic surfaces. We do not claim any originality in this section, although we occasionally sketch proofs of well-known statements in order to verify that these hold in small positive characteristic.
(Cayley-Salmon) \[thm:27\] Every non-singular cubic surface over an algebraically closed field contains exactly $27$ lines.
Every line $\ell$ on the surface meets exactly $10$ other lines, which break up into $5$ pairs $\ell_i$, $\ell^\prime_i$ ($i=1,\dots,5$) such that $\ell$, $\ell_i$ and $\ell_i^\prime$ are coplanar, and $(\ell_i \cup \ell_i^\prime) \cap (\ell_j \cup \ell_j^\prime)=\emptyset$ for $i \ne j$.
For a proof see [@Hartshorne V.4] or [@Shaf Section IV.2].
For now $S$ will denote a smooth cubic surface in ${\mathbb{P}}^3$ over a field $K$, defined by homogeneous cubic polynomial $F \in K[x_0,x_1,x_2,x_3]$.
For a point $P \in S(\overline{K})$, we shall denote the tangent plane to $S$ at $P$ by $\Pi_P$. This is given by $\Pi_P : \nabla{F}(P) \cdot {\mathbf{x}}=0$. We shall write $\Gamma_P$ for the plane curve $S \cdot \Pi_P$. It is easy to check (using the smoothness of $S$) that $\Gamma_P$ does not contain any multiple components. It is a degree $3$ plane curve which is singular at $P$. If $\Gamma_P$ is irreducible, it is nodal or cuspidal at $P$. If $\Gamma_P$ is reducible then it is the union of a line and an irreducible conic, or of three distinct lines.
\[lem:elem\] Let $P \in S(\overline{K})$. The curve $\Gamma_P$ contains every $\overline{K}$-line on $S$ that passes through $P$.
By Euler’s Homogeneous Function Theorem, $P\cdot \nabla{F}(P)=3 \cdot F(P)=0$. The line $\ell$ has a parametrization of the form $s P+t {\mathbf{v}}$ with $(s:t) \in {\mathbb{P}}^1$. Thus the polynomial $F(s P+t{\mathbf{v}})$ vanishes identically. However, coefficient of $t s^{2}$ in this polynomial is $(\nabla{F})(P)\cdot {\mathbf{v}}$. This shows that $\ell$ is also contained in $\Pi_P$, and hence in $\Gamma_p$.
A $\overline{K}$-line $\ell$ is called an [*asymptotic line*]{} (c.f. [@Voloch Section 2]) at $P \in S(\overline{K})$ if $(\ell \cdot S)_P \geq 3$. As $S$ is a cubic surface, it is seen that for an asymptotic line $\ell$ at $P$, either $(\ell \cdot S)_P=3$ or $\ell \subset S$. The asymptotic lines at $P$ are contained in $\Pi_P$. Any line contained in $S$ and passing through $P$ is an asymptotic line through $P$. The number of distinct asymptotic $\overline{K}$-lines at $P$ is either $1$, $2$ or infinity. If $S$ has either $1$ or infinitely many asymptotic lines at $P$ then we shall call $P$ a [*parabolic*]{} point. The case where there are infinitely many asymptotic lines at $P$ is special: in this case $\Gamma_P$ decomposes as a union of three $\overline{K}$-lines lying on $S$ and so the point $P$ is an [*Eckardt*]{} point. If $P$ is parabolic but not Eckardt, the curve $\Gamma_P$ has a cusp at $P$. If $P$ is non-parabolic, then $\Gamma_P$ has a node at $P$.
Next we suppose $P \in S(K)$. Then $\Pi_P$ and $\Gamma_P$ are defined over $K$. Suppose $P$ is non-parabolic. Thus there are precisely two (distinct) asymptotic lines at $P$. If these two lines are individually defined over $K$ then we say that $P$ is [*$K$-hyperbolic*]{}. Otherwise the two lines are defined over a quadratic extension of $K$ and conjugate; in this case we shall say that $P$ is [*$K$-elliptic*]{}. If $K={\mathbb{R}}$ then the terms parabolic, hyperbolic and elliptic agree with their usual meanings in differential geometry: they correspond to points where the Gaussian curvature is respectively $=0$, $>0$ and $<0$.
We shall also need to study the number of parabolic points on a line lying on a cubic surface. Let ${{\mathbb{P}}^3}^*$ be the dual projective space and write $\gamma : S \rightarrow {{\mathbb{P}}^3}^*$ for the [*Gauss map*]{} which sends a point to its tangent plane. A useful characterisation of parabolic points is that they are the points of ramification of the Gauss map [@Voloch Section 2]. If $\ell \subset S$ and $P \in \ell$, then $\ell$ is contained in the tangent plane $\Pi_P$. The family of planes through $\ell$ can be identified with ${\mathbb{P}}^1$ and once such an identification is fixed we let $\gamma_\ell : \ell \rightarrow {\mathbb{P}}^1$ be the map that sends a point on $\ell$ to its tangent plane through $\ell$.
\[lem:paraline\] Let $\ell$ be a $K$-line contained in $S$. Then every $P\in \ell(K)$ is either parabolic or $K$-hyperbolic.
1. If $\operatorname{char}(K) \neq 2$ then $\gamma_\ell$ is separable. Precisely two points $P \in \ell(\overline{K})$ are parabolic, and so there are at most two Eckardt points on $\ell$.
2. If $\operatorname{char}(K)=2$ and $\gamma_\ell$ is separable then there is precisely one point $P \in \ell(\overline{K})$ which is parabolic and so at most one Eckardt point on $\ell$.
3. If $\operatorname{char}(K)=2$ and $\gamma_\ell$ is inseparable then every point $P \in \ell(\overline{K})$ is parabolic and the line $\ell$ contains exactly $5$ Eckardt points.
This is a well-known classical result for cubic surfaces over the reals or complexes; see for example [@Segre pages 103–104]. Let $P \in \ell(K)$. First we would like to show that any $P \in \ell(K)$ is either parabolic or $K$-hyperbolic. Suppose $P$ is not parabolic. Then $S$ has precisely two asymptotic lines passing through $P$. One of these is $\ell$. Since this pair of asymptotic lines must be $K$-rational as a whole, the other asymptotic line is $K$-rational. Thus $P$ is $K$-hyperbolic.
Next we would like to count the number of parabolic points on $\ell$. By a projective transformation defined over $K$ we may suppose that $\ell$ passes through the point $(0:0:0:1)$, that the tangent plane at this point is $x_0=0$ and that the line is $x_0=x_1=0$. Then $F$ has the form $$F=x_0 Q + x_1 R$$ with $Q$, $R$ are homogeneous quadratic forms in $K[x_0,x_1,x_2,x_3]$. Now $\gamma_\ell : \ell \rightarrow {\mathbb{P}}^1$ can be written as $P \mapsto (Q(P) : R(P))$. The fact that $S$ is non-singular implies that $Q$ and $R$ do not simultaneously vanish along the line $\ell$. Thus $\gamma_\ell$ has degree $2$. Suppose first that $\gamma_\ell$ is separable—this is always the case if $\operatorname{char}(K) \ne 2$. Applying the Hurwitz Theorem [@Hartshorne Section IV.2] to $\gamma_\ell$ immediately gives that the ramification divisor has degree $2$. If $\operatorname{char}(K) \ne 2$ then the ramification is tame and so $\gamma_\ell$ is ramified at precisely two distinct $\overline{K}$-points. If $\operatorname{char}(K)=2$, then the ramification is wild and $\gamma_\ell$ is ramified at precisely one point. Parts (i) and (ii) now follow as the parabolic points on $\ell$ are the ramification points of $\gamma_\ell$, and as the Eckardt points are contained among the parabolic points.
Finally suppose that $\operatorname{char}(K)=2$ and $\gamma_\ell$ is inseparable. Then $\gamma_\ell$ is ramified at every point of $\ell$ and so every point is parabolic. To complete the proof of (iii) we must show that there are $5$ Eckardt points on $\ell$. By Theorem \[thm:27\] there are $10$ lines on $S$ that meet $\ell$. Let $\ell^\prime$ be one of these and let $P$ be their point of intersection. Then $\ell$ and $\ell^\prime$ are distinct asymptotic lines to $S$ at $P$. The only way that $P$ can be parabolic is if there is a third line passing through $P$. Thus there are $5$ Eckardt points $\ell$ proving (iii).
Many of the classical notions about cubic surfaces over ${\mathbb{C}}$ break down for cubic surfaces over a field of characteristic $2$. For example, a cubic surface over ${\mathbb{C}}$ has $1$, $2$, $3$, $4$, $6$, $9$, $10$ or $18$ Eckardt points [@Segre Section 100] and any line contains at most two Eckardt points. The following example shows that this need not be the case in characteristic $2$. Take $S$ to be the smooth cubic surface over ${\mathbb{F}}_2$ given by $$S:
x_0^2 x_2 + x_0^2 x_3 + x_0 x_1^2 + x_0 x_1 x_2 + x_0 x_3^2 + x_1^2 x_2
+ x_1 x_2^2.$$ The $27$ lines are rational over ${\mathbb{F}}_{64}$, and $S$ has $13$ Eckardt points. Three of the $27$ lines have $5$ Eckardt points and the remaining all have exactly one.
Proof of Theorem \[thm:mw\] {#sec:mw}
===========================
Throughout this section we shall assume that $\#K\ge 13$.
\[lem:linegen\] Let $\ell$ be a $K$-line on $S$. Let $P \in \ell(K)$ be a point that does not lie on any other line belonging to $S$. Then $$\ell(K) \subseteq \Gamma_P(K) \subseteq \operatorname{Span}(P).$$
By Lemma \[lem:elem\], we know $\ell \subseteq \Gamma_P$. Thus $\Gamma_P=\ell \cup C$ where $C$ is a conic contained in $\Pi_P$. Since $P$ does not lie on any other line belonging to $S$, we know that $C$ is irreducible over $\overline{K}$. Now $P \in \ell \cap C$ since $P$ is either parabolic or $K$-hyperbolic by Lemma \[lem:paraline\]. Thus $\ell\cdot C = P+P^\prime$ where $P^\prime$ is also $K$-rational. Let $Q \in C(K)\backslash \{P,P^\prime\}$. Let $\ell^\prime$ be the $K$-line joining $P$ and $Q$. Then $\ell^\prime \cdot S=2P+Q$ and so $Q \in \operatorname{Span}(P)$. Hence $C(K)\backslash \{P^\prime\} \subseteq \operatorname{Span}(P)$.
Now let $R \in \ell(K)\backslash \{P,P^\prime\}$. Let $Q_1 \in C(K)\backslash \{P,P^\prime\}$ and let $\ell^\prime$ be the $K$-line connecting $R$ and $Q_1$. Then $\ell^\prime \cdot S=R+Q_1+Q_2$ where $Q_2 \in C(K)\backslash \{P,P^\prime\}$. By the above, $Q_1$, $Q_2 \in \operatorname{Span}(P)$. Hence $R \in \operatorname{Span}(P)$. This shows that $\ell(K) \backslash \{P^\prime\}
\subseteq \operatorname{Span}(P)$.
Finally we must show that $P^\prime \in \operatorname{Span}(P)$. There is nothing to prove if $P =P^\prime$. Thus suppose $P \ne P^\prime$. Note that $\Pi_P=\Pi_{P^\prime}$. By the proof of Lemma \[lem:paraline\], the morphism $\gamma_\ell$ that sends points on $\ell$ to their tangent planes has degree $2$, so for any $R \in \ell \backslash \{P,P^\prime\}$, $P^\prime$ will be a non-singular point of $\Gamma_R$. The line $\ell$ meets precisely $10$ other lines lying on $S$ by Theorem \[thm:27\]. The assumption that $\#K \ge 13$ forces the existence of $R \in \ell(K)$, different from $P$ and $P^\prime$, and not lying on any line. Using the above argument with $R$ instead of $P$ we see that $P^\prime \in \operatorname{Span}(R)$. However, $R \in \operatorname{Span}(P)$. This completes the proof.
We now relax the hypotheses of Lemma \[lem:linegen\]
\[lem:linegen2\] Let $\ell$ be a $K$-line on $S$. Let $P \in \ell(K)$ and suppose that $P$ is not an Eckardt point. Then $$\ell(K) \subseteq \Gamma_P(K) \subseteq \operatorname{Span}(P).$$
To ease notation, we write $\ell_1$ for the line $\ell$. If $P$ does not lie on any other line then this follows from Lemma \[lem:linegen\]. Thus we may suppose that $P \in \ell_2$ where $\ell_1 \ne \ell_2$ but is not Eckardt. So $\ell_1$ and $\ell_2$ are distinct asymptotic lines at $P$, and so $P$ is non-parabolic. By Lemma \[lem:paraline\], the point $P$ must be $K$-hyperbolic, and so $\ell_2$ is defined over $K$. Now $\Gamma_P=\ell_1 \cup \ell_2 \cup \ell_3$ where $\ell_3 \subset S$ is a $K$-line not passing through $P$. Write $P_{ij}=\ell_i \cdot \ell_j$; thus $P_{12}=P$ and the points $P_{ij}$ are distinct. Since the field is large enough, there is a point $Q \in \ell_3(K)$ such that $Q \ne P_{13}$, $P_{23}$, and $Q$ does not lie on any other line contained in $S$. By Lemma \[lem:linegen\] we know that $\ell_3(K) \subseteq \operatorname{Span}(Q)$. But if we let $\ell_Q$ be the line joining $Q$ with $P_{12}$, then $\ell_Q\cdot S=2P_{12}+Q$ and so $Q \in \operatorname{Span}(P_{12})$. Thus $\ell_3(K) \subseteq \operatorname{Span}(P_{12})$. Similarly $\ell_1(K) \subseteq \operatorname{Span}(P_{23})$ and $\ell_2(K) \subseteq \operatorname{Span}(P_{13})$. However, $P_{23}$, $P_{13} \in \ell_3(K)$. It follows that $\ell_i(K) \subseteq \operatorname{Span}(P_{12})$ for $i=1$, $2$, $3$. This completes the proof.
\[lem:twolinegen\] Suppose $S$ contains two skew $K$-lines $\ell$, $\ell^\prime$. Then $$\ell^\prime(K) \subset \operatorname{Span}(\ell(K)).$$
As in Section \[sec:class\], let $\gamma_\ell$ be the map that sends a point $P \in \ell$ to the tangent plane $\Pi_P$ to $S$ at $P$. We know by the proof of Lemma \[lem:paraline\] that $\gamma_\ell$ has degree $2$. Since $\ell \subset \Pi_P$ and $\ell$, $\ell^\prime$ are skew, we see that $\ell^\prime \cdot \Pi_P$ is a single point on $\ell^\prime$. Thus we can think of $\gamma_\ell$ as a map $\ell \rightarrow \ell^\prime$ given by $P \mapsto \ell^\prime \cdot \Pi_P$.
We claim the existence of some $P \in \ell(K)$ that is not Eckardt such that $\gamma_\ell(P)$ is not Eckardt. Assume this for the moment. It is clear that $\gamma_\ell(P) \in \Gamma_P(K)$. By Lemma \[lem:linegen2\], we see that $\gamma_\ell(P) \in \operatorname{Span}(P)$ and $\ell^\prime(K) \subseteq \operatorname{Span}(\gamma_\ell(P))$. Thus to complete the proof it is enough to establish our claim.
By Lemma \[lem:paraline\], $\ell^\prime$ contains at most $5$ Eckardt points and so it is enough to show that $$\# \gamma_\ell \left( \ell(K) \backslash \{ \text{Eckardt points}\} \right)
\geq 6.$$ Suppose first that $\gamma_\ell$ is separable. The Eckardt points on $\ell$ are at most $2$ by Lemma \[lem:paraline\]. As $\gamma_\ell$ has degree $2$, $$\# \gamma_\ell \left( \ell(K) \backslash \{ \text{Eckardt points}\} \right)
\geq \frac{\#\ell(K) -2}{2} \geq 6.$$ Finally suppose $\gamma_\ell$ is inseparable. Then $\gamma_\ell : \ell(K) \rightarrow \ell^\prime(K)$ is injective. Now there are $5$ Eckardt points on $\ell$ and $$\# \gamma_\ell \left( \ell(K) \backslash \{ \text{Eckardt points}\} \right)
\geq \#\ell(K) -5 \geq 9.$$
I am grateful to Damiano Testa for pointing out to me that a cubic surface containing skew $K$-lines $\ell$, $\ell^\prime$ is birational to $\ell\times \ell^\prime$ over $K$. This in essence is what the proof of the following lemma is using.
\[lem:birattriv\] Suppose $S$ contains a pair of skew lines $\ell_1$ and $\ell_2$ both defined over $K$. Then $$\operatorname{Span}\left(\ell_1(K)\cup \ell_2(K) \right)=S(K).$$
Let $P$ be a $K$-point on $S$ not belonging to either line; we will show that $P$ belongs to the span of $\ell_1(K) \cup \ell_2(K)$. Let $\Pi_1$ be the unique plane containing $\ell_2$ and $P$, and $\Pi_2$ the unique plane containing $\ell_1$ and $P$. Since $\ell_1$ and $\ell_2$ are skew we know that $\ell_i \not \subset \Pi_i$. Write $Q_i=\ell_i \cap \Pi_i$. Note that $P$, $Q_1$ and $Q_2$ are distinct points on $S$ that also belong to the $K$-line $\ell=\Pi_1 \cap \Pi_2$. Suppose first that $\ell \not \subset S$. Then $\ell\cdot S=P+Q_1+Q_2$. Thus $P \in \operatorname{Span}\left(\ell_1(K)\cup \ell_2(K) \right)$ as required.
Next suppose that $\ell \subset S$. Then $\ell \subset \Gamma_{Q_1}$. If $Q_1$ is not Eckardt, then by Lemma \[lem:linegen2\], $$P \in \ell(K) \subseteq \Gamma_{Q_1}(K) \subseteq \operatorname{Span}(Q_1) \subseteq \operatorname{Span}(\ell_1(K)).$$ Thus we may assume that $Q_1$ is Eckardt. Then $\Gamma_{Q_1}=\ell \cup \ell_1 \cup \ell_3$ where $\ell_3$ is also $K$-rational. Now, as $\ell_1$ and $\ell_2$ are skew, $\ell_2$ meets the plane $\Pi_{Q_1}$ in precisely one point and this is $Q_2$. In particular, $\ell_2$ and $\ell_3$ are skew. Thus $\ell_3(K) \subseteq \operatorname{Span}(\ell_2(K))$ by Lemma \[lem:twolinegen\]. Now let $Q \in \ell_1(K) \backslash \{Q_1\}$. The line connecting $Q$ and $P$ meets $\ell_3$ in a $K$-point $R$, and so $$P \in \operatorname{Span}(\ell_1(K) \cup \ell_3(K)) \subseteq \operatorname{Span}(\ell_1(K) \cup \ell_2(K)).$$
This follows from Lemmas \[lem:linegen2\], \[lem:twolinegen\] and \[lem:birattriv\].
Preliminaries on $H_S(K)$ {#sec:pre}
=========================
We now begin our study of the group $H_S(K)$. Our eventual aim is to prove Theorems \[thm:two\] and \[thm:three\], but we will see other reasons why $H_S(K)$ is a worthwhile object of study.
\[lem:line\] Let $C \subset S$ be either a $K$-line, a smooth plane conic defined over $K$, or an irreducible plane cuspidal or nodal cubic defined over $K$. Suppose $P$, $Q\in S(K)$ are points lying on $C$. Moreover, in the cuspidal or nodal cubic case, suppose that $P$, $Q$ are non-singular points on $C$. Then $[P-Q]=0$ in $H_S(K)$.
Suppose first that $C=\ell$ is a $K$-line lying on $S$ and $P$, $Q$ are $K$-points lying on $\ell$. By definition of $G_S^\prime(K)$, we see that $3P$, $2P+Q \in G_S^\prime(K)$. Thus $[P-Q]=[3P-(2P+Q)]$ is zero in $H_S(K)$.
Next suppose that $C \subset S$ is a smooth plane conic defined over $K$, and let $\Pi$ be the plane containing $C$. Then $S \cap \Pi=C \cup \ell$ where $\ell$ is a $K$-line. Let $P$, $Q$ be $K$-points on $C$. The line joining $P$ and $Q$ meets $\ell$ in a $K$-point $P^\prime$. Likewise, the tangent to $C$ at $Q$ meets $\ell$ in a $K$-point $Q^\prime$. Thus $P+Q+P^\prime$, $2Q+Q^\prime \in G_S^\prime(K)$. Hence $$[P-Q]=[(P+Q+P^\prime)-(2Q+Q^\prime)]+[Q^\prime-P^\prime]=[Q^\prime-P^\prime].$$ But by the previous part, $[Q^\prime]=[P^\prime]$ as both points are on $\ell \subset S$. Hence $[P-Q]=0$.
Finally, suppose $C\subset S$ is a cuspidal or nodal plane cubic defined over $K$, and let $R \in S(K)$ be the singular point on $C$. Let $P$, $Q \in S(K)$ be non-singular points on $C$. By considering the lines joining $P$ with $R$ and $Q$ with $R$, we see that $P+2R$, $Q+2R \in G_S^\prime(K)$. Then $[P-Q]=0$.
The reader will recall that whilst any rational point on a cubic curve can be taken as the zero element, the description of the group law is simpler and more pleasant if this point is a flex. We shall now introduce a cubic surface analogue of flexes. Let $P \in S(K)$. We shall say that $P$ is [*$K$-ternary*]{} if at least one of the asymptotic lines to $S$ through $P$ is defined over $K$.
If $P$ is $K$-hyperbolic then $P$ is $K$-ternary. If $P$ is Eckardt then $P$ is $K$-ternary. If $P$ is parabolic and $\operatorname{char}(K) \ne 2$ then $P$ is $K$-ternary.
The first part follows from the definition of $K$-hyperbolic.
Suppose $P$ is Eckardt. Of course each of the three lines through $P$ contained in $S$ is an asymptotic line. So we may suppose that none of these is $K$-rational. Now let $\ell$ be any $K$-line in $\Pi_P$ through $P$. Then $(\ell \cdot S)_P=3$ and $\ell$ is an asymptotic $K$-line as required.
Finally, suppose $P$ is parabolic and $\operatorname{char}(K) \ne 2$. We may suppose that $P$ is non-Eckardt. Then $S$ has a unique asymptotic line $\ell$ through $P$. The asymptotic lines are obtained by solving a quadratic equation, and as there is exactly one solution and $\operatorname{char}(K) \ne 2$ the solution must be $K$-rational.
\[thm:terngen\] Suppose $P_0 \in S(K)$ is $K$-ternary. Let $B\subseteq S(K)$ such that $\operatorname{Span}(B)=S(K)$. Then the set $\{[P-P_0] : P \in B \}$ generates $H^0_S(K)$. In particular, for any prime $p$, $$r(S,K) \geq \dim_{{\mathbb{F}}_p} H^0_S(K)/p H^0_S(K).$$
Note that if $\ell$ is a $K$-rational line and $\ell\cdot S=P+Q+R$ with $P$, $Q$, $R \in S(K)$ then $P+Q+R$ and $3 P_0$ both belong to $G_S^\prime (K)$ and so $[P-P_0]+[Q-P_0]+[R-P_0]=0$. The first part of the lemma follows easily from this and the fact that $H^0_S(K)$ is generated by elements of the form $[P-P_0]$ with $P \in S(K)$.
For the second part, note that any set which generates $H^0_S(K)$ also generates the ${\mathbb{F}}_p$-vector space $H^0_S(K)/p H^0_S(K)$.
\[thm:line\] Suppose $S$ contains a $K$-rational line $\ell$. Then $H^0_S(K) = H_S(K)[2]$, where $H_S(K)[2]$ is the $2$-torsion subgroup of $H_S(K)$.
Let $Q$ be a $K$-rational point on $\ell$. We claim that $2P+Q \in G^\prime_K$ for all $P \in S(K)$. Assume our claim for a moment. Now $H^0_S(K)$ is generated by classes $[P^\prime - P]$ with $P$, $P^\prime \in S(K)$. By our claim, $2(P^\prime-P)=(2P^\prime+Q)-(2P+Q)$ is an element of degree $0$ in $G^\prime_S(K)$, proving the theorem.
To prove our claim let $P \in S(K)$. If $P \in \ell$ then $2P+Q \in G^\prime_S(K)$ by definition of $G^\prime_S(K)$. Thus suppose $P \notin \ell$ and let $\Pi_P$ be the tangent plane to $S$ at $P$. Now either $\Pi_P$ contains $\ell$ or it meets $\ell$ in precisely one $K$-rational point. In either case, there is some $K$-rational point $Q^\prime \in \Pi_P \cap \ell$. Since $P$ is not on $\ell$, we have $P \ne Q^\prime$. Let $\ell^\prime$ be the unique $K$-line connecting $P$ and $Q^\prime$. Now $\Gamma_P=\Pi_P \cap S$ is singular at $P$. Thus $\ell^\prime$ is tangent to $S$ at $P$, and hence $2P+Q^\prime \in G^\prime_S(K)$. However $Q-Q^\prime \in G^\prime_S(K)$ by Lemma \[lem:line\]. This proves the claim.
The proof of the following theorem is in essence a simplification of the proof of Lemma \[lem:birattriv\].
\[thm:twolines\] Suppose $S$ contains a pair of skew $K$-lines. Then $H_S^0(K)=0$.
This can in fact be easily recovered from Theorem \[thm:mw\] if $\# K \geq 13$. In any case, let $\ell_1$ and $\ell_2$ be a pair of skew $K$-lines contained in $S$. Fix some $Q_i \in \ell_i(K)$. We claim that $P+Q_1+Q_2 \in G_S^\prime(K)$ for every $P \in S(K)$. Let us assume this for the moment. By the proof of Theorem \[thm:line\], we know that $2P+Q_2 \in G_S^\prime(K)$. Hence $P-Q_1 \in G_K^{\prime\prime}$; in other words $P-Q_1 \equiv 0$ in $H_S^0(K)$. Since classes of the form $[P-Q_1]$ generate $H_S^0(K)$, we see that $H_S^0(K)=0$.
It remains to prove our claim. Note, by Lemma \[lem:line\], any two points on $\ell_i$ are equivalent. Moreover, if $P \in \ell_1(K)$ then all we have to show is that $2Q_1+Q_2 \in G_S^\prime(K)$ which is true from the proof of Theorem \[thm:line\]. Thus we may suppose that $P \notin \ell_i$, $i=1$, $2$. Let $\Pi_1$ be the unique plane containing $\ell_2$ and $P$, and $\Pi_2$ the unique plane containing $\ell_1$ and $P$. Since $\ell_1$ and $\ell_2$ are skew we know that $\ell_i \not \subset \Pi_i$. Write $Q_i=\ell_i \cap \Pi_i$. Note that $P$, $Q_1$ and $Q_2$ are distinct points on $S$ that also belong to to the $K$-line $\Pi_1 \cap \Pi_2$. By the definition of $G_S^\prime(K)$ we have $P+Q_1+Q_2 \in G^\prime_S(K)$.
\[cor:ac\] Let $K$ be algebraically closed. Then $H^0_S(K)=0$.
$H_S(K)$ for Local Fields $K$ {#sec:local}
=============================
In this section we let $K$ be a local field. This means that $K={\mathbb{R}}$ or ${\mathbb{C}}$, or that $K$ is a finite extension of ${\mathbb{Q}}_p$ for some prime $p$. The purpose of this section is to prove the following theorem.
\[thm:local\] Let $K$ be a local field and let $S$ be a smooth cubic surface over $K$. Then the following hold.
1. The group $H_S^0(K)$ is finite.
2. The natural map $S(K) \rightarrow H_S(K)$ given by $P \mapsto [P]$ is locally constant, where $S(K)$ has the topology induced by $K$.
Of course, if $S(K)$ is empty there is nothing to prove, so we shall suppose $S(K) \neq \emptyset$. Then $S(K)$ is a $2$-dimensional $K$-manifold, and in particular has $K$-points not lying on the $27$ lines. Such a $K$-point is called a [*general point*]{}.
The proof of Theorem \[thm:local\] requires a series of lemmas. The key lemma is the following.
\[lem:key\] There exists some non-empty $W \subset S(K)$, open in the topology induced by $K$, such that for all $P_1$, $P_2 \in W$ we have $[P_1]=[P_2]$ in $H_S(K)$.
It is well-known that the existence of a general $K$-point makes $S$ unirational, and makes the set $S(K)$ dense in the Zariski topology. One proof of this is found in Manin’s book [@Ma1 Chapter II, Theorem 12.11]. Our lemma follows from a modification, given below, of Manin’s proof.
Let $R$ be a general $K$-point on $S$. Observe that the plane cubic curve $\Gamma_R$ is irreducible, defined over $K$ and singular only at $R$. Let ${\mathcal{T}}_S$ be the projectivized tangent bundle of $S$. In other words, ${\mathcal{T}}_S$ parametrizes pairs $(Q,\ell)$ where $Q$ is a point on $S$ and $\ell$ is a line tangent to $S$ at $Q$. As the bundle ${\mathcal{T}}_S$ is locally trivial, there is a Zariski open subset $U \subset S$ containing $R$ and a local isomorphism $U \times {\mathbb{P}}^1 \rightarrow {\mathcal{T}}_S$ defined over $K$. Remove from $U$ the $27$ lines on $S$. Then the local isomorphism induces a morphism $\phi : U \times {\mathbb{P}}^1 \rightarrow S$ defined over $K$ as follows: if $Q\in U$, $\alpha \in {\mathbb{P}}^1$ and $\ell_\alpha$ is the line tangent to $S$ at $Q$ corresponding to $\alpha$ then let $\phi(Q,\alpha)$ be the third point of intersection of $\ell$ with $S$. Consider the restriction of $\phi$ to $(\Gamma_R \cap U) \times {\mathbb{P}}^1$. The image must be irreducible and contains $\Gamma_Q$ for all $Q \in \Gamma_R \cap U$ (including $\Gamma_R$). If the image is $1$-dimensional then $\Gamma_Q =\Gamma_R$ which is impossible for $Q \ne R$ since $Q$ is non-singular in $\Gamma_R$ but singular in $\Gamma_Q$. This shows that $(\Gamma_R \cap U) \times {\mathbb{P}}^1 \rightarrow S$ is dominant.
Let $V$ be a non-empty subset of $\Gamma_R(K) \cap U(K)$, open in the topology induced by $K$, such that $R \notin V$. Then $\phi(V \times {\mathbb{P}}^1(K))$ contains a non-empty subset $W$ open in $S(K)$. To complete the proof it is enough to show that the map $P \mapsto [P]$ is constant on $\phi(V \times {\mathbb{P}}^1(K))$. Suppose $P_1$, $P_2$ are in this set. So $P_i+2 Q_i = \ell_i \cdot S$ for some $K$-lines $\ell_1$, $\ell_2$ and for some $Q_1$, $Q_2 \in V$. Now $Q_1$, $Q_2 \in \Gamma_R$, but neither is equal to $R$. Thus $[Q_1]=[Q_2]$ by Lemma \[lem:line\]. It follows that $[P_1]=[P_2]$ as desired.
\[lem:local\] The map $S(K) \rightarrow H_S(K)$ given by $P \mapsto [P]$ is locally constant.
By Lemma \[lem:key\], there is some non-empty open $W \subset S(K)$ on which the map $P \mapsto [P]$ is constant. We shall prove the lemma by using secant operations to cover $S(K)$ by translates of $W$.
Let $P_0 \in S(K) \backslash W$. Choose $Q_0 \in W$ such that the $K$-line $\ell$ joining $Q_0$ and $P_0$ is not contained in $S$ and is not tangent to $S$ at either point. Write $R$ for the $K$-point such that $\ell \cdot S=R+Q_0+P_0$. Consider the rational map $$t_R : S \dashrightarrow S, \qquad
\text{$P \mapsto Q$ if $P$, $Q$ and $R$ are collinear}.$$ Clearly $t_R$ restricts to a local homeomorphism of neighbourhoods of $P_0$ and $Q_0$. Thus there is some open $V$ containing $P_0$, contained in the domain of this local homeomorphism, such that $t_R(V) \subset W$. Now let $P \in V$. Then $Q_0+P_0+R \in G_S^\prime(K)$ and $t_R(P)+P+R \in G_S^\prime(K)$. Thus $[Q_0-t_R(P)]+[P_0-P]=0$. But $[t_R(P)]=[Q_0]$ as both $t_R(P)$ and $Q_0$ are in $W$. Thus $[P]=[P_0]$. Hence $P \mapsto [P]$ is constant on the open neighbourhood $V$ of $P_0$.
By Lemma \[lem:local\] there is a covering of $S(K)$ by open sets $U$ such that the function $P \mapsto [P]$ is constant when restricted to $U$. As $S(K)$ is compact, we can assume that we have finitely many such $U$, say $U_1,\dotsc,U_m$. Fix $P_i \in U_i$ for $i=1,\dotsc,m$. The abelian group $H_S^0(K)$ is then generated by the differences $[P_i-P_j]$.
It will be sufficient to show that for any two points $P$, $Q \in S(K)$ the class $[P-Q]$ has finite order. Let $\ell$ be the $K$-line joining $P$ and $Q$. If $\ell \subset S$ then $[P-Q]=0$ by Lemma \[lem:line\], so suppose that $\ell \not \subset S$. Let $\Pi$ be a $K$-plane through $\ell$ that misses all the lines on $S$ and such that the irreducible plane cubic $E =S \cap \Pi$ is non-singular at $P$ and $Q$.
If $E$ is singular, then $[P-Q]=0$ by Lemma \[lem:line\]. Thus we may suppose that $E$ is non-singular. Consider the elliptic curve $(E,Q)$. For an integer $n$, there is a unique $R_n \in E(K)$ such that $n(P-Q) \sim R_n-Q$, where $\sim$ denotes linear equivalence on $E$. Now the group operations on $(E,Q)$ are given by secants and tangents, so we know that $n[P-Q]=[R_n-Q]$. However, from the properties of elliptic curves over local fields, we may choose values $n \neq 0$ so that $R_n$ is arbitrarily close to $Q$. But we already know that if $R_n$ is sufficiently close to $Q$ then $[R_n]=[Q]$. This completes the proof.
$H_S({\mathbb{R}})$ {#sec:real}
===================
Let $S$ be a smooth cubic surface defined over ${\mathbb{R}}$. We quickly summarise some well-known facts; for details see [@PT] and [@SJS]. The cubic surface $S$ contains 3, 7, 15 or 27 real lines, and $S({\mathbb{R}})$ has either one or two connected components. If $S({\mathbb{R}})$ has one connected component then this component is non-convex. If $S({\mathbb{R}})$ has two connected components then one is convex and the other non-convex. The non-convex component contains all the real lines.
\[thm:real\] If $S({\mathbb{R}})$ consists of one connected component then $H_S^0({\mathbb{R}})=0$. If $S({\mathbb{R}})$ consists of two connected components then $H_S^0({\mathbb{R}}) \cong {\mathbb{Z}}/2{\mathbb{Z}}$. This isomorphism may be given explicitly as follows. Let $P_0$ be a point on the non-convex component. The map $$[P-P_0] \mapsto \begin{cases}
\overline{0} & \text{if $P$ is in the non-convex component} \\
\overline{1} & \text{if $P$ is in the convex component}\\
\end{cases}$$ extends to an isomorphism $H_S^0({\mathbb{R}}) \cong {\mathbb{Z}}/2{\mathbb{Z}}$.
By Theorem \[thm:local\], the map $P \mapsto [P]$ is locally constant, and hence constant on each connected component. Now $H_S^0({\mathbb{R}})$ is generated by classes of differences $[P-Q]$, and so if there is only one component then $H_S^0({\mathbb{R}})=0$.
Now suppose $S({\mathbb{R}})$ has two components. To see that the map given in the theorem extends to an isomorphism $H_S^0({\mathbb{R}}) \cong {\mathbb{Z}}/2{\mathbb{Z}}$, it is enough to observe that if $P$, $Q$, $R$ are collinear real points on $S$, then either all three are on the non-convex component, or precisely one is on the non-convex component.
[**Remark.**]{} I am grateful to the referee for the following remark. By a result of Colliot-Thélène and Ischebeck [@CI], the degree $0$ part of $\operatorname{CH}_0(S/{\mathbb{R}})$ is isomorphic $({\mathbb{Z}}/2{\mathbb{Z}})^{s-1}$, where $s$ is the number of real components. It follows from this and Theorem \[thm:real\] that the natural map $H_S({\mathbb{R}}) \rightarrow \operatorname{CH}_0(S/{\mathbb{R}})$ is an isomorphism.
Weak Approximation and $H_S$ {#sec:weak}
============================
In this section $K$ denotes a number field and $\Omega$ the set of places of $K$. Denote the adèles of $K$ by ${\mathbb{A}}_K$. As usual, $S$ is a smooth cubic surface over $K$, but we shall further suppose that $S({\mathbb{A}}_K) \ne \emptyset$. We say $S$ satisfies [*weak approximation*]{} if the image of $S(K)$ in $S({\mathbb{A}}_K)$ is dense. More generally, let $\Sigma$ be a finite subset of $\Omega$, and denote by ${\mathbb{A}}_K^\Sigma$ the adèles of $K$ with the $\Sigma$-components removed. We say that $S$ satisfies [*weak approximation*]{} away from $\Sigma$ if the image of $S(K)$ in $S({\mathbb{A}}_K^\Sigma)$ is dense. If we assume Colliot-Thélène’s Conjecture \[conj\], then it is easy in any given case to write down a finite set of places $\Sigma$ such that $S$ satisfies weak approximation away from $\Sigma$. We shall not do this in general, but only for the surfaces $S_M$ and $S_M^\prime$ in Theorems \[thm:two\] and \[thm:three\].
\[thm:surject\] Let $K$ be a number field and $\Omega$ its places. Let $\Sigma$ be a subset of $\Omega$ and let ${\mathbb{A}}_K^\Sigma$ denote the adèles of $K$ with the $\Sigma$ components removed. Suppose that the image of $S(K)$ in $S({\mathbb{A}}_K^\Sigma)$ is dense. Let $\Delta$ be a finite subset of $\Omega \backslash \Sigma$. Then the diagonal map $$\label{eqn:diag}
H^0_S(K) \rightarrow \prod_{\upsilon \in \Delta} H^0_S(K_\upsilon)$$ is surjective.
The target space of the homomorphism in is generated by elements of the form $\left([P_\upsilon-Q_\upsilon]\right)_{\upsilon\in \Delta}$, thus it is enough to show that such elements are in the image. By the hypotheses, $S(K)$ is dense in $\prod_{\upsilon \in \Delta} S(K_\upsilon)$. Recall, from Theorem \[thm:local\], that the maps $R \rightarrow [R]$ are locally constant on the $S(K_\upsilon)$. Thus choosing $P$, $Q \in S(K)$ that sufficiently approximate $(P_\upsilon)_{\upsilon\in \Delta}$ and $(Q_\upsilon)_{\upsilon\in \Delta}$ will give an element $[P-Q] \in H_S^0(K)$ whose image under is $\left([P_\upsilon-Q_\upsilon]\right)_{\upsilon\in \Delta}$.
At first sight it seems that this theorem enables us to disprove the Mordell-Weil conjecture for cubic surfaces simply by taking the set $\Delta$ to be arbitrarily large and forcing $H_S^0(K)$ to surject onto larger and larger groups. However, extensive—though not systematic—experimentation with cubic surfaces, which we do not describe here, suggests that $H^0_S(K_\upsilon)=0$ if $\upsilon$ is a non-Archimedean place of good reduction for $S$ and $\upsilon \nmid 2$. By a place of good reduction we mean a non-Archimedean place $\upsilon$ such that the polynomial defining $S$ has $\upsilon$-integral coefficients, and the reduction of that polynomial modulo $\upsilon$ defines a smooth cubic surface over the residue field. We remark that for any place $\upsilon$ of good reduction for $S$, it is known (e.g. [@CT2 Theorem A]) that the degree $0$ part of the Chow group vanishes for $S \times K_\upsilon$.
We shall use Theorem \[thm:surject\] to prove Theorems \[thm:two\] and \[thm:three\]. The first step is to prove weak approximation for the surfaces $S_M/{\mathbb{Q}}$ in Theorem \[thm:two\] and weak approximation away from $\{3\}$ for the surfaces $S_M^\prime/{\mathbb{Q}}$ in Theorem \[thm:three\]; in the latter case our result will be conditional on Colliot-Thélène’s Conjecture \[conj\]. To prove weak approximation we introduce the Brauer-Manin obstruction and study it for the surfaces $S_M$ and $S_M^\prime$.
Brauer-Manin Obstruction: A Brief Overview {#sec:BM}
==========================================
To proceed further we need to recall some facts about the Brauer-Manin obstruction; for fuller details see [@Sk Section 5.2]. We continue with the notation of the previous section: $K$ is a number field, ${\mathbb{A}}_K$ the adèles of $K$ and $\Omega$ the set of places of $K$. The [*Hasse reciprocity law*]{} states that the following sequence of abelian groups is exact: $$0 \rightarrow \operatorname{Br}(K) \rightarrow \sum_{{{\upsilon}}\in \Omega} \operatorname{Br}(K_{{\upsilon}}) \rightarrow {\mathbb{Q}}/{\mathbb{Z}}\rightarrow 0.$$ Here the third map is the sum of local invariants $\operatorname{inv}_{{\upsilon}}: \operatorname{Br}(K_{{\upsilon}}) \hookrightarrow {\mathbb{Q}}/{\mathbb{Z}}$. Let $X$ be a smooth, projective and geometrically integral variety over a number field $K$. Let $\operatorname{Br}(X)$ be the Brauer group of $X$ and denote by $\operatorname{Br}_0(X)$ the image of $\operatorname{Br}(K)$ in $\operatorname{Br}(X)$. Consider the pairing $$\langle~,~\rangle : \operatorname{Br}(X) \times X({\mathbb{A}}_K) \rightarrow {\mathbb{Q}}/{\mathbb{Z}},
\qquad
\langle A, ( P_{{\upsilon}}) \rangle =\sum_{{{\upsilon}}\in \Omega} \operatorname{inv}_{{\upsilon}}(A(P_{{\upsilon}})).$$ This is the [*adelic Brauer-Manin pairing*]{} and satisfies the following properties.
1. If $A \in \operatorname{Br}_0(X) \subset \operatorname{Br}(X)$ and $(P_{{\upsilon}}) \in X({\mathbb{A}}_K)$ then $\langle A , (P_{{\upsilon}}) \rangle=0$.
2. If $P \in X(K)$ then $\langle A , P \rangle=0$ for every $A \in \operatorname{Br}(X)$.
3. For any $A \in \operatorname{Br}(X)$, the map $$X({\mathbb{A}}_K) \rightarrow {\mathbb{Q}}/{\mathbb{Z}}, \qquad
(P_{{\upsilon}}) \mapsto \langle A , (P_{{\upsilon}}) \rangle$$ is continuous where ${\mathbb{Q}}/{\mathbb{Z}}$ is given the discrete topology.
We define $$X({\mathbb{A}}_K)^{\operatorname{Br}(X)} =
\{ (P_{{\upsilon}}) \in X({\mathbb{A}}_K) :
\text{ $\langle A , (P_{{\upsilon}}) \rangle=0$ for all $A \in \operatorname{Br}(X)/\operatorname{Br}_0(X)$} \}.$$ By the above we know that $$\overline{X(K)} \subseteq X({\mathbb{A}}_K)^{\operatorname{Br}(X)},$$ where $\overline{X(K)}$ is the closure of $X(K)$. We say that [*the Brauer-Manin obstruction is the only obstruction to weak approximation*]{} if $\overline{X(K)}=X({\mathbb{A}}_k)^{\operatorname{Br}(X)}$.
The Brauer-Manin Obstruction for $S_M$ {#sec:BMSM}
======================================
In this section we prove the following proposition.
\[prop:dense\] Let $p_1,\dots,p_s$ ($s \geq 1$) and $M$ be as in Theorem \[thm:two\]. Let $S=S_M/{\mathbb{Q}}$ be the cubic surface given by $\eqref{eqn:one}$. Then $S$ satisfies weak approximation. In particular, the homomorphism $$H^0_{S}({\mathbb{Q}}) \rightarrow \prod_{p=p_1}^{p=p_s} H^0_S({\mathbb{Q}}_p)$$ is surjective.
To prove this proposition we shall need a theorem of Salberger and Skorobogatov on del Pezzo surfaces of degree $4$. The cubic surface $S_M$ is birational to a degree $4$ del Pezzo $X$ given by the following smooth intersection of two quadrics in ${\mathbb{P}}^4$: $$X_M : \begin{cases}
x^2-xy+y^2+zt=0, \\
z^2+Mw^2 -xt-yt=0.
\end{cases}$$ The map $X_M \dashrightarrow S_M$ is the obvious one $(x,y,z,w,t) \mapsto (x,y,z,w)$.
$\operatorname{Br}(X_M)/\operatorname{Br}_0(X_M)$ is trivial.
To determine $\operatorname{Br}(X_M)/\operatorname{Br}_0(X_M)$ we can use the recipe in [@Swd1] or the more detailed recipe in [@BBFL]. The first step is to write down the $16$ lines on $X_M$. We did this by writing down and solving the equations for the corresponding zero-dimensional Fano scheme (see for example [@EH Section IV.3]). Write $\theta=\sqrt[3]{2}$ and let $\zeta$ be a primitive cube root of unity. There are two Galois orbits of lines. The first orbit has four lines and a representative is $$L_1 :
\begin{cases}
x+\zeta y=0, \\
z+\sqrt{-M} w=0, \\
t=0.\\
\end{cases}$$ The second orbit has 12 lines and a representative is $$L_2 :
\begin{cases}
-\theta z+ \theta \sqrt{-M}w + t=0,\\
3 \theta^2 x + (2\zeta - 2)\theta z + (\zeta + 2) t=0, \\
3 \theta^2 y + (-2 \zeta - 4) \theta z + (-\zeta + 1) t=0.\\
\end{cases}$$ From the size of these orbits we know [@BBFL Proposition 13] that $\operatorname{Br}(X_M)/\operatorname{Br}_0(X_M)$ is trivial.
We shall also need the following theorem.
(Salberger and Skorobogatov [@SS Theorem 6.5]) Let $K$ be a number field and $X$ a del Pezzo surface of degree $4$ over $K$ containing a rational point. Then $X(K)$ is dense in $ X({\mathbb{A}}_K)^{\operatorname{Br}(X)}$.
By the above theorem of Salberger and Skorobogatov we know that $X_M$ satisfies weak approximation. Now $S_M$ is birational to $X_M$, and so by [@SS Lemma 5.5(c)] also satisfies weak approximation. The last part of the proposition follows from Theorem \[thm:surject\].
The Brauer-Manin Obstruction for $S_M^\prime$ {#sec:BMSMd}
=============================================
In this section we prove the following proposition.
\[prop:densed\] Let $p_1,\dots,p_s$ and $M$ be as in Theorem \[thm:three\]. Let $\Sigma=\{3\}$. Let $S=S_M^\prime/{\mathbb{Q}}$ be the cubic surface in and suppose that the Brauer-Manin obstruction is the only obstruction to weak approximation on $S$. Then $S$ satisfies weak approximation away from $\{3\}$. In particular, the homomorphism $$H_S^0({\mathbb{Q}}) \rightarrow \prod_{p=p_1}^{p=p_s} H_S^0({\mathbb{Q}}_p)$$ is surjective.
All the results we need for this proof are due to Colliot-Thélène, Kanevsky and Sansuc [@CKS], though Jahnel’s Habilitation summarizes these results in one convenient theorem [@Jahnel Chapter III, Theorem 6.4]. Indeed, we know that
1. $\operatorname{Br}(S)/\operatorname{Br}_0(S) \cong {\mathbb{Z}}/3{\mathbb{Z}}$. Fix $A \in \operatorname{Br}(S)$ that represents a non-trivial coset of $\operatorname{Br}(S)/\operatorname{Br}_0(S)$.
2. The image of $$\langle~,~\rangle : \operatorname{Br}(S) \times S({\mathbb{A}}_{\mathbb{Q}}) \rightarrow {\mathbb{Q}}/{\mathbb{Z}}$$ is $\frac{1}{3} {\mathbb{Z}}/{\mathbb{Z}}$.
3. The map $$S({\mathbb{Q}}_p) \rightarrow \frac{1}{3} {\mathbb{Z}}/{\mathbb{Z}}, \qquad P \mapsto \operatorname{inv}_p(A,P)$$ is surjective for all $p \mid M$.
Now the strategy is clear. Suppose that $\mathcal{P}=(P_{{\upsilon}}) \in S({\mathbb{A}}_{\mathbb{Q}}^\Sigma)$. Choose $P_3 \in S({\mathbb{Q}}_3)$ such that $$\operatorname{inv}_3(A,P_3)=-\sum_{{{\upsilon}}\neq 3} \operatorname{inv}_{{\upsilon}}(A,P_{{\upsilon}}).$$ Let $\mathcal{P}^\prime \in S({\mathbb{A}}_{\mathbb{Q}})$ be the point obtained from $\mathcal{P}$ by taking $P_3$ to be the component at $3$. Then $\langle A, \mathcal{P}^\prime \rangle=0$. Since $A$ generates $\operatorname{Br}(S)/\operatorname{Br}_0(S)$ we know that $\mathcal{P}^\prime \in S({\mathbb{A}}_{\mathbb{Q}})^{\operatorname{Br}(S)}$. By our assumption that the Brauer-Manin obstruction is the only one to weak approximation we have that $\mathcal{P}^\prime$ is in the closure of the rational points in $S({\mathbb{A}}_{\mathbb{Q}})$. The proposition follows.
A Plane Cubic {#sec:C}
=============
We shall need to study the reduction of the cubic surfaces and at the primes $p \mid M$. We note that modulo $p$ they both reduce to cones over the plane cubic curve $x^3+y^3+z^3=0$. In this section we collect some information we need regarding the Picard group of this cubic curve.
In this section $K$ is a field of characteristic $\neq 3$. Throughout the rest of the paper, $C/{\mathbb{Q}}$ will denote the plane genus $1$ curve given by . Note that with the restriction imposed on the characteristic, $C \times K$ is smooth. Let ${{\mathcal O}}=(1:-1:0) \in C$. We note that ${{\mathcal O}}$ is a flex and so $C$ can be put into Weierstrass form by a projective transformation that sends ${{\mathcal O}}$ to the point at $\infty$ (see for example [@Knapp Proposition 2.14]). It follows for $P$, $Q$, $R \in C(K)$ that $P+Q+R-3{{\mathcal O}}=0$ in $\operatorname{Pic}^0(C\times K)$ if and only if there is $K$-line $\ell$ in ${\mathbb{P}}^2$ such that $\ell \cdot C=P+Q+R$.
\[lem:mod3\] Let $p \equiv 1 \pmod{3}$ be a prime and write $C_p$ for $C \times {\mathbb{F}}_p$. Then $$\dim_{{\mathbb{F}}_3} \operatorname{Pic}^0(C_p)/3\operatorname{Pic}^0(C_p)=2.$$ Moreover, each of the $9$ elements of $\operatorname{Pic}^0(C_p)/3\operatorname{Pic}^0(C_p)$ can be represented by the class of $P-{{\mathcal O}}$ for some $P \in C({\mathbb{F}}_p)$.
$\operatorname{Pic}^0(C_p)$ is a finite abelian group isomorphic to $E({\mathbb{F}}_p)$ where $E$ is the elliptic curve $(C,{{\mathcal O}})$. Hence we have isomorphisms $$\operatorname{Pic}^0(C_p)/3\operatorname{Pic}^0(C_p) \cong \operatorname{Pic}^0(C_p)[3] \cong E({\mathbb{F}}_p)[3],$$ the first of which is of course non-canonical. The assumption that $p \equiv 1 \pmod{3}$ ensures that the nine flex points of $C$ are defined over ${\mathbb{F}}_p$; these have the form $(1 : -\omega :0)$ and permutations of these coordinates, where $\omega^3=1$. Now the $3$-torsion in $\operatorname{Pic}^0(C_p)$ consists of the nine classes $[{{\mathcal O}}^\prime-{{\mathcal O}}]$ where ${{\mathcal O}}^\prime$ is a flex point. Hence $$\operatorname{Pic}^0(C_p)/3\operatorname{Pic}^0(C_p) \cong {\mathbb{Z}}/3{\mathbb{Z}}\oplus {\mathbb{Z}}/3{\mathbb{Z}}.$$ This proves the first part of the lemma. The second part of the lemma follows since every element of $\operatorname{Pic}^0(C_p)$ is the class of $P-{{\mathcal O}}$ for some $P\in C({\mathbb{F}}_p)$.
\[lem:mod2\] Let $p \equiv 1 \pmod{3}$ be a prime such that $2$ is a cube modulo $p$. Then $$\dim_{{\mathbb{F}}_2} \operatorname{Pic}^0(C_p)/2\operatorname{Pic}^0(C_p)=2.$$ Moreover, each of the $4$ elements of $\operatorname{Pic}^0(C_p)/2 \operatorname{Pic}^0(C_p)$ can be represented by the class of $P-{{\mathcal O}}$ for some $P \in C({\mathbb{F}}_p)$.
We can put $E=(C,{{\mathcal O}})$ in Weierstrass form by sending ${{\mathcal O}}$ to the point at infinity. A Weierstrass model is $$y^2 + y = x^3 - 7.$$ The $2$-division polynomial of this model is $4x^3-27$. The assumptions on $p$ ensure that the $2$-division polynomial splits completely over ${\mathbb{F}}_p$; thus $E$ has full $2$-torsion over ${\mathbb{F}}_p$.
Reduction of Lines {#sec:red}
==================
In what follows we would like to conveniently parametrize a ${\mathbb{Q}}_p$-line $\ell$ in ${\mathbb{P}}^3$. Of course if $P$ and $Q$ are two distinct points on $\ell$ then $\ell({\mathbb{Q}}_p)=\{ s P+ t Q : (s:t) \in {\mathbb{P}}^1({\mathbb{Q}}_p)\}$. Now if $S$ is a cubic surface given by $F=0$ then the points of $\ell \cdot S$ correspond to the roots of $F(sP+tQ)$ with the correct multiplicity. The problem with such a parametrization is that it is possible that $\overline{P}=\overline{Q}$ and so we do not obtain a parametrization of $\overline{\ell}$ by reducing the parametrization of $\ell$. In this brief section we indicate how to make a good choice of parametrization. It will be convenient to use the identification of lines in ${\mathbb{P}}^3$ with planes in $4$-dimensional space passing through the origin. Let $V_\ell$ be the $2$-dimensional ${\mathbb{Q}}_p$-subspace of ${\mathbb{Q}}_p^4$ generated by the points of $\ell({\mathbb{Q}}_p)$. Let $W_\ell=V_\ell \cap {\mathbb{Z}}_p^4$. This $W_\ell$ is a ${\mathbb{Z}}_p$-module of rank $2$ and we let ${\mathbf{u}}$, ${\mathbf{v}}$ be a ${\mathbb{Z}}_p$-basis. Then the line $\ell$ can be parametrized as $s {\mathbf{u}}+t {\mathbf{v}}$, and $\overline{\ell}$ as $s \overline{{\mathbf{u}}}+t \overline{{\mathbf{v}}}$. We shall call this a [*good*]{} parametrization for $\ell$. We note that if $P\in \ell({\mathbb{Q}}_p)$ then there is a coprime pair of $p$-adic integers $\lambda$, $\mu$ such that $P=\lambda {\mathbf{u}}+\mu{\mathbf{v}}$; this can be easily seen by regarding $P$ as a primitive element of $W_\ell$.
We now turn our attention to the surfaces of Theorems \[thm:two\] and \[thm:three\]. Let $M$ denote a non-zero squarefree integer and let $S$ be either of the surfaces in or . Let $p \ne 3$ be a prime dividing $M$; we would like to study the group $H_S({\mathbb{Q}}_p)$. Denote the reduction of $S$ modulo $p$ by $\overline{S}$. It is seen that $\overline{S}$ is a cone over $C_p=C \times {\mathbb{F}}_p$, where $C$ is given in , with vertex at $(\overline{0}: \overline{0} : \overline{0} : \overline{1})$. It is convenient to split the set of $p$-adic points on $S$ into subsets of bad and good reduction: $${S^{\mathrm{bd}}}=\{P \in S({\mathbb{Q}}_p) : \overline{P}=(\overline{0}:\overline{0}:\overline{0}:\overline{1})\},
\qquad
{S^{\mathrm{gd}}}=\{P \in S({\mathbb{Q}}_p) : \overline{P} \neq (\overline{0}:\overline{0}:\overline{0}:\overline{1})\}.$$ We note however that ${S^{\mathrm{bd}}}=\emptyset$ in the case of the surface $S_M^\prime$ of . Now we can think of points on ${S^{\mathrm{gd}}}$ as reducing to points on $C({\mathbb{F}}_p)$. We define $\phi : {S^{\mathrm{gd}}}\rightarrow C({\mathbb{F}}_p)$ by $\phi(x:y:z:w)=(\overline{x}:\overline{y}:\overline{z})$ where the four coordinates $x$, $y$, $z$, $w$ are taken to be coprime in ${\mathbb{Z}}_p$.
\[lem:Sdotlred\] Let $M$ denote a non-zero squarefree integer and $S$ either of the surfaces in or . Let $p \neq 3$ be a prime dividing $M$. Let $\ell$ be a ${\mathbb{Q}}_p$-line such that $\ell \cdot S=P_1+P_2+P_3$ with $P_i \in {S^{\mathrm{gd}}}$. Suppose $\overline{\ell} \not \subset \overline{S}$. Then $\phi(P_1)+\phi(P_2)+\phi(P_3) \sim 3 \overline{{{\mathcal O}}}$ in $\operatorname{Pic}(C_p)$.
Let $s {\mathbf{u}}+t{\mathbf{v}}$ be a good parametrization of $\ell$ in the above sense. There are coprime pairs $\lambda_i$, $\mu_i \in {\mathbb{Z}}_p$ for $i=1$, $2$, $3$ such that $P_i=\lambda_i {\mathbf{u}}+ \mu_i {\mathbf{v}}$ for $i=1$, $2$, $3$. Since $\ell \cdot S=P_1+P_2+P_3$ we have $$\label{eqn:Sdotl}
F(s{\mathbf{u}}+t{\mathbf{v}})=\alpha \prod_{i=1}^3(\mu_i s -\lambda_i t)$$ for some $\alpha$ in ${\mathbb{Z}}_p$. Moreover $\alpha \not \equiv 0 \pmod{p}$ as $\overline{\ell} \not \subset \overline{S}$. Write ${\mathbf{u}}=(u_0,u_1,u_2,u_3)$ and ${\mathbf{v}}=(v_0,v_1,v_2,v_3)$ and let ${\mathbf{u}}^\prime=(\overline{u_0},\overline{u_1},\overline{u_2})$ and ${\mathbf{v}}^\prime=(\overline{v_0},\overline{v_1},\overline{v_2})$. We first show that ${\mathbf{u}}^\prime$ and ${\mathbf{v}}^\prime$ are ${\mathbb{F}}_p$-linearly independent. If not, then $\overline{\ell}$ passes through the vertex $(\overline{0}:\overline{0}:\overline{0}:\overline{1})$ as well as the $\overline{P_i}$ which are distinct from the vertex. This forces $\overline{\ell} \subset \overline{S}$ contradicting the lemma’s assumption that $\overline{\ell} \not \subset \overline{S}$. Thus ${\mathbf{u}}^\prime$ and ${\mathbf{v}}^\prime$ are ${\mathbb{F}}_p$-linearly independent.
Let $\ell^\prime \subset {\mathbb{P}}^2$ be the ${\mathbb{F}}_p$ line $s {\mathbf{u}}^\prime +t {\mathbf{v}}^\prime$. Reducing modulo $p$ we instantly see that $\ell^\prime \cdot C_p=\phi(P_1)+\phi(P_2)+\phi(P_3)$ which concludes the proof of the lemma.
A Homomorphism for $H_{S_M}({\mathbb{Q}}_p)$ {#sec:SM}
============================================
In this section $M$ denotes a non-zero squarefree integer, and $S$ the cubic surface denoted by $S_M$ in : $$S : x^3+y^3+z(z^2+M w^2)=0.$$ Let $C$ be the plane cubic curve . Again, let $p$ be a prime divisor of $M$ different from $3$ and let $C_p=C \times {\mathbb{F}}_p$. In this section we shall define a surjective homomorphism $H_S({\mathbb{Q}}_p) \rightarrow \operatorname{Pic}^0(C_p)/2\operatorname{Pic}^0(C_p)$. As previously observed, the reduction of $S$ modulo $p$ is a cone over $C_p$ with vertex at $ (\overline{0}:\overline{0}:\overline{0}:\overline{1})$. What makes the situation here somewhat tricky is that this singular point lifts to $p$-adic (and even rational) points on $S$; for example $(0:0:0:1) \in S({\mathbb{Q}})$ is such a lift. Thus ${S^{\mathrm{gd}}}$ is strictly smaller than $S({\mathbb{Q}}_p)$. We shall extend $\phi:{S^{\mathrm{gd}}}\rightarrow C({\mathbb{F}}_p)$ defined in Section \[sec:red\] to $\phi:S({\mathbb{Q}}_p) \rightarrow C({\mathbb{F}}_p)$ where $\phi(P)=\overline{{{\mathcal O}}}$ for any $P \in {S^{\mathrm{bd}}}$.
\[prop:SM\] Let $\psi : S({\mathbb{Q}}_p) \rightarrow \operatorname{Pic}^0(C_p)$ be given by $\psi(P)=\phi(P)-\overline{{{\mathcal O}}}$. Then $\psi$ induces a well-defined surjective homomorphism $$\psi : H_S({\mathbb{Q}}_p) \rightarrow \operatorname{Pic}^0(C_p)/2 \operatorname{Pic}^0(C_p),
\qquad \psi([P])=\psi(P) \pmod{2 \operatorname{Pic}^0(C_p)}.$$
Before proving Proposition \[prop:SM\] we shall need the following three lemmas.
\[lem:zzero\] Suppose $P\in S({\mathbb{Q}}_p)$ such that $\overline{P}$ lies in the plane $z=\overline{0}$. Then $\psi(P) \in 2 \operatorname{Pic}^0(C_p)$.
If $P \in {S^{\mathrm{bd}}}$ then $\psi(P)=0$. Otherwise, $\phi(P)$ is a flex point on $C_p$. Hence $\psi(P)=\phi(P)-\overline{{{\mathcal O}}}$ is a difference of two flexes and so is an element of order dividing $3$ in $\operatorname{Pic}^0(C_p)$. It follows that $\psi(P) \in 2 \operatorname{Pic}^0(C_p)$: indeed $\psi(P)=-2\psi(P)$.
\[lem:badline\] Let $\ell$ be a ${\mathbb{Q}}_p$-line not contained in $S$. Suppose that $\ell \cdot S=P_1+P_2+P_3$ where $P_i \in S({\mathbb{Q}}_p)$. Suppose moreover that $\overline{\ell} \subset \overline{S}$. Then $\psi_1(P)+\psi_2(P)+\psi_3(P) \in 2\operatorname{Pic}^0(C_p)$.
Let $s {\mathbf{u}}+t{\mathbf{v}}$ be a good parametrization of $\ell$. In particular we know that ${\mathbf{u}}$, ${\mathbf{v}}\in {\mathbb{Z}}_p^4$ and $\overline{{\mathbf{u}}}$, $\overline{{\mathbf{v}}}$ are independent modulo $p$, and that there are coprime pairs $\lambda_i$, $\mu_i$ such that $P_i=\lambda_i {\mathbf{u}}+\mu_i {\mathbf{v}}$. Now $\overline{\ell}$ is contained in $\overline{S}$ and so must pass through the vertex $(\overline{0}:\overline{0} : \overline{0} : \overline{1})$. Thus, applying a unimodular transformation, we may assume that our pair ${\mathbf{u}}$, ${\mathbf{v}}$ have the form $$\label{eqn:uv}
{\mathbf{u}}=(u_1,u_2,u_3,0), \qquad {\mathbf{v}}=(p v_1, p v_2, p v_3, 1), \qquad
u_i,~v_i \in {\mathbb{Z}}_p.$$ Let $F=x^3+y^3+z^3+Mz w^2$. We can also assume that $F({\mathbf{v}}) \ne 0$, by replacing ${\mathbf{v}}$ with ${\mathbf{v}}+\alpha p {\mathbf{u}}$ for an appropriate $\alpha \in {\mathbb{Z}}_p$. It follows that the polynomial $F({\mathbf{u}}+t {\mathbf{v}})$ is of degree $3$ and has precisely $3$ roots in ${\mathbb{Q}}_p$: namely $\mu_i/\lambda_i$ for $i=1,2,3$. However, expanding $F({\mathbf{u}}+t{\mathbf{v}})$ we obtain $$\label{eqn:F}
\sum u_i^3+3p\left(\sum u_i^2 v_i\right) t+
\left(3 p^2 \sum u_i v_i^2+M u_3\right)t^2+
\left(p^3 \sum v_i^3+pM v_3\right)t^3.$$ Write $F({\mathbf{u}}+t{\mathbf{v}})=a_0+a_1 t +a_2 t^2+a_3 t^3$, and let $\alpha_i=\operatorname{ord}_p(a_i)$. Since $\overline{\ell} \subset \overline{S}$ we see that $\alpha_i \geq 1$. Suppose first that $p \nmid u_3$. As $M$ is squarefree and $p \mid M$, we see that $$\alpha_0 \geq 1, \qquad \alpha_1 \geq 1,
\qquad \alpha_2=1, \qquad \alpha_3 \geq 2.$$ We shall need to study the Newton polygon of this polynomial which is the convex hull of the four points $(i,\alpha_i)$; see for example [@Koblitz page 19]. The Newton polygon contains precisely one segment of positive slope joining $(2,1)$ with $(3,\alpha_3)$. The other segments have non-positive slope. Hence the polynomial has precisely one root with negative valuation and two roots in ${\mathbb{Z}}_p$. By reordering the $P_i$ we may suppose that $\mu_1/\lambda_1$ has negative valuation and $\mu_i/\lambda_i \in {\mathbb{Z}}_p$ for $i=2$, $3$. From the expressions $P_i=\lambda_i {\mathbf{u}}+\mu_i {\mathbf{v}}$ we see that $P_1 \in {S^{\mathrm{bd}}}$ and $P_2$, $P_3 \in {S^{\mathrm{gd}}}$. Moreover $\phi(P_2)=\phi(P_3)=(u_1 : u_2 :u_3) \in C({\mathbb{F}}_p)$. Hence $\psi(P_1)+\psi(P_2)+\psi(P_3) \in 2 \operatorname{Pic}^0(C_p)$. Finally we must deal with the case $p \mid u_3$. In this case $\overline{P}_i$ lie in the plane $z=\overline{0}$. The lemma follows from Lemma \[lem:zzero\].
\[lem:ol\] Let $\ell$ be a ${\mathbb{Q}}_p$-line contained in $S$. For every $P\in \ell({\mathbb{Q}}_p)$, we have $\psi(P) \in 2 \operatorname{Pic}^0(C_p)$.
Since $\overline{\ell}$ passes through the vertex of $\overline{S}$ we may parametrize as $s {\mathbf{u}}+t {\mathbf{v}}$ where ${\mathbf{u}}$, ${\mathbf{v}}$ are as in . Now the polynomial in vanishes identically. From the coefficient of $t^2$ we see that $p \mid u_3$. Hence $\overline{\ell}$ lies in the plane $z=\overline{0}$. Now the lemma follows from Lemma \[lem:zzero\].
In fact, more is true: the only ${\mathbb{Q}}_p$-lines on $S$ are contained in the $z=0$ plane, but we do not need this.
Hensel’s Lemma shows that the map $\phi: S^{\mathrm{gd}} \rightarrow C({\mathbb{F}}_p)$ is surjective, and thus $\psi$ is surjective. Therefore, it is enough to show that $\psi(P_1)+\psi(P_2)+\psi(P_3) \in 2 \operatorname{Pic}^0(C_p)$ whenever $P_1$, $P_2$, $P_3 \in S({\mathbb{Q}}_p)$ and
1. either there is a ${\mathbb{Q}}_p$-line $\ell$ not contained in $S$ with $\ell \cdot S=P_1+P_2+P_3$,
2. or there is a ${\mathbb{Q}}_p$-line $\ell$ contained in $S$ with $P_1$, $P_2$, $P_3 \in \ell$.
For (ii) we know by Lemma \[lem:ol\] that $\psi(P_i) \in 2 \operatorname{Pic}^0(C_p)$, so it remains to deal with (i). Thus suppose that $\ell$ is a ${\mathbb{Q}}_p$-line not contained in $S$ and $\ell \cdot S=P_1+P_2+P_3$ where $P_i \in S({\mathbb{Q}}_p)$. If all three $P_i \in {S^{\mathrm{bd}}}$ then the required result follows from the definition of $\psi$. Suppose that at least one $P_i\in {S^{\mathrm{gd}}}$.
- If all three $P_i \in {S^{\mathrm{gd}}}$ and $\overline{\ell}
\not \subset \overline{S}$ then we can conclude using Lemma \[lem:Sdotlred\].
- If all three $P_i \in {S^{\mathrm{gd}}}$ and $\overline{\ell} \subset \overline{S}$ we can conclude using Lemma \[lem:badline\].
We have reduced to the case where at least one of the $P_i$ is in ${S^{\mathrm{gd}}}$ and at least one is in ${S^{\mathrm{bd}}}$. This forces $\overline{\ell} \subset \overline{S}$ and again we can conclude using Lemma \[lem:badline\].
Proof of Theorem \[thm:two\] {#sec:proofone}
============================
In this section we shall put together the results of the previous sections to prove Theorem \[thm:two\]. Thus let $p_1,\dots,p_s$ ($s \geq 1$) be primes such that $p_i \equiv 1 \pmod{3}$ and $2$ is a cube modulo each $p_i$. Let $M=\prod p_i$ and $S=S_M$ be the cubic surface in .
For $p=p_i$, Proposition \[prop:SM\] gives a surjective homomorphism $$\psi : H_S({\mathbb{Q}}_{p}) \rightarrow \operatorname{Pic}^0(C_{p})/2\operatorname{Pic}^0(C_{p}).$$ Now $\psi(1:-1:0:0)=0$, thus the restriction of $\psi$ to $H_S^0({\mathbb{Q}}_p)$ is still surjective. In particular, $$\dim_{{\mathbb{F}}_2} H^0_S({\mathbb{Q}}_p)/2 H^0_S({\mathbb{Q}}_p) \ge \dim_{{\mathbb{F}}_2} \operatorname{Pic}^0(C_{p})/2\operatorname{Pic}^0(C_{p})=2$$ by Lemma \[lem:mod2\]. However, by Proposition \[prop:dense\], the map $$H^0_S({\mathbb{Q}})/2H^0_S({\mathbb{Q}}) \rightarrow \prod_{p=p_1}^{p=p_s} H^0_S({\mathbb{Q}}_p)/2 H^0_S({\mathbb{Q}}_p)$$ is surjective, and so $$\dim_{{\mathbb{F}}_2} H^0_S({\mathbb{Q}})/2 H^0_S({\mathbb{Q}}) \ge 2s.$$ As $S$ contains the ${\mathbb{Q}}$-rational line $x+y=z=0$, we know by Theorem \[thm:line\] that $H^0_S({\mathbb{Q}}) = H_S({\mathbb{Q}})[2]$. Thus $\dim_{{\mathbb{F}}_2} H_S({\mathbb{Q}})[2] \geq 2s$. Now the point $(1:-1:0:0)$ is ${\mathbb{Q}}$-ternary on $S$. Thus we can apply Theorem \[thm:terngen\] obtaining $r(S,{\mathbb{Q}})\geq \dim_{{\mathbb{F}}_2} H_S({\mathbb{Q}})[2] \geq 2s$, which proves Theorem \[thm:two\].
A Homomorphism for $H_{S^\prime_M}({\mathbb{Q}}_p)$ {#sec:SMd}
===================================================
In this section $M>1$ denotes a squarefree integer and $S$ the cubic surface denoted by $S^\prime_M$ in : $$S : x^3+y^3+z^3+Mw^3=0.$$ Let $C$ be the plane cubic curve . Let $p$ be a prime divisor of $M$ different from $3$ and $\phi : S({\mathbb{Q}}_p) \rightarrow C({\mathbb{F}}_p)$ as defined in Section \[sec:red\]; as previously observed for this surface, ${S^{\mathrm{bd}}}=\emptyset$ and so ${S^{\mathrm{gd}}}=S({\mathbb{Q}}_p)$. Let $\psi : S({\mathbb{Q}}_p) \rightarrow \operatorname{Pic}^0(C_p)$ be given by $\psi(P)=\phi(P)-\overline{{{\mathcal O}}}$.
\[prop:SMd\] Let $C_p=C \times {\mathbb{F}}_p$. Then $\psi$ extends uniquely to a well-defined surjective homomorphism $$\psi : H_S({\mathbb{Q}}_p) \rightarrow \operatorname{Pic}^0(C_p)/3 \operatorname{Pic}^0(C_p),
\qquad \psi([P])=\psi(P) \pmod{3 \operatorname{Pic}(C_p)}.$$
Surjectivity follows as in the proof of Proposition \[prop:SM\]. It is therefore enough to show that $\psi(P_1)+\psi(P_2)+\psi(P_3) \in 3 \operatorname{Pic}^0(C_p)$ whenever $P_1$, $P_2$, $P_3 \in S({\mathbb{Q}}_p)$ and
1. either there is a ${\mathbb{Q}}_p$-line $\ell$ not contained in $S$ with $\ell \cdot S=P_1+P_2+P_3$,
2. or there is a ${\mathbb{Q}}_p$-line $\ell$ contained in $S$ with $P_1$, $P_2$, $P_3 \in \ell$.
The second possibility does not arise since none of the $27$ lines on $S$ are defined over ${\mathbb{Q}}_p$. Thus suppose $\ell$ is a ${\mathbb{Q}}_p$-line not contained in $S$ such that $\ell \cdot S=P_1+P_2+P_3$. If $\overline{\ell} \not \subset \overline{S}$ then by Lemma \[lem:Sdotlred\] we know that $\phi(P_1)+\phi(P_2)+\phi(P_3) \sim 3 \overline{{{\mathcal O}}}$ in $\operatorname{Pic}(C_p)$. In this case $\psi(P_1)+\psi(P_2)+\psi(P_3)=0$ in $\operatorname{Pic}^0(C_p)$. Thus we may suppose that $\overline{\ell} \subset \overline{S}$. Hence $\overline{\ell}$ passes through the vertex $(\overline{0}:\overline{0}:\overline{0}:1)$, otherwise its projection on to the $xyz$-projective plane would be a line contained in irreducible curve $C_p$ which is impossible. It follows that $\phi(P_1)=\phi(P_2)=\phi(P_3)$ and thus $\psi(P_1)+\psi(P_2)+\psi(P_3) \in 3 \operatorname{Pic}^0(C_p)$.
Proof of Theorem \[thm:three\] {#sec:prooftwo}
==============================
This follows from Lemma \[lem:mod3\] and Propositions \[prop:densed\] and \[prop:SMd\] in exactly the same way as the proof of Theorem \[thm:two\] (Section \[sec:proofone\]).
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[^1]: The author is supported by an EPSRC Leadership Fellowship
|
---
abstract: 'While the local $L^p$-boundedness of nondegeneral Fourier integral operators is known from the work of Seeger, Sogge and Stein [@SSS], not so many results are available for the global boundedness on $L^p(\R^n)$. In this paper we give a sufficient condition for the global $L^p$-boundedness for a class of Fourier integral operators which includes many natural examples. We also describe a construction that is used to deduce global results from the local ones. An application is given to obtain global $L^p$-estimates for solutions to Cauchy problems for hyperbolic partial differential equations.'
address: ' Michael Ruzhansky: Department of Mathematics Imperial College London 180 Queen’s Gate, London SW7 2AZ, UK Mitsuru Sugimoto: Graduate School of Mathematics Nagoya University Furocho, Chikusa-ku, Nagoya 464-8602, Japan '
author:
- Michael Ruzhansky and Mitsuru Sugimoto
title: |
Global regularity properties for\
a class of Fourier integral operators
---
Introduction {#S1}
============
In this article, we discuss the global $L^p$-boundedness of the Fourier integral operators $$\mathcal P u(x)
=\int_{\R^n}\int_{\R^n} e^{i\phi(x,y,\xi)}
a(x,y,\xi) u(y)\, dy d\xi\quad(x\in \R^n).$$ We always assume that $n\geq 1$ and $1<p<\infty$. Here $\phi(x,y,\xi)$ is a real-valued function that is called a [*phase function*]{} while $a(x,y,\xi)$ is called an [*amplitude function*]{}. Following the theory of Fourier integral operators by Hörmander [@Ho], we originally assume that $\phi(x,y,\xi)$ is positively homogeneous of order $1$ and smooth at $\xi\neq0$, and that $a(x,y,\xi)$ is smooth and satisfies a growth condition in $\xi$ with some $\kappa\in\R$: $$\sup_{(x,y)\in K}\abs{\partial_x^\alpha\partial_y^\beta\partial_\xi^\gamma
a(x,y,\xi)}
\leq C_{\alpha\beta\gamma}^K\jp{\xi}^{\kappa-|\gamma|}\quad
(\forall \alpha, \beta, \gamma)\,;\quad
\langle \xi\rangle=\p{1+|\xi|^2}^{1/2}$$ for any compact set $K\subset\R^n\times\R^n$. Then the operator $\mathcal P$ is just a microlocal expression of the corresponding Lagrangian manifold, and with the local graph condition, it is microlocally equivalent to the special form $$\label{EQ:P}
P u(x)=
\int_{\R^n}\int_{\R^n} e^{i(x\cdot\xi-\varphi(y,\xi))}a(x,y,\xi)u(y)\, dy d\xi$$ by an appropriate microlocal change of variables.
The local $L^p$ mapping properties of Fourier integral operators have been extensively studied, and can be generally summarised as follows:
- $\mathcal P$ is $L^2_{comp}$-$L^2_{loc}$-bounded when $\kappa\leq 0$ (Hörmander [@Ho], Eskin [@Es]);
- $\mathcal P$ is $L^p_{comp}$-$L^p_{loc}$-bounded when $\kappa\leq -(n-1)|1/p-1/2|$, $1<p<\infty$ (Seeger, Sogge and Stein [@SSS]);
- $\mathcal P$ is $H^1_{comp}$-$L^1_{loc}$-bounded when $\kappa\leq -\frac{n-1}{2}$ (Seeger, Sogge and Stein [@SSS]), where here and everywhere $H^1=H^1(\R^n)$ is the Hardy space introduced by Fefferman and Stein [@FS];
- $\mathcal P$ is locally weak $(1,1)$ type when $\kappa\leq -\frac{n-1}{2}$ (Tao [@Tao:weak11]).
The sharpness of order the $-(n-1)|1/p-1/2|$ was shown by Miyachi [@Mi] and Peral [@Pe] (see also [@SSS]). Therefore, the question addressed in this paper is when Fourier integral operators are globally $L^p$-bounded. Although the operator $\mathcal P$ or $P$ is just a microlocal expression of the corresponding Lagrangian manifold due to the Maslov cohomology class (see e.g. Duistermaat [@Duistermaat:FIO-book-1996]), we still regard it as a globally defined operator since it is still important for the applications to the theory of partial differential equations. Indeed, the operator $P$ is used to:
- express solutions to Cauchy problems of hyperbolic equations;
- transform equations to another simpler one (Egorov’s theorem).
For example, the solution to the wave equation $$\left\{
\begin{array}{ll}
\partial_t^2 u(t,x) - \Delta u(t,x) = 0,\ t \in \R,\ x \in \R^n, \\
u(0,x) = g(x),\ \partial_t u(0,x) = 0,\ x \in \R^n,
\end{array}
\right.$$ is expressed as a linear combination of the operators of the form $$P g(x)=\int_{\R^n} \int_{\R^n} e^{i(x\cdot\xi-y\cdot \xi \pm t|\xi|)} g(y) \, dyd\xi.$$ On the other hand, we have the relation $$P\cdot\sigma(D)=\p{\sigma\circ\psi}(D)\cdot P$$ if we take $a(x,y,\xi)=1$ and $\varphi(y,\xi)=y\cdot\psi(\xi)$ so that we can transform[^1] the operator $\sigma(D)$ to $\p{\sigma\circ\psi}(D)$ which might have been very well investigated. Summarising these situations, the typical two types of phase functions for each analysis are $$\text{
(I)\hspace{5mm} $\varphi(y,\xi)=y\cdot\xi+|\psi(\xi)|$,
\hspace{10mm}
(II)\hspace{5mm} $\varphi(y,\xi)=y\cdot\psi(\xi)$,}$$ where $\psi(\xi)$ is a real vector-valued smooth function which is positively homogeneous of order $1$ for large $\xi$. (See Definition \[Def:hom\] for the precise meaning of this terminology).
A global $L^2$-boundedness result of the operator $P$ with the phase function (I) was given by Asada and Fujiwara [@AF] which states it for rather general operators $\mathcal P$:
\[Th:AF\] Let $\phi(x,y,\xi)$ and $a(x,y,\xi)$ be $C^\infty$-functions, and let $$D(\phi):=
\begin{pmatrix}
\partial_x\partial_y\phi&\partial_x\partial_\xi\phi
\\
\partial_\xi\partial_y\phi&\partial_\xi\partial_\xi\phi
\end{pmatrix}.$$ Assume that $|\det D(\phi)|\geq C>0$. Also assume that every entry of the matrix $D(\phi)$, $a(x,y,\xi)$ and all their derivatives are bounded. Then $\mathcal P$ is $L^2(\R^n)$-bounded.
The result of [@AF] was used to construct the solution to the Cauchy problem of Schrödinger equations by means of the Feynman path integrals in Fujiwara [@Fu]. For the operator $P$, the conditions of Theorem \[Th:AF\] are reduced to a global version of the [*local graph condition*]{} $$\label{graphcond}
\abs{\det \partial_y\partial_\xi\varphi(y,\xi)}\geq C>0,$$ and the growth conditions $$\begin{aligned}
&\abs{\partial_y^\alpha\partial_\xi^\beta\varphi(y,\xi)} \le
C_{\alpha\beta}\quad
(\forall\, |\alpha+\beta|\geq2, |\beta|\geq1),
\\
&\abs{\partial_x^\alpha\partial_y^\beta\partial_\xi^\gamma a(x,y,\xi)}
\le C_{\alpha\beta\gamma}\quad
(\forall \alpha, \beta, \gamma),\end{aligned}$$ for all $x,y,\xi\in\R^n$. Note that the phase function (I) satisfies these conditions. We also note that the condition [(\[graphcond\])]{} is required even for the local $L^2$-boundedness of Fourier integral operators of order zero, so it is rather natural to assume it to hold globally on $\R^n$ as well.
We remark that Kumano-go [@Ku] also showed the same conclusion as that of Theorem \[Th:AF\] under weaker conditions on the phase function, namely for $$\abs{\partial_y^\alpha\partial_\xi^\beta (\varphi(y,\xi)-y\cdot\xi)}
\le C_{\alpha\beta}\jp{\xi}^{1-|\beta|}
\quad (\forall \alpha, \beta),$$ with applications to the global $L^2$ estimates for solutions to Cauchy problems of strictly hyperbolic equations.
Unfortunately the phase function (II) does not satisfy the growth condition in Theorem \[Th:AF\] because $\partial_\xi\partial_\xi\varphi$ are usually unbounded. Therefore, another type of conditions was introduced by the authors in [@RS] to obtain the global $L^2$-boundedness for operators with phases of the type (II). Such $L^2$-boundedness results were then used to show global smoothing estimates for dispersive equations in a series of papers [@RS2], [@RS3] and [@RS4]. Thus, the result covering the case (II) is as follows:
\[Th:RS\] Let $\varphi(y,\xi)$ and $a(x,y,\xi)$ be $C^\infty$-functions. Assume [(\[graphcond\])]{}. Also assume that $$\begin{aligned}
&\abs{\partial_y^\alpha\partial_\xi^\beta\varphi(y,\xi)} \leq
C_{\alpha\beta}\langle y\rangle^{1-|\alpha|}\jp{\xi}^{1-|\beta|}\quad
(\forall \alpha, \beta),
\\
&\abs{\partial_x^\alpha\partial_y^\beta\partial_\xi^\gamma a(x,y,\xi)}
\leq C_{\alpha\beta\gamma}\langle y\rangle^{-|\beta|}\quad
(\forall \alpha, \beta, \gamma),\end{aligned}$$ hold for all $x,y,\xi\in\R^n$. Then $P$ is $L^2(\R^n)$-bounded.
As for the global $L^p$-boundedness, Coriasco and Ruzhansky [@CR-CRAS; @CR] established the following result generalising Theorem \[Th:RS\] to the setting of $L^p$-spaces:
\[Th:CR\] Let $\varphi(y,\xi)$ and $a(x,y,\xi)$ be $C^\infty$-functions. Assume that $\varphi(y,\xi)$ is positively homogeneous of order $1$ for large $\xi$ and satisfies [(\[graphcond\])]{}. Also assume that $$\begin{aligned}
&\abs{\partial_y^\alpha\partial_\xi^\beta\varphi(y,\xi)} \leq
C_{\alpha\beta}\langle y\rangle^{1-|\alpha|}\jp{\xi}^{1-|\beta|}\quad
(\forall \alpha, \beta),
\\
&\abs{\partial_x^\alpha\partial_y^\beta\partial_\xi^\gamma a(x,y,\xi)}
\leq C_{\alpha\beta\gamma}
\langle x\rangle^{m_1-|\alpha|}
\langle y\rangle^{m_2-|\beta|}
\jp{\xi}^{-(n-1)|1/p-1/2|-|\gamma|}\quad
(\forall \alpha, \beta, \gamma),\end{aligned}$$ hold for all $x,y,\xi\in\R^n$, and that $a(x,y,\xi)$ vanishes around $\xi=0$. Then $P$ is $L^p(\R^n)$-bounded, for every $1<p<\infty$, provided that $m_1+m_2\leq -n|1/p-1/2|$.
In Theorem \[Th:CR\], the decay of order $-n|1/p-1/2|$ is required for amplitude functions in space variables. It is also shown in [@CR] that this order of decay is in general sharp: otherwise it is possible to construct an example of an operator that is not globally bounded on $L^p(\R^n)$. Thus, in the space $L^p(\R^n)$, in addition to the local loss of regularity of order $(n-1)|1/p-1/2|$, there is also the global loss of weight at infinity of order $n|1/p-1/2|$, and both of these losses are in general sharp. It is possible to improve the order of the weight loss a bit to the order $(n-1)|1/p-1/2|$ in a special case of so-called SG-Fourier integral operators, namely, for operators $$A u(x)=
\int_{\R^n} e^{i\varphi(x,\xi)}a(x,\xi)\widehat{u}(\xi)\, d\xi,$$ with amplitudes satisfying $$\abs{\partial_x^\alpha\partial_\xi^\gamma a(x,\xi)}
\leq C_{\alpha\gamma}\langle x\rangle^{-(n-1)|1/p-1/2|-|\alpha|}
\jp{\xi}^{-(n-1)|1/p-1/2|-|\gamma|}\quad
(\forall \alpha, \gamma).$$ If the function $a(x,\xi)$ vanishes near $\xi=0$, such operators are $L^p(\R^n)$-bounded, see [@CR Theorem 2.6] for the precise formulation.
However, despite the mentioned counter-example to the decay order in space variables, it may be natural to expect some further better properties, particularly for phase functions whose second derivatives $\partial_\xi\partial_\xi\varphi$ in $\xi$ are bounded like in the case (I). For example, for the special case of convolution operators given by $$T u(x)=
\int_{\R^n}\int_{\R^n} e^{i((x-y)\cdot\xi-\varphi(\xi))}a(\xi)u(y)\, dy d\xi,$$ where $\varphi\in C^\infty(\R^n)$ is positively homogeneous of order $1$ for large $\xi$ and $a\in C^\infty(\R^n)$ satisfies $$|\partial^\alpha a(\xi)|\leq C_\alpha\jp{\xi}^{\kappa-|\alpha|},$$ Miyachi [@Mi] showed that for $1<p<\infty$, the operator $T$ is $L^p(\R^n)$-bounded if $\kappa\leq -(n-1)|1/p-1/2|$ under the assumptions that $\varphi>0$ and that the compact hypersurface $$\Sigma=\b{\xi\in\R^n\setminus0\,:\,\varphi(\xi)=1}$$ has non-zero Gaussian curvature. Beals [@Be] and Sugimoto [@Su] discussed the case when $\Sigma$ might have vanishing Gaussian curvature but is still convex. But the local $L^p$-boundedness result by Seeger, Sogge and Stein [@SSS] suggests that it could be possible to remove any geometric condition on $\Sigma$. And indeed, in this paper we establish the following generalised result:
\[main\] Let $\varphi(y,\xi)$ and $a(x,y,\xi)$ be $C^\infty$-functions. Assume that $\varphi(y,\xi)$ is positively homogeneous of order $1$ for large $\xi$ and satisfies [(\[graphcond\])]{}. Also assume that $$\begin{aligned}
&\abs{\partial_y^\alpha\partial_\xi^\beta(y\cdot\xi-\varphi(y,\xi))}
\le
C_{\alpha\beta}\jp{\xi}^{1-|\beta|}\quad
(\forall\, \alpha, |\beta|\geq1),
\\
&\abs{\partial_x^\alpha\partial_y^\beta\partial_\xi^\gamma a(x,y,\xi)}
\leq C_{\alpha\beta\gamma}
\jp{\xi}^{ -(n-1)|1/p-1/2|-|\gamma|}\quad
(\forall \alpha, \beta, \gamma),\end{aligned}$$ hold for all $x,y,\xi\in\R^n$. Then $P$ is $L^p(\R^n)$-bounded, for every $1<p<\infty$.
Compared to Theorem \[Th:CR\], the assumptions on the phase $\varphi(y,\xi)$ as in Theorem \[main\] ensure that no decay of the amplitude $a(x,y,\xi)$ in the space variables is needed for the operator $P$ to be globally bounded on $L^p(\R^n)$.
Theorem \[main\] together with some related results will be restated in Section \[S2\] in a different form (in particular, Theorem \[main\] follows from Corollary \[Cor1\]), emphasising a general construction for deducing global results from the local ones. We now briefly explain the strategy of the global proof. By the interpolation and the duality, the problem is reduced to show the $H^1$-$L^1$-boundedness. To show the $H^1$-$L^1$-boundedness, we use the atomic decomposition of $H^1$: $$f=\sum_{j=1}^\infty\lambda_jg_j,\quad\lambda_j\in\C,\quad g_j: \textrm{ atom}.$$ Here we call a function $g$ on $\R^n$ an atom if there is a ball $B=B_g\subset\R^n$ such that $\supp g\subset B$, $\|g\|_{L^\infty}\leq|B|^{-1}$ and $\int g(x)\,dx=0$. This is the common starting point which is also used to show the known results [@SSS], [@Mi], [@Su], [@CR]. Modifying these results is not so straightforward but we present a new argument which allows to deduce the estimate for large atoms ($|B|\geq1$) from the global $L^2$-boundedness, and for small atoms ($|B|\leq1$) from the local $L^p$-boundedness. More details and proofs are given in Sections \[S3\] and \[S4\].
In Section \[S5\] we give applications of the obtained results to global $L^p$-estimates for solutions of Cauchy problems for hyperbolic partial differential equations. In [@CR], the global $L^p$-boundedness of solutions of such equations was established with a loss of weight at infinity. In Theorem \[THM:hyp1\] we show that this weight loss can be eliminated.
To complement some references on the local and global boundedness properties of Fourier integral operators, we refer to the authors’ paper [@RS-weighted:MN] for the weighted $L^2$- and to Dos Santos Ferreira and Staubach [@Staubach-Dos-Santos-Ferreira:local-global-FIOs-MAMS] for other weighted properties of Fourier integral operators, to Rodríguez-López and Staubach [@Rodriguez-Lopez-Staubach:rough-FIOs] for estimates for rough Fourier integral operators, to [@Ruzhansky:CWI-book] for $L^p$-estimates for Fourier integral operators with complex phase functions, as well as to [@Ruzhansky:FIOs-local-global] for an earlier overview of local and global properties of Fourier integral operators with real and complex phase functions. $L^{p}$-boundedness of bilinear Fourier integral operators has been also investigated, see Hong, Lu, Zhang [@GLU] and references therein.
In this paper we often abuse the notation slightly by writing, for example, $a(x,y,\xi)\in C^\infty$ instead of $a\in C^\infty$, to emphasise the dependence on particular sets of variables. We will also often write $\partial_\xi$ for $\nabla_\xi$.
Main results {#S2}
============
Let $\mathcal P$ be a Fourier integral operator of the form $$\label{FIOp}
\begin{aligned}
&\mathcal{P} u(x)=
\int_{\R^n}\int_{\R^n} e^{i\phi(x,y,\xi)}a(x,y,\xi)u(y)
\, dy d\xi\quad (x\in\R^n),
\\
&\phi(x,y,\xi)=(x-y)\cdot\xi+\Phi(x,y,\xi),
\end{aligned}$$ where $\Phi(x,y,\xi)$ is introduced just for convenience and we do not lose any generality with this notation. We introduce a class for the amplitude $a(x,y,\xi)$.
\[symbol\] For $\kappa\in\R$, $S^\kappa$ denotes the class of smooth functions $a=a(x,y,\xi)\in C^\infty(\R^n\times\R^n\times\R^n)$ satisfying the estimate $$\abs{
\partial^\alpha_x \partial^\beta_y \partial^\gamma_\xi a(x,y,\xi)
}
\leq C_{\alpha\beta\gamma}\jp{\xi}^{\kappa-|\gamma|}$$ for all $x,y,\xi\in\R^n$ and all multi-indices $\alpha,\beta,\gamma$.
We remark that the formal adjoint $\mathcal P^*$ of $\mathcal P$ is of the same form [(\[FIOp\])]{} with the replacement $$\label{adjoint}
\begin{aligned}
&\Phi(x,y,\xi)\longmapsto\Phi^*(x,y,\xi)=-\Phi(y,x,\xi),
\\
&a(x,y,\xi)\longmapsto a^*(x,y,\xi)=\overline{a(y,x,\xi)},
\end{aligned}$$ and $a\in S^\kappa$ is equivalent to $a^*\in S^\kappa$.
We also introduce a notion of the local boundedness of $\mathcal P$. By $\chi_K$ we denote the multiplication by the characteristic function of the set $K\subset\R^n$.
We say that the operator $\mathcal P$ is $H^1_{comp}(\R^n)$-$L^1_{loc}(\R^n)$-bounded if the localised operator $\chi_{K} \mathcal P \chi_{K}$ is $H^1(\R^n)$-$L^1(\R^n)$-bounded for any compact set $K\subset\R^n$. Furthermore, if the operator norm of $\chi_{K_h} \mathcal P \chi_{K_h}$ is bounded in $h\in\R^n$ for the translated set $K_h=\{x+h:x\in K\}$ of any compact set $K\subset\R^n$, we say that the operator $\mathcal P$ is uniformly $H^1_{comp}(\R^n)$-$L^1_{loc}(\R^n)$-bounded.
If we introduce the translation operator $\tau_h:f(x)\mapsto f(x-h)$ and its inverse (formal adjoint) $\tau_h^*=\tau_{-h}$, we have the expression $\chi_{K_h}=\tau_h\chi_{K}\tau_h^*$. Since $L^1$ and $H^1$ norms are translation invariant, $\mathcal P$ is uniformly $H^1_{comp}$-$L^1_{loc}$-bounded if and only if $\chi_{K}(\tau_h^* \mathcal P \tau_h)\chi_{K}$ is $H^1$-$L^1$-bounded for any compact set $K\subset\R^n$ and the operator norm is bounded in $h\in\R^n$. We remark that the operator $\tau_h^*\mathcal{P}\tau_h$ is of the form [(\[FIOp\])]{} with the replacements $$\label{translation}
\begin{aligned}
&\Phi(x,y,\xi)\longmapsto\Phi^h(x,y,\xi)=\Phi(x+h,y+h,\xi),
\\
&a(x,y,\xi)\longmapsto a^h(x,y,\xi)=a(x+h,y+h,\xi).
\end{aligned}$$
Now we are ready to state our main principle:
\[Th:main\] Assume the following conditions:
- $\Phi(x,y,\xi)$ is a real-valued $C^\infty$-function and $\partial^\gamma_\xi\Phi(x,y,\xi)\in S^0$ for $|\gamma|=1$.
- $\mathcal P$ and $\mathcal P^*$ are $L^2(\R^n)$-bounded if $a(x,y,\xi)\in S^{0}$.
- $\mathcal P$ and $\mathcal P^*$ are uniformly $H^1_{comp}(\R^n)$-$L^1_{loc}(\R^n)$-bounded if $a(x,y,\xi)\in S^{-(n-1)/2}$.
Then $\mathcal P$ is $L^p(\R^n)$-bounded if $1<p<\infty$, $\kappa\leq-(n-1)|1/p-1/2|$, and $a(x,y,\xi)\in S^\kappa$.
We remark that assumptions (A1) and (A2) are essentially the requirements for phase functions $\Phi(x,y,\xi)$, and a condition for (A2) is given by Asada and Fujiwara [@AF] or by the authors [@RS], while (A3) is given by Seeger, Sogge and Stein [@SSS]. These conditions will be discussed in Section \[S4\], and here we simply state the final conclusion by restricting our phase functions to the form $$\phi(x,y,\xi)=x\cdot\xi-\varphi(y,\xi)
\quad(\text{in other words
$\Phi(x,y,\xi)=y\cdot\xi-\varphi(y,\xi)$}).$$ We make precise the notion of homogeneity:
\[Def:hom\] We say that [*$\varphi(y,\xi)$ is positively homogeneous of order $1$*]{} if $$\label{EQ:hom}
\varphi(y,\lambda\xi)=\lambda\varphi(y,\xi)$$ holds for all $y\in\R^n$, $\xi\neq0$ and $\lambda>0$. We also say that [*$\varphi(y,\xi)$ is positively homogeneous of order $1$ for large $\xi$*]{} if there exist a constant $R>0$ such that [(\[EQ:hom\])]{} holds for all $y\in\R^n$, $|\xi|\geq R$ and $\lambda\geq1$.
For the operator of the form $$\label{FIOp1}
P u(x)=
\int_{\R^n}\int_{\R^n} e^{i(x\cdot\xi-\varphi(y,\xi))}a(x,y,\xi)u(y)\, dy d\xi
\quad (x\in\R^n)$$ we have:
\[Cor1\] Assume the following conditions:
- $\varphi(y,\xi)$ is a real-valued $C^\infty$-function and $\partial^\gamma_\xi(y\cdot\xi-\varphi(y,\xi))\in S^0$ for $|\gamma|=1$.
- There exists a constant $C>0$ such that $\abs{\det \partial_y\partial_\xi\varphi(y,\xi)}\geq C$ for all $y,\xi\in\R^n$.
- $\varphi(y,\xi)$ is positively homogeneous of order $1$ for large $\xi$.
Then $P$ is $L^p(\R^n)$-bounded if $1<p<\infty$, $\kappa\leq-(n-1)|1/p-1/2|$ and $a(x,y,\xi)\in S^\kappa$.
We can admit positively homogeneous phase functions which might have singularity at the origin for a special kind of operators of the form $$\label{FIOp2}
\mathcal T u(x)=\int_{\R^n} e^{i(x\cdot\xi+\psi(\xi))}a(x,\xi)\widehat{u}(\xi)
\, d\xi\quad (x\in\R^n).$$ For such operators we have
\[Cor2\] Let $1<p<\infty$ and let $\kappa\leq-(n-1)|1/p-1/2|$. Assume that $a=a(x,\xi)\in S^\kappa$ and that $\psi=\psi(\xi)$ is a real-valued $C^\infty$-function on $\R^n\setminus0$ which is positively homogeneous of order $1$. Then $\mathcal T$ is $L^p(\R^n)$-bounded.
The proofs of all results in this section will be given in subsequent sections. The proofs will follow from the global $H^1(\R^n)$-$L^1(\R^n)$-boundedness of the corresponding operators of order $-(n-1)/2$ by interpolation with condition (A2) and by duality. Therefore, among other things, in addition to the $L^p(\R^n)$-boundedness, we will obtain that all the appearing operators of order $-(n-1)/2$ are globally $H^1(\R^n)$-$L^1(\R^n)$-bounded and $L^\infty(\R^n)$-$BMO(\R^n)$-bounded.
Proof of Theorem \[Th:main\] {#S3}
============================
In this section we give the proof of Theorem \[Th:main\]. We only have to show the $L^p$-boundedness of the operator [(\[FIOp\])]{} assuming $a(x,y,\xi)\in S^\kappa$ with the critical case $\kappa=\kappa(p)$, where $$\kappa(p)=-(n-1)|1/p-1/2|.$$ On account of the observation [(\[adjoint\])]{} and the invariance of the assumptions $a=a(x,y,\xi)\in S^\kappa$ and (A1) under such replacement, we can restrict our consideration to the case $1<p<2$ by the duality argument. Furthermore, by assumption (A2) and the complex interpolation argument, we have only to show the $H^1$-$L^1$-boundedness of the operator $\mathcal P$ with $a(x,y,\xi)\in S^{\kappa(1)}$ under the assumptions
- (A1),
- $\mathcal P$ is $L^2$-bounded,
- $\mathcal P$ is uniformly $H^1_{comp}$-$L^1_{loc}$-bounded.
We remark that we still need the global $L^2$-boundedness of $\mathcal P$ to induce the global $H^1$-$L^1$-boundedness form the local one.
First of all, we prepare some useful lemmas. We have (at least formally) the kernel representation $$\label{kernelrep}
\mathcal{P} u(x)=
\int_{\R^n} K(x,y,x-y)u(y)
\, dy,$$ where $$\label{kernel}
K(x,y,z)
=\int_{\R^n}e^{i\{z\cdot\xi+\Phi(x,y,\xi)\}}a(x,y,\xi)\,d\xi.$$ On account of the singularity set $$\Sigma
=\{(x,y,-\partial_\xi\Phi(x,y,\xi))\in\R^n\times\R^n\times\R^n:x,y,\xi\in\R^n\}$$ of the kernel [(\[kernel\])]{}, we introduce the function $$H(x,y,z):=\inf_{\xi\in\R^n}\abs{z+\partial_\xi\Phi(x,y,\xi)}.$$ Then we have $\Sigma=\{(x,y,z)\in\R^n\times\R^n\times\R^n:H(x,y,z)=0 \}
=\bigcap_{d>0}(\Delta_d)^c$, where $$\Delta_d:=\{(x,y,z)\in\R^n\times\R^n\times\R^n:H(x,y,z)\geq d\}.$$ We also introduce $$\begin{aligned}
&\widetilde H(z):=\inf_{x,y\in\R^n}H(x,y,z)=
\inf_{x,y,\xi\in\R^n}\abs{z+\partial_\xi\Phi(x,y,\xi)},
\\
&\widetilde\Delta_d:=\{z\in\R^n:\widetilde H(z)\geq d\}.\end{aligned}$$ Clearly we have the monotonicity of $\Delta_d$ and $\widetilde\Delta_d$ in $d>0$, that is, $\Delta_{d_1}\subset\Delta_{d_2}$, $\widetilde\Delta_{d_1}\subset\widetilde\Delta_{d_2}$ for $d_1\geq d_2\geq0$. In the argument below, we frequently use the quantities $$\begin{aligned}
&M:=\sum_{|\gamma|\leq n+1}\sup_{x,y,\xi\in\R^n}
|\partial^\gamma_\xi a(x,y,\xi)\jp{\xi}^{-(\kappa(1)-|\gamma|)}|,
\\
&N:=\sum_{1\leq|\gamma|\leq n+2}\sup_{x,y,\xi\in\R^n}
|\partial^\gamma_\xi\Phi(x,y,\xi)\jp{\xi}^{-(1-|\gamma|)}|,\end{aligned}$$ which are finite since $a\in S^{\kappa(1)}$ and $\partial_\xi^{\gamma}\Phi\in S^0$ for $|\gamma|=1$ by assumption (A1).
\[Lem:inside\] Let $r>0$. Then for $x\in\widetilde\Delta_{2r}$ and $|y|\leq r$ we have $$\label{HandH}
\widetilde H(x)\leq2H(x,y,x-y)$$ and $(x,y,x-y)\in\Delta_r$.
For $x\in\widetilde\Delta_{2r}$ and $|y|\leq r$, we have $$\begin{aligned}
\widetilde H(x)
&\leq H(x,y,x)\leq|x+\partial_\xi\Phi(x,y,\xi)|
\leq|x-y+\partial_\xi\Phi(x,y,\xi)|+|y|
\\
&\leq|x-y+\partial_\xi\Phi(x,y,\xi)|+\widetilde H(x)/2\end{aligned}$$ since $\widetilde H(x)\geq 2r$, hence we have $\widetilde H(x)\leq2|x-y+\partial_\xi\Phi(x,y,\xi)|$ for all $\xi\in\R^n$ to conclude [(\[HandH\])]{}. Since $\widetilde H(x)\geq 2r$ again, the assertion $(x,y,x-y)\in\Delta_r$ is readily obtained from [(\[HandH\])]{}.
\[Lem:kernel\] The kernel $K(x,y,z)$ is smooth on $\bigcup_{d>0}\Delta_d$, and it satisfies $$\label{boundedness}
\n{H(x,y,z)^{n+1} K(x,y,z)}
_{L^\infty(\Delta_d)}\leq C(n,d,M,N),$$ where $C(n,d,M,N)$ is a positive constant depending only on $n$, $d>0$, $M$ and $N$. The function $\widetilde H(z)$ satisfies $$\label{integrable}
\n{\widetilde H(z)^{-(n+1)}}_{L^1(\widetilde\Delta_d)}
\leq C(n,d,N),$$ where $C(n,d,N)$ is a positive constant depending only on $n$, $d>0$ and $N$.
The expression [(\[kernel\])]{} is justified by the integration by parts, and we have $$K(x,y,z)=\int_{\R^n}e^{i\{z\cdot\xi+\Phi(x,y,\xi)\}}
\left(L^*\right)^{n+1} a(x,y,\xi)\,d\xi,$$ where $L^*$ is the transpose of the operator $$L=\frac{(z+\partial_\xi\Phi)\cdot\partial_\xi}{i|z+\partial_\xi\Phi|^2}.$$ Noticing the relation $d\leq H(x,y,z)\leq|z+\partial_\xi\Phi(x,y,\xi)|$ for $(x,y,z)\in\Delta_d$ and $\xi\in\R^n$, we easily have the property [(\[boundedness\])]{}. On the other hand, we have $$|z|\leq |z+\partial_\xi\Phi(x,y,\xi)|+N$$ for any $x,y\in\R^n$, $\xi\not=0$, hence $|z|\leq \widetilde H(z)+N$. Then for $|z|\geq 2N$ we have $|z|\leq \widetilde H(z)+|z|/2$, hence $|z|\leq 2\widetilde H(z)$, and the property [(\[integrable\])]{} is obtained from it since $$\begin{aligned}
\n{\widetilde H(z)^{-(n+1)}}_{L^1(\widetilde\Delta_d)}
&\leq
\n{\widetilde H(z)^{-(n+1)}}_{L^1(\widetilde\Delta_d\cap\b{|z|\leq2N})}+
\n{\widetilde H(z)^{-(n+1)}}_{L^1(\widetilde\Delta_d\cap\b{|z|\geq2N})}
\\
&\leq
d^{-(n+1)}\n{1}_{L^1(|z|\leq2N)}+
2^{n+1}\n{|z|^{-(n+1)}}_{L^1(|z|\geq2N)}.\end{aligned}$$ The proof is complete.
\[Lem:L1est\] Let $r\geq1$, and let $h\in\R^n$. Suppose $\supp f\subset \{x\in \R^n:|x|\leq r\}$. Then we have $$\left\|\tau_h^*\mathcal{P}\tau_h f\right\|_{L^1(\widetilde\Delta_{2r})}
\leq C(n,M,N)\n{f}_{L^1},$$ where $C(n,M,N)$ is a positive constant depending only on $n$, $M$ and $N$.
For $x\in\widetilde\Delta_{2r}$ and $|y|\leq r$, we have $\widetilde H(x)\leq2H(x,y,x-y)$ and $(x,y,x-y)\in\Delta_r$ by Lemma \[Lem:inside\]. Then from the kernel representation [(\[kernelrep\])]{}, we obtain $$\begin{aligned}
|\mathcal{P} f(x)|
&\leq 2^{n+1}\widetilde H(x)^{-(n+1)}\int_{|y|\leq r}
\abs{H(x,y,x-y)^{n+1}K(x,y,x-y)f(y)}\,dy
\\
&\leq 2^{n+1}\widetilde H(x)^{-(n+1)}
\n{H(x,y,z)^{n+1}K(x,y,z)}_{L^\infty(\Delta_r)}
\n{f}_{L^1}\end{aligned}$$ for $x\in\widetilde\Delta_{2r}$. Hence we have by Lemma \[Lem:kernel\] and the monotonicity of $\Delta_d$ and $\widetilde\Delta_d$ $$\begin{aligned}
\left\|\mathcal{P} f\right\|_{L^1(\widetilde\Delta_{2r})}
&\leq 2^{n+1}
\left\|\widetilde H(x)^{-(n+1)}\right\|_{L^1(\widetilde\Delta_{2r})}
\n{H(x,y,z)^{n+1}K(x,y,z)}_{L^\infty(\Delta_r)}\n{f}_{L^1}
\\
&\leq 2^{n+1}\left\|\widetilde H(x)^{-(n+1)}
\right\|_{L^1(\widetilde\Delta_{2})}
\n{H(x,y,z)^{n+1}K(x,y,z)}_{L^\infty(\Delta_1)}\n{f}_{L^1}
\\
&\leq 2^{n+1}C(n,2,N)\,C(n,1,M,N)\n{f}_{L^1}.\end{aligned}$$ On account of the observation [(\[translation\])]{} and the invariance of the quantities $M$ and $N$ under such replacement, we have the conclusion.
\[Lem:outside\] Let $r\geq1$. Then we have $\R^n\setminus\widetilde\Delta_{2r}\subset\b{z:|z|< \p{2+N}\,r}$.
For $z\in\R^n\setminus\widetilde\Delta_{2r}$, we have $\widetilde H(z)=\inf_{x,y,\xi\in\R^n}\abs{z+\partial_\xi\Phi(x,y,\xi)}
<2r$. Hence, there exist $x_0,y_0,\xi_0\in\R^n$ such that $$|z+\partial_\xi\Phi(x_0,y_0,\xi_0)|<2r.$$ Then we have $$|z|\leq|z+\partial_\xi\Phi(x_0,y_0,\xi_0)|+|\partial_\xi\Phi(x_0,y_0,\xi_0)|
\leq 2r+N\leq \p{2+N}r$$ since $r\geq1$.
Now we are ready to prove the $H^1$-$L^1$-boundedness. We use the characterisation of $H^1$ by the atomic decomposition proved by Coifman and Weiss [@CW]. That is, any $f\in H^1(\R^n)$ can be represented as $$f=\sum_{j=1}^\infty\lambda_jg_j,\quad\lambda_j\in\C,\quad g_j:\textrm{ atom},$$ and the norm $\|f\|_{H^1}$ is equivalent to the norm $\n{\b{\lambda_j}_{j=1}^\infty}_{\ell^1}=\sum^\infty_{j=1}|\lambda_j|$. Here we call a function $g$ on $\R^n$ an atom if there is a ball $B=B_g\subset\R^n$ such that $\supp g\subset B$, $\|g\|_{L^\infty}\leq|B|^{-1}$ ($|B|$ is the Lebesgue measure of the ball $B$) and $\int g(x)\,dx=0$. From this, all we have to show is the estimate $$\left\|\mathcal{P}g\right\|_{L^1(\R^n)}\leq C$$ with a constant $C>0$ for all atoms $g$. By an appropriate translation, it is further reduced to the estimate $$\left\|\tau_h^*\mathcal{P}\tau_hf\right\|_{L^1(\R^n)}\leq C,
\quad f\in \A_r,$$ where $\A_r$ is the set of all functions $f$ on $\R^n$ such that $$\supp f\subset B_r=\{x\in\R^n:|x|\leq r\},\quad\|f\|_{L^\infty}\leq |B_r|^{-1},
\quad\int f(x)\,dx=0.$$ Here an hereafter in this section, $C$ always denotes a constant which is independent of $h\in\R^n$ and $0<r<\infty$.
Suppose $f\in\A_{r}$ with $r\geq 1$. Then we split $\R^n$ into two parts $\widetilde\Delta_{2r}$ and $\R^n\setminus\widetilde\Delta_{2r}$. For the part $\widetilde\Delta_{2r}$, we have by Lemma \[Lem:L1est\] $$\left\|\tau_h^*\mathcal{P} \tau_hf\right\|_{L^1(\widetilde\Delta_{2r})}
\leq C\n{f}_{L^1}\leq C.$$ For the part $\R^n\setminus\widetilde\Delta_{2r}$, we have by Lemma \[Lem:outside\] and the Cauchy-Schwarz inequality $$\begin{aligned}
\left\|\tau_h^*\mathcal{P} \tau_hf\right\|_{L^1(\R^n\setminus\widetilde\Delta_{2r})}
&\leq
\|1\|_{L^2(|x|< \p{2+N}r)}
\left\|\tau_h^*\mathcal{P} \tau_hf\right\|_{L^2(\R^n)}
\\
&\leq
Cr^{n/2}\|f\|_{L^2(\R^n)}
\leq C,\end{aligned}$$ where we have used the assumption that $\mathcal{P}$ is $L^2$-bounded.
Suppose now $f\in\A_{r}$ with $r\leq 1$. then we split $\R^n$ into the parts $\Delta_{2}$ and $\R^n\setminus\Delta_2$. For the part $\Delta_{2}$, we have by Lemma \[Lem:L1est\] with $r=1$ and the inclusion $\supp f\subset B_r\subset B_1$ $$\left\|\tau_h^*\mathcal{P} \tau_hf\right\|_{L^1(\widetilde\Delta_{2})}
\leq C\n{f}_{L^1}\leq C.$$ For the part $\R^n\setminus\Delta_2$, we have by Lemma \[Lem:outside\] $$\begin{aligned}
\left\|\tau_h^*\mathcal{P} \tau_h f\right\|_{L^1(\R^n\setminus\Delta_{2})}
&\leq
\left\|\tau_h^*\mathcal{P} \tau_h f\right\|_{L^1(|x|<2+N)}
\\
&\leq C\n{f}_{H^1}
\leq C,\end{aligned}$$ where we have used the fact that $\mathcal{P}$ is uniformly $H^1_{comp}$-$L^1_{loc}$-bounded. Now the proof of Theorem \[Th:main\] is complete.
Proof of Corollaries \[Cor1\] and \[Cor2\] {#S4}
==========================================
In this section we prove Corollaries \[Cor1\] and \[Cor2\].
Let us induce assumptions (A1)–(A3) of Theorem \[Th:main\] from the assumptions (B1)–(B3) of Corollary \[Cor1\] for the special case $\phi(x,y,\xi)=x\cdot\xi-\varphi(y,\xi)$, in other words, $\Phi(x,y,\xi)=y\cdot\xi-\varphi(y,\xi)$. We remark that (B1) is just an interpretation of assumption (A1).
As for (A2), a sufficient condition for the $L^2$-boundededness of $\mathcal P$ is known from Asada and Fujiwara [@AF], that is, Theorem \[Th:AF\] in Introduction. On account of the observation [(\[adjoint\])]{}, $\mathcal P^*$ is also $L^2$-bounded under the same condition. In particular, (A2) is fulfilled if (B1) and (B2) are satisfied.
A sufficient condition for the $H^1_{comp}$-$L^1_{loc}$-boundedness of $P$ is known by the work of Seeger, Sogge and Stein [@SSS], that is, $P$ is $H^1_{comp}$-$L^1_{loc}$-bounded for $a(x,y,\xi)\in S^{-(n-1)/2}$ if $\varphi(y,\xi)$ is a real-valued $C^\infty$-function on $\R^n\times(\R^n\setminus0)$ and positively homogeneous of order $1$. If we carefully trace the argument in [@SSS], we can say that $\chi_{K} P \chi_{K}$ is $H^1(\R^n)$-$L^1(\R^n)$-bounded for any compact set $K\subset\R^n$ and its operator norm is bounded by a constant depending only on $n$, $K$ and quantities $$\begin{aligned}
&M_\ell=\sum_{|\alpha|+|\beta|+|\gamma|\leq \ell}\sup_{x,y,\xi\in\R^n}
|\partial^\alpha_x\partial^\beta_y
\partial^\gamma_\xi a(x,y,\xi)\jp{\xi}^{(n-1)/2+|\gamma|)}|,
\\
&N_\ell=\sum_{\substack{|\beta|\leq\ell, \\ 1\leq|\gamma|\leq \ell}}
\sup_{\substack{x,y\in\R^n,\\ \xi\neq0}}
|\partial^\beta_y
\partial^\gamma_\xi (y\cdot\xi-\varphi(y,\xi))\abs{\xi}^{-(1-|\gamma|)}|\end{aligned}$$ with some large $\ell$. The same is true for $P^*$ if we trace the argument in [@St2] instead but we require (B2) in this case. Then $P$ and $P^*$ are uniformly $H^1_{comp}$-$L^1_{loc}$-bounded if $M_\ell$ and $N_\ell$ are finite since the quantities $M_\ell$ and $N_\ell$ are invariant under the replacement in [(\[translation\])]{}.
Based on this fact, $P$ and $P^*$ are uniformly $H^1_{comp}$-$L^1_{loc}$-bounded if $a\in S^{-(n-1)/2}$ under the assumptions (B1)–(B3). In fact, if we split $a(x,y,\xi)$ into the sum of $a(x,y,\xi)g(\xi)$ and $a(x,y,\xi)(1-g(\xi))$ with an appropriate smooth cut-off function $g\in C^\infty_0(\R^n)$ which is equal to $1$ near the origin, the terms $P_1$ and $P_1^*$ corresponding to $a(x,y,\xi)(1-g(\xi))$ are uniformly $H^1_{comp}$-$L^1_{loc}$-bounded by the above observation. On the other hand, the terms $P_2$ and $P_2^*$ corresponding to $a(x,y,\xi)g(\xi)$ are $L^1$-bounded (hence uniformly $H^1_{comp}$-$L^1_{loc}$-bounded) because $$\begin{aligned}
&P_2u(x)=\int K(x,y) u(y)\,dy,\quad P_2^*u(x)=\int \overline{K(y,x)} u(y)\,dy,
\\
& K(x,y)
=\int_{\R^n}e^{i(x\cdot\xi-\phi(y,\xi))}
a(x,y,\xi)g(\xi)\,d\xi,\end{aligned}$$ and the integral kernel $K(x,y)$ is integrable in both $x$ and $y$. This fact can be verified by the integration by parts $$\begin{aligned}
K(x,y)
&=(1+|x-y|^2)^{-n} \int_{\R^n}(1-\Delta_\xi)^n e^{i(x-y)\cdot\xi}\cdot
e^{i(y\cdot\xi-\phi(y,\xi))}a(x,y,\xi)g(\xi)\,d\xi
\\
&=(1+|x-y|^2)^{-n}\int_{\R^n} e^{i(x-y)\cdot\xi}\cdot
(1-\Delta_\xi)^n \{e^{i(y\cdot\xi-\phi(y,\xi))}a(x,y,\xi)g(\xi)\}\,d\xi\end{aligned}$$ followed by the the conclusion $$\abs{K(x,y)}\leq C(1+|x-y|^2)^{-n}$$ because of assumptions (B1), $a\in S^{-(n-1)/2}$, and $g\in C^\infty_0$.
As a conclusion, (A3) is fulfilled if (B1)–(B3) are satisfied, and thus the proof of Corollary \[Cor1\] is complete.
Again we spilt the amplitude $a(x,\xi)$ into the sum of $a(x,\xi)g(\xi)$ and $a(x,\xi)(1-g(\xi))$ as in the proof of Corollary \[Cor1\]. We remark that the operator $\mathcal T$ defined by [(\[FIOp2\])]{} is the operator $P$ defined by [(\[FIOp1\])]{} with $\varphi(y,\xi)=y\cdot\xi-\psi(\xi)$ and $a(x,y,\xi)=a(x,\xi)$ independent of $y$. For the term $\mathcal T_1$ corresponding to $a(x,\xi)(1-g(\xi))$, we just apply Corollary \[Cor1\]. For the term $\mathcal T_2$ corresponding to $a(x,\xi)g(\xi)$, we have $$\mathcal T_2u(x)
=
\int_{\R^n} e^{i(x\cdot\xi+\psi(\xi))}a(x,\xi)g(\xi)\widehat u(\xi)\, d\xi
=
a(X,D_x)e^{i\psi(D_x)}g(D_x)u(x).$$ The pseudo-differential operator $a(X,D_x)$ is $L^p$-bounded (see Kumano-go and Nagase [@KN]) and the Fourier multiplier $e^{i\psi(D_x)}g(D_x)$ is also $L^p$-bounded by the Marcinkiewicz theorem (see Stein [@St]) since $\abs{\partial^\alpha \p{e^{i\psi(\xi)}g(\xi)}}
\leq C_\alpha|\xi|^{-|\alpha|}$ for any multi-index $\alpha$. The proof of Corollary \[Cor2\] is complete.
Applications to hyperbolic equations {#S5}
====================================
In this section we briefly outline an application of the obtained results to the global $L^p$-estimates for solutions to the Cauchy problems for strictly hyperbolic partial differential equations. In particular, in [@CR], the global $L^p$-boundedness of solutions of such equations was established with a loss of weight at infinity. In Theorem \[THM:hyp1\] we show that this loss of weight can be eliminated.
For simplicity, we consider equation of the first order $$\left\{ \begin{array}{ll}
(D_t+a(t,x,D_x) u(t,x)=0, & t\not=0, \, x\in\R^n, \\
u(0,x)=f(x),
\end{array} \right.
\label{eq:Cauchy}$$ where, as usual, $D_t=-i\partial_t$ and $D_{x}=-i\partial_{x}$. We assume that the symbol $a(t,x,\xi)$ is a classical symbol with real-valued principal part such that $$\label{EQ:apps-symbol}
|\partial_t^k \partial_x^\beta \partial_\xi^\alpha a(t,x,\xi)|\leq C_{k\alpha\beta}
\jp{\xi}^{1-|\alpha|}$$ holds for all $x,\xi\in\R^n$, all $t\in [0,T]$ for some $T>0$, and all $k,\alpha,\beta$, with constants $C_{k\alpha\beta}$ independent of $t,x,\xi$.
We consider strictly hyperbolic equations which means that the principal symbol of $a(t,x,\xi)$ is real-valued.
We note that following Kumano-go [@Kumano-go:BOOK-pseudos] we can extend the conclusions below also to higher order equations, especially if we impose appropriate conditions on lower order terms to achieve the perfect diagonalisation of the corresponding hyperbolic system to keep the phase function in the required form, similarly to the SG-case as in Coriasco [@Coriasco:SG-FIOs-II].
First we note that it was shown by Seeger, Sogge and Stein [@SSS] that if we have the Sobolev space data $f\in L^{p}_{\alpha+(n-1)|1/p-1/2|}(\R^n)$, for some $\alpha\in\R$, then for each fixed time $t$ the solution satisfies $u(t,\cdot)\in L^p_{\alpha}(\R^n)$ locally, $1<p<\infty$. Moreover, this order is sharp for every $t$ in the complement of a discrete set in $\R$ provided that $a$ is elliptic in $\xi$.
Let us now outline that Theorem \[main\] implies that this result holds globally on $\R^n$. Under the assumption [(\[EQ:apps-symbol\])]{}, it follows from Kumano-go [@Kumano-go:BOOK-pseudos Ch. 10, §4] that for sufficiently small times, the solution $u(t,x)$ to the Cauchy problem [(\[eq:Cauchy\])]{} can be constructed as a Fourier integral operator in the form [(\[EQ:P\])]{}. Moreover, it follows from [@Kumano-go:BOOK-pseudos Ch. 10, Theorem 4.1] that the phase and the amplitude of the propagator satisfy assumptions of Theorem \[main\]. Consequently, we obtain:
\[THM:hyp1\] Let the symbol $a(t,x,\xi)$ satisfy conditions [(\[EQ:apps-symbol\])]{}. Let $1<p<\infty$. If $f$ is such that $f\in L^p_{(n-1)|1/p-1/2|}(\R^n)$, then for each $t\in [0,T]$, the solution $u(t,x)$ of the Cauchy problem [(\[eq:Cauchy\])]{} satisfies $u(t,\cdot)\in L^p(\R^n)$. Moreover, for every $\alpha\in\R$ and $m\in\R$, there is $C_T>0$ such that we have the estimate $$\label{EQ:est-hyp-2}
\|u(t,\cdot)\|_{L^p_{\alpha}(\R^n)}\leq C_T
\|f\|_{L^p_{\alpha+(n-1)|1/p-1/2|}(\R^n)},$$ for all $t\in [0,T]$ and all $f$ such that the right hand side norm is finite.
In particular, Theorem \[THM:hyp1\] eliminates the weight loss in the global estimates estimates for solutions as it was obtained in [@CR Theorems 5.1 and 5.2]. The transition between Sobolev spaces for obtaining estimate [(\[EQ:est-hyp-2\])]{} for all $\alpha$ can be done by using the global calculus of Fourier integral operators developed by the authors in [@RS-weighted:MN].
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[^1]: Microlocally, this idea was explored by Duistermaat and Hörmander [@Duistermaat-Hormander:FIOs-2] in a variety of problems, while in the global analysis it was applied by the authors in [@RS2; @RS4] to the study of the global smoothing estimates.
|
---
author:
- |
M. J. Gillan$^{1,2,3}$, F. R. Manby$^{5}$, M. D. Towler$^{4,6,7}$ and D. Alfè$^{1,2,3,4}$\
$^1$London Centre for Nanotechnology, Gordon St, London WC1H 0AH, UK\
$^2$Dept of Physics and Astronomy, UCL, Gower St, London WC1E 6BT, UK\
$^3$Thomas Young Centre, UCL, Gordon St, London WC1H 0AH, UK\
$^4$Dept of Earth Sciences, UCL, Gower St, London WC1E 6BT, UK\
$^5$Centre for Computational Chemistry, School of Chemistry, University of Bristol,\
Bristol BS8 1TS, UK\
$^6$TCM Group, Cavendish Laboratory, University of Cambridge,\
Madingley Rd, Cambridge CB3 0HE, UK\
$^7$Apuan Alps Centre for Physics, via del Collegio 22, Vallico Sotto\
Fabbriche di Vallico 55020, Italy
title: Assessing the accuracy of quantum Monte Carlo and density functional theory for energetics of small water clusters
---
Introduction {#sec:intro}
============
Water in its many forms is one of the most intensively studied of all substances, because of its relevance to so many different scientific and technological fields. But in spite of many decades of effort, a fully comprehensive account of the energetics of water systems at the molecular level is still lacking. Density functional theory (DFT) is very important in water studies, since it is readily applied to the bulk liquid [@laasonen93; @tuckerman95; @sprik96; @silvistrelli99] and solid [@lee92; @hamann97; @singer05; @dekoning06] and their surfaces [@kuo04; @pan08], as well as to solutions [@marx97; @tateyama05], and to interfaces with other materials [@liu09; @liu10]. Unfortunately, DFT does not yet give satisfactory overall accuracy [@grossman04; @allesch04; @fernandez-serra04; @vandevondele05; @lee06; @todorova06; @guidon08], and the past few years have seen strenuous efforts to analyse and remedy the deficiencies of current DFT approximations [@jwang11; @santra11; @mogelhoj11]. Wavefunction-based molecular quantum chemistry techniques, particularly MP2 (second-order Møller-Plesset) and CCSD(T) (coupled-cluster singles and doubles with perturbative triples), the latter often referred to as the “gold standard”, can reliably deliver much higher accuracy than DFT. These quantum chemistry techniques have been extensively used to study water clusters [@xantheas94; @xantheas95; @pedulla96; @burnham99; @burnham02; @klopper00; @tschumper02; @xu04; @zhao05; @anderson06; @olson07; @dahlke08; @bates09; @yoo10] and to develop parameterised interaction models [@popkie73; @matsuoka76; @kim94a; @hodges97; @mas03; @huang06; @bukowski07; @huang08; @fanourgakis08; @szalewicz09; @kumar10; @ywang11]. However, the cost of obtaining this high accuracy increases dramatically with system size. The direct application of correlated quantum chemistry methods to bulk solid and liquid water is still in its infancy [@erba09; @oneill11]. Recently, quantum Monte Carlo (QMC) methods, and in particular diffusion Monte Carlo (DMC) [@hammond94; @anderson99; @nightingale99; @foulkes01; @needs10] have begun to be applied to both cluster and bulk forms of water [@santra11; @gurtubay07; @santra08] and the results indicate that DMC is much more accurate than DFT for these systems as well as for weak non-covalent interactions in other molecular systems [@korth08; @ma09]. Our purpose in this paper is to present a systematic comparison of the accuracy of DMC and DFT approximations for a wide range of configurations of small H$_2$O clusters, ranging from the monomer to the hexamer. The use of thermal ensembles of configurations and the many-body analysis of errors in the total energy are important themes of the work.
The errors of DFT are already troublesome for the H$_2$O monomer, and it has been shown recently [@santra09] that some common exchange-correlation functionals give rather poor accuracy for its distortion energy, underestimating the O-H bond-stretch energy. Difficulties with the energetics of the dimer are well known [@xantheas95; @tsuzuki01], with concerns about short-range exchange-repulsion and intermediate-range hydrogen bonding, as well as the lack of long-range dispersion in standard forms of DFT. Many-body interactions are also known to play an important role in water energetics [@xantheas94; @pedulla96; @bukowski07; @fanourgakis08; @szalewicz09; @kumar10; @hankins70; @white90], with induction effects due to the dipolar and higher multipolar polarisabilities of the monomer generally regarded as the dominant contribution, so that inaccurate DFT predictions of these polarisabilities may be problematic. In order to describe the subtle balance between these different interaction mechanisms without adjustment of empirical parameters, a good description of all of them is highly desirable.
The many-body expansion of the total energy of a system of molecules gives a helpful way of analysing the different types of interaction in water [@xantheas94; @pedulla96; @hankins70], and has also played a key role in the construction of interaction models fitted to molecular quantum chemistry calculations [@ywang11]. In this expansion, the total energy of a system of $N$ molecules is expressed as $$E = E^{(1)}+E^{(2)}+E^{(3)}+\cdots,
\label{eqn:many-body-expansion}$$ where $$\begin{aligned}
E^{(1)} &=& \sum_i E(i),\\
E^{(2)} &=& \sum_{i<j} \delta E(i,j),\\
E^{(3)} &=& \sum_{i<j<k} \delta E(i,j,k)\;,\end{aligned}$$ etc. Throughout the paper we use the notation $E(1,2,\ldots)$ to denote the energy of the system composed of the listed monomers, and the 2- and 3-body energy increments are given by the usual formulae: $$\begin{aligned}
\delta E(i,j) &=& E(i,j)- E(i) - E(j) \label{eq:2bodinc}\\
\delta E(i,j,k) &=& E(i,j,k) - \delta E(i,j) - \delta E(i,k) -
\delta E(j,k) - E(i) - E(j) - E(k). \notag\end{aligned}$$ We use the notation $E^{(>2)}$ to denote all of the beyond-2-body effects, i.e. $E^{(>2)} = E-E^{(1)}-E^{(2)}$.
By calculating the DMC energy for sets of configurations of H$_2$O clusters from the monomer to the hexamer, and by comparing these energies with many-body decompositions of both benchmark CCSD(T) energies and of energies given by various DFT approximations, we will attempt to probe the accuracy of DMC for the 1-, 2- and beyond-2-body parts of the energy. The evidence to be presented will indicate that DMC is very accurate for all the main components of the energy, while the DFT approximations we study all encounter difficulties with one or more of them. An important feature of this work is that many of our configuration sets are random statistical samples created by drawing clusters from a classical simulation of liquid water based on an interaction model for flexible monomers. A recent investigation of monomer and dimer energetics was based on a similar idea [@santra09].
Our work builds on very extensive earlier work on water clusters in which DFT has been compared with accurate molecular quantum chemistry calculations [@xu04; @anderson06; @dahlke08; @santra08; @santra09; @ireta04; @santra07; @su04], sometimes employing the many-body expansion, though rather little of that work has been done on the kind of random statistical samples that we emphasise here. We also build on previous DMC work on clusters [@gurtubay07; @santra08], bulk water [@grossman05] and ice structures [@santra11]. The DMC work already reported on clusters [@gurtubay07; @santra08] investigated only a single configuration of the dimer and some of the equilibrium isomers of the hexamer, but was vital in giving evidence for the accuracy of DMC for water systems. Very recent work on the cohesive energy and relative energies, equilibrium volumes and transition pressures of a number of ice structures [@santra11] demonstrated the remarkable accuracy of DMC ($\sim 0.2$ m$E_{\rm h} \simeq
5$ meV per monomer) in the bulk. Combining the evidence from that work on ice with the present evidence on clusters, we shall argue that DMC can now be regarded as a valuable tool which will be able to provide benchmark energies for larger clusters, for statistical samples of configurations of liquid water under a range of conditions, and for more complex systems such as ice surfaces.
Techniques {#sec:techniques}
==========
Quantum Monte Carlo methods have been described in detail in many previous papers [@hammond94; @anderson99; @nightingale99; @foulkes01; @needs10; @towler11; @austin12]. The main technique used here is diffusion Monte Carlo (DMC), which is a stochastic technique for obtaining the ground-state energy of a general many-electron system. We recall that it works by propagating an initial trial many-electron wavefunction in imaginary time, with the time-dependent part of the wavefunction represented by an evolving population of walkers. Although DMC is in principle exact, practical calculations generally employ the fixed-node approximation [@fixed-node] and the pseudopotential locality [@pseudo-locality] approximation. Errors arising from both these approximations are greatly reduced by improving the accuracy of the trial wavefunction, and we do this by variance minimisation [@variance-min] within variational Monte Carlo. (The alternative procedure of energy minimisation is favoured by some research groups.) All our DMC calculations are made with the [casino]{} code [@casino].
We use conventional Slater-Jastrow wavefunctions with a Jastrow factor containing electron-nucleus, electron-electron and electron-nucleus-electron terms, each of which depends on variational parameters [@drummond04]. We use Dirac-Fock pseudopotentials [@dirac-fock-pseudo], with the oxygen pseudopotential having a He core with core radius $0.4$ Å and the hydrogen pseudopotential having core radius $0.26$ Å. The single-electron orbitals in the trial wavefunction are obtained from plane-wave LDA calculations performed with the PWSCF code [@pwscf]. We use LDA to generate the single-electron orbitals, because the evidence indicates that this usually gives more accurate many-electron trial functions than other DFT approximations or Hartree-Fock [@ma09]. These DFT calculations are performed on water clusters in large cubic boxes having a length of $40$ a.u. in periodic boundary conditions, with $\Gamma$-point sampling and a plane-wave cut-off of $300$ Ry. The resulting single-electron orbitals are then re-expanded in B-splines [@alfe04] and the DMC calculations are performed with free (as opposed to periodic) boundary conditions. The B-spline grid has the natural spacing $a = \pi / G_{\rm max}$, where $G_{\rm max}$ is the plane-wave cut-off wavevector.
Since DMC is a stochastic technique, its computed energies suffer from a statistical error, which is inversely proportional to the square root of configuration-space sampling points, so that it is desirable to increase the number of walkers and the imaginary-time duration of the run. Since the recent DMC work on ice structures [@santra11] suggests that DMC is capable of an accuracy of $0.1 - 0.2$ m$E_{\rm h}$/monomer or better for water systems, we generally aim to run the DMC calculations long enough to reduce the statistical error below this tolerance. For the imaginary-time propagation of DMC, we use a time-step of $\delta t = 0.005$ a.u. This value was chosen after tests with smaller $\delta t$ down to $0.001$ a.u., which showed that with $\delta t = 0.005$ a.u. the total energy of the H$_2$O monomer is converged to within $0.4$ m$E_{\rm h}$. This tolerance is outside our target of $0.1$ $E_{\rm h}$ per monomer, but we are concerned in this work with energy [*differences*]{}, and we expect that all the relevant differences will be converged with respect to time-step to very much better than $0.4$ m$E_{\rm h}$ with $\delta t = 0.005$ a.u. Our DMC results on the H$_2$O monomer (Sec. \[sec:monomer\]) will confirm this. As further confirmation, we have also made tests on the energy differences between dimer configurations with $\delta t$ ranging from $0.001$ to $0.005$ a.u. and we find a variation of no more than $0.1$ m$E_{\rm h}$.
All the present DMC calculations were performed on the JaguarPF supercomputer at Oak Ridge National Laboratory, which at the time of the calculations consisted of 224,000 cores organised into 12-core shared-memory nodes. The parallel implementation of the [casino]{} code distributes walkers over cores. Because the walker populations fluctuate, we find that reasonable load-balancing is achieved only if the mean number of walkers on each core is at least 10. The total number of walkers and hence the optimal number of cores depend strongly on the number of atoms in the system and the required statistical accuracy, but in general we find it convenient to use up to about $50,000$ cores. The parallel scaling of our [casino]{} implementation on JaguarPF is excellent, becoming essentially perfect for very large numbers of walkers. For the statistical samples of configurations used in the present work, it is efficient to run a number of DMC calculations simultaneously.
Our absolute standard of accuracy throughout this work is the coupled-cluster approximation CCSD(T) in the complete basis-set (CBS) limit. Of course, this limit cannot be attained, but our convergence criterion is that residual basis-set errors should be less than the threshold of $0.1$ m$E_{\rm h}$/monomer mentioned above. In order to achieve this tolerance, we use the identity $$E ( {\rm CCSD(T)} ) = E ( {\rm MP2} ) + E ( \Delta {\rm CCSD(T)} ) \; ,$$ where $E ( \Delta {\rm CCSD(T)} ) \equiv E ( {\rm CCSD(T)} ) - E ( {\rm MP2} )$ is the change of the correlation energy when CCSD(T) is used in place of MP2. This allows us to use different basis sets for $E ( {\rm MP2} )$ and $E ( \Delta {\rm CCSD(T)} )$, exploiting the fact that the size of basis set needed to converge $E ( \Delta {\rm CCSD(T)} )$ is less than that needed to converge $E ( {\rm CCSD(T)} )$ itself. This approach is sometimes referred to as the “focal point” scheme [@tschumper02; @east93; @allinger97]. All our molecular quantum-chemistry calculations are performed with the [molpro]{} code [@molpro; @molpro:2011].
For all the benchmark calculations, we use the Dunning augmented correlation-consistent basis sets aug-cc-pVXZ [@dunning89; @kendall92], with X the cardinality, referred to here simply as AVXZ. The basis-set convergence of Hartree-Fock (HF) energies is rapid and unproblematic, and that of correlation energies can be greatly accelerated by the use of F12 (explicitly correlated) methods [@kutzelnigg85; @kutzelnigg91; @klopper06]. The F12 method is available for both MP2 and CCSD(T) in the [molpro]{} code [@werner07; @adler07], and we use it in all the present calculations. In addition, the efficiency of HF and MP2 calculations, and of F12 contributions to MP2 and CCSD(T), is greatly enhanced by using density fitting (DF) [@werner07; @manby03; @werner03; @polly04], the errors incurred being typically a few $\mu E_{\rm h}$, and so completely negligible for present purposes.
Since the many-body decomposition (Eq. (\[eqn:many-body-expansion\])) plays a key role in our analysis, we need the many-body components of the benchmark CCSD(T) energy for every configuration studied. For the 1-body energy $E^{(1)}$, rather than using CCSD(T) itself, we prefer to use the more accurate Partridge-Schwenke (PS) energy function [@partridge97]. This is an elaborate parameterised function fitted to a very large number of extremely accurate quantum chemistry calculations spanning a wide range of monomer geometries. The function, denoted here by $E^{(1)} ( {\rm PS} )$, has been shown to have spectroscopic accuracy, and it can therefore be regarded as essentially exact for present purposes. When we refer to CCSD(T) benchmarks, what we actually mean is $E^{(1)} ( {\rm PS} ) + E^{(2)} ( {\rm CCSD(T)} ) + \ldots$. The many-body decomposition is an additional aid to basis-set convergence, with smaller basis sets sufficing for high-order interaction terms. As a further important point of technique, we systematically use the counterpoise method [@boys70; @helgaker00] to reduce basis-set superposition error in all our calculations; so for example to calculate a 2-body energy contribution $\delta E(i,j) = E ( i, j ) - E( i ) - E( j )$ the full dimer basis sets is used for each of the three contributions. For completeness, we note that we do not include core correlation or relativistic corrections. These corrections were studied for the H$_2$O dimer by Tschumper [*et al.*]{} [@tschumper02] and found to be typically a few $\mu E_{\rm h}$.
To conclude our summary of techniques, we note briefly how we have performed the DFT calculations. We use the same Dunning correlation-consistent AVXZ basis sets that we used for the benchmark calculations, and we find that the differences between successive cardinalities decrease in essentially the same way for all the functionals as in the HF calculations, so that convergence of the total energy to our tolerance of $0.1$ m$E_{\rm h}$/monomer is easy to achieve. In computing the many-body components of the energy, we use exactly the same counterpoise methods outlined above for the benchmark calculations.
DMC and DFT compared with benchmarks {#sec:DMC_QC}
====================================
We will start by studying a thermal sample of configurations of the H$_2$O monomer drawn from a classical molecular dynamics (m.d.) simulation of the bulk liquid based on the flexible [amoeba]{} model [@ren03; @ren04]. The comparisons will allow us to assess the accuracy of DMC and DFT approximations for the 1-body energy $E^{(1)}$. For the dimer, we will gain insight into the 2-body energy by comparing DMC and DFT energies with benchmarks for two sets of configurations: first, the 10 stationary points which have been extensively studied in earlier work [@tschumper02; @anderson06; @huang06; @huang08; @smith90]; and second, a thermal sample of $198$ configurations drawn from the bulk liquid. We then move on to comparisons for thermal samples of the trimer, tetramer, pentamer and hexamer, which allow us to see how well the different methods account for the beyond-2-body energy $E^{(>2)}$. At the end of the Section, we will present results for the many-body decomposition of the energy of a number of stationary points of the hexamer, for which QMC results have been reported in earlier work [@santra08].
Since all our thermal samples of cluster configurations were drawn from the same [amoeba]{} m.d. simulation, we summarise the relevant details here. Monomer flexibility is one of the important features of the [amoeba]{} model [@ren03; @ren04], whose parameters are adjusted to fit selected [*ab initio*]{} and experimental data. The model also accounts for many-body interactions through distributed polarisabilities of the monomers. It is known to give a rather good description of the radial distribution functions and the self-diffusion coefficient of liquid water over quite a range of conditions, including ambient. The m.d. simulation run from which we drew the configurations was performed on a system of 216 water molecules in periodic boundary conditions at the ambient density of $1.0$ gm/cm$^3$ and room temperature (300 K). This way of making thermal samples is motivated by our long-term aim of obtaining accurate descriptions of the energetics of condensed phases of water in thermal equilibrium. To this end, it is natural to demand that the methods we use should be accurate for cluster geometries typical of those found in the condensed phases of interest. Naturally, we have to bear in mind always that the mean and rms errors we find on comparing a chosen technique with benchmark energies for a thermal sample are not absolute quantities, but will depend on the way the sample was constructed.
The monomer {#sec:monomer}
-----------
All our calculations on the monomer were performed on a set of 100 configurations drawn at random from the [amoeba]{} m.d. simulation. The mean value of the O-H bond length in this sample was 0.968 Å, and the probability distribution of bond-lengths was roughly Gaussian, with an rms fluctuation about the mean of 0.019 Å; the minimum and maximum O-H bond lengths occurring in the sample were 0.913 and 1.020 Å. The mean and rms fluctuation of the H-O-H bond angle were $105.3^\circ$ and $3.7^\circ$ respectively, with the minimum and maximum angles being 97.0$^\circ$ and 117.0$^\circ$.
We assess the errors of DMC and DFT approximations for the monomer by comparing their energies with the essentially exact Partridge-Schwenke energy function $E^{(1)} ( {\rm PS} )$ referred to in Sec. \[sec:techniques\]. The equilibrium monomer geometry according to PS has bond lengths of $0.95865$ Å and a bond angle of $104.348^\circ$. It is convenient to take this as the “standard” geometry of the isolated monomer, so that when we refer to the energy of the monomer in any geometry computed with a given method we will always mean the energy of that geometry minus the energy in the PS equilibrium geometry computed with the same method. This means that the monomer energy with PS itself is by definition non-negative, but with a DFT approximation or with DMC it can be negative, since the minimum-energy configurations with these methods will generally differ from the PS equilibrium geometry.
Our DMC calculations on the 100 monomer configurations were performed as described in Sec. \[sec:techniques\]. In our production results, the Jastrow factor was not re-optimised for each geometry. We used a smaller set of 25 configurations to test the effect of re-optimising it, and we could not detect any significant changes of energy due to re-optimisation. For every geometry, the DMC calculations were continued long enough to reduce the rms statistical errors to $30$ $\mu E_{\rm h}$. We show in Fig. \[fig:monomer\_DMC-ps\_DFT-ps\] $E(\text{DMC})-E(\text{PS})$ plotted against $E(\text{PS})$. The distortion energy itself covers a range up to $\sim 5$ m$E_{\rm h}$ ($140$ meV), and we see that DMC errors are only a tiny fraction of this. In fact, for the given thermal sample, the mean DMC error and its rms fluctuation about the mean are $10$ and $40$ $\mu E_{\rm h}$ (Table \[tab:monomer\_energy\]).
We have made comparisons against PS also for DFT, using the PBE, BLYP, B3LYP and PBE0 exchange-correlation functionals and the AV5Z basis. Our convergence tests using AVQZ indicate mean (rms) errors due to basis-set incompleteness of around 4 (10) $\mu E_{\rm h}$ for all functionals, suggesting that any residual basis-set errors beyond AV5Z will be much smaller than $10$ $\mu E_{\rm h}$ ($0.27$ meV).
The differences $E(\text{DFT})-E(\text{PS})$ are also displayed in Fig. \[fig:monomer\_DMC-ps\_DFT-ps\], and the mean and rms fluctuations are recorded in Table \[tab:monomer\_energy\]. We see from this that PBE and BLYP give very poor results, and their negative deviations from the PS benchmarks imply that the energy needed to distort the monomer is considerably underestimated by both approximations, as already noted in a recent paper by Santra [*et al.*]{} [@santra09]. The B3LYP functional is much better, but PBE0 is the clear winner out of the functionals examined, with mean and rms errors of only $-10$ and $80$ $\mu E_{\rm h}$. The good performance of PBE0 for the monomer was also reported by Santra [*et al.*]{} [@santra09]. However, DMC is markedly superior, and it appears that residual DMC errors in the 1-body energy can safely be neglected in cluster and bulk systems, at least so long as the variation of the bond lengths and bond angle are not too much larger than those treated here.
The dimer {#sec:dimer}
---------
### The stationary points {#sec:stationary}
The geometries of the 10 dimer stationary points are depicted in many previous papers (see e.g. Ref. [@tschumper02]), and there is a standard numbering, which we follow here. We have worked with two closely related sets of configurations for the stationary points. The configurations of Tschumper [*et al.*]{} [@tschumper02] were used for the tests we performed to check that our techniques can deliver dimer energies with CCSD(T) within $100$ $\mu E_{\rm h}$ of the CBS limit. However, because of the way the project developed, the DMC and DFT calculations were performed on a slightly different configuration set due to the Bowman group [@huang08], and we used exactly the same techniques tested on the Tschumper set to produce our CCSD(T) benchmarks for the Bowman set.
Our benchmarks for the total dimer energy are represented as $$\begin{aligned}
E = E^{(1)}(\mathrm{PS})
+ E^{(2)}(\text{MP2-F12/AV5Z})
+ E^{(2)}(\Delta\text{CCSD(T)-F12/AVQZ})
\label{eq:E1E2dimer}\end{aligned}$$ with full counterpoise for all energies. Our basis-set tests show that with AVQZ the HF and MP2-F12 correlation energies have residual basis-set errors of less than $\sim 15$ and $\sim 10$ $\mu E_{\rm h}$ respectively, while with AVTZ the basis-set errors in $E^{(2)} ( \Delta {\rm CCSD(T)-F12} )$ are less than $\sim 30$ $\mu E_{\rm h}$. For the present calculations on the stationary points, we have gone beyond this so that the remaining basis-set errors should be much less than those just quoted, and our comparisons with the energies reported by Tschumper [*et al.*]{} confirm this.
Our DMC calculations on the total energies of the stationary points are computed as described in Sec. \[sec:techniques\]. The runs were continued long enough to reduce the DMC statistical error to $77$ $\mu E_{\rm h}$. Since we focus here on the energies of the stationary points relative to that of the global minimum, we have extended the run on the global minimum so that its rms statistical error is only $44$ $\mu E_{\rm h}$. The relative DMC energies are compared with the CCSD(T) benchmarks in Table \[tab:sp\_bench\_DMC\_DFT\], and we see that they agree in all but two cases within better than $100$ $\mu E_{\rm h}$ ($2.7$ meV), and in those two cases the DMC errors are still less than 170 $\mu E_{\rm h}$.
We already know from earlier work [@anderson06] that DFT predictions of the energies of the stationary points suffer from much larger errors than the DMC errors just mentioned, but we considered it worthwhile to calculate our own values using PBE, BLYP, B3LYP and PBE0. (We note that the present DFT energies are all calculated at exactly the same geometries, rather than at the stationary-point geometries that would be given by the DFTs themselves, and this should be borne in mind when comparing with earlier DFT results.) Our tests of basis-set convergence for DFT show that with AV5Z the relative energies of the stationary points are converged to better than $20$ $\mu E_{\rm h}$. We report values of these relative energies for the Bowman geometries in Table \[tab:sp\_bench\_DMC\_DFT\]. We see that the DFT approximations overestimate all the energy differences between the stationary points and the global minimum, with many of the DFT errors being greater than $0.5$ m$E_{\rm h}$ and a few being as much as $1.0$ m$E_{\rm h}$, in general agreement with earlier work [@anderson06]. The hybrid functionals B3LYP and PBE0 perform slightly better than non-hybrid PBE and BLYP, but there is not a great deal to choose between them. The individual monomers have almost exactly the same geometries at the 10 stationary points, so that DFT errors in the monomer distortion energies play almost no role here, and any disagreements with the benchmarks are due almost entirely to the 2-body energies.
We conclude from these comparisons that DMC gives better (in most cases, much better) agreement with the benchmarks than any of the DFT approximations we have studied, with DMC errors being typically five times smaller than than those of DFT.
### A thermal sample of dimer configurations {#sec:dimer_thermal}
The thermal sample was produced by drawing 198 configurations at random from the [amoeba]{} simulation, with O-O distances included out to $7.5$ Å but with a bias towards shorter distances. To construct the CCSD(T) benchmark energies for this set, we followed the procedure outlined above for the stationary points. The benchmark energy is computed as $$\begin{aligned}
E &=& E^{(1)}(\text{PS}) +
E^{(2)}(\text{MP2-F12/AVQZ})
+ E^{(2)}(\Delta\text{CCSD(T)-F12/AVTZ}).\end{aligned}$$ Because the configuration sample is reasonably large, we are able to analyse the statistics of basis-set differences for the three components. It turns out that the differences depend rather uniformly on $R_{\rm O O}$, providing a simple scheme for partially correcting residual basis-set errors in the MP2-F12 and $\Delta$CCSD(T)-F12 correlation contributions. Tests indicate that our benchmark dimer energies are within $\sim 20$ $\mu E_{\rm h}$ of the CBS limit of CCSD(T).
The DMC calculations were performed in the same way as for the stationary points, with the single-electron orbitals and Jastrow factor constructed as described in Sec. \[sec:techniques\] and the time-step chosen as before to have the value $0.005$ a.u. For every configuration, the DMC run was continued long enough to reduce the statistical error on the total energy to $60$ $\mu E_{\rm h}$. To characterise the performance of DMC, we show in Fig. \[fig:dimer\_tot\_errors\] the energy difference DMC-bench plotted as a function of $R_{\rm O O}$. The differences are typically on the order of $100$ $\mu E_{\rm h}$, and there is no obvious dependence of their magnitude on $R_{\rm O O}$. Quantitatively, the mean value of $E ( {\rm DMC} ) - E ( {\rm bench} )$ over the 198-configuration sample is $20$ $\mu E_{\rm h}$ and the rms fluctuation is $0.10$ m$E_{\rm h}$. Since the statistical errors of the Monte Carlo sampling in our DMC calculations are $\sim 60$ $\mu E_{\rm h}$, this means that there are statistically significant deviations of the DMC energies from the CCSD(T) benchmarks, but they appear to be no more than $\sim 0.1$ m$E_{\rm h}$ ($2.7$ meV).
Our DFT calculations on the thermal sample were done both as direct calculations of the total energy using AV5Z basis set, and also by separating the total energy into 1- and 2-body parts, using the AV5Z basis for $E^{(1)}$ and AVQZ with counterpoise for $E^{(2)}$. A comparison of the direct total energies calculated using AVQZ and AV5Z basis sets shows mean and rms differences of the energies that are less than $25$ $\mu E_{\rm h}$ for all the functionals. The total energies obtained in the direct calculations differ from those obtained by adding the 1- and 2-body energies by at most $50$ $\mu E_{\rm h}$. The differences $E ( {\rm DFT} ) - E ( {\rm bench} )$ for the dimer total energies are plotted against $R_{\rm O O}$ in Fig. \[fig:dimer\_tot\_errors\]. It is immediately clear that the DFT errors are considerably greater than those of DMC, with BLYP and PBE being much inferior to B3LYP and PBE0. The mean values of the DFT-bench differences and their rms fluctuations about these means are reported in Table \[tab:dimer\_errors\].
An important theme of the present work is the separation of the energy into its many-body parts. We have already seen that some of the DFT approximations suffer from large 1-body errors, so it is natural to ask how they perform when these errors are corrected. If we separate the total energy $E({\rm DFT})$ computed with a given DFT into its 1-body and 2-body parts, and we then replace the 1-body energy by its Partridge-Schwenke value $E^{(1)} ( {\rm PS} )$, we obtain an approximation denoted here by DFT-$\Delta_1$. Clearly, for dimers the errors of DFT-$\Delta_1$ are entirely 2-body errors. In Fig. \[fig:dimer\_2b\_errors\], we compare the total-energy differences $E ( {\rm DFT} \mbox{-} \Delta_1 ) - E ( {\rm bench} )$ as a function of $R_{\rm OO}$ with the differences $E ( {\rm DMC} ) - E ( {\rm bench} )$. The mean and rms values of these differences are reported in Table \[tab:dimer\_errors\]. We see that BLYP is very poor indeed, being much too repulsive over the whole range $2.5 - 4.5$ Å, and B3LYP suffers from the same problem, though its errors are smaller. The PBE0 approximation is much better, though it still too repulsive. Best of all these DFTs is PBE. The DMC errors are even small than those of PBE.
These findings are generally in line with what is known from previous DFT and DMC work on the binding energy of the dimer. It is well known that BLYP seriously underbinds, that B3LYP underbinds somewhat less, and that PBE0 and PBE give good binding energies, with PBE being almost exactly correct. It is also known that DMC gives a rather accurate value of the dimer binding energy [@gurtubay07]. The present results enlarge the picture by showing that these errors in the binding energy at the global minimum can be seen as part of general trends over a range of O-O distances.
The trimer {#sec:trimer}
----------
The trimer is important, because it is the smallest cluster for which we can probe beyond-2-body interactions. We created a thermal sample of trimer configurations using the same [amoeba]{} simulation of liquid water as before, drawing 50 trimer geometries at random, with the condition that all three O-O distances must be less than $4.5$ Å.
Our method for obtaining basis-set converged CCSD(T) energies is a straightforward extension of what we outline above for the dimer. The total trimer energy is decomposed as $$\begin{aligned}
E
&=&
E^{(1)} + E^{(2)} +
E^{(3)}(\text{MP2-F12/AVQZ}) +
E^{(3)}(\Delta\text{CCSD(T)-F12/AVTZ})
\label{eq:E3trimer}\end{aligned}$$ where the terms $E^{(1)} + E^{(2)}$ are treated exactly as in Eq. \[eq:E1E2dimer\]. We find that the Hartree-Fock component $E^{(3)} ( {\rm HF} )$ converges very rapidly with basis set: the mean difference and the rms fluctuation for AVQZ $-$ AVTZ are only $1.0$ and $1.1$ $\mu E_{\rm h}$. The same is true for the correlation part of $E^{(3)} (\text{MP2-F12} )$, for which the corresponding values are $1.0$ and $1.6$ $\mu E_{\rm h}$. For $E^{(3)} ( \Delta \text{CCSD(T)-F12} )$, we have results only for AVTZ. However, our calculations on the dimer samples showed that the AVQZ $-$ AVTZ difference for CCSD(T)-F12 was very similar to that of MP2-F12, and we assume the same to be true for $E^{(3)}$. From the evidence we have obtained, the energy expression in Eq. \[eq:E3trimer\] should be well within $50$ $\mu E_{\rm h}$ of CCSD(T)/CBS.
Our DMC calculations on the 50-configuration trimer sample follow exactly the methods outlined in Sec. \[sec:techniques\]. The duration of the DMC runs was long enough to reduce the statistical error on the total energy to $77$ $\mu E_{\rm h}$. We show in Fig. \[fig:tr\_errors\_DMC\_DFT\] the errors $E ( {\rm DMC} ) - E ( {\rm bench} )$ of the DMC total energy of the trimers plotted against the benchmark total energy. We see that the errors and their fluctuations are very small, their mean value and rms fluctuation over the 50-configuration set being $0.30$ m$E_{\rm h}$ (8.1 meV) and $0.13$ m$E_{\rm h}$ ($3.5$ meV). (Note that these values refer to the [*total*]{} energy, not the energy per monomer.)
DFT calculations of the total trimer energy are straightforward, and our tests indicate that the total energy relative to that of three monomers in the PS reference geometry is converged to within $0.15$ m$E_{\rm h}$ with AVQZ basis sets. It is useful to have the many-body decomposition, and we use $$E ( {\rm DFT} ) = E^{(1)} ( {\rm DFT/AV5Z} ) + E^{(2)} ( {\rm DFT/AVQZ} ) +
E^{(3)} ( {\rm DFT/AVTZ} ),$$ again with full counterpoise correction. The two ways of calculating the total trimer energies agree to within mean and rms differences of $120$ and $56$ $\mu E_{\rm h}$ respectively. As we show in Fig. \[fig:tr\_errors\_DMC\_DFT\], the errors of the DFT total energy with PBE, BLYP, B3LYP and PBE0 are much greater than those of DMC: the mean and rms deviations (Table \[tab:compare\_DFT-n\]) are typically five times greater than the DMC values.
It is now interesting to analyse how much of the DFT errors come from 1-, 2- and 3-body parts of the energy. Here we follow the approach of Taylor *et al.* [@taylor12], correcting the low-order many-body contributions to DFT. For example, the effect of 1-body errors can be eliminated by using the energy expression $$\begin{aligned}
E ( \text{DFT-}{\ensuremath{\Delta_1}}) &=& E ( \text{DFT} ) +
E^{(1)} ( \text{bench} ) - E^{(1)} ( \text{DFT} ) \\
&\equiv&
E^{(1)} ( \text{bench} ) + E^{(2)} ( \text{DFT} ) + E^{(3)} ( \text{DFT} ).\end{aligned}$$
The mean and rms deviations $E ( \text{DFT-}{\ensuremath{\Delta_1}}) - E ( \text{bench} ) =
[ E^{(2)} ( \text{DFT} ) - E^{(2)} ( \text{bench} ) ] +
[ E^{(3)} ( \text{DFT} ) - E^{(3)} ( \text{bench} ) ]$ are reported in Table \[tab:compare\_DFT-n\]. As expected from the results of Sec. \[sec:monomer\], the rms fluctuations of $E ( \text{DFT-{\ensuremath{\Delta_1}}} ) - E ( \text{bench} )$ are very much reduced for PBE and BLYP, because their 1-body energies are poor; the mean value of the deviation is improved for PBE but worsened for BLYP, again as expected. On the other hand for B3LYP and particularly for PBE0, correction of the 1-body errors makes little difference.
We can go further by correcting both the 1-body and the 2-body DFT energies, thus obtaining a scheme that we denote by DFT-[$\Delta_{12}$]{}. The trimer energy in this scheme is $E ( \text{DFT-{\ensuremath{\Delta_{12}}}} ) =
E ( \text{DFT-{\ensuremath{\Delta_1}}} ) + [ E^{(2)} ( \text{bench} ) - E^{(2)} (
\text{DFT} ) ]$, which is the same as $E^{(1)} ( \text{bench} ) +
E^{(2)} ( \text{bench} ) + E^{(3)} ( \text{DFT} )$. The deviations $E ( \text{DFT-{\ensuremath{\Delta_{12}}}} ) - E ( \text{bench} ) =
E^{(3)} ( \text{DFT} ) - E^{(3)} ( \text{bench} )$ now come entirely from DFT errors in the 3-body part. The mean and rms values of these deviations are reported in Table \[tab:compare\_DFT-n\]. Not surprisingly, these values are very small, so that 3-body effects are quite well represented by all the DFT approximations.
It is clear that the trimer is really too small to yield very interesting conclusions about the accuracy of DFT compared with DMC for beyond-2-body interactions, because there is only a single 3-body term in the total energy. However, as we go to larger clusters, the number of beyond-2-body interactions increases rapidly, so that more interesting comparisons can be made. We therefore turn next to benchmark, DMC and DFT calculations on the tetramer, pentamer and hexamer.
Thermal samples of the tetramer, pentamer and hexamer {#sec:tet_pent_hex}
-----------------------------------------------------
Our procedures for the tetramer, pentamer and hexamer follow quite closely those outlined above for the smaller clusters. To generate the samples for cluster size $N \ge 4$, we repeatedly planted a sphere of chosen radius $R ( N )$ at a random position at random time-steps of the [amoeba]{} simulations, and if the number of molecules inside the sphere was equal to $N$ we accepted these molecules as a sample configuration. (For this purpose, a molecule was counted as being inside the sphere if the O position was inside the sphere.) In making the sample of configurations for a given $N$, we chose $R ( N )$ so that the mean number of molecules found inside the sphere was close to $N$. In this way, we formed samples of 25 configurations each for $N = 4$, $5$ and $6$.
Our benchmark energies were computed as $$\begin{aligned}
E
&=&
E^{(1)}(\text{PS}) +
E^{(2)}(\text{MP2-F12/AV5Z}) +
E^{(2)}(\Delta\text{CCSD(T)-F12/AVTZ}) \\
&+&
E^{(3)}(\text{MP2-F12/AVTZ}) +
E^{(3)}(\Delta\text{CCSD(T)-F12/AVDZ}) \\&+&
E^{(>3)}(\text{MP2-F12/AVDZ})
\label{eq:Egeneral}\end{aligned}$$ for the tetramer and pentamer, but for the hexamer we use $E^{(2)}(\text{MP2-F12/AVQZ})$ in place of $E^{(2)}(\text{MP2-F12/AV5Z})$. Contributions up to 3-body were treated using the counterpoise correction as described above; higher-order terms were computed using full counterpoise for the entire cluster; for example, in computing 4-, 5- and 6-body terms for the hexamer, we used the full basis set of the entire cluster for every contribution.
The DMC calculations for the 25-configuration samples of the tetramer, pentamer and hexamer followed exactly the same procedures as before, and the runs were continued until the statistical errors on the total energy were reduced to $0.13$, $0.14$ and $0.17$ m$E_{\rm h}$ for the tetramer, pentamer and hexamer respectively. As an example of the very close agreement between DMC and the CCSD(T) benchmarks, we show in Fig. \[fig:pe\_errors\_DMC\_DFT\] the total-energy difference DMC-benchmark for the pentamer configurations plotted against the total energy itself. The mean value and the rms fluctuations of this difference are reported for all the clusters in Table \[tab:compare\_DFT-n\]. We see from this that the DMC errors are on the same scale of $\sim 0.2$ m$E_{\rm h}$/monomer or less that characterise the recently reported DMC values of the absolute and relative energies of various ice structures [@santra11].
The DFT energies of the $N \ge 4$ cluster configurations were all computed as sums of many-body contributions. For the 1-, 2- and 3-body energies, we employed AV5Z, AVQZ and AVTZ basis sets respectively, with full counterpoise for all dimers and trimers for the 2- and 3-body energies. In the $(n \ge 4)$-body parts, we found it more convenient to use full counterpoise for the entire cluster, as we did for the CCSD(T) benchmarks, and for this purpose we used AVDZ basis sets. To cross-check the total energies obtained in this way, we also computed them directly (i.e. without many-body decomposition), using AVQZ basis sets.
As an example of DFT performance, we show in Fig. \[fig:pe\_errors\_DMC\_DFT\] the differences DFT-benchmark for the pentamer sample plotted against the benchmark total energies. The mean values of these differences and their rms fluctuations are reported for all the larger clusters in Table \[tab:compare\_DFT-n\]. It is evident that the accuracy of DMC is very much greater than any of the DFT approximations. As might be expected from our results for the smaller clusters, BLYP is very poor, having rms deviations from the benchmarks that are about 10 times the size of those with DMC, and its mean deviations are also large. For all the clusters, PBE is somewhat better and B3LYP still better, but best of all is PBE0, whose rms errors are a little over 2.5 times those of DMC.
In our discussion of DFT errors for the trimer (Sec. \[sec:trimer\]), we pointed out the possibility of correcting DFT first for 1-body errors and then for both 1-body and 2-body errors, these two levels of corrected DFT being denoted by DFT-[$\Delta_1$]{} and DFT-[$\Delta_{12}$]{}. Since we have all the $n$-body parts of both benchmark and DFT energies of the tetramer, pentamer and hexamer, we can make the same analysis for them. We report in Table \[tab:compare\_DFT-n\] the mean and rms deviations of the DFT-[$\Delta_1$]{} and DFT-[$\Delta_{12}$]{} energies away from the benchmarks. As expected, correction of the 1-body energy substantially reduces the rms errors of PBE and BLYP, because these DFTs have quite large 1-body errors, but it makes rather little difference in the case of B3LYP and PBE0, because their 1-body errors are small. Interestingly, this correction considerably worsens the mean BLYP errors, because in the uncorrected version there is a partial cancellation of errors between the 1- and 2-body parts. Correcting for both 1- and 2-body errors, the approximations suffer only from $( n \ge 3)$-body errors, which we also report in Table \[tab:compare\_DFT-n\]. Clearly, these errors are extremely small. Indeed, the corrected DFTs B3LYP-[$\Delta_{12}$]{}and PBE0-[$\Delta_{12}$]{}are even slightly better than DMC.
The comparisons we have presented demonstrate the high accuracy of DMC, but they also indicate that the 1-body, 2-body and beyond-2-body parts of the total energy are individually well described by DMC. However, there is another aspect of DMC predictions that is worth examining. If we judged solely by our comparisons for the thermal samples, we would infer that all the main errors of DFT are in the 1- and 2-body parts, so that once these are corrected we get approximations that are as good as DMC. However, this inference is not true in general, as we will show next by a many-body analysis of the stable isomers of the hexamer.
Stable isomers of the hexamer {#sec:hexamer}
-----------------------------
The global- and local-minimum structures of the H$_2$O hexamer have long played a role in the understanding of water energetics, because their relative energies are determined by a rather delicate interplay between different kinds of interactions [@olson07; @dahlke08; @santra08; @bates09; @kumar10; @kim94b; @liu96; @xantheas02]. The isomers we will be concerned with here are the prism, cage, book and ring, whose geometries have been presented in many previous papers (see e.g. Ref. [@santra08]). The atomic coordinates that we use here are the MP2/AVTZ-optimised structures of Santra [*et al.*]{} [@santra08]. The more open structures, such as the ring, favour hydrogen bonding with OH$\cdots$O angles that maximise the strengths of individual hydrogen bonds. In the more compact structures, including the prism and cage, the total number of hydrogen bonds is greater, but the OH$\cdots$O angles are less favourable. The book structure is a compromise betweeen the two kinds. Coupled-cluster CCSD(T) calculations near the basis-set limit leave no doubt that the more compact structures are more stable, the consensus being that the prism is the global minimum, with the cage slightly above it [@bates09]. The ring is less stable by $\sim 3.0$ m$E_{\rm h}$ in the total energy, and the book has an intermediate energy. DMC calculations give the correct energy ordering and energy differences that agree closely with the CCSD(T) values [@santra08], and we noted in the Introduction that this is one of the key pieces of evidence for the accuracy of DMC. Most of the conventional DFT approximations give the wrong energy ordering, with BLYP and B3LYP making the ring the global minimum, and PBE giving this honour to the book [@dahlke08]. The reasons for this have been widely discussed, and a many-body analysis has already been used to identify the cause of the problems, the suggestion being that the lack of long-range dispersion is responsible [@santra08]. A detailed break-down of the contributions to the relative energies of isomers of the hexamer has also been reported by Wang [*et al.*]{} [@wang10]. We find it worthwhile to revisit this question, because DMC can now be compared with more accurate CCSD(T) results than were available before and because our own many-body analysis indicates that the lack of dispersion may not be the only cause of DFT errors.
We report in Table \[tab:hex\_isomer\] our MP2 and CCSD(T) results for the prism, cage, book and ring isomers, computed in all cases with the MP2-optimized structures given by Santra [*et al.*]{} [@santra08], which are also the structures used in their DMC calculations. Our MP2 energies come from direct calculations on the entire hexamer, using AVQZ basis sets with F12. The CCSD(T) energies reported in the Table, do not, however, come from direct calculations on the cluster; instead, we compute $$E = E(\text{MP2-F12/AVQZ}) + \sum_{i=1}^3E^{(i)}(\Delta\text{CCSD(T)-F12})$$ As shown in the Table, our MP2 energy differences between the isomers agree very closely with earlier highly converged results, and our CCSD(T) energy differences also agree very well with recent CCSD(T) results close to the CBS limit. In agreement with previous work, we find that on going from MP2 to CCSD(T) the energy difference between the prism and the cage increases significantly, and the energy of the ring above the prism increases by $\sim 1.0$ m$E_{\rm h}$. The Table also gives the DMC energy differences of Santra [*et al.*]{}, and we note that they agree very closely with the CCSD(T) results reported here. Importantly, DMC is closer to the CCSD(T) energies than to those from MP2. Our own PBE, BLYP, B3LYP and PBE0 energies computed with AV5Z basis sets are also included in the Table. We fully confirm the conclusions from previous work that these DFTs give completely erroneous trends, wrongly predicting that the less compact isomers are more stable than the compact ones. The quantitative errors of the DFT energy differences are substantial, with the energy difference between the ring and the prism being in error by as much as $5$ m$E_{\rm h}$ in some cases.
To understand the origin of the erroneous DFT predictions, we compare in Fig. \[fig:he\_det2b\_3b\_bench\_dft\_1-4\] the 2-body and 3-body energies from benchmark CCSD(T) and from DFT. The 2-body benchmark energies were obtained using MP2-F12/AVQZ and $\Delta$CCSD(T)-F12/AVTZ, while for the 3-body benchmarks we used AVTZ for all parts. The DFT 2- and 3-body energies were computed using AVQZ and AVTZ respectively. It is clear from the 2-body results that BLYP and B3LYP predict much too weak a lowering of 2-body energy as we go from the ring to the cage and prism, while PBE is rather accurate and PBE0 is less accurate than PBE but better than BLYP and B3LYP. (The difference of 2-body energies of ring and prism is $10.0$ m$E_{\rm h}$ according to CCSD(T), but is only $3.4$ and $4.7$ m$E_{\rm h}$ with BLYP and B3LYP respectively, so that BLYP is in error by a factor of 3 and B3LYP by a factor of 2.) This is what we should expect from the 2-body energies presented in Sec. \[sec:dimer\_thermal\], since BLYP and B3LYP are systematically too repulsive over a rather wide range of distances, while the errors of PBE and PBE0 are much smaller. By contrast, in the 3-body energy the situation is reversed, with B3LYP now being almost perfect and BLYP only a little worse, while PBE and PBE0 are both rather poor. (The ring-prism difference of 3-body energy is $5.01$ m$E_{\rm h}$ according to CCSD(T), but PBE and PBE0 give $8.38$ and $6.97$ m$E_{\rm h}$ respectively.) We expect from this that if we correct the DFT approximations for their 1- and 2-body errors, thus obtaining what we referred to earlier as DFT-[$\Delta_{12}$]{}, then BLYP-[$\Delta_{12}$]{} and B3LYP-[$\Delta_{12}$]{} should agree rather well with CCSD(T) and DMC, while PBE-[$\Delta_{12}$]{}and PBE0-[$\Delta_{12}$]{} should be less good. This expectation is confirmed by our DFT-[$\Delta_{12}$]{}energies reported in Table \[tab:hex\_isomer\]. If we now correct also for 3-body errors, we would expect this to give little further improvement for BLYP and B3LYP, but substantial improvements for PBE and PBE0, and the DFT-[$\Delta_{123}$]{} results of Table \[tab:hex\_isomer\] confirm this.
These comparisons are very useful for our assessment of the accuracy of DMC, because they indicate that DMC must be accurate not only for the 2-body energy, as we already know from Sec. \[sec:dimer\], but also for the 3-body energy. We can draw this conclusion because PBE and DMC have very similar, and very small errors for 2-body energy, but DMC is very much better than PBE for the hexamer isomers, and we have traced the main cause of this to the 3-body energy.
Discussion and conclusions {#sec:discussion}
==========================
The main aims of the present work have been to assess the accuracy of diffusion Monte Carlo (DMC) for water clusters by comparing with quantum chemistry benchmarks, and to investigate how well it overcomes the deficiencies of common DFT approximations. We noted in the Introduction that any method that is intended to give an accurate description of cluster and bulk water and ice systems across a wide range of conditions must be accurate for all the key components of the energy, including the distortion energy of the H$_2$O monomer, the 2-body interactions that determine the energetics of the water dimer, and the many-body contributions arising from polarisability and other mechanisms that are known to be crucial for larger clusters and for the bulk liquid and solid phases. Our comparisons with well converged CCSD(T) energies for both statistical samples of configurations and in some cases for sets of stable isomers show that DMC gives all the main components of the energy rather accurately, while the standard DFT approximations that we have studied encounter problems with one or more of these components. We have emphasised the importance of studying random thermal samples, since these allow us to characterise the accuracy of QMC and DFT approximations across an entire domain of configurations, rather than at a small number of special configurations. At the same time, we have noted that the mean and rms errors of any given approximation will depend on the choice of thermal sample.
The importance of achieving an accurate description of monomer energetics was emphasised in a recent paper of Santra [*et al.*]{} [@santra09], who showed that some commonly used DFT functionals gives a rather poor description of the distortion energy, making bond-stretching too easy. We have confirmed this on a thermal sample of 100 distorted monomer configurations drawn from a classical m.d. simulation of liquid water performed with the flexible [amoeba]{} interaction model. We found poor accuracy with PBE and BLYP, better accuracy with B3LYP and excellent accuracy with PBE0, in agreement with Ref. [@santra09]. The accuracy of DMC turns out to be even better than PBE0. It has been suggested that the excessive deformability of the H$_2$O monomer with PBE and BLYP may be a significant factor in their rather poor predictions for bulk water. The very high accuracy of DMC for the monomer means that it does not suffer from such problems.
Our calculations on the 10 stationary points of the H$_2$O dimer provide important evidence that the accuracy of DMC for the 2-body energy of water systems is also very good. All the energies come in the correct order, though we recall that this is also achieved by most DFT approximations. Much more significant is the very close agreement with highly converged CCSD(T) benchmarks for the energies relative to the global minimum, which are almost all reproduced by DMC to within $0.1$ m$E_{\rm h}$ ($2.7$ meV). By contrast, DFT errors for the relative energies are typically $0.5$ – $1.0$ m$E_{\rm h}$, and no DFT approximation that we are aware of comes near the accuracy of DMC. We note that these comparisons give a direct test of the 2-body energy, since the monomer distortion energies at the 10 stationary points are extremely small.
More relevant to bulk-phase water are our comparisons between DMC, DFT and CCSD(T) benchmarks for a large random thermal sample of dimer configurations drawn from the [amoeba]{} m.d. simulation of bulk water. This sample is large enough for us to examine errors as a function of O-O distance, and we have seen that DMC reproduces the benchmarks accurately and consistently throughout the range $2.5$ – $7.0$ Å that we have examined. The DMC errors barely exceed the statistical errors of the Monte Carlo sample of the DMC calculations themselves, the mean and rms deviation of the DMC energy from the CCSD(T) benchmarks being 0.018 and 0.102 m$E_{\rm h}$ (0.5 and 2.8 meV). These errors are of about the size that might be expected from previous DMC work on the H$_2$O dimer. On the other hand, the errors of the total dimer energy with all the DFT approximations examined here are very much greater (see Table \[tab:dimer\_errors\]). However, in the case of PBE and BLYP, the errors in the monomer distortion energy contribute significantly. In practical calculations using these approximations on clusters or bulk systems, it would be perfectly straightforward to correct for these 1-body errors, simply by adding the difference between the essentially exact Partridge-Schwenke and the DFT distortion energies. If we do this for our dimer samples, then we obtain corrected DFT approximations which we refer to as DFT-[$\Delta_1$]{}, whose errors are solely in the 2-body part. We have seen that the 2-body energy of BLYP is much too repulsive and B3LYP suffers from the same problem, as would be expected from the substantial underbinding of the dimer with BLYP and B3LYP. However, for PBE, the 2-body energy turns out to be very good, its quality being comparable with that of DMC, so that if we simulated the dimer with PBE corrected for 1-body errors, rather accurate results would be obtained; the approximation PBE0-[$\Delta_1$]{} is also quite respectable.
For the larger clusters from the trimer to the hexamer, we have followed a similar procedure, drawing sets of configurations at random from the [amoeba]{} m.d. simulation and comparing DMC, DFT and benchmark CCSD(T) energies for these samples, taking care as usual that the total energies with DFT and CCSD(T) are basis-set converged to $\sim 0.1$ m$E_{\rm h}$ or better. For these clusters, the thermal samples are smaller than for the dimer, consisting of 50 configurations for the trimer and 25 each for the tetramer, pentamer and hexamer. Since the errors in either or both of the 1-body and 2-body components with the DFT approximations are considerably larger than those of DMC, we expect that DMC will substantially outperform them for these large clusters, and this is indeed what we find. However, this is not the whole story, because it is possible that some of the problems encountered by DFT approximations in treating bulk liquid water and ice may be associated with many-body (i.e. beyond-2-body) components of the energy, perhaps because their description of the relevant polarisabilities is inadequate. We have therefore tried to test whether DMC also outperforms DFT for these many-body contributions.
One way we have used to test the quality of the DMC beyond-2-body energy is based on making a many-body decomposition of the DFT total energy for our thermal samples of the trimer and higher clusters, and then to replace the DFT 1- and 2-body energies by the benchmark energies (i.e. Partridge-Schwenke for 1-body and near-CBS CCSD(T) for 2-body). Any remaining errors in the resulting corrected versions of DFT, which we refer to as DFT-[$\Delta_{12}$]{}, are then due entirely to errors in the beyond-2-body energy. If we then compare with DMC, we are putting DMC to an extreme and certainly unfair test, since it has to compete unaided against massively assisted DFT. Remarkably, DMC survives even this rather well, having errors in its total energy that are still smaller than or comparable with the errors in the beyond-2-body energy for the DFT approximations.
We have noted that the relative energies of the well-known isomers of the H$_2$O hexamer also provide an excellent way of testing the beyond-2-body energy of DMC. The point here is that all the DFT approximations we examined give completely erroneous energy orderings of these isomers. It has been shown in earlier work [@santra08] that DMC gives the energy differences between these isomers in excellent agreement with CCSD(T), and we confirmed this here by comparing with the improved CCSD(T) energies now available. As also pointed out earlier, a many-body decomposition of the DFT and CCSD(T) energies allows one to determine where the DFT errors come from. We have presented our own many-body analysis showing that for some of the DFTs (e.g. BLYP) the errors lie mainly in the 2-body part, whereas in others (e.g. PBE) the 2-body component is accurate, but there are substantial errors in the beyond-2-body components. Since we know that DMC gives an accurate account of the 2-body component, these comparisons confirm its accuracy also for the beyond-2-body components.
It is intriguing that the isomers of the hexamer reveal the superiority of DMC over some of the DFTs for beyond-2-body energy in a much clearer way than the thermal sample of hexamer configurations. The implication of this is that thermal samples of cluster configurations drawn from a realistic model of the liquid do not necessarily suffice for a full assessment of the errors of approximate methods. It is instructive to note that if an isolated hexamer in free space were simulated in thermal equilibrium using one of our DFT approximations, a completely erroneous distribution consisting mainly of ring-like structures would be observed, whereas if the simulation were performed with DMC (assuming this to be feasible), a much more realistic distribution consisting mainly of compact structures would be observed. However, the thermal sample of configurations generated by DMC would not suffice to assess fully the errors of DFT, because these errors only become apparent for thermal samples that include both open and compact structures. Similarly in assessing DFT errors using thermal samples relevant to the liquid, it seems desirable to use a wider range of configurations than those that occur commonly in the real liquid. This is an important matter for future study.
The present comparative study on water clusters, taken together with the recently demonstrated high accuracy of DMC for the energetics of several ice crystal structures [@santra11], indicates that DMC has the accuracy needed to serve as a benchmarking tool for water systems across a wide range of conditions. Its advantages over correlated quantum chemistry techniques are that its scaling with system size is much more favourable, convergence to the basis-set limit is easily achieved, it is straightforward to apply to periodic systems, and its parallel scaling on large supercomputers is essentially perfect. In the immediate future, we plan to use DMC to obtain benchmark energies for thermal samples of much larger water clusters than those studied here. These benchmarks will then be used to test DFT approximations. We expect to find ways of separating both the benchmark and the DFT total energies into their 1-, 2- and beyond-2-body contributions, as we have done here, so that we can analyse the origin of DFT errors. It should also be possible to do the same for bulk water itself. As we noted in the Introduction, DMC calculations on thermal samples of periodic liquid water configurations were already demonstrated some years ago [@grossman05], so that the technical feasibility of what we suggest is not in doubt. The possibility of tuning DFT approximations to reproduce DMC and quantum chemistry benchmarks for a range of water systems is also an interesting possibility for the future. It is worth adding that tests of DMC against CCSD(T) for larger clusters should be possible in the future, because the availability of quantum chemistry codes that can be run on very large parallel computers is already making it possible to perform coupled-cluster calculations on much larger molecular systems than before [@yoo10].
In conclusion, we have shown that the accuracy of DMC for thermal and other configuration samples of H$_2$O clusters from the monomer to the hexamer is excellent, the errors of DMC being typically $0.1 - 0.2$ m$E_{\rm h}$ ($2.5 - 5.0$ meV) per molecule. This error is much smaller than the typical DFT errors, and this, together with other evidence for the accuracy of DMC, supports its reliability as a source of benchmarks for testing and calibrating DFT. It is desirable to extend the tests of DMC accuracy to larger clusters, and we have indicated how this might be possible. We have also suggested how DMC calculations can be used to improve the understanding of both liquid and solid bulk water.
Acknowledgments {#acknowledgments .unnumbered}
===============
The work of MDT is supported by EPSRC grant EP/I030131/1, and that of FRM by EPSRC grant EP/F000219/1. DMC calculations were performed on the Oak Ridge Leadership Computing Facility, located in the National Center for Computational Sciences at Oak Ridge National Laboratory, which is supported by the Office of Science of the Department of Energy under contract DE-AC05-00OR22725 (USA).
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Tables {#tables .unnumbered}
======
mean rms
------- --------- --------
DMC $0.01$ $0.04$
PBE $-0.37$ $0.54$
BLYP $-0.45$ $0.63$
B3LYP $-0.14$ $0.20$
PBE0 $-0.01$ $0.08$
: Mean values and rms fluctuations of DMC and DFT errors of monomer energy for a thermal sample of 100 configurations (see text). The Partridge-Schwenke (PS) energy function is used as the “exact” energy, and the energy zero for DMC and DFT approximations is taken to be the energy in the PS equilibrium geometry. Units: m$E_{\rm h}$.[]{data-label="tab:monomer_energy"}
s.p. CCSD(T) DMC PBE BLYP B3LYP PBE0
------ --------- ------- ------- ------- ------- -------
1 0.000 0.000 0.000 0.000 0.000 0.000
2 0.783 0.866 0.904 0.838 0.834 0.867
3 0.910 1.001 1.288 1.194 1.040 1.085
4 1.115 1.250 1.707 1.814 1.690 1.627
5 1.519 1.445 2.374 2.406 2.208 2.187
6 1.603 1.505 2.707 2.676 2.382 2.401
7 2.893 2.863 3.544 3.534 3.447 3.395
8 5.663 5.739 5.869 5.698 5.881 5.883
9 2.840 2.910 3.502 3.485 3.371 3.304
10 4.307 4.471 4.838 4.591 4.601 4.654
: Comparison of DMC energies and DFT energies given by the PBE, BLYP, B3LYP and PBE0 functionals with CCSD(T) benchmarks for the 10 stationary points of the H$_2$O dimer. For each set of energies, the zero of energy has been taken so that the energy of the global minimum geometry is equal to zero. Numbering of stationary points follows that of previous authors (see e.g. Refs. [@tschumper02]). Energy units: m$E_{\rm h}$.[]{data-label="tab:sp_bench_DMC_DFT"}
method mean error rms error
---------------------- ------------ -----------
DMC 0.018 0.102
PBE -0.681 0.783
BLYP -0.155 1.102
B3LYP 0.180 0.463
PBE0 0.093 0.177
PBE-[$\Delta_1$]{} 0.056 0.144
BLYP-[$\Delta_1$]{} 0.736 0.679
B3LYP-[$\Delta_1$]{} 0.469 0.371
PBE0-[$\Delta_1$]{} 0.111 0.133
: DMC and DFT mean and rms fluctuation of errors of the total energy for thermal sample of 198 dimer configurations. In the case of DFT, the approximations denoted by PBE-[$\Delta_1$]{} etc represent the total energy after correction for errors of the 1-body energy (see text). Units: m$E_{\rm h}$.[]{data-label="tab:dimer_errors"}
[lrrr]{} DFT & 1c[DFT]{} & 1c[DFT-[$\Delta_1$]{}]{} & 1c[DFT-[$\Delta_{12}$]{}]{}\
\
PBE&$-$0.034 (0.376)& 0.152 (0.137)& 0.062 (0.049)\
BLYP&0.517 (0.413)& 1.101 (0.256)& $-$0.023 (0.039)\
B3LYP&0.542 (0.175)& 0.709 (0.147)& $-$0.011 (0.028)\
PBE0&0.215 (0.132)& 0.212 (0.125)& 0.031 (0.026)\
\
\
PBE&$-$0.163 (0.231)& 0.230 (0.143)& 0.110 (0.070)\
BLYP&0.861 (0.392)& 1.331 (0.335)& $-$0.041 (0.036)\
B3LYP&0.709 (0.219)& 0.859 (0.206)& $-$0.019 (0.025)\
PBE0&0.272 (0.126)& 0.282 (0.119)& 0.056 (0.038)\
\
\
PBE&$-$0.153 (0.282)& 0.276 (0.125)& 0.156 (0.077)\
BLYP&1.191 (0.335)& 1.703 (0.215)& $-$0.101 (0.050)\
B3LYP&0.902 (0.168)& 1.060 (0.152)& $-$0.056 (0.035)\
PBE0&0.297 (0.121)& 0.304 (0.131)& 0.075 (0.042)\
\
\
PBE&$-$0.068 (0.266)& 0.318 (0.134)& 0.178 (0.077)\
BLYP&1.298 (0.376)& 1.754 (0.304)& $-$0.107 (0.053)\
B3LYP&0.989 (0.180)& 1.123 (0.175)& $-$0.058 (0.037)\
PBE0&0.367 (0.114)& 0.367 (0.112)& 0.088 (0.041)\
\
method prism cage book ring
-------------------------- -------- ------------------- ------------------- -------------------
MP2 $0.00$ $0.08$ $0.51$ $1.84$
MP2$^\dagger$ $0.00$ $0.10$ $0.53$ $1.93$
CCSD(T) $0.00$ $0.39$ $1.12$ $2.70$
CCSD(T)$^\dagger$ $0.00$ $0.40$ $1.15$ $2.87$
DMC $0.00$ $0.53$ $0.90$ $2.45$
PBE $0.00$ $-0.55$ ($-0.95$) $-1.99$ ($-3.14$) $-1.45$ ($-4.32$)
PBE-[$\Delta_{12}$]{} $0.00$ $0.00$ ($-0.40$) $-0.78$ ($-1.93$) $-0.20$ ($-3.07$)
PBE-[$\Delta_{123}$]{} $0.00$ $0.35$ ($-0.05$) $1.34$ ($0.19$) $3.16$ ($0.29$)
BLYP $0.00$ $-0.94$ ($-1.34$) $-3.47$ ($-4.62$) $-3.90$ ($-6.77$)
BLYP-[$\Delta_{12}$]{} $0.00$ $0.04$ ($-0.36$) $0.47$ ($-0.68$) $2.64$ ($-0.23$)
BLYP-[$\Delta_{123}$]{} $0.00$ $0.01$ ($-0.39$) $0.22$ ($-0.93$) $1.72$ ($-1.15$)
B3LYP $0.00$ $-0.69$ ($-1.09$) $-2.65$ ($-3.80$) $-2.98$ ($-5.85$)
B3LYP-[$\Delta_{12}$]{} $0.00$ $0.05$ ($-0.35$) $0.41$ ($-0.74$) $2.17$ ($-0.70$)
B3LYP-[$\Delta_{123}$]{} $0.00$ $0.04$ ($-0.36$) $0.49$ ($-0.66$) $2.04$ ($-0.83$)
PBE0 $0.00$ $-0.45$ ($-0.85$) $-1.76$ ($-2.91$) $-1.65$ ($-4.52$)
PBE0-[$\Delta_{12}$]{} $0.00$ $0.13$ ($-0.27$) $0.04$ ($-1.11$) $0.96$ ($-1.91$)
PBE0-[$\Delta_{123}$]{} $0.00$ $0.30$ ($-0.10$) $1.21$ ($0.06$) $2.92$ ($0.05$)
: Total energies of selected isomers of the water hexamer relative to that of the prism, calculated by different methods. In all cases, the geometry of the isomer is the relaxed geometry given by MP2 calculations with the AVTZ basis, as given in the Supplementary Information of Ref. [@santra08]. All energies were calculated in the present work, except for the MP2 and CCSD(T) energies marked with $\dagger$ from Ref. [@bates09] and the DMC energies from Ref. [@santra08]. Entries DFT-$n$ with $n = 2$ and $3$ are DFT energies corrected for 1- and 2-body errors, and corrected for 1-, 2- and 3-body errors respectively. Values in parentheses represent errors compared with the CCSD(T) energies from Ref. [@bates09]. Energy units: m$E_{\rm h}$.[]{data-label="tab:hex_isomer"}
Figures {#figures .unnumbered}
=======
![Errors of DFT and DMC distortion energy of the H$_2$O monomer for a thermal sample of 100 configurations (see text). Quantities shown are deviations of calculated energies from Partridge-Schwenke (PS) benchmark values with PBE (black pluses), BLYP (purple triangles), B3LYP (green diamonds) and PBE0 (red crosses) and with DMC (black squares) plotted against the PS distortion energy itself. Units: m$E_{\rm h}$[]{data-label="fig:monomer_DMC-ps_DFT-ps"}](monomer.pdf){width="0.8\linewidth"}
![Errors of DMC and DFT approximations relative to CCSD(T) benchmarks for total energies of thermal sample of 198 dimer configuration, plotted vs O-O distance. Symbols represent PBE (black pluses), BLYP (purple triangles), B3LYP (green diamonds) and PBE0 (red crosses) and DMC (black squares). Units: m$E_{\rm h}$.[]{data-label="fig:dimer_tot_errors"}](dimer.pdf){width="0.8\linewidth"}
![Errors of DFT approximations for total energies of thermal sample of 198 dimer configurations when 1-body part is corrected by replacing the DFT 1-body energy by the essentially exact Partridge-Schwenke function. As in Fig. \[fig:dimer\_tot\_errors\], errors are relative to CCSD(T) benchmarks and are plotted vs O-O distance. Symbols represent PBE (black pluses), BLYP (purple triangles), B3LYP (green diamonds) and PBE0 (red crosses). Errors of DMC (black squares) are shown for comparison. Units: m$E_{\rm h}$. []{data-label="fig:dimer_2b_errors"}](dimer1.pdf){width="0.8\linewidth"}
![Errors of DFT and DMC total energy of the H$_2$O trimer for a thermal sample of 50 configurations drawn from a classical simulation of liquid water (see text). Quantities shown are deviations of calculated energies from CCSD(T) benchmark energies near the basis-set limit, with PBE (black pluses), BLYP (purple triangles), B3LYP (green diamonds) and PBE0 (red crosses) and with DMC (black squares) plotted against the benchmark energy itself. Units: m$E_{\rm h}$[]{data-label="fig:tr_errors_DMC_DFT"}](trimer.pdf){width="0.8\linewidth"}
![Errors of DFT and DMC total energy of the H$_2$O pentamer for a thermal sample of 25 configurations drawn from a classical simulation of liquid water (see text). Quantities shown are deviations of calculated energies from CCSD(T) benchmark energies near the basis-set limit, with PBE (black pluses), BLYP (purple triangles), B3LYP (green diamonds) and PBE0 (red crosses) and with DMC (black squares) plotted against the benchmark energy itself. Units: m$E_{\rm h}$[]{data-label="fig:pe_errors_DMC_DFT"}](pentamer.pdf){width="0.8\linewidth"}
![Comparison of DFT values of 2-body energies (upper panel) and 3-body energies (lower panel) of four isomers of the H$_2$O hexamer with benchmark values from CCSD(T). Numbering of isomers is prism: 1, cage: 2, book: 3, ring: 4. Units: m$E_{\rm h}$[]{data-label="fig:he_det2b_3b_bench_dft_1-4"}](isomers.pdf){width="0.8\linewidth"}
|
---
abstract: 'We report density functional theory (DFT) investigation of $B$-site doped CaFeO$_3$, a prototypical charge-ordered perovskite. At 290 K, CaFeO$_3$ undergoes a metal-insulator transition and a charge disproportionation reaction 2Fe$^{4+}$$\rightarrow$Fe$^{5+}$+Fe$^{3+}$. We observe that when Zr dopants occupy a (001) layer, the band gap of the resulting solid solution increases to 0.93 eV due to a 2D Jahn-Teller type distortion, where FeO$_6$ cages on the $xy$ plane elongate along $x$ and $y$ alternatively between neighboring Fe sites. Furthermore, we show that the rock-salt ordering of the Fe$^{5+}$ and Fe$^{3+}$ cations can be enhanced when the $B$-site dopants are arranged in a (111) plane due to a collective steric effect that facilitates the size discrepancy between the Fe$^{5+}$O$_6$ and Fe$^{3+}$O$_6$ octahedra and therefore gives rise to a larger band gap. The enhanced charge disproportionation in these solid solutions is verified by rigorously calculating the oxidation states of the Fe cations with different octahedral cage sizes. We therefore predict that the corresponding transition temperature will increase due to the enhanced charge ordering and larger band gap. The compositional, structural and electrical relationships exploited in this paper can be extended to a variety of perovskites and non-perovskite oxides providing guidance in structurally manipulating electrical properties of functional materials.'
author:
- Lai Jiang
- 'Diomedes Saldana-Greco'
- 'Joseph T. Schick'
- 'Andrew M. Rappe'
title: 'Enhanced charge ordering transition in doped CaFeO$_3$ through steric templating'
---
Introduction
============
The perovskite ($AB$O$_3$) family of materials has been paid considerable attention both in experimental and theoretical studies due to their flexible and coupled compositional, structural, electrical and magnetic properties [@Bellaiche00p5427; @Saito04p84; @Bilc06p147602]. Such flexibility arises from the structural building blocks — the corner-connected $B$O$_6$ octahedra, where $B$ is usually a transition metal. Typical structural variations from the cubic structure include rotation and tilting of the octahedra [@Glazer72p3384], off-centering of the $A$ and/or $B$ cations (pseudo Jahn-Teller effect) [@Qi10p134113], and expansion/contraction of the $B$O$_6$ octahedral cages [@Thonhauser06p2121061]. While the first two distortions are ubiquitous, the cooperative octahedral breathing distortion is rather rare in perovskites with a single $B$ cation composition. Such breathing distortion, resulting from the alternation of elongation and contraction of the $B$-O bonds between neighboring $B$O$_6$ cages, is usually concomitant with the charge ordering of the $B$ cations and a corresponding metal-insulator transition. CaFeO$_3$ is a typical perovskite material exhibiting such charge ordering transition [@Woodward00p844]. At room temperature, the strong covalency in the Fe $e_g$ - O 2$p$ interaction leads to a $\sigma^*$ band and electron delocalization which gives rise to metallic conductivity in CaFeO$_3$. Near 290 K, a second-order metal-insulator transition (MIT) occurs which reduces the conductivity dramatically [@Kawasaki98p1529]. The Mössbauer spectrum of low temperature CaFeO$_3$ has revealed the presence of two chemically distinct Fe sites (with different hyperfine fields) present in equal proportion [@Takano77p923]. This indicates that the Fe cations undergo charge disproportionation 2Fe$^{4+}$$\rightarrow$Fe$^{5+}$+Fe$^{3+}$ below the transition temperature. The origin of the charge ordering transition is usually attributed to Mott insulator physics, where the carriers are localized by strong electron-lattice interactions [@Millis98p147; @Takano77p923; @Ghosh05p245110; @Woodward00p844]. More recently, it has been debated whether the difference in charge state resides on the $B$ cations or as holes in the oxygen $2p$ orbitals [@Yang05p10A312; @Akao03p156405; @Mizokawa00p11263], and several computational studies showed that the magnetic configuration, in addition to structural changes, plays a vital role in stabilizing the charge ordered state in CaFeO$_3$ [@Mizokawa98p1320; @Ma11p224115; @Cammarata12p195144]. Nevertheless, the amplitude of the cooperative breathing mode is a key indicator of the magnitude of electron trapping and band gap opening in MIT. Conversely because the MIT is sensitive to lattice distortion, structural manipulation such as cation doping and epitaxial strain can be exploited to control the electrical properties of this family of oxide systems.
In this study, we examine the structural and electrical properties in $B$-cation doped CaFeO$_3$ with density functional theory (DFT). Various dopant cations, concentrations, and arrangements have been tested. Dopants of different sizes are tested, and alignments of pairs of dopants along different crystallographic planes are examined. To confirm the presence of charge ordering in (111) doped CaFeO$_3$, we also carried out rigorous oxidation state calculations for Fe cations in different octahedral cages based on their wave function topologies [@Jiang12p166403]. Through examination of these model systems, we assess the extent to which the structure-coupled electronic transition in doped oxide materials like CaFeO$_3$ can be influenced via doping to enhance band gap tunability, which in turn controls the MIT temperature.
Methodology
===========
Our DFT calculations are performed using the norm-conserving nonlocal pseudopotential plane-wave method [@Payne92p1045]. The pseudopotentials [@Rappe90p1227] are generated by the <span style="font-variant:small-caps;">Opium</span> package [@OPIUM] with a 50 Ry plane-wave energy cutoff. Calculations are performed with the <span style="font-variant:small-caps;">Quantum-Espresso</span> package [@Giannozzi09p395502] using the local density approximation [@Perdew81p5048] with the rotationally invariant effective Hubbard $U$ correction [@Johnson98p15548] of 4 eV on the Fe $d$ orbitals [@Fang01p180407; @Cammarata12p195144] for the exchange-correlation functional. In case of Ni and Ce doping, we applied $U$ = 4.6 eV [@Cococcioni05p035105] and $U$ = 5 eV [@Loschen07p035115] for Ni $d$ and Ce $f$, respectively. Calculations are performed on a $4\times4\times4$ Monkhorst-Pack $k$-point grid [@Monkhorst76p5188] with electronic energy convergence of $1\times10^{-8}$ Ry, force convergence threshold of $2\times10^{-4}$ Ry/Å, and pressure convergence threshold of 0.5 kbar. For polarization calculations a $4\times6\times12$ $k$-point grid is used, where the densely sampled direction is permuted in order to obtain all three polarization components. Different spin orderings for pure CaFeO$_3$ are tested to find the magnetic ground state, and subsequent solid solution calculations all start with that magnetic ground state.
Results and discussion
======================
Ground state of CaFeO$_3$ and CaZrO$_3$
---------------------------------------
To identify the correct spin ordering in pure CaFeO$_3$, we performed relaxations on both high temperature metallic orthorhombic $Pbnm$ [@Takano77p923; @Ghosh05p245110; @Woodward00p844; @Kanamaru70p257] and low temperature semiconducting monoclinic $P2_1/n$ [@Saha-Dasgupta05p045143] structures with common magnetic orderings commensurate with the $2\times2\times2$ supercells, as shown in Fig. \[fig:CFO\] (note that diamagnetic (DM) ordering is not included in the figure).
![(a) Low temperature $P2_1/n$ structure of CaFeO$_3$ with two symmetry-distinct Fe cation sites color coded. The spin ordering of the Fe cations are (b) ferromagnetic (FM), (c) A-type anti-ferromagnetic (AFM), (d) C-type anti-ferromagnetic and (e) G-type anti-ferromagnetic.[]{data-label="fig:CFO"}](CFO.pdf){width="\textwidth"}
[ X X X X X X X X]{} & & $E$ (eV) & $m_1$ ($\mu_\mathrm{B}$) & $m_2$ ($\mu_\mathrm{B}$) & $M$ ($\mu_\mathrm{B}$) & $V_1$ (Å$^3$) & $V_2$ (Å$^3$)\
& DM & 6.64 & N/A & N/A & N/A & 8.40 & 8.40\
& FM & 0.07 & 3.38 & 3.38 & 4.00 & 8.94 & 8.94\
& A-AFM & 0.43 & 3.29 & -3.29 & 0.00 & 9.00 & 9.00\
& C-AFM & 0.58 & 3.24 & -3.24 & 0.00 & 8.92 & 8.92\
& G-AFM & 0.96 & 3.44 & -3.51 & 0.06 & 9.52 & 8.47\
& DM & 6.64 & N/A & N/A & N/A & 8.40 & 8.40\
& FM & 0 & 3.13 & 3.60 & 4.00 & 9.16 & 8.75\
& A-AFM & 0.32 & 3.69 & -3.69 & 0.00 & 9.57 & 8.40\
& C-AFM & 0.58 & 3.24 & -3.24 & 0.00 & 8.92 & 8.92\
& G-AFM & 0.81 & 3.85 & -2.39 & 1.00 & 10.05 & 8.13\
\[tab:CFO\]
From the results in Table. \[tab:CFO\], we can see that both high temperature and low temperature ferromagnetic CaFeO$_3$ relax to the ferromagnetic ground states. Note that an additional magnetic phase transition is experimentally observed for CaFeO$_3$ at 15 K, where it adopts an incommensurate magnetic structure with a modulation vector \[$\delta$, 0, $\delta$\] ($\delta\approx0.32$, and reciprocal lattice vectors as basis) [@Woodward00p844]. Since DFT calculates 0 K internal energy, the ferromagnetic ground state represents a reasonable approximation of the spin-spin interactions within a unit cell given the relatively long spin wave length and low experimental crossover temperature to FM. The volumes of the two FeO$_6$ cages are equivalent in the high temperature metallic phase, as expected. The low temperature ground state has a cage size difference $\Delta V =0.41 $Å$^3$, indicating some degree of charge ordering. However, the projected density of states (PDOS) of the $P2_1/n$ ground state CaFeO$_3$ in Fig. \[fig:PDOS\]a shows that although there are separate gaps in each spin channel, the valence band edge in the majority spin touches the conduction band edge in the minority spin, resulting in zero total gap. The absence of band gap and the weak charge ordering is a result of the underestimation of band gaps in DFT [@Kohn65pA1133] due to its unphysical electron delocalization, This result is common in Mott insulators with partially filled $d$ orbitals and is in agreement with another DFT study of CaFeO$_3$ [@Yang05p10A312]. Even though DFT does not predict the correct electronic ground state of CaFeO$_3$, it is however indicative of the sensitive nature of the CaFeO$_3$ band gap as it can be easily influenced when Fe $d$ orbital filling is varied by structural or other perturbations. Moreover, the different $\Delta V$ between high temperature and low temperature of FeO$_6$ ground state suggests that the structural aspect of the MIT can be modeled reasonably well by DFT.
![Projected density of states of (a) CaFeO$_3$ (left) and (b) CaZrO$_3$.[]{data-label="fig:PDOS"}](CFO_CZO_pdos.pdf){width="\textwidth"}
For comparison, we calculated the PDOS of CaZrO$_3$ relaxed from experimental $Pbmn$ structure [@Levin03p170], shown also in Fig. \[fig:PDOS\]b. Because Zr$^{4+}$ has empty $4d$ orbitals, the fraction of Zr $d$ states in the valence band is negligible compared to O $p$ states, and a wide charge-transfer gap of 3.82 eV occurs. Since we expect to exploit the size effect of the dopants like Zr to influence the electronic property of CaFeO$_3$, we expect that the nature of Fe spin-spin interaction is not greatly affected by doping. Therefore in the following study we continue to use FM as the starting magnetic configuration for relaxations of the doped materials.
CaFeO$_3$-CaZrO$_3$ solid solutions with $2\times2\times2$ super cell
---------------------------------------------------------------------
To test how Zr doping influences the structural and electrical properties of CaFeO$_3$, we performed relaxations and subsequent band gap calculations of CaFeO$_3$-CaZrO$_3$ solid solutions. We employ a $2\times2\times2$ super cell and explore all possible $B$-site cation combinations. All the solid solutions tested turn out to be metallic except one, which has a gap of 0.93 eV. The insulating solid solution has a cation arrangement with four Zr cations on the (001) plane, making it a layered structure along \[001\]. Interestingly, instead of a breathing-mode charge disproportionation, this structure has a 2D Jahn-Teller type distortion, and all four Fe cations are in the same chemical environment. As shown in Fig. \[fig:JT\], the Fe-O bond lengths in the $xy$ plane are 2.16 Å and 1.83 Å in each FeO$_6$, with the orientation alternating between neighbors. The shorter Fe-O bond length is essentially the same as that in high temperature CaFeO$_3$. The consequence of the addition of larger Zr$^{4+}$ cations ($r=0.72$ Å) compared to Fe$^{4+}$ ($r=0.59$ Å) is that when Zr cations occupy an entire (001) plane the in-plane lattice is expanded from 3.70 Å to 3.88 Å, elongating the Fe-O bonds and enabling the 2D Jahn-Teller type distortion.
![(a) Crystal structure of Ca(Fe$_{1/2}$Zr$_{1/2}$)O$_3$ and (b) top view of the ZrO$_2$ layer showing the 2D Jahn-Teller type distortion.[]{data-label="fig:JT"}](CFZO_JT.pdf){width="\textwidth"}
From the PDOS of the solid solution in Fig. \[fig:CFZO\_PDOS\]a we can see that like pure CaFeO$_3$ both the valence and the conduction edges are of Fe $3d$ and O $2p$ characters, with virtually no Zr contribution. In charge ordering MIT, the delocalized electrons on Fe$^{4+}$ transfer to neighboring Fe$^{4+}$, making Fe$^{3+}$/Fe$^{5+}$ pairs with the valence and conduction bands located on different cations, concomitant with FeO$_6$ cage size changes. The band gap in case of charge ordering therefore depends on the energy difference between the $e_g$ orbitals in Fe$^{3+}$ and Fe$^{5+}$, which in turn is affected by the crystal field splitting energy caused by the oxygen ligands. On the other hand, as illustrated in Fig. \[fig:CFZO\_PDOS\]b, the solid solution band gap is caused by the removal of degeneracy in the $e_g$ orbitals and is controlled by the difference in energy between the two $e_g$ orbitals on the same Fe$^{4+}$ cation. Since the $e_g$ gap splitting is a result of the Fe-O bond length difference, it is easier to tune by applying either chemical pressure or biaxial strain to change the in-plane lattice constant, whereas the charge ordering mechanism requires the control of individual FeO$_6$ octahedral sizes to change the relative energy of $e_g$ orbitals between two Fe sites. Nevertheless, this CaFeO$_3$-CaZrO$_3$ solid solution demonstrates that when arranged in a particular way, in this case on (001) plane, the size effect of the large Zr cation can cause a cooperative steric effect on the structure and affect the electrical properties of CaFeO$_3$, opening up the band gap via a completely different mechanism.
![(a) PDOS of of Ca(Fe$_{1/2}$Zr$_{1/2}$)O$_3$ and (b) illustration of band gap formation in CaFeO$_3$ (left) and Ca(Fe$_{1/2}$Zr$_{1/2}$)O$_3$ (right). []{data-label="fig:CFZO_PDOS"}](CFZO_PDOS.pdf){width="\textwidth"}
CaFeO$_3$ with dopants on the (111) plane
-----------------------------------------
As discussed in the previous section, simply doping Zr into $2\times2\times2$ CaFeO$_3$ does not increase the band gap except for one case where 2D Jahn-Teller instead of breathing mode serves to make the system insulating. In a perovskite system with rock-salt ordered alternating $B$ cations, such as CaFeO$_3$, one $B$ cation type occupies entire (111) planes and the other type occupies its neighbors in all directions. Since Zr cation is larger than Fe cation, to fully utilize its steric effect to distinguish Fe$^{3+}$ from Fe$^{5+}$, it follows that Zr should replace a full Fe$^{3+}$ plane to increase the $B$O$_6$ size on the plane, maximizing its utility by enhancing the cage size difference. A schematic of the (111) doping strategy and the influence of the dopants on their neighboring planes is shown in Fig. \[fig:CFZO\_111\]b.
![(a) Crystal structure of $\sqrt{2}\times\sqrt{6}\times2\sqrt{3}$ CaFeO$_3$ supercell doped with one layer of Zr on the (111) plane. The average $B$O$_6$ octahedron size is listed on the side. (b) Schematic of how a layer of dopants with larger ionic radius exerts a cooperative size effect on the neighboring layers and enhances the existing charge ordering.[]{data-label="fig:CFZO_111"}](CFZO_111.pdf){width="\textwidth"}
Following this logic, we perform calculations with $\sqrt{2}\times\sqrt{6}\times2\sqrt{3}$ CaFeO$_3$ super cell, which has six (111) FeO$_2$ layers stacked perpendicularly, as the parent material. One layer of Fe’s is replaced with Zr’s and the structural is relaxed. The final structure is shown in Fig. \[fig:CFZO\_111\]a, as well as the average $B$O$_6$ cage size of each layer. Clearly the introduction of a (111) Zr layer drives the charge disproportionation of Fe$^{4+}$ by exerting chemical pressure on both sides of the layer and favoring the FeO$_6$ on the two adjacent planes to be smaller and become Fe$^{5+}$. The second next neighboring layers in turn have more room to expand and favor larger Fe$^{3+}$. The size difference between the largest and the smallest FeO$_6$ cages ($\Delta V =0.58$Å$^3$) in this structure is an enhancement compared to pure CaFeO$_3$ $\Delta (V=0.41$Å$^3$), which suggests the presence of a stronger charge ordering and a wider band gap. However electronic structure calculation shows that this solid material is metallic as well. The reason that the seemingly more charge ordered system still does not possess a gap can be attributed to the supercell employed. By using a unit cell with six (111) layers and replacing only one layer of Fe with Zr, structurally the remaining five Fe layers are disturbed by the large Zr layer as expected. However the charge disproportionation reaction 2Fe$^{4+}$$\rightarrow$Fe$^{5+}$+Fe$^{3+}$ cannot proceed to completion, because it requires an even number of Fe layers. Therefore with one layer of dopants there will always be Fe$^{4+}$ “leftovers” that render the whole system metallic.
To resolve the issue of odd number of Fe layers we introduce another layer of +4 dopants with smaller ionic radius than Fe. For simplicity we denote a solid solution in this case by listing the $B$ cations in each of its six (111) layers, with dopant elements in bold. For example, the previously discussed one layer Zr-doped solid solution would be denoted as **Zr**FeFeFeFeFe. The presence of two dopant layers provides both positive and negative chemical pressure to expand Fe$^{3+}$ and contract Fe$^{5+}$. These two dopant layers are separated by an even number of Fe layers so that the FeO$_6$ size alternation is enhanced. An odd number of Fe layers in between the dopant layers would disrupt and impede the size modulation period. Relaxations are performed on **ZrNi**FeFeFeFe and **Zr**FeFe**Ni**FeFe, as well as **CeNi**FeFeFeFe and **Ce**FeFe**Ni**FeFe. The average FeO$_6$ cage size per layer is listed in Table \[tab:size\_111\], along with the maximum cage size difference $\Delta V$ and the corresponding band gap of each solid solution. It can be seen that with only Zr as dopant, the $\Delta V$ is significantly smaller than those with two layers of dopants, and $\Delta V$ correlates with the band gap. The Ce-containing solid solutions have a larger $\Delta V$ compared to the Zr-containing ones, due to the larger size of Ce. It also shows that when the larger dopant and the smaller dopant layers are adjacent, the resulting $\Delta V$ is larger than when they are two Fe layers apart, this is due to the lack of symmetry of the former configuration where the absence of a mirror plane perpendicular to the $z$ axis allows for the FeO$_6$ close to the dopant layers to further expand or contract compared to the ones that are not neighbors of the dopant layers. In the latter configuration, symmetry guarantees that octahedra on either side of the dopant layer are deformed equally.
[X X X X X X X X X X]{} & $V_1$ & $V_2$ & $V_3$ & $V_4$ & $V_5$ & $V_6$ & $\Delta V$ & $E_g$\
**Zr**FeFeFeFeFe & & Zr & 8.81 & **9.33** & **8.75** & 9.29 & 8.83 & 0.58 & 0\
**ZrNi**FeFeFeFe & & Zr & Ni & 9.52 & 8.53 & **9.59** & **8.27** & 1.32 & 0.49\
**Zr**FeFe**Ni**FeFe & & Zr & 8.32 & 9.43 & Ni & **9.45** & **8.28** & 1.17 & 0.11\
**CeNi**FeFeFeFe & & Ce & Ni & **9.77** & 8.61 & 9.69 & **8.29** & 1.48 & 0.83\
**Ce**FeFe**Ni**FeFe & & Ce & **8.39** & **9.75** & Ni & 9.73 & 8.41 & 1.36 & 0.53\
\[tab:size\_111\]
From Fig. \[fig:gap\] we can see that with increasing difference in FeO$_6$ size, the band gap of the corresponding solid solutions increases accordingly. This relationship demonstrates the coupling between structural and electrical properties as larger FeO$_6$ size discrepancy indicates stronger and more complete charge disproportionation. As illustrated in Fig. \[fig:CFZO\_PDOS\]b, when charge ordering is the band gap opening mechanism, the gap size depends on the crystal field splitting energy difference between Fe$^{3+}$ and Fe$^{5+}$. A larger FeO$_6$ cage size difference means that the O $2p$ - Fe $3d$ repulsion difference is also larger between the two Fe sites. This causes the energy difference of the $e_g$ orbitals in the two sites to increase and the band gap to increase as well. Using linear regression we estimate that the chemical pressure excerted on the band gap by the volume difference in this type of solid solutions is quite large at about 370 GPa, in accordance with the effective band gap tuning. Since the transition to metal occurs when thermally activated electrons have enough energy to cross the band gap and flow between the two Fe sites to make them indistinguishable, we believe that by (111) doping the MIT temperature of CaFeO$_3$ can be increased, making devices based on it more operable at room temperature.
![Band gap $E_g$ of the (111) doped CaFeO$_3$ solid solutions increases with the corresponding maximum FeO$_6$ size difference $\Delta V$. This gives an effective chemical pressure on the band gap of 2.30 eV/Å$^3$ or 370 GPa.[]{data-label="fig:gap"}](gap.pdf){width="\textwidth"}
To investigate the layered nature of the solid solutions, we use **ZrNi**FeFeFeFe as an example and plot the projected density of states of it in Fig. \[fig:layered\_PDOS\] in a layer resolved fashion. Each of the six panels in Fig. \[fig:layered\_PDOS\] represents a layer of Ca$B$O$_3$, and the relative position of the panels corresponds to the that of the six layers in the crystal. It can be seen clearly that for the four layers containing Fe ions, the first and the third layers have more majority spin Fe $d$ in the valence band, while the second and fourth layers have more majority spin Fe $d$ in the valence band. This difference is consistent with the fact that Fe$^{3+}$ has more filled $d$ orbitals than Fe$^{5+}$ and supports our prediction that the doubly doped (111) layered CaFeO$_3$ has an enhanced charge ordering due to the strong modulation of the FeO$_6$ cage volume.
![Layer resolved projected density of states of **ZrNi**FeFeFeFe. Each of the six panels represents a layer of Ca$B$O$_3$, and the relative position of the panels corresponds to the that of the six layers in the crystal.[]{data-label="fig:layered_PDOS"}](layered_DOS.pdf){width="\textwidth"}
To further verify the charge disproportionation mechanism, we performed oxidation state calculations of the Fe cations in **ZrNi**FeFeFeFe. We employed an unambiguous oxidation state definition [@Jiang12p166403] based on wave function topology, whereby moving a target ion to its image site in an adjacent cell through an insulating path and calculating the polarization change during the process, the number of electrons that accompany the moving nucleus can be calculated. The oxidation state obtained this way is guaranteed to be an integer and is unique for an atom in a given chemical environment, not dependent on other factors such as charge partitioning or the choice of orbital basis. In Fig. \[fig:ox\] we show how the quantity $N = \Delta\vec{P}\cdot\vec{R}/\vec{R}^2$ changes as each Fe cation is moved along an insulating path to the next cell, which is equivalent to the oxidation state of the cation. The two Fe ions with larger cages are confirmed to be Fe$^{3+}$ and the ones with smaller cages are Fe$^{5+}$. This proves that charge ordering occurs in this material and causes band gap opening. Note that the oxidation states calculated are not directly related to the charges localized around the Fe sites, which has been been shown to change insignificantly upon oxidation reaction in some cases [@Sit11p12136]. In fact the Bader charge [@Bader90] of the Fe cations are 1.76 and 1.73 for the smaller and larger FeO$_6$ cages, respectively, which shows minuscule differences between Fe sites that are in significantly different chemical environments in terms of oxygen ligand attraction.
The (111) doping strategy shows that the size difference between Fe$^{3+}$ and Fe$^{5+}$ can be exploited and reinforced by selectively replacing layers of Fe$^{3+}$ with Fe$^{5+}$ with atoms of even larger or smaller size, respectively, to enhance charge ordering and the insulating character of the CaFeO$_3$ system.
![Oxidation state $N$ of the four Fe cations. $\lambda$ denotes the reaction coordinate of moving the Fe ion sublattice to the neighboring, cell and the change in $N=\Delta\frac{\vec{P}\cdot\vec{R}}{\vec{R^2}}$ from $\lambda=0$ to $\lambda=1$ corresponds to the oxidation state of that Fe ion.[]{data-label="fig:ox"}](ox.pdf){width="\textwidth"}
Conclusions
===========
We have demonstrated that for prototypical charge ordering perovskite CaFeO$_3$, the band gap of the insulating state can be engineered by $B$-site cation doping and structural manipulation. For the dopant atoms to exert significant influence on the parent material, it is favorable to arrange them in a way that their size effects are cooperative and synergistic, producing a collective steric effect and greatly altering the structural and electrical properties. When doped on the (001) plane with larger Zr cations, the in-plane lattice constant expands and supports a 2D Jahn-Teller type distortion, where each FeO$_6$ has two distinct Fe-O bond length in the $xy$ plane. Such distortion removes the degeneracy of the two $e_g$ orbitals on each Fe$^{4+}$ and opens up a band gap (not caused by charge ordering) of 0.93 eV. On the other hand, to enhance the weak charge ordering in pure CaFeO$_3$, we discovered that including two types of dopants on the (111) plane can increase the FeO$_6$ cage size difference and enhance the charge ordering. Using Zr or Ce to replace the larger Fe$^{3+}$ and Ni to replace the smaller Fe$^{5+}$ increases the band gap up to 0.83 eV. The degree of charge ordering is closely related to the magnitude of FeO$_6$ cage size difference. We used the rigorous definition of oxidation state to verify that in the latter case the band gap opening mechanism is indeed charge disproportionation, as the oxidation states of the larger and smaller Fe cations are calculated to be +3 and +5, respectively. Our results show that the structural and electrical properties of CaFeO$_3$ are coupled, and simple steric effects can enhance charge ordering transition and alter the band gap of the material greatly when the dopant atoms are placed to act cooperatively. Lastly by enhancing the charge ordering via doping, we predict that the MIT temperature of CaFeO$_3$ can also be increased to a temperature more suitable for practical device operation.
L. J. was supported by the Air Force Office of Scientific Research under Grant No. FA9550-10-1-0248. D. S. G. was supported by the Department of Energy Office of Basic Energy Sciences under Grant No. DE-FG02-07ER15920. J. T. S. was supported by a sabbatical granted by Villanova University. A. M. R. was supported by the Office of Naval Research under Grant No. N00014-12-1-1033. Computational support was provided by the High Performance Computing Modernization Office of the Department of Defense, and the National Energy Research Scientific Computing Center of the Department of Energy.
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---
abstract: 'We introduce the class of MAT-free hyperplane arrangements which is based on the Multiple Addition Theorem by Abe, Barakat, Cuntz, Hoge, and Terao. We also investigate the closely related class of MAT2-free arrangements based on a recent generalization of the Multiple Addition Theorem by Abe and Terao. We give classifications of the irreducible complex reflection arrangements which are MAT-free respectively MAT2-free. Furthermore, we ask some questions concerning relations to other classes of free arrangements.'
address:
- 'Michael Cuntz, Institut für Algebra, Zahlentheorie und Diskrete Mathematik, Fakultät für Mathematik und Physik, Leibniz Universität Hannover, Welfengarten 1, D-30167 Hannover, Germany'
- 'Paul Mücksch, Fakultät für Mathematik, Ruhr-Universität Bochum, D-44780 Bochum, Germany'
author:
- 'M. Cuntz'
- 'P. Mücksch'
title: 'MAT-free reflection arrangements'
---
Introduction
============
A hyperplane arrangement $\Ac$ is a finite set of hyperplanes in a finite dimensional vector space $V \cong \KK^\ell$. The intersection lattice $L(\Ac)$ of $\Ac$ encodes its combinatorial properties. It is a main theme in the study of hyperplane arrangements to link algebraic properties of $\Ac$ with the combinatorics of $L(\Ac)$.
The algebraic property of *freeness* of a hyperplane arrangement $\Ac$ was first studied by Saito [@Saito80_LogForms] and Terao [@Terao1980_FreeI]. In fact, it turns out that freeness of $\Ac$ imposes strong combinatorial constraints on $L(\Ac)$: by Terao’s Factorization Theorem [@OrTer92_Arr Thm. 4.137] its characteristic polynomial factors over the integers. Conversely, sufficiently strong conditions on $L(\Ac)$ imply the freeness of $\Ac$. One of the main tools to derive such conditions is Terao’s Addition-Deletion Theorem \[Thm\_Add\_Del\]. It motivates the class of *inductively free* arrangements (see Definition \[Def\_IF\]). In this class the freeness of $\Ac$ is combinatorial, i.e. it is completely determined by $L(\Ac)$ (cf. Definition \[Def\_CombClass\]). Recently, a remarkable generalization of the Addition-Deletion theorem was obtained by Abe. His Division Theorem [@Abe16_DivFree Thm. 1.1] motivates the class of *divisionally free* arrangements. In this class freeness is a combinatorial property too.
Despite having these useful tools at hand, it is still a major open problem, known as Terao’s Conjecture, whether in general the freeness of $\Ac$ actually depends only on $L(\Ac)$, provided the field $\KK$ is fixed (see [@Zieg90_MatrFree] for a counterexample when one fixes $L(\Ac)$ but changes the field). We should also mention at this point the very recent results by Abe further examining Addition-Deletion constructions together with divisional freeness [@Abe2018_DelThm_Combinatorics], [@Abe18_AddDel_Combinatorics].
A variation of the addition part of the Addition-Deletion theorem \[Thm\_Add\_Del\] was obtained by Abe, Barakat, Cuntz, Hoge, and Terao in [@ABCHT16_FreeIdealWeyl]: the Multiple Addition Theorem \[Thm\_MAT\] (MAT for short). Using this theorem, the authors gave a new uniform proof of the Kostant-Macdonald-Shapiro-Steinberg formula for the exponents of a Weyl group. In the same way the Addition-Theorem defines the class of inductively free arrangements, it is now natural to consider the class ${{\mathfrak{MF}}}$ of those free arrangements, called *MAT-free*, which can be build inductively using the MAT (Definition \[Def\_MATfree\]). It is not hard to see (Lemma \[Lem\_MATComb\]) that MAT-freeness only depends on $L(\Ac)$. In this paper, we investigate classes of MAT-free arrangements beyond the classes considered in [@ABCHT16_FreeIdealWeyl].
Complex reflection groups (classified by Shephard and Todd [@ST_1954_fcrg]) play an important role in the study of hyperplane arrangements: many interesting examples and counterexamples are related or derived from the reflection arrangement $\Ac(W)$ of a complex reflection group $W$. It was proven by Terao [@Terao1980_FreeUniRefArr] that reflection arrangements are always free. There has been a series of investigations dealing with reflection arrangements and their connection to the aforementioned combinatorial classes of free arrangements (e.g. [@BC12_CoxCrystIndFree], [@HoRoe15_IndFreeRef], [@Abe16_DivFree]). Therefore, it is natural to study reflection arrangements in conjunction with the new class of MAT-free arrangements.
Our main result is the following.
\[Thm\_matref\] Except for the arrangement $\Ac(G_{32})$, an irreducible reflection arrangement is MAT-free if and only if it is inductively free. The arrangement $\Ac(G_{32})$ is inductively free but not MAT-free. Thus every reflection arrangement is MAT-free except the reflection arrangements of the imprimitive reflection groups $G(e,e,\ell)$, $e>2$, $\ell>2$ and of the reflection groups $$G_{24}, G_{27}, G_{29}, G_{31}, G_{32}, G_{33}, G_{34}.$$
A further generalization of the MAT \[Thm\_MAT\] was very recently obtained by Abe and Terao [@AbeTer18_MultAddDelRes]: the Multiple Addition Theorem 2 \[Thm\_MAT2\] (MAT2 for short). Again, one might consider the inductively defined class of arrangements which can be build from the empty arrangement using this more general tool, i.e. the class ${{\mathfrak{MF}}}'$ of *MAT2-free* arrangements (Defintion \[Def\_MAT2free\]). By definition, this class contains the class of MAT-free arrangements. Regarding reflection arrangements we have the following:
\[Thm\_mat2free\] Let ${\mathcal{A}}= \Ac(W)$ be an irreducible reflection arrangement. Then $\Ac$ is MAT2-free if and only if it is MAT-free.
In contrast to (irreducible) reflection arrangements, in general the class of MAT-free arrangements is properly contained in the class of MAT2-free arrangements (see Proposition \[Propo\_MATproperSubclassMAT2\]).
Based on our classification of MAT-free (MAT2-free) reflection arrangements and other known examples ([@ABCHT16_FreeIdealWeyl], [@CRS17_IdealIF]) we arrive at the following question:
\[Ques\_MATIF\] Is every MAT-free (MAT2-free) arrangement inductively free?
In [@CRS17_IdealIF] the authors proved that all ideal subarrangements of a Weyl arrangement are inductively free by extensive computer calculations. A positive answer to Question \[Ques\_MATIF\] would directly imply their result and yield a uniform proof (cf. [@CRS17_IdealIF Rem. 1.5(d)]).
Looking at the class of divisionally free arrangements which properly contains the class of inductively free arrangements [@Abe16_DivFree Thm. 4.4] a further natural question is:
\[Ques\_MATDF\] Is every MAT-free (MAT2-free) arrangement divisionally free?
This article is organized as follows: in Section \[Sec\_freeArr\] we briefly recall some notions and results about hyperplane arrangements and free arrangements used throughout our exposition. In Section \[Sec\_MAT\] we give an alternative characterization of MAT-freeness and two easy necessary conditions for MAT/MAT2-freeness. Furthermore, we comment on the relation of the two classes ${{\mathfrak{MF}}}$ and ${{\mathfrak{MF}}}'$ and on the product construction. Section \[Sec\_ProofImprim\] and Section \[Sec\_ProofPrim\] contain the proofs of Theorem \[Thm\_matref\] and Theorem \[Thm\_mat2free\]. In the last Section \[Sec\_Remarks\] we comment on Question \[Ques\_MATIF\] and further problems connected with MAT-freeness.
Acknowledgments {#acknowledgments .unnumbered}
---------------
We thank Gerhard R[ö]{}hrle for valuable comments on an earlier draft of our manuscript.
Hyperplane arrangements and free arrangements {#Sec_freeArr}
=============================================
Let $\Ac$ be a hyperplane arrangement in $V \cong \KK^\ell$. If $\Ac$ is empty, then it is denoted by $\Phi_\ell$.
The *intersection lattice* $L(\Ac)$ of $\Ac$ consists of all intersections of elements of $\Ac$ including $V$ as the empty intersection. Indeed, with the partial order by reverse inclusion $L(\Ac)$ is a geometric lattice [@OrTer92_Arr Lem. 2.3]. The *rank* $\operatorname{rk}(\Ac)$ of $\Ac$ is defined as the codimension of the intersection of all hyperplanes in $\Ac$.
If $x_1,\ldots,x_\ell$ is a basis of $V^*$, to explicitly give a hyperplane we use the notation $(a_1,\ldots,a_\ell)^\perp := \ker(a_1x_1+\ldots+a_\ell x_\ell)$.
\[Def\_CombClass\] Let ${{\mathfrak{C}}}$ be a class of arrangements and let ${\mathcal{A}}\in {{\mathfrak{C}}}$. If for all arrangements $\Bc$ with $L(\Bc) \cong L(\Ac)$, (where $\Ac$ and $\Bc$ do not have to be defined over the same field), we have ${\mathcal{B}}\in {{\mathfrak{C}}}$, then the class ${{\mathfrak{C}}}$ is called *combinatorial*.
If ${{\mathfrak{C}}}$ is a combinatorial class of arrangements such that every arrangement in ${{\mathfrak{C}}}$ is free than ${\mathcal{A}}\in {{\mathfrak{C}}}$ is called *combinatorially free*.
For $X \in L(\Ac)$ the *localization* $\Ac_X$ of $\Ac$ at X is defined by: $$\Ac_X := \{ H \in {\mathcal{A}}\mid X \subseteq H \},$$ and the *restriction* $\Ac^X$ of $\Ac$ to $X$ is defined by: $$\Ac^X := \{ X\cap H \mid H \in {\mathcal{A}}\setminus \Ac_X \}.$$
Let $\Ac_1$ and $\Ac_2$ be two arrangements in $V_1$ respectively $V_2$. Then their *product* $\Ac_1 \times \Ac_2$ is defined as the arrangement in $V = V_1 \oplus V_2$ consisting of the following hyperplanes: $$\Ac_1 \times \Ac_2 := \{H_1 \oplus V_2 \mid H_1 \in \Ac_1 \} \cup \{V_1 \oplus H_2 \mid H_2 \in \Ac_2 \}.$$ We note the following facts about products (cf. [@OrTer92_Arr Ch. 2]):
- $| \Ac_1\times\Ac_2| = |\Ac_1| + |\Ac_2|$.
- $L(\Ac_1\times\Ac_2) = \{X_1 \oplus X_2 \mid X_1 \in L(\Ac_1)$ and $X_2 \in L(\Ac_2)\}$.
- $(\Ac_1\times\Ac_2)^X = \Ac_1^{X_1} \times \Ac_2^{X_2}$ if $X = X_1\oplus X_2$ with $X_i \in L(\Ac_i)$.
Let $S = S(V^*)$ be the symmetric algebra of the dual space. We fix a basis $x_1,\ldots,x_\ell$ for $V^*$ and identify $S$ with the polynomial ring $\KK[x_1,\ldots,x_\ell]$. The algebra $S$ is equipped with the grading by polynomial degree: $S = \bigoplus_{p\in \ZZ} S_p$, where $S_p$ is the set of homogeneous polynomials of degree $p$ ($S_p = \{0\}$ for $p < 0$).
A $\KK$-linear map $\theta:S\to S$ which satisfies $\theta(fg) = \theta(f)g + f\theta(g)$ is called a $\KK$-*derivation*. Let $\operatorname{Der}(S)$ be the $S$-module of $\KK$-derivations of $S$. It is a free $S$-module with basis $D_1,\ldots,D_\ell$ where $D_i$ is the partial derivation ${\partial/\partial x_{i}}$. We say that $\theta \in \operatorname{Der}(S)$ is *homogeneous of polynomial degree* $p$ provided $\theta = \sum_{i=1}^\ell f_i D_i$ with $f_i \in S_p$ for each $1 \leq i \leq \ell$. In this case we write $\operatorname{pdeg}{\theta} = p$. We obtain a $\ZZ$-grading for the $S$-module $\operatorname{Der}(S)$: $\operatorname{Der}(S) = \bigoplus_{p \in \ZZ} \operatorname{Der}(S)_p$.
For $H \in \Ac$ we fix $\alpha_H \in V^*$ with $H = \ker(\alpha_H)$. The *module of $\Ac$-derivations* is defined by $$ D(\Ac) := \{ \theta \in \operatorname{Der}(S) \mid \theta(\alpha_H) \in {\alpha_H}S \text{ for all } H \in \Ac\}.$$ We say that $\Ac$ is *free* if the module of $\Ac$-derivations is a free $S$-module.
If $\Ac$ is a free arrangement we may choose a homogeneous basis $\{ \theta_1, \ldots, \theta_\ell \}$ for $D(\Ac)$. Then the polynomial degrees of the $\theta_i$ are called the *exponents* of $\Ac$ and they are uniquely determined by $\Ac$, [@OrTer92_Arr Def. 4.25]. We write $\exp(\Ac) := (\operatorname{pdeg}{\theta_1},\ldots$, $\operatorname{pdeg}{\theta_\ell})$. Note that the empty arrangement $\Phi_\ell$ is free with $\exp(\Phi_\ell)=(0,\ldots,0)\in\ZZ^\ell$. If $d_1,\ldots,d_\ell \in \ZZ$ with $d_1 \leq d_2 \leq \ldots \leq d_\ell$ we write $(d_1,\ldots,d_\ell)_\leq$.
The notion of freeness is compatible with products of arrangements:
\[Prop\_ProdFree\] Let $\Ac=\Ac_1\times \Ac_2$ be a product of two arrangements. Then $\Ac$ is free if and only if both $\Ac_1$ and $\Ac_2$ are free. In this case if $\exp(\Ac_i)=(d^i_1,\ldots,d^i_{\ell_i})$ for $i=1,2$ then $$\exp(\Ac) = (d^1_1,\ldots,d^1_{\ell_1},d^2_1,\ldots,d^2_{\ell_2}).$$
The following theorem provides a useful tool to prove the freeness of arrangements.
\[Thm\_Add\_Del\] Let $\Ac$ be a hyperplane arrangement and $H_0 \in \Ac$. We call $(\Ac,\Ac'=\Ac\setminus \{H_0\},\Ac''=\Ac^{H_0})$ a *triple of arrangements*. Any two of the following statements imply the third:
1. $\Ac$ is free with $\exp(\Ac) = (b_1,\ldots,b_{l-1},b_\ell)$,
2. $\Ac'$ is free with $\exp(\Ac') = (b_1,\ldots,b_{\ell-1},b_\ell-1)$,
3. $\Ac''$ is free with $\exp(\Ac'') = (b_1,\ldots,b_{\ell-1})$.
The preceding theorem motivates the following definition.
\[Def\_IF\] The class ${{\mathfrak{IF}}}$ of *inductively free* arrangements is the smallest class of arrangements which satisfies
1. the empty arrangement $\Phi_ \ell$ of rank $\ell$ is in ${{\mathfrak{IF}}}$ for $\ell \geq 0$,
2. if there exists a hyperplane $H_0 \in \Ac$ such that $\Ac'' \in {{\mathfrak{IF}}}$, $\Ac' \in {{\mathfrak{IF}}}$, and $\exp(\Ac'') \subset \exp(\Ac')$, then $\Ac$ also belongs to ${{\mathfrak{IF}}}$.
Here $(\Ac,\Ac',\Ac'') = (\Ac,\Ac\setminus \{H_0\},\Ac^{H_0})$ is a triple as in Theorem \[Thm\_Add\_Del\].
The class ${{\mathfrak{IF}}}$ is easily seen to be combinatorial [@CHo15_FreeNotRecFree Lem. 2.5].
The following result was a major step in the investigation of freeness properties for reflection arrangements.
\[Thm\_ClassIFReflArr\] For W a finite complex reflection group, the reflection arrangement $\Ac(W)$ is inductively free if and only if $W$ does not admit an irreducible factor isomorphic to a monomial group $G(r,r,\ell)$ for $r, \ell ≥ 3$, $G_{24}$, $G_{27}$, $G_{29}$, $G_{31}$, $G_{33}$, or $G_{34}$.
Let $\Ac$ be an arrangement with $|\Ac|=n$. We say that $\Ac$ has a *free filtration* if there are subarrangements $$\emptyset = \Ac_0 \subsetneq \Ac_1 \subsetneq \cdots \subsetneq \Ac_{n-1} \subsetneq \Ac_n = \Ac$$ such that $|\Ac_i| = i$ and $\Ac_i$ is free for all $1 \leq i \leq n$.
Very recently, Abe [@Abe18_AddDel_Combinatorics] introduced the class ${{\mathfrak{AF}}}$ of *additionally free* arrangements. Arrangements in ${{\mathfrak{AF}}}$ are by definition exactly the arrangements admitting a free filtration. Furthermore, it is a direct consequence of [@Abe18_AddDel_Combinatorics Thm. 1.4] that the class ${{\mathfrak{AF}}}$ is combinatorial.
Multiple Addition Theorem {#Sec_MAT}
=========================
The following theorem presented in [@ABCHT16_FreeIdealWeyl] is a variant of the addition part ((2) and (3) imply (1)) of Theorem \[Thm\_Add\_Del\].
\[Thm\_MAT\] Let $\Ac'$ be a free arrangement with $\exp(\Ac')=(d_1,\ldots,d_\ell)_\le$ and $1 \le p \le \ell$ the multiplicity of the highest exponent, i.e., $$d_{\ell-p} < d_{\ell-p+1} =\cdots=d_\ell=:d.$$ Let $H_1,\ldots,H_q$ be hyperplanes with $H_i \not \in \Ac'$ for $i=1,\ldots,q$. Define $$\Ac''_j:=(\Ac'\cup \{H_j\})^{H_j}=\{H\cap H_{j} \mid H\in \Ac'\}, \quad j=1,\ldots,q.$$ Assume that the following three conditions are satisfied:
- $X:=H_1 \cap \cdots \cap H_q$ is $q$-codimensional.
- $X \not \subseteq \bigcup_{H \in \Ac'} H$.
- $|\Ac'|-|\Ac''_j|=d$ for $1 \le j \le q$.
Then $q \leq p$ and $\Ac:=\Ac' \cup \{H_1,\ldots,H_q\}$ is free with $\exp(\Ac)=(d_1,\ldots,d_{\ell-q},d+1,$ $\ldots,d+1)_\le$. \[MAT\]
Note that in contrast to Theorem \[Thm\_Add\_Del\] no freeness condition on the restriction is needed to conclude the freeness of $\Ac$ in Theorem \[Thm\_MAT\]. The MAT motivates the following definition.
\[Def\_MATfree\] The class ${{\mathfrak{MF}}}$ of *MAT-free* arrangements is the smallest class of arrangements subject to
- $\Phi_\ell$ belongs to ${{\mathfrak{MF}}}$, for every $\ell \ge 0$;
- if $\Ac' \in {{\mathfrak{MF}}}$ with $\exp(\Ac')=(d_1,\ldots,d_\ell)_\le$ and $1 \le p \le \ell$ the multiplicity of the highest exponent $d=d_\ell$, and if $H_1,\ldots,H_q$, $q\le p$ are hyperplanes with $H_i \not \in \Ac'$ for $i=1,\ldots,q$ such that:
- $X:=H_1 \cap \cdots \cap H_q$ is $q$-codimensional,
- $X \not \subseteq \bigcup_{H \in \Ac'} H$,
- $|\Ac'|-|(\Ac'\cup \{H_j\})^{H_j}|=d$, for $1 \le j \le q$,
then $\Ac:=\Ac' \cup \{H_1,\ldots,H_q\}$ also belongs to ${{\mathfrak{MF}}}$ and has exponents $\exp(\Ac) = (d_1,\ldots,d_{\ell-q},d+1,\ldots,d+1)_\le$.
Abe and Terao [@AbeTer18_MultAddDelRes] proved the following generalization of Theorem \[Thm\_MAT\]:
\[Thm\_MAT2\] Assume that $\Ac'$ is a free arrangement with $\exp(\Ac')=(d_1,d_2,\ldots,d_\ell)_\le$. Let $$t :=
\begin{cases}
\min\{i \mid d_i \neq 0\} & \text{if } \Ac'\neq \Phi_\ell \\
0 & \text{if } \Ac'=\Phi_\ell
\end{cases}.$$ For $H_s,\ldots,H_\ell \notin \Ac$ with $s > t$, define $\Ac_j'':=(\Ac' \cup \{H_j\})^{H_j}$, $\Ac:=\Ac' \cup \{H_s,\ldots,H_\ell\}$ and assume the following conditions:
- $X:=\bigcap_{i=s}^\ell H_i$ is $(\ell-s+1)$-codimensional,
- $X \not \subset \bigcup_{K \in \Ac'}K$, and
- $|\Ac'| -|\Ac_j''|=d_j$ for $j=s,\ldots,\ell$.
Then $\Ac$ is free with exponents $(d_1,d_2,\ldots,d_{s-1},d_{s}+1,\ldots,d_\ell+1)_\leq$. Moreover, there is a basis $\theta_1,\theta_2,\ldots,\theta_{s-1},\eta_s,\ldots,\eta_\ell$ for $D(\Ac')$ such that $\deg \theta_i=d_i,\ \deg \eta_j=d_j$, $\theta_i \in D(\Ac)$ and $\eta_j \in D({\mathcal{A}}\setminus \{H_j\})$ for all $i$ and $j$.
This in turn motivates:
\[Def\_MAT2free\] The class ${{\mathfrak{MF}}}'$ of *MAT2-free* arrangements is the smallest class of arrangements subject to
- $\Phi_\ell$ belongs to ${{\mathfrak{MF}}}'$, for every $\ell \ge 0$;
- if $\Ac' \in {{\mathfrak{MF}}}'$ with $\exp(\Ac')=(d_1,d_2,\ldots,d_\ell)_\le$ and if $H_s,\ldots,H_\ell$ are hyperplanes with $H_i \not \in \Ac'$ for $i=s,\ldots,\ell$, where $$s >
\begin{cases}
\min\{i \mid d_i \neq 0\} & \text{if } \Ac'\neq \Phi_\ell \\
0 & \text{if } \Ac'=\Phi_\ell
\end{cases},$$ and with
- $X:=H_s \cap \cdots \cap H_\ell$ is $(\ell-s+1)$-codimensional,
- $X \not \subseteq \bigcup_{H \in \Ac'} H$,
- $|\Ac'|-|(\Ac'\cup \{H_j\})^{H_j}|=d_j$ for $s \le j \le \ell$,
then $\Ac:=\Ac' \cup \{H_s,\ldots,H_\ell\}$ also belongs to ${{\mathfrak{MF}}}'$ and has exponents $\exp(\Ac) = (d_1,\ldots,d_{s-1},d_s+1,\ldots,d_\ell+1)_\le$.
We note the following:
\[Rem\_MATexpMAT2\]
1. We have ${{\mathfrak{MF}}}\subseteq {{\mathfrak{MF}}}'$.
2. If $\Ac$ is a free arrangement with $\exp(\Ac) = (0,\ldots,0,1,\ldots,1,d,\ldots,d)_\le$, i.e.$\Ac$ has only two distinct exponents $\neq 0$, then it is clear from the definitions that $\Ac$ is MAT2-free if and only if $\Ac$ is MAT-free.
\[Exam\_rk2\_boolean\_WeylMAT\]
1. If $\operatorname{rk}(\Ac)=2$ then $\Ac$ is MAT-free and therefore MAT2-free too.
2. Every ideal subarrangement of a Weyl arrangement is MAT-free and therefore also MAT2-free, [@ABCHT16_FreeIdealWeyl].
\[Lem\_MATComb\] The classes ${{\mathfrak{MF}}}$ and ${{\mathfrak{MF}}}'$ are combinatorial.
The class of all empty arrangements is combinatorial and contained in ${{\mathfrak{MF}}}$. Let ${\mathcal{A}}\in {{\mathfrak{MF}}}$ (${\mathcal{A}}\in {{\mathfrak{MF}}}'$). Since conditions (1)–(3) in Defintion \[Def\_MATfree\] (respectively Defintion \[Def\_MAT2free\]) only depend on $L(\Ac)$ the claim follows. See also [@AbeTer18_MultAddDelRes Thm. 5.1].
If an arrangement $\Ac$ is MAT-free, the MAT-steps yield a partition of $\Ac$ whose dual partition gives the exponents of $\Ac$. Vice versa, the existence of such a partition suffices for the MAT-freeness of the arrangement:
\[Lem\_MATPart\] Let $\Ac$ be an $\ell$-arrangement. Then $\Ac$ is MAT-free if and only if there exists a partition $\pi = (\pi_1|\cdots|\pi_n)$ of $\Ac$ where for all $0 \leq k \leq n-1$,
- $\operatorname{rk}(\pi_{k+1}) = \vert \pi_{k+1} \vert$,
- $\cap_{H \in \pi_{k+1}} H = X_{k+1} \nsubseteq \bigcup_{H' \in \Ac_k}H'$ where $\Ac_k = \bigcup_{i=1}^k \pi_i$,
- $\vert \Ac_k \vert - \vert (\Ac_k \cup \{H\})^H \vert = k$ for all $H \in \pi_{k+1}$.
In this case $\Ac$ has exponents $\exp(\Ac) = (d_1,\ldots,d_\ell)_\le$ with $d_i = |\{k \mid |\pi_k|\geq \ell-i+1 \}|$.
This is immediate from the definition.
If $\pi$ is a partition as in Lemma \[Lem\_MATPart\] then $\pi$ is called an *MAT-partition* for $\Ac$.
If we have chosen a linear ordering ${\mathcal{A}}= \{ H_1,\ldots,H_m\}$ of the hyperplanes in $\Ac$, to specify the partition $\pi$, we give the corresponding ordered set partition of $[m] = \{1,\ldots,m\}$.
\[Exam\_IFbnMAT2\] Supersolvable arrangements, a proper subclass of inductively free arrangements [@OrTer92_Arr Thm. 4.58], are not necessarily MAT2-free: an easy calculation shows that the arrangement denoted $\Ac(10,1)$ in [@Grue09_SimplArr] is supersolvable but not MAT2-free. In particular $\Ac(10,1)$ is neither MAT-free.
Restrictions of MAT2-free (MAT-free) arrangements are not necessarily MAT2-free (MAT-free):
\[Exam\_ResNotMAT2\] Let ${\mathcal{A}}= \Ac(E_6)$ be the Weyl arrangement of the Weyl group of type $E_6$. Then $\Ac$ is MAT-free by Example \[Exam\_rk2\_boolean\_WeylMAT\](2). Let $H \in \Ac$. A simple calculation (with the computer) shows that $\Ac^H$ is not MAT2-free.
We have two simple necessary conditions for MAT-freeness respectively MAT2-freeness. The first one is:
\[Lem\_AResnMATFree\] Let $\Ac$ be a non-empty MAT2-free arrangement with exponents $\exp(\Ac)$ $= (d_1,\ldots,d_\ell)_\leq$. Then there is an $H \in \Ac$ such that $\vert {\mathcal{A}}\vert - \vert \Ac^H \vert = d_\ell$. In particular, the same holds, if $\Ac$ is MAT-free.
By definition there are $H_q,\ldots,H_\ell \in \Ac$, $2\leq q$ such that $\Ac' := {\mathcal{A}}\setminus \{H_q,\ldots,H_\ell \}$ is MAT2-free. Furthermore by condition (1) the hyperplanes $H_q,\ldots,H_\ell$ are linearly independent. Let $H := H_\ell$. By condition (2), we have $X = \cap_{i=q}^\ell H_i \nsubseteq \cup_{H' \in \Ac'} H'$ and thus $|\Ac^{H}| = |(\Ac'\cup \{H\})^H| + \ell-q$. Now $$\vert \Ac' \vert - \vert (\Ac'\cup \{H\})^H \vert = d_\ell-1$$ by condition (3) and hence $$\vert {\mathcal{A}}\vert - \vert \Ac^H \vert = |\Ac'|+\ell-q+1 - \vert (\Ac'\cup \{H\})^H \vert - \ell+ q = d_\ell.$$
The second one is:
\[Lem\_MAT\_free\_filtration\] Let $\Ac$ be an MAT2-free arrangement. Then $\Ac$ has a free filtration, i.e. $\Ac$ is additionally free. In particular, the same is true, if $\Ac$ is MAT-free.
Let $\Ac$ be MAT2-free. Then by definition there are $H_q,\ldots,H_\ell \in \Ac$ such that $\Ac' := {\mathcal{A}}\setminus \{H_q,\ldots,H_\ell\}$ is MAT2-free and conditions (1)–(3) are satisfied. Set ${\mathcal{B}}:= \{H_q,\ldots,H_\ell\}$. By [@AbeTer18_MultAddDelRes Cor. 3.2] for all ${\mathcal{C}}\subseteq \Bc$ the arrangement $\Ac' \cup \Cc$ is free. Hence by induction $\Ac$ has a free filtration.
An MAT2-free but not MAT-free arrangement {#an-mat2-free-but-not-mat-free-arrangement .unnumbered}
-----------------------------------------
We now provide an example of an arrangement which is MAT2-free but not MAT-free.
\[Exam\_MAT2bnMAT\] Let $\Ac$ be the arrangement defined by $$\begin{aligned}
{\mathcal{A}}:= \{ &H_1,\ldots,H_{10} \} \\
:= \{ &(1,0,0)^\perp, (0,1,0)^\perp, (0,0,1)^\perp, (1,1,0)^\perp, (1,2,0)^\perp, (0,1,1)^\perp, \\
&(1,3,0)^\perp, (1,1,1)^\perp, (2,3,0)^\perp, (1,3,1)^\perp \}.\end{aligned}$$ It is not hard to see that $\Ac$ is inductively free (actually supersolvable) with $\exp(\Ac) = (1,4,5)$.
\[Prop\_ExamMAT2\] The arrangement $\Ac$ from Example \[Exam\_MAT2bnMAT\] is MAT2-free.
Let $\Bc_1=\{H_1,H_2,H_3\}$, $\Bc_2=\{H_4\}$, $\Bc_3=\{H_5,H_6\}$, $\Bc_4=\{H_7,H_8\}$, $\Bc_5 = \{H_9,H_{10}\}$, and $\Ac_k = \cup_{i=1}^k \Bc_i$ for $1 \leq k \leq 5$. It is clear that $\Ac_1$ is MAT2-free. A simple linear algebra computation shows that the addition of $\Bc_{i+1}$ to $\Ac_{i}$ for $1 \leq i \leq 4$ satisfies Condition (1)–(3) of Definition \[Def\_MAT2free\]. Hence ${\mathcal{A}}= \Ac_5$ is MAT2-free.
\[Prop\_ExamNotMAT\] The arrangement $\Ac$ from Example \[Exam\_MAT2bnMAT\] is not MAT-free.
Suppose $\Ac$ is MAT-free and $\pi = (\pi_1,\ldots,\pi_5)$ is an MAT-partition. Since $\exp(\Ac) = (1,4,5)$ the last block $\pi_5$ has to be a singleton, i.e. $\pi_5 = \{H\}$. By Condition (3) of Lemma \[Lem\_MATPart\] we have $|\Ac^H| = 5$ and the only hyperplane with this property is $H_9 = (2,3,0)^\perp$. Similarly $\pi_4$ can only contain one of $H_3,H_6,H_8,H_{10}$. But looking at their intersections we see that all of the latter are contained in another hyperplane of $\Ac$, e.g.$H_3 \cap H_8 \subseteq H_4$. This contradicts Condition (2). Hence $\Ac$ is not MAT-free.
As a direct consequence we get:
\[Propo\_MATproperSubclassMAT2\] We have $${{\mathfrak{MF}}}\subsetneq {{\mathfrak{MF}}}'.$$
Products of MAT-free and MAT2-free arrangements {#products-of-mat-free-and-mat2-free-arrangements .unnumbered}
-----------------------------------------------
As for freeness in general (Proposition \[Prop\_ProdFree\]), the product construction is compatible with the notion of MAT-freeness:
\[Thm\_ProdMAT\] Let ${\mathcal{A}}= \Ac_1 \times \Ac_2$ be a product of two arrangements. Then ${\mathcal{A}}\in {{\mathfrak{MF}}}$ if and only if $\Ac_1 \in {{\mathfrak{MF}}}$ and $\Ac_2 \in {{\mathfrak{MF}}}$.
Assume $\Ac_i$ is an arrangement in the vector space $V_i$ of dimension $\ell_i$ for $i=1,2$. We argue by induction on $|\Ac|$. If $|\Ac|=0$, i.e. $\Ac_1 = \Phi_{\ell_1}$, and $\Ac_2 = \Phi_{\ell_2}$ then the statement is clear. Assume $\Ac_1$ is MAT-free with $\exp(\Ac_1) = (d^1_1,\ldots,d^1_{\ell_1})_\le$ and $\Ac_2$ is MAT-free with $\exp(\Ac_1) = (d^2_1,\ldots,d^2_{\ell_2})_\le$. Then without loss of generality $d:=d^1_{\ell_1} \geq d^2_{\ell_2}$. Let $q_i$ be the multiplicity of the exponent $d$ in $\exp(\Ac_i)$ for $i=1,2$ (note that $q_2=0$ if $d>d^2_{\ell_2}$). Then since $\Ac_i$ is MAT-free there are hyperplanes $\{H^i_1,\ldots,H^i_{q_i}\} \subseteq \Ac_i$ such that $\Ac_i' := \Ac_i \setminus \{H^i_1,\ldots,H^i_{q_i}\}$ is MAT-free, i.e. they satisfy Conditions (1)–(3) from Definition \[Def\_MATfree\]. Now by the induction hypothesis $\Ac' = \Ac_1' \times \Ac_2'$ is MAT-free and clearly $\{H^1_1\oplus V_2,\ldots,H^1_{q_1}\oplus V_2\} \cup \{V_1\oplus H^2_1,\ldots,V_1\oplus H^2_{q_2}\}$ satisfy Conditions (1)–(3). Hence $\Ac$ is MAT-free.
Conversely assume $\Ac$ is MAT-free with $\exp(\Ac)=(d_1,\ldots,d_\ell)_\leq$. By Proposition \[Prop\_ProdFree\] both factors $\Ac_1$ and $\Ac_2$ are free with $\exp(\Ac_i) = (d^i_1,\ldots,d^i_{\ell_i})_\leq$ and without loss of generality $d_\ell=d^1_{\ell_1}\geq d^2_{\ell_2}$. Assume further that $q_i$ is the multiplicity of $d_\ell$ in $\exp(\Ac_i)$ and $q$ is the multiplicity of $d_\ell$ in $\exp(\Ac)$, i.e. $q = q_1+q_2$. There are hyperplanes $\{H_1,\ldots,H_q\} \subset \Ac$ such that $\Ac' = {\mathcal{A}}\setminus \{H_1,\ldots,H_q\}$ is MAT-free with $\exp(\Ac') = (d_1,\ldots,d_{\ell-q},d_{\ell-q+1}-1,\ldots,d_\ell-1)_\le$, and Conditions (1)–(3) are satisfied. We may further assume that $H_i = H^1_i \oplus V_2$ for $1\leq i \leq q_1$ and $H_j = V_1 \oplus H^2_{j-q_1}$ for $q_1+1 \leq j \leq q$. Let $\Ac_i' = \Ac_i \setminus \{H^i_1,\ldots,H^i_{q_i}\}$ for $i=1,2$. Note that if $d_\ell > d^2_{\ell_2}$ we have $q_2 = 0$ and $\Ac_2' = \Ac_2$. But at least we have $\Ac_1' \subsetneq \Ac_1$. Then $\Ac' = \Ac_1' \times \Ac_2'$, $|\Ac'| < |\Ac|$ and by the induction hypothesis $\Ac_1'$ and $\Ac_2'$ are MAT-free and Conditions (1) and (2) are clearly satified for $\Ac_i'$ and $\{H^i_1,\ldots,H^i_{q_i}\}$. But since $$\begin{aligned}
d_\ell-1 =\, &|\Ac'| - |(\Ac'\cup\{H_i\})^{H_i}| \\
=\, &|\Ac_1'| + |\Ac_2'| - (|(\Ac_1\cup \{H^1_i\})^{H^1_i}| + |\Ac_2'|) \\
=\, &|\Ac_1'| - |(\Ac_1\cup \{H^1_i\})^{H^1_i}|\end{aligned}$$ for $1\leq i \leq q_1$ and $$\begin{aligned}
d_\ell-1 =\, &|\Ac'| - |(\Ac'\cup\{H_j\})^{H_j}| \\
=\, &|\Ac_1'| + |\Ac_2'| - (|(\Ac_1\cup \{H^2_{j-q_1}\})^{H^2_{j-q_1}}| + |\Ac_2'|) \\
=\, &|\Ac_1'| - |(\Ac_1\cup \{H^2_{j-q_1}\})^{H^2_{j-q_1}}|\end{aligned}$$ for $q_1+1 \leq j \leq q_2$, Condition (3) is also satisfied for $\Ac_1'$ and $\Ac_2'$. Hence both $\Ac_1$ and $\Ac_2$ are MAT-free.
Altenatively, one can prove Theorem \[Thm\_ProdMAT\] by observing that MAT-Partitions for $\Ac_1$ and $\Ac_2$ are directly obtained from an MAT-Partition for $\Ac$: take the non-empty factors of each block in the same order, and vise versa: take the products of the blocks of partitions for $\Ac_1$ and $\Ac_2$.
\[Rem\_ReducibleMAT\] Thanks to the preceding theorem, our classification of MAT-free irreducible reflection arrangements proved in the next 2 sections gives actually a classification of all MAT-free reflection arrangements: a reflection arrangement $\Ac(W)$ is MAT-free if and only if it has no irreducible factor isomorphic to one of the non-MAT-free irreducible reflection arrangements listed in Theorem \[Thm\_matref\].
In contrast to MAT-freeness, the weaker notion of MAT2-freeness is not compatible with products as the following example shows:
Let $\Ac_1$ be the MAT2-free but not MAT-free arrangement of Example \[Exam\_MAT2bnMAT\] with exponents $\exp(\Ac_1) = (1,4,5)$. Let $\zeta = \frac{1}{2}(-1+i\sqrt{3})$ be a primitive cube root of unity, and let $\Ac_2$ be the arrangement defined by the following linear forms: $$\begin{aligned}
\Ac_2 := \{ &H^2_1,\ldots,H^2_{10}\} \\
:= \{ &(1,0,0)^\perp,(0,1,0)^\perp,(0,0,1)^\perp,(1,-\zeta,0)^\perp,(1,0,-\zeta)^\perp \\
&(1,-\zeta^2,0)^\perp,(1,0,-\zeta^2)^\perp,(1,-1,0)^\perp,(1,0,-1)^\perp, (0,1,-\zeta)^\perp \}.\end{aligned}$$ A linear algebra computation shows that $\pi = (1,2,3|4,5|6,7|8,9|10)$ is an MAT-partition for $\Ac_2$. In particular $\Ac_2$ is MAT2-free with $\exp(\Ac_2) = (1,4,5)$.
Now by Proposition \[Prop\_ProdFree\] the product ${\mathcal{A}}:= \Ac_1 \times \Ac_2$ is free with $\exp(\Ac) =(1,1,$ $4,4,5,5)$. Suppose $\Ac$ is MAT2-free. Then either there are hyperplanes $H_1 \in \Ac_1$ and $H_2 \in \Ac_2$ such that $\Ac' = \Ac_1' \times \Ac_2'$ is MAT2-free with exponents $\exp(\Ac')=(1,1,4,4,4,4)$ where $\Ac_i' = \Ac_i \setminus \{H_i\}$. Or there are hyperplanes $H^1_1,H^1_2 \in \Ac_1$, $H^2_1,H^2_2\in \Ac_2$ such that $\Ac' = \Ac_1' \times \Ac_2'$ is MAT2-free with exponents $\exp(\Ac')=(1,1,3,3,4,4)$ where $\Ac_i' = \Ac_i \setminus \{H^i_1,H^i_2\}$.
In the first case $\Ac'$ is actually MAT-free by Remark \[Rem\_MATexpMAT2\]. But then by Theorem \[Thm\_ProdMAT\] $\Ac_2'$ is MAT-free and $\Ac_2$ is MAT-free too which is a contradiction.
In the second case $H^1_1\oplus V_2,H^1_2\oplus V_2, V_1 \oplus H^2_1, V_1 \oplus H^2_2$ satisfy Condition (1)–(3) of Defintion \[Def\_MAT2free\]. But by Condition (3) we have $$|\Ac_1'| - |(\Ac_1'\cup\{H^1_1\})^{H^1_1}| = 4$$ and $$|\Ac_1'| - |(\Ac_1'\cup\{H^1_2\})^{H^1_2}| = 3.$$ But an easy calculation shows that there are no two hyperplanes in $\Ac_1$ with this property and which also satisfy Condition (2)–(3). This is a contradiction and hence ${\mathcal{A}}= \Ac_1 \times \Ac_2$ is not MAT2-free.
MAT-free imprimitive reflection groups {#Sec_ProofImprim}
======================================
Let $x_1,\ldots,x_\ell$ be a basis of $V^*$. Let $\zeta = \exp(\frac{2\pi i}{r})$ ($r \in \NN$) be a primitive $r$-th root of unity. Define the linear forms $\alpha_{ij}(\zeta^k) \in V^*$ by $$\alpha_{ij}(\zeta^k) = x_i - \zeta^k x_j$$ and the hyperplanes $$H_{ij}(\zeta^k) = \ker(\alpha_{ij}(\zeta^k)).$$ for $1 \leq i,j \leq \ell$ and $1 \leq k \leq r$. Then the reflection arrangement of the imprimitive complex reflection group $G(r,1,\ell)$ can be defined by: $$\Ac(G(r,1,\ell)) = \{\ker(x_i) \mid 1 \leq i \leq \ell\} \dot{\cup} \{ H_{ij}(\zeta^k) \mid 1\leq i < j \leq \ell,\, 1\leq k \leq r\}.$$
\[Prop\_Monr1lMATFree\] Let $\Ac=\Ac(G(r,1,\ell))$. Let $$\pi_{11} := \{\ker(x_i) \mid 1 \leq i \leq \ell \},$$ and $$\pi_{ij} := \{H_{(i-1)k}(\zeta^j) \mid i \leq k \leq \ell \},$$ for $2\leq i \leq \ell$, $1\leq j \leq r$. Then $$\begin{aligned}
\pi = \, &(\pi_{ij})_{\substack{1 \leq i \leq \ell, \\ 1\leq j \leq m_i}},\
m_i = \begin{cases}
1 & \quad \text{for } i=1\\
r & \quad \text{for } 2\leq i \leq \ell
\end{cases} \\
= \, &(\pi_{11}|\pi_{21}|\cdots|\pi_{2r}|\cdots|\pi_{\ell r})\end{aligned}$$ is an MAT-partition of $\Ac$. In particular ${\mathcal{A}}\in {{\mathfrak{MF}}}$ with exponents $$\exp(\Ac) = (1,r+1,2r+1,\ldots,(l-1)r +1).$$
We verify Conditions (1)–(3) from Lemma \[Lem\_MATPart\] in turn.
Let $$\Ac_{ij} := (\bigcup_{\substack{1 \leq a \leq i-1, \\ 1 \leq b \leq m_a}} \pi_{ab}) \cup (\bigcup_{1\leq b \leq j} \pi_{ib})$$ and $$\Ac_{ij}' := (\bigcup_{\substack{1 \leq a \leq i-1, \\ 1 \leq b \leq m_a}} \pi_{ab}) \cup (\bigcup_{1\leq b \leq j-1} \pi_{ib}).$$
For $\pi_{11}$ we clearly have $\vert \pi_{11} \vert = \operatorname{rk}(\pi_{11}) = \ell$. Similarly for $2 \leq i \leq \ell$, $1 \leq j \leq r$ we have $\vert \pi_{ij} \vert = \operatorname{rk}(\pi_{ij}) = \ell-i+1$ since all the defining linear forms $\alpha_{(i-1)k}(\zeta^j)$ ($i \leq k \leq \ell$) for the hyperplanes in $\pi_{ij}$ are linearly independent. Thus Condition (1) holds.
Furthermore, the forms $\{\alpha_{ac}(\zeta^b)\} \dot{\cup} \{\alpha_{(i-1)k}(\zeta^j) \mid i \leq k \leq \ell \}$ are linearly independent for all $1 \leq a \leq i-1$, $1 \leq b \leq j-1$, and $a+1 \leq c \leq \ell$, i.e. $\cap_{H \in \pi_{ij}}H =: X_{ij} \nsubseteq H$ for all $H \in \Ac_{ij}'$. Hence Condition (2) is also satisfied.
To verify Condition (3) let $H=H_{(i-1)k}(\zeta^j) \in \pi_{ij}$ for a fixed $1 \leq k \leq r$. We show $$\vert \Ac_{ij}'\vert - (j+(i-2)r) = \vert(\Ac_{ij}')^H \vert.$$ Let $H_a' := H_{(i-1)k}(\zeta^a) \in \Ac_{ij}'$, $1 \leq a \leq j-1$. Then $${\mathcal{B}}:= (\Ac_{ij}')_{H\cap H_a'} = \{\ker(x_{i-1}),\ker(x_k)\} \dot{\cup} \{ H'_b \mid 1 \leq b \leq j-1 \},$$ and $\operatorname{rk}(\Bc)=2$. So all $H' \in \Bc$ give the same intersection with $H$ and $\vert {\mathcal{B}}\vert = j + 1$. For $H' = H_{a(i-1)}(\zeta^b) \in \Ac_{ij}'$ with $a \leq i-2$, and $1 \leq b \leq r$ we have $${\mathcal{C}}:= (\Ac_{ij}')_{H\cap H'} = \{H', H_{ak}(\zeta^(j+b))\},$$ $\vert {\mathcal{C}}\vert = 2$ and there are exactly $(i-2)r$ such $H'$. All other $H'' \in \Ac_{ij}'$ intersect $H$ simply. Hence $$\begin{aligned}
\vert (\Ac_{ij}')^H) \vert &= \vert \Ac_{ij}' \vert - (|\Bc|-1) - (i-2)r(|\Cc|-1) \\
&= \vert \Ac_{ij}' \vert - j - (i-2)r,\end{aligned}$$ or $\vert \Ac_{ij}' \vert - \vert (\Ac_{ij}')^H) \vert = \sum_{a=1}^{i-1} m_i + (j-1)$. This finishes the proof.
\[Prop\_MoneelNMATFree\] Let ${\mathcal{A}}= \Ac(G(r,r,\ell))$ ($r, \ell \geq 3$). Then $\Ac$ is not MAT2-free. In particular $\Ac$ is not MAT-free.
By [@OrTer92_Arr Prop. 6.85] the arrangement $\Ac$ is free with $\exp(\Ac) = (d_1,\ldots,d_{\ell}) = (1,r+1,2r+1,\ldots,(\ell-2)r+1,(\ell-1)(r-1))$. In particular we have $(\ell-1)(r-1) = d_\ell$ and $\vert {\mathcal{A}}\vert = \frac{\ell(\ell-1)}{2}r$. But for all $H \in \Ac$ by [@OrTer92_Arr Prop. 6.82, 6.85] we have $\vert \Ac^H \vert = \frac{(\ell-1)(\ell-2)}{2}r +1$. Hence $\vert {\mathcal{A}}\vert - \vert \Ac^H \vert = (\ell-1)r -1 \neq d_\ell$ and by Lemma \[Lem\_AResnMATFree\] the arrangement $\Ac$ is not MAT2-free.
Let ${\mathcal{A}}= \Ac(W)$ be the reflection arrangement of the imprimitive complex reflection group $W = G(r,e,\ell)$ ($r, \ell \geq 3$). Then $\Ac$ is MAT-free if and only if it is MAT2-free if and only if $e\neq r$.
Since ${\mathcal{A}}= \Ac(G(r,1,\ell))$ if and only if $r\neq e$, this is Proposition \[Prop\_Monr1lMATFree\] and Proposition \[Prop\_MoneelNMATFree\].
MAT-free exceptional complex reflection groups {#Sec_ProofPrim}
==============================================
To prove the MAT-freeness of one of the following reflection arrangements, we explicitly give a realization by linear forms.
First note that if $W$ is an exceptional Weyl group, or a group of rank $\leq2$, then by Example \[Exam\_rk2\_boolean\_WeylMAT\] $\Ac(W)$ is MAT-free.
Let $\Ac$ be the reflection arrangement of the reflection group $H_3$ (Shephard-Todd: $G_{23}$). Then $\Ac$ is MAT-free. In particular $\Ac$ is MAT2-free.
Let $\tau = \frac{1+\sqrt{5}}{2}$ be the golden ratio and $\tau' = 1/\tau$ its reciprocal. The arrangement $\Ac$ can be defined by the following linear forms: $$\begin{aligned}
{\mathcal{A}}= \{ &H_1,\ldots,H_{15} \} \\
= \{ &(1,0,0)^\perp, (0,1,0)^\perp, (0,0,1)^\perp, (1,\tau,\tau')^\perp, (\tau',1,\tau)^\perp, (\tau,\tau',1)^\perp, \\
&(1,-\tau,\tau')^\perp, (\tau',1,-\tau)^\perp, (-\tau,\tau',1)^\perp, (1,\tau,-\tau')^\perp, (-\tau',1,\tau)^\perp, \\
&(\tau,-\tau',1)^\perp, (1,-\tau,-\tau')^\perp, (-\tau',1,-\tau)^\perp, (-\tau,-\tau',1)^\perp \}.\end{aligned}$$ With this linear ordering of the hyperplanes the partition $$\pi = (13,14,15|10,12|5,6|4,11|8,9|7|3|2|1)$$ satisfies Conditions (1)–(3) of Lemma \[Lem\_MATPart\] as one can verify by an easy linear algebra computation. Hence $\pi$ is an MAT-partition and $\Ac$ is MAT-free.
Let $\Ac$ be the reflection arrangement of the complex reflection group $G_{24}$. Then $\Ac$ is not MAT2-free. In particular $\Ac$ is not MAT-free.
The arrangement $\Ac$ is free with $\exp(\Ac)=(1,9,11)$ and $|\Ac|-|\Ac^H| = 13$ for all $H \in \Ac$ by [@OrTer92_Arr Tab. C.5]. Hence by Lemma \[Lem\_AResnMATFree\] $\Ac$ is not MAT2-free.
Let $\Ac$ be the reflection arrangement of the complex reflection group $G_{25}$. Then $\Ac$ is MAT-free. In particular $\Ac$ is MAT2-free.
Let $\zeta = \frac{1}{2}(-1+i\sqrt{3})$ be a primitive cube root of unity. The reflecting hyperplanes of $\Ac$ can be defined by the following linear forms (cf. [@LehTay09_UnReflGrps Ch. 8, 5.3]): $$\begin{aligned}
{\mathcal{A}}= \{ &H_1,\ldots,H_{12} \} \\
= \{ &(1,0,0)^\perp, (0,1,0)^\perp, (0,0,1)^\perp, (1,1,1)^\perp, (1,1,\zeta)^\perp, (1,1,\zeta^2)^\perp, \\
&(1,\zeta,1)^\perp, (1,\zeta,\zeta)^\perp, (1,\zeta,\zeta^2)^\perp, (1,\zeta^2,1)^\perp, (1,\zeta^2,\zeta)^\perp, (1,\zeta^2,\zeta^2)^\perp \}.\end{aligned}$$ With this linear ordering of the hyperplanes the partition $$\pi = (7,4,3|8,5|9,6|2,1|10|11|12)$$ satisfies the three conditions of Lemma \[Lem\_MATPart\] as one can easily verify by a linear algebra computation. Hence $\pi$ is an MAT-partition and $\Ac$ is MAT-free.
Let $\Ac$ be the reflection arrangement of the complex reflection group $G_{26}$. Then $\Ac$ is MAT-free. In particular $\Ac$ is MAT2-free.
Let $\zeta = \frac{1}{2}(-1+i\sqrt{3})$ be a primitive cube root of unity. The reflection arrangement $\Ac$ is the union of the reflecting hyperplanes of $\Ac(G_{25})$ and $\Ac(G(3,3,3))$ (cf. [@LehTay09_UnReflGrps Ch. 8, 5.5]). In particular the hyperplanes contained in $\Ac$ can be defined by the following linear forms: $$\begin{aligned}
{\mathcal{A}}= \{ &H_1,\ldots,H_{21} \} \\
= \{ &(1,0,0)^\perp, (0,1,0)^\perp, (0,0,1)^\perp, (1,1,1)^\perp, (1,1,\zeta)^\perp, (1,1,\zeta^2)^\perp, \\
&(1,\zeta,1)^\perp, (1,\zeta,\zeta)^\perp, (1,\zeta,\zeta^2)^\perp, (1,\zeta^2,1)^\perp, (1,\zeta^2,\zeta)^\perp, (1,\zeta^2,\zeta^2)^\perp, \\
&(1,-\zeta,0)^\perp, (1,-\zeta^2,0)^\perp, (1,-1,0)^\perp, (1,0,-\zeta)^\perp, (1,0,-\zeta^2)^\perp, \\
&(1,0,-1)^\perp, (0,1,-\zeta)^\perp, (0,1,-\zeta^2)^\perp, (0,1,-1)^\perp \}.
\end{aligned}$$ With this linear ordering of the hyperplanes the partition $$\pi = (12,19,20|16,18|13,15|17,21|10,14|6,11|8,9|7|5|4|3|2|1)$$ satisfies the three conditions of Lemma \[Lem\_MATPart\] as one can verify by a standard linear algebra computation. Hence $\pi$ is an MAT-partition and $\Ac$ is MAT-free.
Let $\Ac$ be the reflection arrangement of the complex reflection group $G_{27}$. Then $\Ac$ is not MAT2-free. In particular $\Ac$ is not MAT-free.
The arrangement $\Ac$ is free with $\exp(\Ac)=(1,19,25)$ and $|\Ac|-|\Ac^H| = 29$ for all $H \in \Ac$ by [@OrTer92_Arr Tab. C.8]. Hence by Lemma \[Lem\_AResnMATFree\] $\Ac$ is not MAT2-free.
Let $\Ac$ be the reflection arrangement of the reflection group $H_4$ (Shephard-Todd: $G_{30}$). Then $\Ac$ is MAT-free. In particular $\Ac$ is MAT2-free.
Let $\tau = \frac{1+\sqrt{5}}{2}$ be the golden ratio and $\tau' = 1/\tau$ its reciprocal. The arrangement $\Ac$ can be defined by the following linear forms: $$\begin{aligned}
{\mathcal{A}}= \{ &H_1,\ldots,H_{60} \} \\
= \{ &(1,0,0,0)^\perp, (0,1,0,0)^\perp, (0,0,1,0)^\perp, (0,0,0,1)^\perp, (1,\tau,\tau',0)^\perp, \\
&(1,0,\tau,\tau')^\perp, (1,\tau',0,\tau)^\perp, (\tau,1,0,\tau')^\perp, (\tau',1,\tau,0)^\perp, (0,1,\tau',\tau)^\perp, \\
&(\tau,\tau',1,0)^\perp, (0,\tau,1,\tau')^\perp, (\tau',0,1,\tau)^\perp, (\tau,0,\tau',1)^\perp, (\tau',\tau,0,1)^\perp, \\
&(0,\tau',\tau,1)^\perp, (-1,\tau,\tau',0)^\perp, (1,-\tau,\tau',0)^\perp, (1,\tau,-\tau',0)^\perp, (-1,0,\tau,\tau')^\perp, \\
&(1,0,-\tau,\tau')^\perp, (1,0,\tau,-\tau')^\perp, (-1,\tau',0,\tau)^\perp, (1,-\tau',0,\tau)^\perp, (1,\tau',0,-\tau)^\perp, \\
&(-\tau,1,0,\tau')^\perp, (\tau,-1,0,\tau')^\perp, (\tau,1,0,-\tau')^\perp, (-\tau',1,\tau,0)^\perp, (\tau',-1,\tau,0)^\perp, \\
&(\tau',1,-\tau,0)^\perp, (0,-1,\tau',\tau)^\perp, (0,1,-\tau',\tau)^\perp, (0,1,\tau',-\tau)^\perp, (-\tau,\tau',1,0)^\perp, \\
&(\tau,-\tau',1,0)^\perp, (\tau,\tau',-1,0)^\perp, (0,-\tau,1,\tau')^\perp, (0,\tau,-1,\tau')^\perp, (0,\tau,1,-\tau')^\perp, \\
&(-\tau',0,1,\tau)^\perp, (\tau',0,-1,\tau)^\perp, (\tau',0,1,-\tau)^\perp, (-\tau,0,\tau',1)^\perp, (\tau,0,-\tau',1)^\perp, \\
&(\tau,0,\tau',-1)^\perp, (-\tau',\tau,0,1)^\perp, (\tau',-\tau,0,1)^\perp, (\tau',\tau,0,-1)^\perp, (0,-\tau',\tau,1)^\perp, \\
&(0,\tau',-\tau,1)^\perp, (0,\tau',\tau,-1)^\perp, (1,1,1,1)^\perp, (-1,1,1,1)^\perp, (1,-1,1,1)^\perp, \\
&(1,1,-1,1)^\perp, (1,1,1,-1)^\perp, (-1,-1,1,1)^\perp, (-1,1,-1,1)^\perp, (-1,1,1,-1)^\perp \}.\end{aligned}$$ With this linear ordering of the hyperplanes the partition $$\begin{aligned}
\pi =
( &\,31, 43, 48, 54
| 29, 38, 51
| 23, 34, 58
| 18, 20, 25
| 17, 59, 60 \\
&| 21, 47, 52
| 39, 41, 44
| 26, 32, 49
| 30, 35, 40
| 2, 3, 42
| 33, 46, 50 \\
&| 4, 37
| 27, 57
| 19, 24
| 55, 56
| 10, 22
| 12, 45
| 16, 28
| 15, 36 \\
&| 53
| 14
| 13
| 11
| 9
| 8
| 7
| 6
| 5
| 1 )\end{aligned}$$ satisfies Conditions (1)–(3) of Lemma \[Lem\_MATPart\] as one can verify with a linear algebra computation. Hence $\pi$ is an MAT-partition and $\Ac$ is MAT-free. In particular $\Ac$ is MAT2-free.
We recall the following result about free filtration subarrangements of $\Ac(G_{31})$:
Let ${\mathcal{A}}:= \Ac(G_{31})$ be the reflection arrangement of the finite complex reflection group $G_{31}$. Let $\tilde{\Ac}$ be a minimal (w.r.t. the number of hyperplanes) free filtration subarrangement. Then $\tilde{\Ac} \cong \Ac(G_{29})$.
\[Coro\_G31\_no\_free\_filtration\] Let $\Ac$ be the reflection arrangement of one of the complex reflection groups $ G_{29}$ or $G_{31}$. Then $\Ac$ has no free filtration.
Let $\Ac$ be the reflection arrangement of one of the complex reflection groups $G_{29}$ or $G_{31}$. Then $\Ac$ is not MAT2-free. In particular $\Ac$ is not MAT-free.
By Corollary \[Coro\_G31\_no\_free\_filtration\] both arrangements have no free filtration and hence are not MAT2-free by Lemma \[Lem\_MAT\_free\_filtration\].
Let $\Ac$ be the reflection arrangement of the complex reflection group $G_{32}$. Then $\Ac$ is not MAT-free and also not MAT2-free.
Up to symmetry of the intersection lattice there are exactly 9 different choices of a basis, where a basis is a subarrangement ${\mathcal{B}}\subseteq \Ac$ with $|\Bc|=r(\Bc)=r(\Ac)=4$. Suppose that $\Ac$ is MAT-free. Then the first block in an MAT-partition for $\Ac$ has to be one of these bases. But a computer calculation shows that non of these bases may be extended to an MAT-partition for $\Ac$. Hence $\Ac$ is not MAT-free. A similar but more cumbersome calculation shows that $\Ac$ is also not MAT2-free.
Let $\Ac$ be the reflection arrangement of one of the complex reflection group $G_{33}$ or $G_{34}$. Then $\Ac$ is not MAT2-free. In particular $\Ac$ is not MAT-free.
First, let ${\mathcal{A}}= \Ac(G_{33})$. Then $\exp(\Ac) = (1,7,9,13,15)$ by [@OrTer92_Arr Tab. C.14]. But $|\Ac|-|\Ac^H| = 17$ for all $H \in \Ac$ also by [@OrTer92_Arr Tab. C.14]. So $\Ac$ is not MAT2-free by Lemma \[Lem\_AResnMATFree\].
Similarly ${\mathcal{A}}= \Ac(G_{34})$ is free with $\exp(\Ac) = (1,13,19,25,31,37)$ by [@OrTer92_Arr Tab. C.17] and $|\Ac|-|\Ac^H| = 41$ for all $H \in \Ac$. Hence $\Ac$ is not MAT2-free by Lemma \[Lem\_AResnMATFree\].
Comparing with Theorem \[Thm\_ClassIFReflArr\] finishes the proofs of Theorem \[Thm\_matref\] and Theorem \[Thm\_mat2free\].
Further remarks on MAT-freeness {#Sec_Remarks}
===============================
In their very recent note [@HogeRoehrle19_ConjAbeAF] Hoge and Röhrle confirmed a conjecture by Abe [@Abe18_AddDel_Combinatorics] by providing two examples $\Bc$, $\Dc$ of arrangements, related to the exceptional reflection arrangement $\Ac(E_7)$, which are additionally free but not divisionally free and in particular also not inductively free. The arrangements have exponents $\exp(\Bc) = (1,5,5,5,5,5,5)$ and $\exp(\Dc) = (1,5,5,5,5)$. Since both arrangements have only 2 different exponents by Remark \[Rem\_MATexpMAT2\] they are MAT-free if and only if they are MAT2-free. Now a computer calculation shows that both arrangements are not MAT-free and hence also not MAT2-free. In particular they provide no negative answer to Question \[Ques\_MATIF\] and Question \[Ques\_MATDF\].
Several computer experiments suggest that similar to the poset obtained from the positive roots of a Weyl group giving rise to an MAT-partition (cf. Example \[Exam\_rk2\_boolean\_WeylMAT\]) MAT-free arrangements might in general satisfy a certain poset structure:
\[Prob\_MATPoset\] Can MAT-freeness be characterized by the existence of a partial order on the hyperplanes, generalizing the classical partial order on the positive roots of a Weyl group?
Recall that by Example \[Exam\_ResNotMAT2\] the restriction $\Ac^H$ is in general not MAT-free (MAT2-free) if the arrangement $\Ac$ is MAT-free (MAT2-free). But regarding localizations there is the following:
\[Prob\_MATLocal\] Is $\Ac_X$ MAT-free (MAT2-free) for all $X \in L(\Ac)$ provided $\Ac$ is MAT-free (MAT2-free)?
Last but not least, related to the previous problem, our investigated examples suggest the following:
\[Prob\_MaxExpLocal\] Suppose $\Ac'$ and ${\mathcal{A}}= \Ac' \cup \{H\}$ are free arrangements such that $\exp(\Ac') = (d_1,\ldots,d_\ell)_\le$ and $\exp(\Ac) = (d_1,\ldots,d_{\ell-1},d_\ell+1)_\le$. Let $X \in L(\Ac)$ with $X \subseteq H$. By [@OrTer92_Arr Thm. 4.37] both localizations $\Ac'_X$ and $\Ac_X$ are free. If $\exp(\Ac'_X) = (c_1,\ldots,c_r)_\le$ is it true that $\exp(\Ac) = (c_1,\ldots,c_{r-1},c_r+1)_\le$, i.e. if we only increase the highest exponent is the same true for all localizations?
Note that the answer is yes if we only look at localizations of rank $\leq 2$. Our proceeding investigation of Problem \[Prob\_MATPoset\] suggests that this should be true at least for MAT-free arrangements. Furthermore, a positive answer to Problem \[Prob\_MaxExpLocal\] would imply (with a bit more work) a positive answer to Problem \[Prob\_MATLocal\].
[[Abe]{}18b]{}
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abstract: 'We extend Walsh’s theory of martingale measures in order to deal with hyperbolic stochastic partial differential equations that are second order in time, such as the wave equation and the beam equation, and driven by spatially homogeneous Gaussian noise. For such equations, the fundamental solution can be a distribution in the sense of Schwartz, which appears as an integrand in the reformulation of the s.p.d.e. as a stochastic integral equation. Our approach provides an alternative to the Hilbert space integrals of Hilbert-Schmidt operators. We give several examples, including the beam equation and the wave equation, with nonlinear multiplicative noise terms.'
author:
- |
Robert C. Dalang$^1$ and Carl Mueller$^2$\
\
Département de Mathématiques\
Ecole Polytechnique Fédérale\
1015 Lausanne, Switzerland\
E-mail: [email protected]\
\
Department of Mathematics\
University of Rochester\
Rochester, NY 14627 USA\
E-mail: [email protected]
title: 'Some non-linear s.p.d.e.’s that are second order in time'
---
\[stat\][Example]{} \[section\] \[stat\][Theorem]{} \[stat\][Proposition]{} \[stat\][Corollary]{} \[stat\][Theorem]{} \[stat\][Lemma]{} \[stat\][Remark]{} \[section\]
Introduction
============
The study of stochastic partial differential equations (s.p.d.e.’s) began in earnest following the papers of Pardoux [@par72], [@par75], [@par77], and Krylov and Rozovskii [@kr81], [@kr82]. Much of the literature has been concerned with the heat equation, most often driven by space-time white noise, and with related parabolic equations. Such equations are first order in time, and generally second order in the space variables. There has been much less work on s.p.d.e.’s that are second order in time, such as the wave equation and related hyperbolic equations. Some early references are Walsh [@wal86], and Carmona and Nualart [@cn88a], [@cn88b]. More recent papers are Mueller [@mue97], Dalang and Frangos [@df98], and Millet and Sanz-Solé [@mill99].
For linear equations, the noise process can be considered as a random Schwartz distribution, and therefore the theory of deterministic p.d.e.’s can in principle be used. However, this yields solutions in the space of Schwartz distributions, rather than in the space of function-valued stochastic processes. For linear s.p.d.e.’s such as the heat and wave equation driven by space-time white noise, this situation is satisfactory, since, in fact, there is no function-valued solution when the spatial dimension is greater than $1$. However, since non-linear functions of Schwartz distributions are difficult to define (see however Oberguggenberger and Russo [@or98]), it is difficult to make sense of non-linear s.p.d.e.’s driven by space-time white noise in dimensions greater than $1$.
A reasonable alternative to space-time white noise is Gaussian noise with some spatial correlation, that remains white in time. This approach has been taken by several authors, and the general framework is given in Da Prato and Zabczyk [@dz92]. However, there is again a difference between parabolic and hyperbolic equations: while the Green’s function is smooth for the former, for the latter it is less and less regular as the dimension increases. For instance, for the wave equation, the Green’s function is a bounded function in dimension 1, is an unbounded function in dimension 2, is a measure in dimension 3, and a Schwartz distribution in dimensions greater than 3.
There are at least two approaches to this issue. One is to extend the theory of stochastic integrals with respect to martingale measures, as developed by Walsh [@wal86], to a more general class of integrands that includes distributions. This approach was taken by Dalang [@dal99]. In the case of the wave equation, this yields a solution to the non-linear equation in dimensions 1, 2 and 3. The solution is a [*random field,*]{} that is, it is defined for every $(t,x) \in {\mathbf{R}}_+ \times {\mathbf{R}}^d$. Another approach is to consider solutions with values in a function space, typically an $L^2$-space: for each fixed $t \in {\mathbf{R}}_+$, the solution is an $L^2$-function, defined for almost all $x \in {\mathbf{R}}^d$. This approach has been taken by Peszat and Zabczyk in [@pz00] and Peszat [@pes02]. In the case of the non-linear wave equation, this approach yields function-valued solutions in all dimensions. It should be noted that the notions of [*random field*]{} solution and [*function-valued*]{} solution are [*not*]{} equivalent: see Lévèque [@Lev01].
In this paper, we develop a general approach to non-linear s.p.d.e’s, with a focus on equations that are second order in time, such as the wave equation and the beam equation. This approach goes in the direction of unifying the two described above, since we begin in Section \[sec2\] with an extension of Walsh’s martingale measure stochastic integral [@wal86], in such a way as to integrate processes that take values in an $L^2$-space, with an integral that takes values in the same space. This extension defines stochastic integrals of the form $$\int_0^t\int_{{\mathbf{R}}^d} G(s,\cdot-y) Z(s,y)\, M(ds,dy),$$ where $G$ (typically a Green’s function) takes values in the space of Schwartz distributions, $Z$ is an adapted process with values in $L^2({\mathbf{R}}^d)$, and $M$ is a Gaussian martingale measure with spatially homogeneous covariance.
With this extended stochastic integral, we can study non-linear forms of a wide class of s.p.d.e.’s, that includes the wave and beam equations in all dimensions, namely equations for which the p.d.e. operator is $$\frac{\partial^2 u}{\partial t^2} + (-1)^k \Delta^{k} u,$$ where $k \geq 1$ (see Section \[sec3\]). Indeed, in Section \[sec4\] we study the corresponding non-linear s.p.d.e.’s. We only impose the minimal assumptions on the spatial covariance of the noise, that are needed even for the linear form of the s.p.d.e. to have a function-valued solution. The non-linear coefficients must be Lipschitz and vanish at the origin. This last property guarantees that with an initial condition that is in $L^2({\mathbf{R}}^d)$, the solution remains in $L^2({\mathbf{R}}^d)$ for all time.
In Section \[sec5\], we specialize to the wave equation in a weighted $L^2$-space, and remove the condition that the non-linearity vanishes at the origin. Here, the compact support property of the Green’s function of the wave equation is used explicitly. We note that Peszat [@pes02] also uses weighted $L^2$ spaces, but with a weight that decays exponentially at infinity, whereas here, the weight has polynomial decay.
Extensions of the stochastic integral {#sec2}
=====================================
In this section, we define the class of Gaussian noises that drive the s.p.d.e.’s that we consider, and give our extension of the martingale measure stochastic integral.
Let ${\cal{D}}({\mathbf{R}}^{d+1})$ be the topological vector space of functions $\varphi \in C^\infty_0({\mathbf{R}}^{d+1})$, the space of infinitely differentiable functions with compact support, with the standard notion of convergence on this space (see Adams [@ada75], page 19). Let $\Gamma$ be a non-negative and [*non-negative definite*]{} (therefore symmetric) tempered measure on ${\mathbf{R}}^d$. That is, $$\int_{{\mathbf{R}}^d} \Gamma (dx)\, (\varphi \ast \tilde{\varphi})(x) \geq 0,
\qquad \mbox{for all } \varphi \in {\cal{D}}({\mathbf{R}}^d),$$ where $\tilde{\varphi}(x) = \varphi(-x)$, “$\ast$" denotes convolution, and there exists $r > 0$ such that $$\label{2.1}
\int_{{\mathbf{R}}^d} \Gamma(dx)\, \frac{1}{(1+\vert x \vert^2)^r} < \infty.$$ We note that if $\Gamma(dx) = f(x) dx$, then $$\int_{{\mathbf{R}}^d} \Gamma(dx)(\varphi \ast \tilde{\varphi})(x)
= \int_{{\mathbf{R}}^d} dx \int_{{\mathbf{R}}^d} dy \, \varphi (x) f(x-y) \varphi(y)$$ (this was the framework considered in Dalang [@dal99]). Let ${\mbox{${\mathcal S}$}}({\mathbf{R}}^d)$ denote the Schwartz space of rapidly decreasing $C^\infty$ test functions, and for $\varphi \in {\mbox{${\mathcal S}$}}({\mathbf{R}}^d)$, let ${\mbox{${\mathcal F}$}}\varphi$ denote the Fourier transform of $\varphi$: $${\mbox{${\mathcal F}$}}\varphi(\eta) = \int_{{\mathbf{R}}^d} \exp(-i\, \eta \cdot x) \varphi(x)\, dx.$$ According to the Bochner-Schwartz theorem (see Schwartz [@sch66], Chapter VII, Théorème XVII), there is a non-negative tempered measure $\mu$ on ${\mathbf{R}}^d$ such that $\Gamma = {\mbox{${\mathcal F}$}}\mu,$ that is $$\label{2.2}
\int_{{\mathbf{R}}^d} \Gamma (dx)\, \varphi(x) = \int_{{\mathbf{R}}^d} \mu (d\eta)\, {\mbox{${\mathcal F}$}}\varphi(\eta),
\qquad \mbox{for all } \varphi \in {\mbox{${\mathcal S}$}}({\mathbf{R}}^d).$$
[**Examples.**]{} (a) Let $\delta_0$ denote the Dirac functional. Then $\Gamma(dx) = \delta_0 (x)\, dx$ satisfies the conditions above.
\(b) Let $0 < \alpha < d$ and set $f_\alpha(x) = \vert x \vert^{-\alpha}$, $x \in {\mathbf{R}}^d$. Then $f_\alpha = c_\alpha {\mbox{${\mathcal F}$}}f_{d-\alpha}$ (see Stein [@ste70], Chapter V §1, Lemma 2(a)), so $\Gamma(dx) = f_\alpha(x)\, dx$ also satisfies the conditions above.
Let $F = (F(\varphi),\ \varphi \in {\cal{D}}({\mathbf{R}}^{d+1}))$ be an $L^2(\Omega, {\mbox{${\mathcal G}$}}, P)$-valued mean zero Gaussian process with covariance functional $$(\varphi, \psi) \mapsto E(F(\varphi) F(\psi))
= \int_{{\mathbf{R}}^d} \Gamma(dx)\, (\varphi \ast \tilde{\psi})(x).$$ As in Dalang and Frangos [@df98] and Dalang [@dal99], $\varphi \mapsto F(\varphi)$ extends to a worthy martingale measure $(t, A) \mapsto M_t(A)$ (in the sense of Walsh [@wal86], pages 289-290) with covariance measure $$Q([0, t] \times A \times B) = \langle M(A), M(B)\rangle_t
= t \int_{{\mathbf{R}}^d} \Gamma (dx) \int_{{\mathbf{R}}^d} dy\, 1_A(y)\, 1_B(x+y)$$ and dominating measure $K \equiv Q,$ such that $$F(\varphi) = \int_{{\mathbf{R}}_+} \int_{{\mathbf{R}}^d} \varphi(t, x) M(dt, dx),
\qquad \mbox{for all } \varphi \in {\cal{D}}({\mathbf{R}}^{d+1}).$$ The underlying filtration is $({\mbox{${\mathcal F}$}}_t = {\mbox{${\mathcal F}$}}_t^0 \vee {\cal{N}},\ t \geq 0)$, where $${\mbox{${\mathcal F}$}}^0_t= \sigma (M_s(A), \ s \leq t, \ A \in {\cal{B}}_b({\mathbf{R}}^d)),$$ ${\cal{N}}$ is the $\sigma$-field generated by $P$-null sets and ${\cal{B}}_b({\mathbf{R}}^d)$ denotes the bounded Borel subsets of ${\mathbf{R}}^d$.
Recall [@wal86] that a function $(s, x, \omega) \mapsto g(s, x; \omega)$ is termed [*elementary*]{} if it is of the form $$g(s, x; \omega) = 1_{]a, b]}(s) 1_A(x) X(\omega),$$ where $0 \leq a < b$, $A \in {\cal{B}}_b({\mathbf{R}}^d)$ and $X$ is a bounded and ${\mbox{${\mathcal F}$}}_a$-measurable random variable. The $\sigma$-field on ${\mathbf{R}}_+ \times {\mathbf{R}}^d \times \Omega$ generated by elementary functions is termed the [*predictable*]{} $\sigma$-field.
Fix $T > 0$. Let ${\cal{P}}_+$ denote the set of predictable functions $(s, x; \omega) \mapsto g(s, x; \omega)$ such that $\Vert g \Vert_+ < \infty$, where $$\Vert g \Vert^2_+ = E\left(\int_0^T ds \int_{{\mathbf{R}}^d} \Gamma(dx)
\int_{I\!\!R^d} dy\, \vert g (s, y) g(s, x + y)\vert\right).$$ Recall [@wal86] that ${\cal{P}}_+$ is the completion of the set of elementary functions for the norm $\Vert \cdot \Vert_+$.
For $g \in {\cal{P}}_+$, Walsh’s stochastic integral $$M_t^g(A) = \int_0^t ds \int_A g(s, x) M(ds,dx)$$ is well-defined and is a worthy martingale measure with covariation measure $$Q_g([0, t] \times A \times B)
= \int_0^t ds \int_{{\mathbf{R}}^d} \Gamma (dx) \int_{{\mathbf{R}}^d} dy\, 1_A(y) 1_B(x+y)\, g(s, y) g(s, x+y)$$ and dominating measure $$K_g([0, t] \times A\times B) = \int_0^t ds \int_{{\mathbf{R}}^d} \Gamma(dx)
\int_{{\mathbf{R}}^d} dy\, 1_A(y) 1_B(x + y)\, \vert g(x, y) g(s, x+y)\vert.$$
For a deterministic real-valued function $(s, x) \mapsto g(s, x)$ and a real-valued stochastic process $(Z(t, x),\ (t, x) \in {\mathbf{R}}_+ \times {\mathbf{R}}^d)$, consider the following hypotheses ($T>0$ is fixed).
- For $0 \leq s \leq T$, $g(s,\cdot) \in C^\infty({\mathbf{R}}^d)$, $g(s,\cdot)$ is bounded uniformly in $s$, and ${\mbox{${\mathcal F}$}}g(s,\cdot)$ is a function.
- For $0 \leq s \leq T$, $Z(s, \cdot) \in C_0^\infty({\mathbf{R}}^d)$ a.s., $Z(s, \cdot)$ is ${\mbox{${\mathcal F}$}}_s$-measurable, and in addition, there is a compact set $K \subset {\mathbf{R}}^d$ such that supp $Z(s,\cdot) \subset K$, for $0 \leq s \leq T$. Further, $s \mapsto Z(s,\cdot)$ is mean-square continuous from $[0,T]$ into $L^2({\mathbf{R}}^d)$, that is, for $s \in [0,T]$, $$\lim_{t \to s} E\left(\Vert Z(t,\cdot) - Z(s,\cdot) \Vert^2_{L^2({\mathbf{R}}^d)}\right) = 0.$$
- $I_{g, Z} < \infty$, where $$I_{g, Z} = \int_0^T ds \int_{{\mathbf{R}}^d} d \xi\, E(\vert {\mbox{${\mathcal F}$}}Z(s, \cdot)(\xi)\vert^2) \int_{{\mathbf{R}}^d}
\mu(d \eta)\, \vert {\mbox{${\mathcal F}$}}g(s, \cdot)(\xi-\eta)\vert^2.$$
Under hypotheses (G1), (G2) and (G3), for all $x \in {\mathbf{R}}^d$, the function defined by $(s, y; \omega) \mapsto g(s, x-y) Z(s, y; \omega)$ belongs to ${\cal{P}}_+$, and so $$v_{g, Z} (x) = \int^T_0 \int_{{\mathbf{R}}^d} g(s, x-y) Z(s, y) M(ds, dy)$$ is well-defined as a (Walsh-) stochastic integral. Further, a.s., $x \mapsto v_{g,Z}(x)$ belongs to $L^2({\mathbf{R}}^d)$, and $$\label{2.3}
E\left(\Vert v_{g,Z} \Vert_{L^2({\mathbf{R}}^d)}^2\right) = I_{g,Z}.$$ \[rlem1\]
[[Proof. ]{}]{}Observe that $\Vert g(\cdot, x-\cdot) Z(\cdot, \cdot)\Vert_+^2$ is equal to $$E\left(\int_0^T ds \int_{{\mathbf{R}}^d} \Gamma (dz) \int_{{\mathbf{R}}^d} dy\,
\vert g(s, x-y) Z(s, y) g(s, x - y-z) Z(s, y + z)\vert\right).$$ Because $g(s,\cdot)$ is bounded uniformly in $s$ by (G1), this expression is bounded by a constant times $$\begin{aligned}
&& E\left(\int_0^T ds \int_{{\mathbf{R}}^d} \Gamma (dz) \int_{{\mathbf{R}}^d} dy\, \vert Z(s, y) Z(s, y + z)\vert\right) \\
&&\qquad = E\left(\int_0^T ds \int_{{\mathbf{R}}^d} \Gamma (dz)\,
(\vert Z(s,\cdot)\vert \ast \vert \tilde{Z}(s,\cdot)\vert)(-z) \right).\end{aligned}$$ By (G2), the inner integral can be taken over $K - K = \{z - y: z \in K,\ y \in K\}$, and the sup-norm of the convolution is bounded by $\Vert Z(s,\cdot) \Vert^2_{L^2({\mathbf{R}}^d)}$, so this is $$\leq E\left(\int_0^T ds\, \Vert Z(s,\cdot) \Vert^2_{L^2({\mathbf{R}}^d)} \Gamma (K - K)\right)
= \int_0^T ds\, \Vert Z(s,\cdot) \Vert^2_{L^2({\mathbf{R}}^d)}\, \Gamma (K - K) < \infty,$$ by (\[2.1\]) and the fact that $s \mapsto E(\Vert Z(s,\cdot) \Vert^2_{L^2({\mathbf{R}}^d)})$ is continuous by (G2). Therefore, $v_{g,Z}(x)$ will be well-defined provided we show that $(s,y,\omega) \mapsto g(s,x-y) Z(s,y;\omega)$ is predictable, or equivalently, that $(s,y,\omega) \mapsto Z(s,y;\omega)$ is predictable.
For this, set $t^n_j = j T 2^{-n}$ and $$Z_n(s,y) = \sum_{j=0}^{2^n-1} Z(t^n_j, x)\, 1_{]t^n_j, t^n_{j+1}]}(s).$$ Observe that $$\begin{aligned}
\Vert Z_n \Vert_+^2 &=& E\left(\sum_{j=0}^{2^n-1} \int_{t^n_j}^{t^n_{j+1}} ds
\int_{{\mathbf{R}}^d} \Gamma(dx)\, (\vert Z(t^n_j,\cdot)\vert \ast
\vert \tilde Z(t^n_j,\cdot)\vert)(-x)\right) \\
&\leq& T 2^{-n} \sum_{j=0}^{2^n-1} E(\Vert Z(t^n_j,\cdot)\Vert^2_{L^2({\mathbf{R}}^d)})\, \Gamma (K - K) \\
&<& \infty.\end{aligned}$$ Therefore, $Z_n \in {\cal{P}}_+$, since this process, which is adapted, continuous in $x$ and left-continuous in $s$, is clearly predictable. Further, $$\begin{aligned}
&& E\left(\int_0^T ds \int_{{\mathbf{R}}^d} \Gamma(dx)\, (\vert Z(s,\cdot) - Z_n(s,\cdot)\vert) \ast
(\vert \tilde Z(s,\cdot) - \tilde Z_n(s, \cdot)\vert)(-x)\right) \\
&&\qquad\qquad \leq \int_0^T ds\, E(\Vert Z(s,\cdot) - Z_n(s,\cdot)\Vert^2_{L^2({\mathbf{R}}^d)})
\, \Gamma (K - K).\end{aligned}$$ The integrand converges to $0$ and is uniformly bounded over $[0,T]$ by (G2), so this expression converges to $0$ as $n \to \infty$. Therefore, $Z$ is predictable.
Finally, we prove (\[2.3\]). Clearly, $E(\Vert v_{g, Z} \Vert^2_{L^2({\mathbf{R}}^d)})$ is equal to $$E\left(\int_{{\mathbf{R}}^d} dx\, \left(\int_0^T \int_{{\mathbf{R}}^d} g(s, x-y) Z(s, y) M(ds, dy)\right)^2\right).$$ Since the covariation measure of $M$ is $Q$, this equals $$\label{2.4}
E\left(\int_{{\mathbf{R}}^d} dx \int_0^T ds \int_{{\mathbf{R}}^d} \Gamma(dz) \int_{{\mathbf{R}}^d} dy
\, g(s, x-y) Z (s, y) g(s, x-z-y) Z(s, z+y)\right).$$ The inner integral is equal to $(g(s, x-\cdot) Z(s, \cdot)) \ast (\tilde{g}(s, x - \cdot) \tilde{Z}(s, \cdot))(-z)$, and since this function belongs to ${\mbox{${\mathcal S}$}}({\mathbf{R}}^d)$ by (G1) and (G2), (\[2.4\]) equals $$E\left(\int_{{\mathbf{R}}^d} dx \int_0^T ds \int_{{\mathbf{R}}^d} \mu (d \eta)\,
\vert {\mbox{${\mathcal F}$}}(g(s, x - \cdot) Z(s, \cdot)) (\eta) \vert^2\right),$$ by (\[2.2\]). Because the Fourier transform takes products to convolutions, $${\mbox{${\mathcal F}$}}(g(s, x - \cdot) Z(s, \cdot))(\eta) = \int_{{\mathbf{R}}^d} d \xi^\prime
e^{i \xi^\prime \cdot x} {\mbox{${\mathcal F}$}}g(s, \cdot)(-\xi^\prime) {\mbox{${\mathcal F}$}}Z (s, \cdot)(\eta - \xi^\prime),$$ so, by Plancherel’s theorem, $$\label{2.5}
\int_{{\mathbf{R}}^d} dx\, \vert {\mbox{${\mathcal F}$}}(g(s, x-\cdot) Z(s, \cdot))(\eta) \vert^2 = \int_{{\mathbf{R}}^d}
d \xi^\prime\, \vert {\mbox{${\mathcal F}$}}g(s, \cdot)(-\xi^\prime) {\mbox{${\mathcal F}$}}Z(s, \cdot) (\eta - \xi^\prime)\vert^2.$$ The minus can be changed to plus, and using the change of variables $\xi = \eta + \xi^\prime$ ($\eta$ fixed), we find that (\[2.3\]) holds. ${\quad \vrule height7.5pt width4.17pt depth0pt}$
An alternative expression for $I_{g,Z}$ is $$I_{g,Z} = E\left(\int_0^T ds \int_{I\!\!R^d} \mu(d \eta)\,
\Vert g(s, \cdot) \ast (\chi_\eta(\cdot)Z(s, \cdot)) \Vert_{L^2({\mathbf{R}}^d)}^2\right),$$ where $\chi_\eta(x) = e^{i \eta \cdot x}$. Indeed, notice that (\[2.5\]) is equal to $$\int d \xi^\prime\, \vert {\mbox{${\mathcal F}$}}g(s, \cdot)(\xi^\prime) {\mbox{${\mathcal F}$}}(\chi_\eta(\cdot)Z(s, \cdot))(\xi^\prime)\vert^2,$$ which, by Plancherel’s theorem, is equal to $$\Vert g(s, \cdot) \ast (\chi_\eta(\cdot)Z(s, \cdot)) \Vert^2_{L^2({\mathbf{R}}^d)}.$$
Fix $(s, x) \mapsto g(s, x)$ such that (G1) holds. Consider the further hypotheses:
- $\int_0^T ds\, \sup_\xi \int_{{\mathbf{R}}^d} \mu(d\eta)\,
\vert {\mbox{${\mathcal F}$}}g(s,\cdot)(\xi - \eta)\vert^2 < \infty$,
- For $0 \leq s \leq T$, $Z(s,\cdot) \in L^2({\mathbf{R}}^d)$ a.s., $Z(s,\cdot)$ is ${\mbox{${\mathcal F}$}}_s$-measurable, and $s \mapsto Z(s,\cdot)$ is mean-square continuous from $[0,T]$ into $L^2({\mathbf{R}}^d)$.
Fix $g$ such that (G1) and (G4) hold. Set $${\mbox{${\mathcal P}$}}= \{Z \mbox{ : (G5) holds}\}.$$ Define a norm $\Vert \cdot \Vert_g$ on ${\mbox{${\mathcal P}$}}$ by $$\Vert Z \Vert_g^2 = I_{g,Z}.$$ We observe that by (G4) and (G5) (and Plancherel’s theorem), $I_{g,Z} \leq \tilde{I}_{g,Z} < \infty$, where $$\tilde{I}_{g,Z} = \int_0^T ds\, E(\Vert Z(s,\cdot)\Vert^2_{L^2({\mathbf{R}}^d)})
\left(\sup_\xi \int_{{\mathbf{R}}^d} \mu(d\eta)\,
\vert {\mbox{${\mathcal F}$}}g(s,\cdot)(\xi - \eta)\vert^2\right).$$ Let $${\mbox{${\mathcal E}$}}= \{Z \in {\mbox{${\mathcal P}$}}: \mbox{ (G2) holds}\}.$$ By Lemma \[rlem1\], $Z \mapsto v_{g,Z}$ defines an isometry from $({\mbox{${\mathcal E}$}},\Vert \cdot \Vert_g)$ into $L^2(\Omega \times {\mathbf{R}}^d, dP \times dx)$. Therefore, this isometry extends to the closure of $({\mbox{${\mathcal E}$}},\Vert \cdot \Vert_g)$ in ${\mbox{${\mathcal P}$}}$, which we now identify.
${\mbox{${\mathcal P}$}}$ is contained in the closure of $({\mbox{${\mathcal E}$}},\Vert \cdot \Vert_g)$. \[rlem3\]
[[Proof. ]{}]{}Fix $\psi \in C_0^\infty({\mathbf{R}}^d)$ such that $\psi \geq 0$, the support of $\psi$ is contained in the unit ball of ${\mathbf{R}}^d$ and $\int_{{\mathbf{R}}^d} \psi(x)\, dx = 1$. For $n \geq 1$, set $$\psi_n(x) = n^d \psi(nx).$$ Then $\psi_n \to \delta_0$ in ${\mbox{${\mathcal S}$}}({\mathbf{R}}^d)$ and ${\mbox{${\mathcal F}$}}\psi_n(\xi) = {\mbox{${\mathcal F}$}}\psi(\xi/n)$, therefore $\vert {\cal{F}} \psi_n(\cdot)\vert$ is bounded by 1.
Fix $Z \in {\cal{P}}$, and show that $Z$ belongs to the completion of ${\mbox{${\mathcal E}$}}$ in $\Vert \cdot \Vert_g$. Set $$Z_n(s,x) = Z(s, x) 1_{[-n,n]^d}(x)\quad \mbox{ and }\quad Z_{n,m}(s, \cdot) = Z_n(s, \cdot) \ast \psi_m.$$ We first show that $Z_{n,m} \in {\mbox{${\mathcal E}$}}$, that is, (G2) holds for $Z_{n,m}$. Clearly, $Z_{n,m}(s, \cdot) \in C_0^\infty({\mathbf{R}}^d)$, $Z_{n,m}(s, \cdot)$ is ${\mbox{${\mathcal F}$}}_s$-measurable by (G5), and there is a compact set $K_{n,m} \subset {\mathbf{R}}^d$ such that supp $Z_{n,m}(s, \cdot) \subset K$, for $0 \leq s \leq T$. Further, $$\Vert Z_{n,m}(t,\cdot) - Z_{n,m}(s,\cdot) \Vert^2_{L^2({\mathbf{R}}^d)} \leq \Vert
Z_n(t,\cdot) - Z_n(s,\cdot) \Vert^2_{L^2({\mathbf{R}}^d)} \leq \Vert Z(t,\cdot) -
Z(s,\cdot) \Vert^2_{L^2({\mathbf{R}}^d)},$$ so $s \mapsto Z_{n,m}(s, \cdot)$ is mean-square continuous by (G5). Therefore, $Z_{n,m} \in {\mbox{${\mathcal E}$}}$.
We now show that for $n$ fixed, $\Vert Z_n - Z_{n,m}\Vert_g \to 0$ as $m \to \infty$. Clearly, $$I_{g, Z_n-Z_{n, m}} = \int_0^T ds \int_{{\mathbf{R}}^d} d \xi\, E(\vert {\mbox{${\mathcal F}$}}Z_n(s,
\cdot)\vert^2) \ \vert 1 - {\mbox{${\mathcal F}$}}\psi_m(\xi)\vert^2 \int_{{\mathbf{R}}^d} \mu(d
\eta)\, \vert {\mbox{${\mathcal F}$}}g(s, \cdot) (\xi - \eta)\vert^2.$$ Because $\vert 1 - {\mbox{${\mathcal F}$}}\psi_m(\xi)\vert^2 \leq 4$ and $$\begin{aligned}
I_{g, Z_n} &\leq& \int_0^T ds\, E\left(\int_{{\mathbf{R}}^d} d\xi\, \vert {\mbox{${\mathcal F}$}}Z_n(s,\cdot)\vert^2\right)
\left(\sup_\xi \int_{{\mathbf{R}}^d} \mu(d \eta)\, \vert {\mbox{${\mathcal F}$}}g(s, \xi -n)\vert^2 \right) \\
&=& \tilde{I}_{g, Z_n} \leq \tilde{I}_{g, Z} < \infty,\end{aligned}$$ we can apply the Dominated Convergence Theorem to see that for $n$ fixed, $$\lim_{m \to \infty} \Vert Z_n - Z_{n,m}\Vert_g
= \lim_{m \to \infty} \sqrt{I_{g, Z_n-Z_{n, m}}} = 0.$$ Therefore, $Z_n$ belongs to the completion of ${\mbox{${\mathcal E}$}}$ in $\Vert\cdot \Vert_g$. We now show that $\Vert Z-Z_n \Vert_g \to 0$ as $n \to \infty$. Clearly, $$\begin{aligned}
\Vert Z-Z_n \Vert_g^2 &=& I_{g, Z-Z_n} \leq \tilde{I}_{g, Z-Z_n} \\
&=& \int_0^T ds\,
E\left(\Vert (Z - Z_n)(s,\cdot) \Vert^2_{L^2({\mathbf{R}}^d)}\right)
\left(\sup_\xi \int_{{\mathbf{R}}^d} \mu(d \eta)\, \vert {\mbox{${\mathcal F}$}}g(s, \xi -n)\vert^2 \right).\end{aligned}$$ Because $$\Vert Z-Z_n \Vert^2_{L^2({\mathbf{R}}^d)} \leq (\Vert Z \Vert_{L^2({\mathbf{R}}^d)}
+ \Vert Z_n \Vert_{L^2({\mathbf{R}}^d)})^2 \leq 4 \Vert Z \Vert_{L^2({\mathbf{R}}^d)}^2,$$ and $\tilde{I}_{g, Z} < \infty$, the Dominated Convergence Theorem implies that $$\lim_{n \to \infty} \Vert Z - Z_n\Vert_g = 0,$$ and therefore $Z$ belongs to the completion of ${\mbox{${\mathcal E}$}}$ in $\Vert\cdot \Vert_g$. Lemma \[rlem3\] is proved. ${\quad \vrule height7.5pt width4.17pt depth0pt}$
Lemma \[rlem3\] allows us to define the stochastic integral $v_{g,Z} = g \cdot M^Z$ provided $g$ satisfies (G1) and (G4), and $Z$ satisfies (G5). The key property of this stochastic integral is that $$E\left(\Vert v_{g,Z} \Vert_{L^2({\mathbf{R}}^d)}^2\right) = I_{g,Z}.$$ \[rrem4\]
We now proceed with a further extension of this stochastic integral, by extending the map $g \mapsto v_{g,Z}$ to a more general class of $g$.
Fix $Z \in {\mbox{${\mathcal P}$}}$. Given a function $s \mapsto G(s) \in {\mbox{${\mathcal S}$}}'({\mathbf{R}}^d)$, consider the two properties:
- For all $s \geq 0$, ${\mbox{${\mathcal F}$}}G(s)$ is a function and $$\int_0^T ds\, \sup_\xi \int_{{\mathbf{R}}^d} \mu(d \eta)\, \vert {\mbox{${\mathcal F}$}}G(s, \xi - \eta)\vert^2 < \infty.$$
- For all $\psi \in C_0^\infty({\mathbf{R}}^d)$, $\sup_{0 \leq s \leq T} G(s) \ast \psi$ is bounded on ${\mathbf{R}}^d$.
Set $${\mbox{${\mathcal G}$}}= \{s \mapsto G(s): \mbox{ (G6) and (G7) hold}\},$$ and $${\mbox{${\mathcal H}$}}= \{s \mapsto G(s): G(s) \in C^\infty({\mathbf{R}}^d) \mbox{ and (G1)and (G4) hold}\}.$$ Clearly, ${\mbox{${\mathcal H}$}}\subset {\mbox{${\mathcal G}$}}$. For $G \in {\mbox{${\mathcal G}$}}$, set $$\Vert G \Vert_Z = \sqrt{I_{G,Z}}.$$ Notice that $I_{G,Z} \leq \tilde{I}_{G,Z} < \infty$ by (G5) and (G6). By Remark \[rrem4\], the map $G \mapsto v_{G,Z}$ is an isometry from $({\mbox{${\mathcal H}$}}, \Vert \cdot \Vert_Z)$ into $L^2(\Omega \times {\mathbf{R}}^d, dP \times dx)$. Therefore, this isometry extends to the closure of $({\mbox{${\mathcal H}$}}, \Vert \cdot \Vert_Z)$ in ${\mbox{${\mathcal G}$}}$.
${\mbox{${\mathcal G}$}}$ is contained in the closure of $({\mbox{${\mathcal H}$}}, \Vert \cdot \Vert_Z)$. \[rlem5\]
[[Proof. ]{}]{}Fix $s \mapsto G(s)$ in ${\mbox{${\mathcal G}$}}$. Let $\psi_n$ be as in the proof of Lemma \[rlem3\]. Set $$G_n(s, \cdot) = G(s) \ast \psi_n(\cdot).$$ Then $G_n(s, \cdot) \in C^\infty({\mathbf{R}}^d)$ by [@sch66], Chap.VI, Thm.11 p.166. By (G6), ${\mbox{${\mathcal F}$}}G_n(s,\cdot) = {\mbox{${\mathcal F}$}}G(s) \cdot {\mbox{${\mathcal F}$}}\psi_n$ is a function, and so by (G7), (G1) holds for $G_n$. Because $\vert {\mbox{${\mathcal F}$}}\psi_n\vert \leq 1$, (G4) holds for $G_n$ because it holds for $G$ by (G6). Therefore, $G_n \in {\mbox{${\mathcal H}$}}$.
Observe that $$\begin{aligned}
\Vert G - G_n\Vert_Z^2 &=& I_{G-G_n,Z} \\
&=& \int^T_0 ds \int_{{\mathbf{R}}^d} d \xi\,
E(\vert {\mbox{${\mathcal F}$}}Z(s, \cdot)(\xi)\vert^2 \int_{{\mathbf{R}}^d} \mu(d \eta)\,
\vert {\mbox{${\mathcal F}$}}G(s,\cdot) (\xi- \eta) \vert^2 \vert 1 - {\mbox{${\mathcal F}$}}\psi_n (\xi-\eta) \vert^2\end{aligned}$$ The last factor is bounded by $4$, has limit $0$ as $n \to \infty$, and $I_{G,Z} < \infty$, so the Dominated Convergence Theorem implies that $$\lim_{n \to \infty} \Vert G - G_n\Vert_Z =0.$$ This proves the lemma. ${\quad \vrule height7.5pt width4.17pt depth0pt}$
Fix $Z$ such that (G5) holds, and $s \mapsto G(s)$ such that (G6) and (G7) hold. Then the stochastic integral $v_{G,Z} = G \cdot M^Z$ is well-defined, with the isometry property $$E\left(\Vert v_{G,Z} \Vert _{L^2({\mathbf{R}}^d)}^2\right) = I_{G,Z}.$$ \[rthm6\]
It is natural to use the notation $$v_{G,Z} = \int^T_0 ds \int_{{\mathbf{R}}^d} G(s,\cdot - y) Z(s,y) M(ds,dy),$$ and we shall do this in the sequel.
[Proof of Theorem \[rthm6\].]{} The statement is an immediate consequence of Lemma \[rlem5\]. ${\quad \vrule height7.5pt width4.17pt depth0pt}$
Fix a deterministic function $\psi \in L^2({\mathbf{R}}^d)$ and set $$X_t = \left\langle \psi,\ \int_0^t\int_{{\mathbf{R}}^d} G(s,\cdot-y) Z(s,y)\,
M(ds,dy)\right\rangle_{L^2({\mathbf{R}}^d)}.$$ It is not difficult to check that $(X_t,\ 0 \leq t \leq T)$ is a (real-valued) martingale.
Examples {#sec3}
========
In this section, we give a class of examples to which Theorem \[rthm6\] applies. Fix an integer $k \geq 1$ and let $G$ be the Green’s function of the p.d.e. $$\label{3.1}
\frac{\partial^2 u}{\partial t^2} + (-1)^k \Delta^{(k)} u = 0.$$ As in [@dal99], Section 3, ${\mbox{${\mathcal F}$}}G(t)(\xi)$ is easily computed, and one finds $${\mbox{${\mathcal F}$}}G(t)(\xi) = \frac{\sin(t\vert \xi \vert^k)}{\vert \xi \vert^k}.$$ According to [@dal99], Theorem 11 (see also Remark 12 in that paper), the linear s.p.d.e $$\label{3.2}
\frac{\partial^2u}{\partial t^2} + (-1)^k \Delta^{(k)}u = {\dot F}(t, x)$$ with vanishing initial conditions has a process solution if and only if $$\int_0^T ds \int_{I\!\!R^d} \mu (d \xi)\, \vert {\mbox{${\mathcal F}$}}G (s) \vert^2 < \infty,$$ or equivalently, $$\label{3.3}
\int_{{\mathbf{R}}^d} \mu (d \xi)\, \frac{1}{(1+\vert \xi \vert^2)^k} < \infty.$$ It is therefore natural to assume this condition in order to study non-linear forms of (\[3.2\]).
In order to be able to use Theorem \[rthm6\], we need the following fact.
Suppose (\[3.3\]) holds. Then the Green’s function $G$ of equation (\[3.1\]) satisfies conditions (G6) and (G7). \[rlem6\]
[[Proof. ]{}]{}We begin with (G7). For $\psi \in C_0^\infty ({\mathbf{R}}^d)$, $$\begin{array}{ll}
\Vert G(s) \ast \psi \Vert_{L^\infty({\mathbf{R}}^d)} &\leq \Vert {\mbox{${\mathcal F}$}}(G(s)
\ast \psi)\Vert_{L^1({\mathbf{R}}^d)} = \displaystyle\int_{{\mathbf{R}}^d}
\frac{\vert \sin(s \vert \xi \vert^k) \vert}{\vert \xi \vert^k} \vert {\mbox{${\mathcal F}$}}\psi(\xi)\vert\, d \xi\\
&\leq s \displaystyle\int_{{\mathbf{R}}^d} \vert {\mbox{${\mathcal F}$}}\psi (\xi)\vert\, d \xi < \infty,
\end{array}$$ so (G7) holds.
Turning to (G6), we first show that $$\label{3.4}
\langle \chi_\xi G_{d,k} , \Gamma \rangle
= \left\langle \frac{1}{(1+ \vert \xi - \cdot \vert^2)^k}, \mu \right\rangle,$$ where $$0 \leq G_{d,k}(x) = \frac{1}{\gamma(k)} \int_0^\infty e^{-u} u^{k-1} p(u,x)\,
du,$$ $\gamma(\cdot)$ is Euler’s Gamma function and $p(u, x)$ is the density of a $N(0, uI)-$ random vector (see [@san00], Section 5). In particular, $${\mbox{${\mathcal F}$}}G_{d,k} (\xi) = \frac{1}{(1+\vert \xi \vert^2)^k},$$ and it is shown in [@kar00] and [@san00] that $$\label{3.5}
\langle G_{d,k}, \Gamma \rangle = \langle (1+\vert \cdot \vert^2)^{-k}, \mu \rangle,$$ and the right-hand side is finite by (\[3.3\]). However, the proofs in [@san00] and [@kar00] use monotone convergence, which is not applicable in presence of the oscillating function $\chi_\xi$. As in [@kar00], because $e^{-t \vert \cdot \vert^2}$ has rapid decrease, $$\left\langle \frac{e^{-t \vert \cdot \vert^2}}{(1+ \vert \xi - \cdot \vert^2)^k},
\mu \right\rangle = \left\langle {\mbox{${\mathcal F}$}}\left(\frac{e^{-t \vert \cdot \vert^2}}
{(1 + \vert \xi - \cdot \vert^2)^k}\right), \mu \right\rangle
= \langle p(t, \cdot) \ast (\chi_\xi G_{d,k}), \Gamma \rangle.$$ Notice that $G_{d,k} \geq 0$, and so $$\vert p(t, \cdot) \ast (\chi_\xi G_{d,k})\vert \leq p(t, \cdot) \ast G_{d,k}
\leq e^T G_{d, k}$$ by formula (5.5) in [@san00], so we can use monotone convergence in the first equality below and the Dominated Convergence Theorem in the third equality below to conclude that $$\begin{array}{lll}
\left\langle \frac{1}{(1+ \vert \xi - \cdot \vert^2)^k}, \mu \right\rangle
&=& \lim_{t \downarrow 0} \left\langle \frac{e^{-t \vert \cdot \vert^2}}
{(1+ \vert \xi - \cdot \vert^2)^k} , \mu\right\rangle
= \lim_{t \downarrow 0} \langle p(t, \cdot) \ast (\chi_\xi G_{d,k})), \Gamma \rangle \\
&& \\
&=& \langle \lim_{t \downarrow 0} (p(t, \cdot) \ast (\chi_\xi G_{d,k})), \Gamma \rangle
= \langle \chi_\xi G_{d, k}, \Gamma \rangle,
\end{array}$$ which proves (\[3.4\]). Because $G_{d,k} \geq 0$, $$\label{3.6}
\sup_\xi\, \langle \chi_\xi G_{d,k}, \Gamma\rangle \leq \langle G_{d,k}, \Gamma\rangle < \infty$$ by (\[3.5\]) and (\[3.3\]). The lemma is proved. ${\quad \vrule height7.5pt width4.17pt depth0pt}$
A non-linear s.p.d.e {#sec4}
====================
Let $\alpha: {\mathbf{R}}\to {\mathbf{R}}$ be a Lipschitz function such that $\alpha(0) = 0$, so that there is a constant $K > 0$ such that for $u, u_1, u_2 \in {\mathbf{R}}$, $$\label{4.1}
\vert \alpha(u)\vert \leq K \vert u \vert \qquad
\mbox{and} \qquad \vert \alpha(u_1) - \alpha(u_2) \vert \leq K \vert u_1 - u_2 \vert.$$ Examples of such functions are $\alpha(u) = u$, $\alpha(u) = \sin(u)$, or $\alpha(u) = 1-e^{-u}$.
Consider the non-linear s.p.d.e. $$\label{4.2}
\frac{\partial^2 }{\partial t^2}u(t,x) + (-1)^k \Delta^{(k)}u(t,x) = \alpha(u(t, x)) F(t, x),$$ $$u(0, x) = v_0(x),\qquad \frac{\partial}{\partial t} u(0, x) = \tilde{v}_0(x)$$ where $v_0 \in L^2({\mathbf{R}}^d)$ and $\tilde{v}_0 \in H^{-k}({\mathbf{R}}^d)$, the Sobolev space of distributions such that $$\Vert \tilde{v}_0 \Vert^2_{H^{-k}({\mathbf{R}}^d)}
\stackrel{\mbox{\scriptsize def}}{=} \int_{{\mathbf{R}}^d} d\xi\,
\frac{1}{(1+\vert \xi \vert^2)^k} \vert {\mbox{${\mathcal F}$}}\tilde{v}_0(\xi)\vert^2 < \infty.$$ We say that a process $(u(t, \cdot),\ 0 \leq t \leq T)$ with values in $L^2({\mathbf{R}}^d)$ is a solution of (\[4.2\]) if, for all $t \geq 0$, a.s., $$\label{4.3}
u(t, \cdot) = \frac{d}{dt} G(t) \ast v_0 + G(t) \ast \tilde{v}_0
+ \int_0^t \int_{{\mathbf{R}}^d} G(t-s, \cdot -y) \alpha(u(s, y)) M(ds, dy),$$ where $G$ is the Green’s function of (\[3.1\]). The third term is interpreted as the stochastic integral from Theorem \[rthm6\], so $(u(s, \cdot))$ must be adapted and mean-square continuous from $[0, T]$ into $L^2({\mathbf{R}}^d)$.
Suppose that (\[3.3\]) holds. Then equation (\[4.2\]) has a unique solution $(u(t, \cdot),\ 0 \leq t \leq T)$. This solution is adapted and mean-square continuous. \[rthm7\]
[[Proof. ]{}]{}We will follow a standard Picard iteration scheme. Set $$u_0(t, \cdot) = \frac{d}{dt} G(t) \ast v_0 + G(t) \ast \tilde{v}_0.$$ Notice that $v_0(t, \cdot) \in L^2({\mathbf{R}}^d)$. Indeed, $$\begin{aligned}
\label{4.4}
\left\Vert \frac{d}{dt} G(t) \ast v_0\right\Vert_{L^2({\mathbf{R}}^d)}
&=& \left\Vert {\mbox{${\mathcal F}$}}\frac{d}{dt} G(t) \cdot {\mbox{${\mathcal F}$}}v_0\right\Vert_{L^2({\mathbf{R}}^d)}
= \int_{{\mathbf{R}}^d} \sin^2(t\vert \xi \vert^k) \vert {\mbox{${\mathcal F}$}}v_0 (\xi)\vert^2 d\xi\\
&\leq& \Vert v_0\Vert_{L^2({\mathbf{R}}^d)},\nonumber\end{aligned}$$ and one checks similarly that $\Vert G(t) \ast \tilde{v}_0 \Vert_{L^2({\mathbf{R}}^d)} \leq \Vert \tilde{v} \Vert_{H^{-k}}$. Further, $t \mapsto u_0(t, \cdot)$ from $[0, T]$ into $L^2({\mathbf{R}}^d)$ is continuous. Indeed, $$\lim_{t \to s} \left\Vert \frac{d}{dt}G(t)
\ast v_0 - \frac{d}{dt} G(s) \ast v_0 \right\Vert_{L^2({\mathbf{R}}^d} = 0,$$ as is easily seen by proceeding as in (\[4.4\]) and using dominated convergence. Similarly, $$\lim_{t \to s} \Vert G(t) \ast \tilde{v}_0 - G(s) \ast \tilde{v}_0\Vert = 0.$$
For $n \geq 0$, assume now by induction that we have defined an adapted and mean-square continuous process $(u_n(s, \cdot),\ 0 \leq s \leq T)$ with values in $L^2({\mathbf{R}}^d),$ and define $$\label{4.5}
u_{n+1}(t, \cdot) = u_0 (t, \cdot) + v_{n+1}(t, \cdot),$$ where $$\label{4.6}
v_{n+1} (t, \cdot) = \int_0^t \int_{{\mathbf{R}}^d} G(t-s, \cdot -y) \alpha(v_n(s, g)) M(ds, dy).$$ We note that $(\alpha(u_n(s, \cdot)),\ 0 \leq s \leq T)$ is adapted and mean-square continuous, because by (\[4.1\]), $$\Vert \alpha(u_n(s, \cdot)) - \alpha(u_n(t, \cdot))\Vert_{L^2({\mathbf{R}}^d)}
\leq K \Vert u_n(s, \cdot) - u_n(t, \cdot) \Vert_{L^2({\mathbf{R}}^d)},$$ so the stochastic integral in (\[4.6\]) is well-defined by Lemma \[rlem6\] and Theorem \[rthm6\].
Set $$\label{4.7}
J(s) = \sup_\xi \int_{{\mathbf{R}}^d} \mu(d \eta)\, \vert {\mbox{${\mathcal F}$}}G(s, \cdot)(\xi - \eta)\vert^2.$$ By (\[3.3\]), (\[3.5\]) and (\[3.6\]), $\sup_{0 \leq s \leq T} J(s)$ is bounded by some $C < \infty$, so by Theorem \[rthm6\] and using (\[4.1\]), $$\begin{aligned}
\nonumber
E(\Vert u_{n+1}(t, \cdot) \Vert^2_{L^2({\mathbf{R}}^d)}) &\leq& 2 \Vert u_0 (t, \cdot) \Vert^2_{L^2({\mathbf{R}}^d)}
+ 2 \int_0^t ds\, E(\Vert \alpha(u_n(s, \cdot))\Vert^2_{L^2({\mathbf{R}}^d)}) J(t-s)\\
&\leq& 2 \Vert u_0 (t, \cdot)\Vert_{L^2({\mathbf{R}}^d)}^2 + 2 K C \int_0^t ds\, E(\Vert u_n(s, \cdot)\Vert^2_{L^2({\mathbf{R}}^d)}).
\label{4.8}\end{aligned}$$ Therefore, $u_{n+1}(t, \cdot)$ takes its values in $L^2({\mathbf{R}}^d)$. By Lemma \[rlem7\] below, $(u_{n+1}(t, \cdot),\ 0 \leq t \leq T)$ is mean-square continuous and this process is adapted, so the sequence $(u_n,\ n \in {\mathbf{N}})$ is well-defined. By Gronwall’s lemma, we have in fact $$\sup_{0 \leq t \leq T} \sup_{n \in {\mathbf{N}}} E(\Vert u_n(t, \cdot) \Vert^2_{L^2({\mathbf{R}}^d)}) < \infty.$$ We now show that the sequence $(u_n(t, \cdot),\ n \geq 0)$ converges. Let $$M_n(t) = E(\Vert u_{n+1} (t, \cdot) - u_n(t, \cdot) \Vert^2_{L^2({\mathbf{R}}^d)}).$$ Using the Lipschitz property of $\alpha(\cdot), (\ref{4.5})$ and $(\ref{4.6})$, we see that $$M_n(t) \leq K C \int_0^t ds\, M_{n-1}(s).$$ Because $\sup_{0 \leq s \leq T} M_0(s) < \infty$, Gronwall’s lemma implies that $$\sum_{n=0}^\infty M_n(t)^{1/2} < \infty.$$ In particular, $(u_n(t, \cdot),\ n \in {\mathbf{N}})$ converges in $L^2(\Omega \times {\mathbf{R}}^d, dP \times dx)$, uniformly in $t \in [0, T]$, to a limit $u(t, \cdot)$. Because each $u_n$ is mean-square continuous and the convergence is uniform in $t$, $(u(t, \cdot),\ 0 \leq t \leq T)$ is also mean-square continuous, and is clearly adapted. This process is easily seen to satisfy (\[4.3\]), and uniqueness is checked by a standard argument. ${\quad \vrule height7.5pt width4.17pt depth0pt}$
The following lemma was used in the proof of Theorem \[rthm7\].
Each of the processes $(u_n(t, \cdot),\ 0 \leq t \leq T)$ defined in (\[4.5\]) is mean-square continuous. \[rlem7\]
[[Proof. ]{}]{}Fix $n \geq 0$. It was shown in the proof of Theorem \[rthm7\] that $t \mapsto u_0(t, \cdot)$ is mean-square continuous, so we establish this property for $t \mapsto v_{n+1}(t, \cdot)$, defined in $(\ref{4.6})$. Observe that for $h > 0$, $$E(\Vert v_{n+1}(t+h, \cdot) - v_{n+1} (t, \cdot) \Vert^2_{L^2({\mathbf{R}}^d)}) \leq 2 (I_1 + I_2),$$ where $$\begin{aligned}
I_1 &=& E\left(\left\Vert \int_t^{t+h} \int_{{\mathbf{R}}^d} G(t+h-s, \cdot - y)
\alpha(u_n(s, y)) M(ds, dy)\right\Vert^2_{L^2({\mathbf{R}}^d)}\right),\\
I_2 &=& E\left(\left\Vert \int_0^t \int_{{\mathbf{R}}^d}
(G(t+h-s, \cdot -y) - G(t-s, \cdot -y)) \alpha(u_n(s, y))
M(ds dy) \right\Vert^2_{L^2({\mathbf{R}}^d)}\right).\end{aligned}$$ Clearly, $$I_1 \leq K^2 \int_t^{t+h} ds\, E( \Vert u_n(s, \cdot) \Vert^2_{L^2({\mathbf{R}}^d)}) J(t+h-s),$$ while $$I_2 = \int_0^t ds \int_{{\mathbf{R}}^d} d \xi\, \vert {\mbox{${\mathcal F}$}}(\alpha(u_n(s, \cdot)))(\xi)\vert^2
\int_{{\mathbf{R}}^d} \mu(d \eta) \left(\frac{\sin(t+h-s) \vert \xi - \eta \vert^k)
- \sin((t-s) \vert \xi - \eta \vert^k)}{\vert \xi - \eta \vert^k}\right)^2.$$ The squared ratio is no greater than $$4 \left( \frac{\sin(h \vert \xi - \eta \vert^k)}
{\vert \xi - \eta \vert^k}\right)^2
\leq \frac{C}{(1 + \vert \xi - \eta \vert^2)^k}.$$ It follows that $I_2$ converges to $0$ as $h \to 0$, by the dominated convergence theorem, and $I_1$ converges to $0$ because the integrand is bounded. This proves that $t \mapsto v_{n+1}(t, \cdot)$ is mean-square right-continuous, and left-continuity is proved in the same way. ${\quad \vrule height7.5pt width4.17pt depth0pt}$
The wave equation in weighted $L^2$-spaces {#sec5}
==========================================
In the case of the wave equation (set $k=1$ in (\[4.2\])), we can consider a more general class of non-linearities $\alpha(\cdot)$ than in the previous section. This is because of the compact support property of the Green’s function of the wave equation.
More generally, in this section, we fix $T > 0$ and consider a function $s \mapsto G(s) \in {\cal{S}}'({\mathbf{R}}^d)$ that satisfies (G6), (G7) and, in addition,
- There is $R > 0$ such that for $0 \leq s \leq T$, supp $G(s) \subset B(0, R)$.
Fix $K > d$ and let $\theta : {\mathbf{R}}^d \to {\mathbf{R}}$ be a smooth function for which there are constants $0 < c < C$ such that $$c(1 \wedge \vert x \vert^{-K} ) \leq \theta(x) \leq C(1 \wedge \vert x \vert^{-K}).$$ The weighted $L^2$-space $L^2_\theta$ is the set of measurable $f: {\mathbf{R}}^d \to {\mathbf{R}}$ such that $\Vert f \Vert_{L_\theta^2} < \infty$, where $$\Vert f \Vert_{L_\theta^2}^2 = \int_{{\mathbf{R}}^d} f^2(x) \theta(x)\, dx.$$ Let $H_n = \{x \in {\mathbf{R}}^d: n R \leq \vert x \vert < (n+1)R\}$, set $$\Vert f \Vert_{L^2(H_n)} = \left(\int_{H_n} f^2(x)\, dx\right)^{1/2},$$ and observe that there are positive constants, which we again denote $c$ and $C$, such that $$\label{5.1}
c \sum_{n=0}^\infty n^{-K} \Vert f \Vert^2_{L^2(H_n)}
\leq \Vert f \Vert^2_{L_\theta^2}
\leq C \sum_{n=0}^\infty n^{-K} \Vert f \Vert^2_{L^2(H_n)}.$$ For a process $(Z(s, \cdot),\ 0 \leq s \leq T)$, consider the following hypothesis:
- For $0 \leq s \leq T$, $Z(s, \cdot) \in L_\theta^2$ a.s., $Z(s, \cdot)$ is ${\mbox{${\mathcal F}$}}_s$-measurable, and $s \mapsto Z(s, \cdot)$ is mean-square continuous from $[0, T]$ into $L^2_\theta$.
Set $$\begin{array}{lll}
{\mbox{${\mathcal E}$}}_\theta = \{Z: &\mbox{ (G9) holds, and there is } K \subset {\mathbf{R}}^d \mbox{ compact such}\\
& \mbox{that for } 0 \leq s \leq T, \mbox{ supp} \, Z(s, \cdot) \subset K \}.
\end{array}$$ Notice that for $Z \in {\mbox{${\mathcal E}$}}_\theta$, $Z(s, \cdot) \in L^2({\mathbf{R}}^d)$ because $\theta(\cdot)$ is bounded below on $K$ by a positive constant, and for the same reason, $s \mapsto Z(s, \cdot)$ is mean-square continuous from $[0, T]$ into $L^2({\mathbf{R}}^d)$. Therefore, $$v_{G,Z} = \int_0^T \int_{{\mathbf{R}}^d} G(s, \cdot - y) Z(s, y) M(ds, dy)$$ is well-defined by Theorem \[rthm6\].
For $G$ as above and $Z \in {\mbox{${\mathcal E}$}}_\theta$, $v_{G, Z} \in L^2_\theta$ a.s. and $$E(\Vert v_{G, Z} \Vert^2_{L^2_\theta}) \leq \int_0^T ds\, \Vert Z(s, \cdot) \Vert^2_{L_\theta^2} J(s),$$ where $J(s)$ is defined in $(\ref{4.7})$. \[rlem8\]
[[Proof. ]{}]{}We assume for simplicity that $R = 1$ and $K$ is the unit ball in ${\mathbf{R}}^d$. Set $D_0 = H_0 \cup H_1$ and, for $n \geq 1$, set $D_n = H_{n-1} \cup H_n \cup H_{n+1}$ and $Z_n(s, \cdot) = Z(s, \cdot) 1_{D_n}(\cdot)$. By (\[5.1\]), then (G8), $$\begin{aligned}
\Vert v_{G,Z} \Vert^2_{L_\theta^2} &\leq& \sum_{n=0}^\infty n^{-K} \Vert v_{G,Z} \Vert^2_{L^2(H_n)}
= \sum_{n=0}^\infty n^{-K} \Vert v_{G,Z_n} \Vert^2_{L^2(H_n)}\\
&\leq& \sum_{n=0}^\infty n^{-K} \Vert v_{G,Z_n} \Vert^2_{L^2({\mathbf{R}}^d)}.\end{aligned}$$ Therefore, by Theorem \[rthm6\], $$\begin{aligned}
E(\Vert v_{G,Z} \Vert^2_{L^2_\theta}
&\leq& \sum_{n=0}^\infty n^{-K} \int_0^T ds\, E(\Vert Z_n(s, \cdot) \Vert^2_{L^2(I\!\!R^d)}) J(s)\\
&=& \sum_{n=0}^\infty n^{-K} \int_0^T ds\, E(\Vert Z(s, \cdot) \Vert^2_{L^2(D_n)}) J(s)\\
&\leq& C \int_0^T ds\, \sum_{n=0}^\infty n^{-K}\, E(\Vert Z(s, \cdot) \Vert^2_{L^2(H_n)}) J(s)\\
&\leq& C \int_0^T ds\, E( \Vert Z(s, \cdot) \Vert^2_{L^2_\theta})J(s).\end{aligned}$$ This proves the lemma. ${\quad \vrule height7.5pt width4.17pt depth0pt}$
For a process $(Z(s, \cdot))$ satisfying (G9), let $\Vert Z \Vert_\theta =(I^\theta_{G, Z})^{\frac{1}{2}}$, where $$I_{G, Z}^\theta = \int_0^T ds\, \Vert Z(s, \cdot) \Vert_{L_\theta^2}^2 J(s).$$ Because $s \mapsto \Vert Z(s, \cdot) \Vert_{L_\theta^2}^2$ is bounded, $I^\theta_{G,Z} < \infty$ provided (G6) holds. Therefore, $\Vert Z \Vert_\theta$ defines a norm, and by Lemma \[rlem8\], $Z \mapsto v_{G,Z}$ from ${\mbox{${\mathcal E}$}}_\theta$ into $L^2(\Omega \times {\mathbf{R}}^d, dP \times \theta(x)dx)$ is continuous. Therefore this map extends to the closure of ${\mbox{${\mathcal E}$}}_\theta$ for $\Vert \cdot \Vert_\theta$, which we now identify.
Consider a function $s \mapsto G(s) \in {\cal{S}}'({\mathbf{R}}^d)$ such that (G6), (G7) and (G8) hold. Let $(Z(s, \cdot),\ 0 \leq s \leq T)$ be an adapted process with values in $L_\theta^2$ that is mean-square continuous from $[0, T]$ into $L^2_\theta$. Then $Z$ is in the closure of ${\mbox{${\mathcal E}$}}_\theta$ for $\Vert \cdot \Vert_\theta$, and so the stochastic integral $v_{G, Z}$ is well-defined, and $$\label{5.2}
E(\Vert v_{G,Z} \Vert^2_{L^2_\theta}) \leq I^\theta_{G, Z}.$$ \[rthm8\]
[[Proof. ]{}]{}Set $Z_n(s, \cdot) = Z(s, \cdot) 1_{[-n,n]^d}(\cdot)$. Then $(Z_n)$ satisfies (G9) and belongs to ${\mbox{${\mathcal E}$}}_\theta$. Because $\Vert Z_n(s, \cdot)\Vert_{L^2_\theta}\leq \Vert Z(s, \cdot) \Vert_{L^2_\theta}$ and $I_{G,Z}^\theta< \infty$, the dominated convergence theorem implies that $\lim_{n \to \infty} \Vert Z-Z_n\Vert_\theta = 0$, so $Z$ is in the closure of ${\mbox{${\mathcal E}$}}_\theta$ for $\Vert \cdot \Vert_\theta$, and $(\ref{5.2})$ holds by Lemma \[rlem8\]. ${\quad \vrule height7.5pt width4.17pt depth0pt}$
We now use this result to obtain a solution to the following stochastic wave equation: $$\label{5.3}
\frac{\partial^2}{\partial t^2}u(t,x) - \Delta u(t,x) = \alpha(u(t, x)) {\dot F}(t, y),$$ $$u(0, x) = v_0(x),\qquad \frac{\partial u}{\partial t} (0, x) = \tilde{v}_0(x).$$ We say that a process $(u(t, \cdot),\ 0 \leq t \leq T)$ with values in $L^2_\theta$ is a solution of $(\ref{5.3})$ if $(u(t, \cdot))$ is adapted, $t \mapsto u(t, \cdot)$ is mean-square continuous from $[0, T]$ into $L^2_\theta$ and $$\label{5.4}
u(t, \cdot) = \frac{d}{dt} G(t) \ast v_0 + G(t) \ast \tilde{v}_0
+ \int_0^t \int_{{\mathbf{R}}^d} G(t-s, \cdot -y) \alpha(u(s, y)) M(ds, dy),$$ where $G$ is the Green’s function of the wave equation. In particular, ${\mbox{${\mathcal F}$}}G(s) (\xi) = \vert \xi \vert^{-1} \sin(t \vert \xi \vert)$ and (G6), (G7) and (G8) hold provided $(\ref{3.3})$ holds with $k=1$. Therefore, the stochastic integral in $(\ref{5.4})$ is well-defined by Theorem \[rthm8\].
Suppose $$\int_{{\mathbf{R}}^d} \mu(d\xi)\, \frac{1}{1+\vert \xi\vert^2} < \infty,$$ $v_0 \in L^2({\mathbf{R}}^d)$, $\tilde{v}_0 \in H^{-1}({\mathbf{R}}^d)$, and $\alpha(\cdot)$ is a globally Lipschitz function. Then (\[5.3\]) has a unique solution in $L^2_\theta$. \[rthm9\]
[[Proof. ]{}]{}The proof follows that of Theorem \[rthm7\], so we only point out the changes relative to the proof of that theorem. Because $\alpha(\cdot)$ is globally Lipschitz, there is $K > 0$ such that for $u, u_1, u_2 \in {\mathbf{R}}$, $$\vert \alpha (u) \vert \leq K(1+\vert u\vert)\qquad
\mbox{ and }\qquad \vert \alpha(u_1) - \alpha(u_2) \vert \leq K \vert u_1-u_2\vert.$$ Using the first of these inequalities, $(\ref{4.8})$ is replaced by $$E(\Vert u_{n+1} (t, \cdot) \Vert^2_{L^2_\theta}) \leq 2 \Vert u_0(t, \cdot)
\Vert_{L^2({\mathbf{R}}^d)} + 2 KC \int_0^t ds\, (1 + E(\Vert u_n(s, \cdot) \Vert^2_{L_\theta^2})).$$ Therefore $u_{n+1}(t, \cdot)$ takes its values in $L^2_\theta$. The remainder of the proof is unchanged, except that $\Vert \cdot \Vert_{L^2({\mathbf{R}}^d)}$ must be replaced by $\Vert \cdot \Vert_{L_\theta^2}$. This proves Theorem \[rthm9\]. ${\quad \vrule height7.5pt width4.17pt depth0pt}$
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|
---
abstract: 'We show effects of the event-by-event fluctuation of the initial conditions (IC) in hydrodynamic description of high-energy nuclear collisions on some observables. Such IC produce not only fluctuations in observables but, due to their bumpy structure, several non-trivial effects appear. They enhance production of isotropically distributed high-$p_T$ particles, making $v_2$ smaller there. Also, they reduce $v_2$ in the forward and backward regions where the global matter density is smaller, so where such effects become more efficacious. They may also produce the so-called [*ridge effect*]{} in the two large-$p_T$ particle correlation.'
address:
- 'Instituto de Física/USP, C.P. 66318, 05314-970 São Paulo - SP, Brazil'
- 'Instituto de Física/UFRJ, C.P. 68528, 21945-970 Rio de Janeiro - RJ, Brazil'
author:
- 'Y. Hama, R.P.G. Andrade, F. Grassi, W. Qian'
- 'T. Kodama'
title: |
FLUCTUATION OF THE INITIAL CONDITIONS AND\
ITS CONSEQUENCES ON SOME OBSERVABLES [^1]
---
Introduction
============
Hydrodynamics is one of the main tools for studying the collective flow in high-energy nuclear collisions. In this approach, it is assumed that, after a complex process involving microscopic collisions of nuclear constituents, at a certain early instant a hot and dense matter is formed, which would be in local thermal equilibrium. After this instant, the system would expand hydrodynamically, following the well known set of differential equations.
The initial conditions (IC) for the hydrodynamic expansion are usually parametrized as smooth distributions of thermodynamic quantities and four-velocity. However, since our systems are small, important [*event-by-event fluctuations*]{} are expected in the IC of the real collisions. Moreover, each set of IC should presents strongly [*inhomogeneous structure*]{}. Our purpose here is to discuss some of the effects caused by such fluctuating and bumpy IC on observables.
Event-by-event fluctuating hydrodynamics
========================================
The main tool for our study is called NeXSPheRIO. It is a junction of two computational codes: NeXus and SPheRIO. The NeXus code [@nexus] is used to compute the IC. It is a microscopic model based on the Regge-Gribov theory and, once a pair of incident nuclei and their incident energy are chosen, it can produce, in the event-by-event basis, detailed space distributions of energy-momentum tensor, baryon-number, strangeness and charge densities, at a given initial time $\tau=\sqrt{t^2-z^2}\sim1\,$fm. We show in Fig.\[ic\] an example of such a fluctuating event, produced by NeXus event generator, for central Au + Au collision at $200\,A\,$GeV. As seen, the energy-density distribution is highly irregular and in a transverse plane (left panel) it presents several high-density blobs, whereas in a longitudinal plane (right panel) it presents a baton-like structure. When averaged over many events, these bumps disappear completely giving smooth IC, as those commonly used in hydro calculations. However, this bumpy structure gives important effects as we will show below. The SPheRIO code [@sp1; @hks], based on Smoothed Particle Hydrodynamics (SPH) algorithm [@sph], is well suited to computing the evolution of such systems, so complex as the one shown in Fig.\[ic\].
![Fluctuating IC, produced by NEXUS generator, for the most central Au+Au collisions at 200$\,$A GeV. Left: Energy density distribution is plotted in the $\eta=0$ plane. Right: Corresponding plot in the $x=2$ fm plane.[]{data-label="ic"}](ci_xy.eps "fig:"){width="5.5cm"} ![Fluctuating IC, produced by NEXUS generator, for the most central Au+Au collisions at 200$\,$A GeV. Left: Energy density distribution is plotted in the $\eta=0$ plane. Right: Corresponding plot in the $x=2$ fm plane.[]{data-label="ic"}](ci_ye.eps "fig:"){width="6.5cm"}
Effects of bumpy and fluctuating IC
===================================
In a previous paper [@granular], we discussed some effects of the bumpy structure of IC, as shown in Fig.\[ic\], on $p_T$ spectra and $v_2$ coefficient. The main consequence of baton structure[^2] is a violent and cylindrically isotropic expansion of the batons, which produces additional isotropic high-$p_T$ particles.
Transverse-momentum spectra
---------------------------
As clearly shown in Fig \[dndpt\], this implies that high-$p_T$ part of the $p_T$ spectrum becomes enhanced as compared with the smooth averaged IC case, making it more concave and closer to data.
![Charged-particle $p_T$ distributions (including those from resonance decays) computed in two different ways. The solid line indicates result for fluctuating IC, whereas the dotted line the one for the averaged IC. Data points [@PHOBOS1] are also plotted for comparison.[]{data-label="dndpt"}](dndpt1.eps){width="7.cm"}
$p_T$ dependence of $\langle v_2\rangle$
----------------------------------------
As for the anisotropy of the transverse flow, we illustrate in Fig.\[v2pt\] that the elliptic-flow coefficient $v_2$ suffers reduction as we go to high-$p_T$ region, due to the additional high-$p_T$ isotropic component included now. The result is closer to the available data.
![$p_T$ dependence of $\langle v_2\rangle$ in the centrality window and $\eta$ interval as indicated, compared with data [@PHOBOS2]. The solid line indicates result for fluctuating IC, whereas the dotted one that for the averaged IC. The curves are averages over PHOBOS centrality sub-intervals with freeze-out temperatures as indicated. []{data-label="v2pt"}](v2pt.eps){width="6.5cm"}
$\eta$ dependence of $\langle v_2\rangle$
-----------------------------------------
As for the $\eta$ dependence of $v_2\,$, we know that the average matter density decreases as $\vert\eta\vert$ increases as reflected in the $\eta$ distribution of charged particles, so the isotropic expansion effect of the batons becomes more efficacious there and, therefore, $v_2\,$ suffers a considerable reduction in those regions.
![$\eta$ dependence of $\langle v_2\rangle$ for three centrality windows. The solid lines indicate results for fluctuating IC, whereas the dotted lines the ones for the averaged IC. Data points[@PHOBOS2] are also plotted for comparison. $T_{fo}$ has been taken as in Fig.\[v2pt\].[]{data-label="v2eta"}](v2eta.eps){width="6.5cm"}
$v_2$ fluctuation
-----------------
The IC fluctuation also implies a large $v_2$ fluctuation. In preliminary works [@preliminary] on Au+Au collisions at $130A$ GeV, we showed that this actually happens. The results, with QGP included, have indeed been confirmed in recent experiments[@STAR; @PHOBOS3]. More recent computations for Au+Au at $200A\,$GeV [@fv2] gave similar results. In Fig. \[flv2\], we compare the latter with those data.
![$\sigma_{v_2}/\langle v_2\rangle$ computed for Au+Au collisions at $200\,A\,$GeV, compared with data. Upper: $\sigma_{v_2}/\langle v_2\rangle$ is given as function of the impact parameter $\langle b\rangle$ and compared with the STAR data [@STAR]. Lower: the same results are expressed as function of participant nucleon number $N_p$ and compared with the PHOBOS data, [@PHOBOS3]. ](flv2.eps){width="6.5cm"}
\[flv2\]
Ridge effect
------------
Another effect, which is produced naturally by the longitudinal baton structure of IC, as shown in Fig.\[ic\], is the so-called [*ridge phenomenon*]{} which has been experimentally seen in high-$p_T$ nearside correlations [@ridge]. Since batons in fluctuating IC which are close to the surface produce longitudinally correlated high-$p_T$ particles, always in the same side of the hot matter, the ridge structure naturally appears. This is shown in Fig.\[ridge\], as computed with NeXSPheRIO by J. Takahashi [*et al.*]{} [@jun] The ridge phenomenon has been discussed also in connection with glasma flux tubes [@mclerran].
![Two high-$p_T$ particle correlation, computed with event-by-event fluctuating IC. [@jun][]{data-label="ridge"}](ridge_FIC.eps){width="6.cm"}
Summary
=======
In this paper, we presented several consequences of event-by-event fluctuating IC with baton structure, computed with NeXSPheRIO code. The main results are: 1) The baton structure of IC produces more concave $p_T$ spectra, as compared with smooth IC; 2) It reduces $\langle v_2\rangle$ both in the high-$p_T$ and large-$|\eta|$ regions; 3) It also produces the [*ridge structure*]{} in the nearside correlations; 4) Large $v_2$ fluctuations occurs, in good agreement with data.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was partially supported by FAPESP, CNPq, FAPERJ and PRONEX. We are grateful to Jun Takahashi for providing us with Fig.\[ridge\]
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[^1]: Presented at IV Workshop on Particle Correlations and Femtoscopy (WPCF2008), Krakow (Poland), September 11-14, 2008.
[^2]: In that paper, we have considered only the transverse structure of IC and, then called it [*granular*]{}, but all the discussions remain valid replacing granular by [*baton*]{} there.
|
---
abstract: 'The paper studies some properties of the ring of integer-valued quasi-polynomials. On this ring, theory of generalized Euclidean division and generalized greatest common divisor are presented. Applications to finite simple continued fraction expansion of rational numbers and Smith normal form of integral matrices with an integer parameter are also given.'
address: |
Department of Mathematics, Harbin Institute of Technology\
Harbin, 150001, P. R. China
author:
- Nan Li
- Sheng Chen
title: 'On the Ring of Integer-valued Quasi-polynomials'
---
,
ring of integer-valued quasi-polynomials, generalized Euclidean division, continued fraction, Smith normal form, integer parameter
11A05;13F20;05A15
\[\]
Introduction
============
Integer-valued quasi-polynomials occur naturally in graded algebras (see [@Atiyah]) and enumerative combinatorics (see [@stanley]), such as the Ehrhart quasi-polynomial of a rational polytope. In this article, we study division theory of the ring $R$ of all integer-valued quasi-polynomials and its applications.
Obviously $\mathbb{Z}\varsubsetneq \mathbb{Z}[x]\varsubsetneq R $, where $\mathbb{Z}[x]$ is a unique factorization domain(UFD) but not an Euclidean Domain. It turns out that $R$ is neither a domain nor a Noetherian ring, but every finitely generated ideal in $R$ is a principal ideal(see Corollary \[cor1\] ). However, we can develop theory of generalized Euclidean division and greatest common divisor(GCD) on $R$, which has close relations to Euclidean division and GCD theory on $\mathbb{Z}$ pointwisely(i.e., through evaluation).
The organization of the paper is as follows. In Section 2, we will give the definition of integer-valued quasi-polynomial (see Definition $\ref{2. 0. 002}$) and prove some elementary results. In Section 3, we will present generalized Euclidean division on $R$. The relation between this generalized division and Euclidean division on $\mathbb{Z}$ is considered in Remark $\ref{2. 0. 01}$. In Section 4, we will present generalized GCD theory through successive generalized Euclidean divisions. Then, in Section 5 we will develop generalized GCD theory through pointwisely GCD over $\mathbb{Z} $ and prove the equivalent of the generalized GCD developed in Section 4 and 5.(see Theorem $\ref{2. 0. 015}$). In Section 6, we will give some applications of our generalized Euclidean division and GCD theory. We will show that expanding rational functions $h(x)=\frac{f(x)}{g(x)}$, where $f(x), g(x)\in \mathbb{Z}[x]$, into finite simple continued fractions for every $x\in \mathbb{Z}$, the numbers of terms for $h(x)$ are uniformly bounded (see Theorem $\ref{finite}$). Applications in Smith normal form for integral matrices with an integer parameter are also given( see Theorem $\ref{3. 0. 6}$).
Most of our proofs are simple but constructive. We have implemented all of the related algorithms in Maple and would like to send the Maple files to the interested readers upon request.
Ring of integer-valued Quasi Polynomial
=======================================
We begin with two simple lemmas.
\[2. 0. 000\] Suppose that $f(x)\in \mathbb{Q}[x]$ and for every $n\in
\mathbb{Z}$, $f(n)\in \mathbb{Z}$. Let $a$ be a positive integer such that $af(x)\in \mathbb{Z}[x]$. Then for every $i=0, 1,
\cdots, a-1$, we have $$g_i(x)=f(ax+i)\in \mathbb{Z}[x]$$
Let $f(x)=b_lx^{l}+b_{l-1}x^{l-1}+\cdots +b_1x+b_0$. Then for $i=0, 1, \cdots, a-1$, using binomial expansion theorem, we have $$\begin{aligned}
\label{2. 0. 010}
% \nonumber to remove numbering (before each equation)
f(ax+i) &=& b_l(ax+i)^{l}+b_{l-1}(ax+i)^{l-1}+\cdots+b_1(ax+i)+b_0 \\
&=& b_l[(ax)^{l}+\cdots+C_{l}^{1}(ax)i^{l-1}+i^{l}]\\&&+b_{l-1}[(ax)^{l-1}+\cdots+C_{l-1}^{1}(ax)i^{l-2}+i^{l-1}]\\&&+\cdots+b_1(ax+i)+b_0\end{aligned}$$ So there exists $h_i(x)\in \mathbb{Q}[x]$ such that $$f(ax+i)=ah_i(x)+f(i)$$ Since $af(x)\in \mathbb{Z}[x]$, we get $ah_i(x)\in \mathbb{Z}[x]$. For $i=0, 1, \cdots, a-1$, $f(i)\in
\mathbb{Z}$ and thus we have $g_i(x)=f(ax+i)\in \mathbb{Z}[x]$.
\[2. 0. 001\]For a function $f:\mathbb{Z}\to\mathbb{Z}$, the following two conditions are equivalent:
1. there exists $T\in \mathbb{N}$ and $f_i(x)\in \mathbb{Q}[x]$ ($i=0, 1, \cdots, T-1$) such that for any $m\in \mathbb{Z}$, if $n=Tm+i$, we have $f(n)=f_i(m)$;
2. there exists $T\in \mathbb{N}$ and $f_i(x)\in \mathbb{Z}[x]$($i=0, 1, \cdots, T-1$) such that for any $m\in
\mathbb{Z}$, if $n=Tm+i$, we have $f(n)=f_i(m)$.
It suffices to show that $(1)$ implies $ (2)$. Suppose that there exists $T_0\in \mathbb{N}$ and $g_i(x)\in \mathbb{Q}[x]$($i=0, 1,
\cdots, T_0-1$) such that for every $n\in \mathbb{Z}$, when $n=T_0k+i$, we have $f(n)=g_i(k)$. Choose an integer $a$ large enough such that for each $i\in \{0, 1, \cdots, T_0-1\}$, $ag_i(x)\in \mathbb{Z}[x]$. Define $$% \nonumber to remove numbering (before each equation)
f_{ij}: \mathbb{Z} \rightarrow \mathbb{Z}:\\
m \longmapsto g_{i}(am+j)$$ where $i=0, 1, \cdots, T_0-1$ and $j=0, 1, \cdots, a-1$. Since $ag_i(x)\in \mathbb{Z}[x]$ and $g_i(j)=f(Tj+i)\in \mathbb{Z}$, by Lemma $\ref{2. 0. 000}$, for each $j\in\{0, 1, \cdots,
a-1\}$, we have $f_{ij}(x)\in \mathbb{Z}[x]$. Now let $T=aT_0$. For $l=jT_0+i\in \{0, 1, \cdots, T-1\}$, define $f_l(x)=f_{ij}(x)$. Then $f(x)$ satisfies condition (2).
\[2. 0. 002\]We call function $f(x)$ an *integer-valued quasi-polynomial* if it satisfies the equivalent conditions in Lemma $\ref{2. 0. 001}$. If $T\in \mathbb{N}$ and $f_i(x)\in
\mathbb{Z}[x]$ ($i=0, 1, \cdots, T-1$) satisfy condition 2 in Lemma $\ref{2. 0. 001}$, we call $(T, \{f_i(x)\}_{i=0}^{T-1})$ a *representation* of $f(x)$ and write $$f(x)=(T, \{f_i(x)\}_{i=0}^{T-1})$$We call $\max\{degree(f_i(x))|
i=0, 1, \cdots, T-1\}$ and $T$ the degree and period of this representation respectively.
We can see that degree of $f(x)$ is independent from its representations. In fact, let $(T_1, \{f_i(x)\}_{i=0}^{T_1-1})$ and $(T_2,
\{g_j(x)\}_{j=0}^{T_2-1})$ be two representations of $f(x)$. For each $s\in \{0, 1, \cdots, \frac{T_1T_2}{d}-1\}$, if $s=T_1s_1+i=T_2s_2+j$, where $s_1, s_2\in \mathbb{Z} $, $ 0
\leq i< T_1 $ and $ 0 \leq j< T_2 $, then $\frac{T_1T_2}{d}x+s=T_1(\frac{T_2}{d}x+s_1)+i=T_2(\frac{T_1}{d}x+s_2)+j$, and therefore we have $$f_i(\frac{T_2}{d}x+s_1)=g_j(\frac{T_1}{d}x+s_2)$$ Hence it is not difficult to check that $$max\{degree(f_i(x))|
i=0, 1, \cdots, T_1-1\}=max\{degree(g_i(x))| i=0, 1, \cdots,
T_2-1\}$$ The periods are not unique. However, we have the following result.
\[2. 0. 011\] Let $f(x)$ be an integer-valued quasi-polynomial and $ \Omega $ be the set of positive periods $T$ satisfying the equivalent conditions in Lemma $\ref{2. 0. 001}$. Define a partial order $\preccurlyeq\, \, $ on $ \Omega $ as follows: $\, T_1\preccurlyeq T_2$ if and only if $T_1\mid T_2$. Then $ \Omega $ has a smallest item, which will be called the least positive period of $f(x)$.
It is sufficient for us to prove that if $f(x)\neq{0}$ and $ T_1, T_2\in
\Omega $, then $d=gcd_{\mathbb{Z}}(T_1, T_2)\in
\Omega $, i.e., if $f(x)=(T_1, \{f_i(x)\}_{i=0}^{T_1-1})=(T_2,
\{g_j(x)\}_{j=0}^{T_2-1})\neq 0$, then $f(x)$ has a representation with period $d$. Suppose that $k$ is the degree of $f(x)$. Let $$f(T_1x+i) =f_i(x)=a_ix^{k}+\cdots$$ and $$f(T_2x+j) = g_j(x)=b_jx^{k}+\cdots$$
Now we are going to show $d\in \Omega $ by induction on $k$. For $k=0$, the assertion reduces to $f(n)=f(n+d)$, for every $ n\in
\mathbb{Z}$. Note that there exist $u, v\in \mathbb{Z}$ such that $T_1u+T_2v=d$. Since $ T_1, T_2\in
\Omega $, for every $x\in
\mathbb{Z}$, we have $f(x+d)=f(x+T_1u+T_2v)=f(x+T_2v)=f(x)$, as desired.
Suppose the assertion true for $k-1$ and prove it for $k$. Assume that $s\in \{0, 1, \cdots, \frac{T_1T_2}{d}-1\}$, and $s=T_1s_1+i=T_2s_2+j$, where $s_1, s_2\in \mathbb{Z} $, $ 0
\leq i< T_1 $ and $ 0 \leq j< T_2 $. Then since $\frac{T_1T_2}{d}x+s=T_1(\frac{T_2}{d}x+s_1)+i=T_2(\frac{T_1}{d}x+s_2)+j$, we have $$f_i(\frac{T_2}{d}x+s_1)=g_j(\frac{T_1}{d}x+s_2)$$ Take the $k$-th order derivative to the above equation, we get $$k!(\frac{T_2}{d})^{k}a_i =f_i^{(k)}(\frac{T_2}{d}x+s_1) =
g_j^{(k)}(\frac{T_1}{d}x+s_2) =k!(\frac{T_1}{d})^{k}b_j$$ The left and right of the identity are all integer constant function with periods $T_1$ and $T_2$ respectively. By the assertion for $k=0$, when $i\equiv j\, \, (mod\, \, T)$, we have $$(\frac{T_2}{d})^{k}a_i =(\frac{T_2}{d})^{k}a_j= (\frac{T_1}{d})^{k}b_i=(\frac{T_1}{d})^{k}b_j$$ Define $$h:\mathbb{Z} \rightarrow \mathbb{Z}:n \mapsto a_im^{k}$$ where $n=T_1m+i$, $i=0, 1, \cdots, T_1-1$. Let $l(x)=f(x)-h(x)$. Then $l(x)\in R$ has periods $T_1$, $T_2$ and $k-1$ is its degree. By induction hypothesis, $l(x)\in
R$ and has a period $d$. Note that $h(x)$ has a period $d$. Consequently, $f(x)=l(x)+h(x)$ has a period $d$, as desired.
\[2. 0. 003\] In the remainder of this paper, we often use the following fact. Let $f(x), g(x)$ be two integer-valued quasi-polynomials with periods, say, $T_1, T_2$ respectively. Then $f(x), g(x)$ have a common period $T=lcm(T_1, T_2)$.
\[R\]The set of all integer-valued quasi-polynomials, denoted by $R$, with pointwisely defined addition and multiplication, is a commutative ring with identity.
Let $\Gamma$ be the set of functions $f:\mathbb{Z}\to\mathbb{Z}$. It is obvious that $1\in R$ and with pointwisely defined addition and multiplication, $\Gamma$ is a commutative ring with identity $1$. Therefore, it is sufficient for us to prove $R$ is a subring of $\Gamma$. In fact, by remark $\ref{2. 0. 003}$, we know that subtraction and multiplication are closed in $R$. The proof is completed.
\[2. 0. 004\]
1. Let $f(x)\in R$ and $f(x)\neq 0$. Then f(x) is a zero divisor if and only if it has a representation $(T, \{f_i(x)\}_{i=0}^{T-1})$ and there exists $i\in\{0, 1, \cdots, T-1\}$ such that $f_i(x)=0$.
2. Let $f(x)\in R$. Then f(x) is invertible if and only if it has a representation $(T, \{f_i(x)\}_{i=0}^{T-1})$ such that for $i\in\{0, 1, \cdots, T-1\}$, $f_i(x)=1$ or $f_i(x)=-1$.
\(1) Suppose $f(x)g(x)=0$ where $g(x)\in R- \{0\}$. Let $T$ be a common period of $f(x)$ and $g(x)$ such that $f(x)=(T,
\{f_i(x)\}_{i=0}^{T-1})$ and $g(x)=(T, \{g_i(x)\}_{i=0}^{T-1})$ (see Remark $\ref{2. 0. 003}$). Since $g(x)\neq 0$, there exists $i_0\in \{0, 1, \cdots, T-1\}$ with $g_{i_0}(x)\neq 0$ and thus $f_{i_0}(x)= 0$.
Conversely, suppose that there exists $i_0\in
\{0, 1, \cdots, T-1\}$ with $f_{i_0}(x)=0$. Then define $g(x)\in
R$ as follows: if $n=Tm+i_0$, $g(n)=g_{i_0}(m)=m$, and else $g(n)=0$. Obviously, we have $f(x)g(x)=0$. Note that $f(x)\neq
0$. Hence $f(x)$ is a zero divisor in $R$.
\(2) Assume that there exists $g(x)\in R-{0}$ with $f(x)g(x)=1$. Let $T$ be a common period of $f(x)$ and $g(x)$, such that $f(x)=(T, \{f_i(x)\}_{i=0}^{T-1})$ and $g(x)=(T,
\{g_i(x)\}_{i=0}^{T-1})$. Since for each $i\in \{0, 1, \cdots,
T-1\}$, $f_i(x)g_i(x)=1$ and $f_i(x), g_i(x)\in \mathbb{Z}[x]$. It follows that $f_i(x)=g_i(x)=1$ or $-1$. The reverse part is trivial.
\[2. 0. 005\]The ring $R$ is not Noetherian.
Let$f_1(n)=n$ and $$f_2(n)=\left\{\begin{array}{ll}
\frac{n}{2}, & \hbox{$n=2m$} \\
n, & \hbox{$n=2m+1$}
\end{array}
\right. ,
f_3(n)=\left\{
\begin{array}{ll}
\frac{n}{4}, & \hbox{$n=4m$} \\
\frac{n}{2}, & \hbox{$n=4m+2$} \\
n, & \hbox{else}
\end{array}
\right. , \cdots
f_k(n)=\left\{\begin{array}{ll}
\frac{f_{k-1}(n)}{2}, & \hbox{$n=2^{k-1}m$} \\
f_{k-1}(n), & \hbox{else}
\end{array}
\right. \cdots$$ Let $$g_2(n)=\left\{\begin{array}{ll}
2, & \hbox{$n=2m$} \\
1, & \hbox{$n=2m+1$}
\end{array}
\right. , g_3(n)=\left\{\begin{array}{ll}
2, & \hbox{$n=4m$} \\
1, & \hbox{else}
\end{array}
\right. , \cdots
g_k(n)=\left\{\begin{array}{ll}
2, & \hbox{$n=2^{k-1}m$} \\
1, & \hbox{else}
\end{array}
\right. \cdots$$ Note that $f_{k-1}(n)=f_k(n)g_k(n)$. We have $(f_{k-1}(n))\varsubsetneq (f_{k}(n))$, where $k=2, 3, \cdots$. Thus $R$ does not satisfy ascending chain condition(acc) on ideals in $R$, i.e., it is not Noetherian(cf. [@Ring]).
Generalized Euclidean Division
==============================
\[2. 0. 2\] Let $r(x)\in R$ and $r(x)=(T,
\{r_i(x)\}_{i=0}^{T-1})$. We shall say $r(x)$ is nonnegative and write $r(x)\succcurlyeq 0$, if it satisfies the following equivalent conditions ：
1. for every $i=0, 1, \cdots, T-1$, $r_i(x)=0$ or its leading coefficient is positive;
2. there exists $C\in \mathbb{Z}$, such that for every integer $n>C$, we have $r(n) \geqslant 0$.
We shall say $r(x)$ is strictly positive and write $r(x)\succ 0$, if $r(x)=(T,
\{r_i(x)\}_{i=0}^{T-1})$ satisfies the following condition: $(1^{'})$ for every $i=0, 1, \cdots, T-1$, the leading coefficient of $r_i(x)$ is positive. [**We write $
f(x)\preccurlyeq g(x)$ if $ g(x)- f(x)\succcurlyeq 0$**]{}.
Because of the existence of zero divisor, the “order” in $R$ defined above is not a partial order, i.e., $r(x)\in R$ satisfies both $r(x)\succcurlyeq 0$ and $r(x)\preccurlyeq 0$ does not imply $r(x)=0$. More precisely, $r(x)\succcurlyeq 0$ implies $r(x)$ falls in one of the following three cases: (a) $r(x)\succ 0$, (b) $r(x)=
0$, (c) $r(x)$ is a nonnegative zero divisor, where case $(c)$ is impossible if $r(x)\in \mathbb{Z}[x]$(by Proposition $\ref{2. 0.
004}$). Thus the following definition is [**well defined**]{}.
.
\[2. 0. 20\] Let $r(x) \in \mathbb{Z}[x] $, define a function $| . |$ as follows: $\mathbb{Z}[x]\rightarrow \mathbb{Z}[x]$: $$|r(x)|=\left\{
\begin{array}{ll}
r(x), & if\, \, \hbox{$r(x)\succ 0$} \\
-r(x), & if \, \, \hbox{$r(x)\prec 0$} \\
0, & if \, \, \hbox{$r(x)=0$}
\end{array}
\right.$$
\[2. 0. 3\] Let $f(x), g(x)\in \mathbb{Z}[x]$ and $g(x)\neq 0$. Then there exist unique $P(x), r(x)\in R$ such that $$f(x)=P(x)g(x)+r(x)\quad\, where\quad
0\preccurlyeq r(x)\prec| g(x)|$$ In this situation, we write $P(x)=quo(f(x), g(x))$ and $r(x)=rem(f(x),g(x))$.
*Step 1*: We first prove for the existence of $P(x)$ and $r(x)$. If $f(x)=0$, then put $P(x)=0, r(x)=0$. If $f(x)\neq 0$, then let $$f(x)=a_kx^{k}+a_{k-1}x^{k-1}+\cdots+a_1x+a_0$$ and $$g(x)=b_lx^{l}+b_{l-1}x^{l-1}+\cdots+b_1x+b_0$$ where $a_k\neq 0$ and $b_l\neq 0$.
When $k<l$, define $P(x), r(x)$ as follows:
1. if $f(x)\succ 0$, then P(x)=0, r(x)=f(x);
2. if $f(x)\prec 0$ and $g(x)\prec 0$, then P(x)=1, r(x)=f(x)-g(x);
3. if $f(x)\prec 0$ and $g(x)\succ 0$, then P(x)=-1, r(x)=f(x)+g(x).
It is easy to check they satisfy the remainder conditions in the result. When $k\geqslant l$, we claim that there exist $P(n),
r(n)\in R$ with a period $T=b_l^{k-l}$ and prove it by induction on $k-l$. This assertion is trivial for $k-l=0$. Suppose that it is true for $k-l=h$ and prove it for $k-l=h+1$. Now for $i=0, \cdots,
b_l-1$, when $n=b_lm+i$, define $$h_i(m)=f(b_lm+i)=a_kb_l^{k}m^{k}+h_{i, k-1}(m)$$ and $$l_i(m)=g(b_lm+i)=b_l^{l+1}m^{l}+l_{i, l-1}(m)$$ where $h_{i, k-1}(x)$ and $ l_{i, l-1}(x)\in \mathbb{Z}[x]$ have degrees $k-1$ and $l-1$ respectively. Since for every $m\in
\mathbb{Z}$ $$a_kb_l^{k}m^{k}=a_kb_{l}^{h}\, m^{h+1}\, l_i(m)-a_kb_l^{h}\, m^{h+1}\, l_{i,l-1}(m)$$ we have $$h_i(m)=a_kb_{l}^{h}\, m^{h+1}\, l_i(m)+S_{i, k-1}(m)$$ where$$S_{i, k-1}(m)=h_{i, k-1}(m)-a_kb_l^{h}\, m^{h+1}\,
l_{i，l-1}(m)$$ Note that $S_{i, k-1}(x)$ and $l_i(x)\in
\mathbb{Z}[x]$ have degrees $k-1$ and $l$ respectively. Thus by induction hypothesis, both $$P_i(x)=quo(S_{i, k-1}(x), l_i(x))\in R$$ and $$r_i(x)=rem(S_{i, k-1}(x), l_i(x))\in R$$ have a period $b_l^{h}$. For $i=0, \cdots, b_l-1$, when $n=b_lm+i$, take $
% \nonumber to remove numbering (before each equation)
P(n) =a_kb_l^{h}m^{h+1}+P_i(m)$ and $
r(n) =r_i(m)$. Then it is easy to check that $P(x)=quo(f(x),g(x))$ and $r(x)=rem(f(x),g(x))$ are elements in $R$ with a period $b_l^{h+1}$, as desired.
*Step 2*: Now we turn to the proof for the uniqueness of $P(x), r(x)$. Suppose that there exist $P_1(x), P_1(x), r_1(x), r_2(x)\in R$ such that$$f(x)=P_i(x)g(x)+r_i(x), \, 0\preccurlyeq r_i(x)\prec
|g(x)|\,
\, \, i=1, 2$$ Let $T$ be a common period for $P_1(x),P_2(x),r_1(x)$ and $r_2(x)$. For $j=0, 1,
\cdots, T-1$), when $n=Tm+j$, we have $P_i(n)=P_{ij}(m)$, $r_i(n)=r_{ij}(m), \, \, i=1, 2$. Thus $$(P_{1j}(m)-P_{2j}(m))g(Tm+j)=r_{1j}(m)-r_{2j}(m)$$
By Definition $\ref{2. 0. 2}$, there exists an integer $C$ such that for every integer $m>C$, we have $0\leqslant
r_{ij}(m)<|g(Tm+j)|$. So $$|r_{1j}(m)-r_{2j}(m)|<|g(Tm+j)|$$ For every $m\in \mathbb{Z}$, we have $P_{1j}(m)-P_{2j}(m)\in
\mathbb{Z}$. Therefore, we have $r_{1j}(x)=r_{2j}(x)$ and $P_{1j}(x)=P_{2j}(x)$ for every $j=0, 1, \cdots, T-1$.
\[2. 0. 01\]This division above, which will be called generalized Euclidean division over $\mathbb{Z}[x]$, almost coincides with division in $\mathbb{Z}$ pointwisely in the following sense. Let $f(x), g(x)\in \mathbb{Z}[x]$ and $$f(x)=P_1(x)g(x)+r_1(x), \, where \, 0 \preccurlyeq r_1(x)\prec |g(x)|$$By Definition $\ref{2. 0. 2}$, the inequality $0\leqslant
r_1(x)<|g(x)|$ provides us an integer $C_1$, such that for all $n
>C_1$, $$\big[\frac{f(n)}{g(n)}\big]=P_1(n)\, , \, \big\{\frac{f(n)}{g(n)}\big\}g(n)=r_1(n)$$ Put $f_1(x)=f(-x)$ and $g_1(x)=g(-x)$. Since $$f_1(x)=P_2(x)g_1(x)+r_2(x), \, 0\preccurlyeq r_2(x)\prec |g_1(x)|$$ there exists an integer $C_2$, such that for all $n>C_2$, $0\leqslant r_2(n)<|g_1(n)|$. Note that$$f(-x)=g(-x)P_2(x)+r_2(x)$$Put $y=-x$, we have $$f(y)=g(y)P_2(-y)+r_2(-y)$$For every integer $y<-C_2$, we get $0\leqslant r_2(-y)<|g(y)|$. Thus for every integer $n<-C_2$, we have $$\big[\frac{f(n)}{g(n)}\big]=P_2(-n), \, \, \, \, \big\{\frac{f(n)}{g(n)}\big\}g(n)=r_2(-n)$$
In the special case where $r_1(x)=0$, we also have $r_2(x)=0$. Then for [**every** ]{} $n\in \mathbb{Z}$, $$r_1(n)=r_2(n)=\big\{\frac{f(n)}{g(n)}\big\}g(n)=0$$
\[eg1\]The following is an example to illustrate the relation between generalized division on $\mathbb{Z}[x]$ and on $\mathbb{Z}$. When $n>1$, $$\big[\frac{n^{2}}{2n+1}\big]=\left\{
\begin{array}{ll}
m-1, & \hbox{$n=2m$;} \\
m-1, & \hbox{$n=2m-1$. }
\end{array}
\right.$$ $$\big\{\frac{n^{2}}{2n+1}\big\}(2n+1)=\left\{
\begin{array}{ll}
3m+1, & \hbox{$n=2m$;} \\
m, & \hbox{$n=2m-1$. }
\end{array}
\right.$$ When $n<-1$, $$\big[\frac{n^{2}}{2n+1}\big]=\left\{
\begin{array}{ll}
m, & \hbox{$n=2m$;} \\
m, & \hbox{$n=2m-1$. }
\end{array}
\right.$$ $$\big\{\frac{n^{2}}{2n+1}\big\}(2n+1)=\left\{
\begin{array}{ll}
-m, & \hbox{$n=2m$;} \\
-3m+1, & \hbox{$n=2m-1$. }
\end{array}
\right.$$
So far, we have defined generalized Euclidean Division on $\mathbb{Z}[x]$. Now we consider the case when $f(x),g(x)\in R$. Suppose that $T_0$ is the least common period of $f(x)$, $g(x)$, such that $f(x)=(T_0, \{f_i(x)\}_{i=0}^{T_0-1})$ and $g(x)=(T_0,
\{g_i(x)\}_{i=0}^{T_0-1})$. Based on generalized division on $\mathbb{Z}[x]$ (in Theorem $\ref{2. 0. 3}$), we can define $quo(f(x), g(x))$ and $rem(f(x), g(x))$ as follows (denoted by $P(x)$ and $r(x)$ respectively): when $n=Tm+i$ $$P(n)=\left\{
\begin{array}{ll}
quo(f_i(m), g_i(m)), & \hbox{ if $g_i(m)\neq 0$} \\
0, & if \hbox{$g_i(m)=0$}
\end{array}
\right.
r(n)=\left\{
\begin{array}{ll}
rem(f_i(m), g_i(m)), & \hbox{if $g_i(m)\neq 0$} \\
f_i(m), &if \hbox{$g_i(m)=0$}
\end{array}
\right.$$
Then it is easy to check that $P(x), r(x)\in R$ and $$\label{fg}
f(x)=R(x)g(x)+r(x)$$ This will be called the generalized Euclidean algorithm in the ring of integer-valued quasi-polynomials.
Generalized GCD through Generalized Euclidean Division
======================================================
When studying Euclidean division in $\mathbb{Z}$, we can develop GCD theory and related algorithm (see [@Bhu]). Its counterpart in $R$ is generalized GCD through generalized Euclidean division.
\[2. 0. 5\]Suppose that $f(x), g(x)\in R$ and for every $n\in \mathbb{Z}$, $g(n)\neq 0$. Then by Remark $\ref{2. 0. 01}$ and generalized Euclidean division, the following two statements are equivalent:
1. rem($f(x), g(x)$)=0;
2. for every $x\in \mathbb{Z}$, $g(x)$ is a divisor of $f(x)$.
If the two conditions are satisfied, we shall call $g(x)$ a divisor of $f(x)$ and write $g(x)\mid f(x)$.
By the equivalence of the above two statements, similar to the situation in $\mathbb{Z}$, we have the following proposition.
\[2. 0. 6\]Let $g(x), f(x)\in $R. If $f(x)\mid g(x)$ and $g(x)\mid f(x)$, we have $f(x)=\varepsilon g(x)$, where $\varepsilon $ is an invertible element in $R$.
\[2. 0. 006\]\[GCD\]Let $f_1(x), f_2(x), \cdots, f_s(x), d(x)\in
R$.
1. We call $d(x)$ a common divisor of $f_1(x), f_2(x), \cdots, f_s(x)$, if we have $d(x)\mid f_k(x)$ for every $k=1, 2, \cdots, s$.
2. We call $d(x)$ a greatest common divisor of $f_1(x), f_2(x), \cdots,
f_s(x)$ if $d(x)$ is a common divisor of $f_1(x), f_2(x), \cdots,
f_s(x)$ and for any common divisor $p(x)\in R$ of $f_1(x), f_2(x),
\cdots, f_s(x)$, we have $p(x)\mid d(x)$.
\[2. 0. 014\] Suppose that both $d_1(x)$ and $ d_2(x)$ are greatest common divisors of $f_1(x), f_2(x), \cdots, f_s(x)$. Then we have $d_1(x)\mid d_2(x)$ and $d_2(x)\mid d_1(x)$. Thus, by Proposition $\ref{2. 0. 6}$, we have $d_1(x)=\varepsilon d_2(x)$, where $\varepsilon $ is an invertible element in $R$. So we have a unique GCD $d(x)\in R$ for $f_1(x), f_2(x), \cdots, f_s(x)$ such that $d(x)\succ 0$ or $d(x)=0$ and write it as $ggcd(f_1(x), f_2(x),
\cdots, f_s(x))$.
\[2. 0. 007\]Let $f_1(x), f_2(x), \cdots, f_s(x)\in R$. Then $$ggcd(f_1(x), f_2(x), \cdots, f_s(x))=ggcd(r(x), f_2(x), \cdots, f_s(x))$$where $r(x)=rem(f_1(x), f_2(x))$.
By ($\ref{fg}$), $f_1(x)=f_2(x)P_1(x)+r(x) $, we have the set of common divisors of $f_1(x), f_2(x), \cdots, f_s(x)$ and that of $r(x),
f_2(x), \cdots, f_s(x)$ are the same. The proof is completed.
\[2. 0. 008\]Let $f_0(x), g_0(x)\in R$. By generalized Euclidean division, define $f_k(x), g_k(x)(k\in \mathbb{N}-\{0\})$ recursively as follows: $$f_{k}(x)=g_{k-1}(x), \, \, \, \, g_{k}(x)=rem(f_{k-1}(x), g_{k-1}(x))$$ Then there exists $k_0\in \mathbb{N}-\{0\}$ such that: $rem(f_{k_0}(x), g_{k_0}(x))=0$.
When $k\geqslant 1$, if $g_k(x)=rem(f_{k-1}(x), g_{k-1}(x))\neq
0$. Let $(T, \{g_{ki}(x)\}_{i=0}^{T-1})$ be a representation of $g_{k}(x)$. Define $$\begin{aligned}
% \nonumber to remove numbering (before each equation)
g_{ki}(x)&=& a_lx^{l}+a_{l-1}x^{l-1}+\cdots, +a_1x+a_0 \\
g_{k-1, i}(x)&=& b_sx^{s}+b_{s-1}x^{s-1}+\cdots, +b_1x+b_0=g_{k-1}(Tx+i)
\end{aligned}$$ where $a_l, b_s\neq 0$. Since $0\preccurlyeq g_{ki}(x)\prec |g_{k-1, i}(x)|$, there are four cases:
1. $l=s$ and $a_l<|b_s|$;
2. $l=s$ and $a_l=|b_s|$;
3. $l<s$;
4. $g_{ki}(x)=0$.
By the generalized Euclidean algorithm in Section 3 and the remainder condition $0 \preccurlyeq rem(f(x), g(x))\prec|g(x)|$. We can reduce case (1) to case (2), (3) or (4), reduce (2) to (3) or (4) and reduce (3) to (4). For case (4), however, by generalized Euclidean division, if $g_{ki}(x)=0$, we have, for every $t\geqslant k$, $g_{ti}(x)=0$. Therefore, for all the four cases, we can find $k_0\in \mathbb{N}$ such that: $rem(f_{k_0}(x),
g_{k_0}(x))=0$.
\[2. 0. 0099\]Let $f_1(x), f_2(x), \cdots, f_s(x)\in R$. Then there exist $d(x), u_i(x)\in R$ ($ i=1, 2, \cdots, s$), such that $d(x)=ggcd(f_1(x), f_2(x), \cdots, f_s(x))$ and $$f_1(x)u_1(x)+f_2(x)u_2(x)+\cdots+f_s(x)u_s(x)=d(x)$$
To prove this statement, we apply induction on $s$. It is trivial when $s=1$. Suppose the statement true for $s=l$. Then for $s=l+1$, by induction hypothesis, there exist $d_0(x), d(x),
u_0(x), u_1(x), \cdots, u_{l+1}(x), v_1(x), v_2(x)\in R$ such that$$d_0(x)=ggcd(f_1(x), f_2(x))=v_1(x)f_1(x)+v_2(x)f_2(x)$$and $$d(x)=ggcd(d_0(x), f_3(x), \cdots, f_{l+1}(x))=u_0(x)d_0(x)+u_3(x)f_3(x)+\cdots+u_{l+1}(x)f_{l+1}(x)$$ By Lemma $\ref{2. 0. 007}$ and Lemma $\ref{2. 0. 008}$, we get $$ggcd(f_1(x), f_2(x), f_3(x), \cdots, f_{l+1}(x))=ggcd(0, d_0(x), f_3(x), \cdots, f_{l+1}(x))=d(x)$$ Let $u_1(x)=v_1(x)u_0(x)$ and $ u_2(x)=v_2(x)u_0(x)$. Then we have $u_1(x), u_2(x), \cdots, u_{l+1}(x)\in R$ such that $$d(x)=u_1(x)f_1(x)+u_2(x)f_2(x)+u_3(x)f_3(x)+\cdots+u_{l+1}(x)f_{l+1}(x)$$
\[cor1\]Every finitely generated ideal in $R$ is a principal ideal.
The result follows readily from Theorem \[2. 0. 0099\].
GGCD through Pointwise GCD
===========================
Now we are going to study generalized GCD by considering pointwisely defined GCD in $\mathbb{Z}$ of $f_1(x), f_2(x), \cdots,
f_s(x)$($x\in \mathbb{Z}$), which will give us a more efficient algorithm for generalized GCD of elements in $R$ than successive divisions in practice.
Let $f_k(x)\in \mathbb{Z}[x]$, $k=1, 2, \cdots, s$. For any $n\in \mathbb{Z}$, denote the GCD of $f_1(n), f_2(n), \cdots,
f_s(n)$ by $gcd_{\mathbb{Z}}$ $(f_1(n), f_2(n), \cdots, f_s(n))$. Denote the greatest common factor as polynomials over $\mathbb{Z}$ by $gcd_{\mathbb{Z}[x]}(f_1(x), f_2(x), \cdots, f_s(x))$.
\[2. 0. 012\]Let $f_i(x)\in \mathbb{Z}[x]$ and $gcd_{\mathbb{Z}[x]}(f_1(x), f_2(x), \cdots, f_s(x))=1$. Then there exists $a_0\in \mathbb{N}$, such that for all $n \equiv i\,
(mod\, a_0)$, $$gcd_{\mathbb{Z}}(f_1(n), f_2(n), \cdots, f_s(n))=gcd_{\mathbb{Z}}(f_1(i), f_2(i),
\cdots, f_s(i))$$ where $i=0, 1, \cdots, a_0-1$.
First, there exist $u_k(x)\in \mathbb{Q}[x](k=1, 2, \cdots, s)$ such that $\sum_{k=1}^{s}f_k(x)u_k(x)=1$. Choose $a_0\in \mathbb{N}$ large enough such that for all $k=1, 2, \cdots, s$, $w_{k}(x)=a_0u_k(x)\in \mathbb{Z}[x]$. Since $\sum_{k=1}^{s}f_k(x)w_{k}(x)=a_0$, we have, for every $ n\in \mathbb{Z}$, $$\begin{aligned}
gcd_{\mathbb{Z}}(f_1(n), f_2(n), \cdots, f_s(n))\mid a_0\label{a0}
\end{aligned}$$ Then we will show that for every $ n\in \mathbb{Z}$, if $n \equiv i(mod a_0), i=0, 1, \cdots, a_0-1$, we will have $$gcd_{\mathbb{Z}}(f_1(n), f_2(n), \cdots, f_s(n))=gcd_{\mathbb{Z}}(f_1(i), f_2(i), \cdots, f_s(i))$$ For $i=0, 1, \cdots, a_0-1$ and any $m\in \mathbb{Z}$, let $$\begin{aligned}
% \nonumber to remove numbering (before each equation)
d_i&=& gcd_{\mathbb{Z}}(f_1(i), f_2(i), \cdots, f_s(i)) \label{di}\\
d_{im} &=& gcd_{\mathbb{Z}}(f_1(a_0m+i), f_2(a_0m+i), \cdots, f_s(a_0m+i))\label{dim}
\end{aligned}$$
For each $k=1, 2, \cdots, s$, expand $f_k(a_0m+i)$ by binomial expansion theorem. By Lemma $\ref{2. 0. 000}$, we get polynomials $h_{ik}(x)\in \mathbb{Z}[x]$ such that $$\begin{aligned}
f_k(n)=f_k(a_0m+i)=a_0h_{ik}(m)+f_k(i)\label{a1}
\end{aligned}$$ By ($\ref{di}$), we have $d_i\mid
f_k(i)$ for every $k=1, 2, \cdots, s$. By ($\ref{a0}$), we have $d_i\mid a_0$. Therefore, by ($\ref{a1}$), for all $m\in \mathbb{Z}$, we have $d_i\mid
f_k(a_0m+i)$. Then from ($\ref{dim}$), we know $d_i\mid
d_{im}$. Similarly, $d_{im}\mid
d_i$. The proof is completed.
\[2. 0. 013\]Let $f_1(x), f_2(x), \cdots, f_s(x)\in R$ and for every $n\in \mathbb{Z}, \sum_{i=1}^{s}f_i(n)^{2}\neq
0$. Define a function $g(x)$ as follows: $$\begin{aligned}
% \nonumber to remove numbering (before each equation)
g: \mathbb{Z}& \rightarrow & \mathbb{Z} \\
n &\longmapsto & gcd_{\mathbb{Z}}(f_1(n), f_2(n), \cdots, f_s(n))\end{aligned}$$Then $g(x)\in R$.
Let $T$ be a common period of $f_1(x), f_2(x), \cdots, f_s(x)$. Then there exist $f_{ij}(x)\in
\mathbb{Z}[x]$, $i=1, 2, \cdots, s$, $j=0, 1, \cdots, T-1$ such that if $n=Tm+j$, $f_i(n)=f_{ij}(m)$. Put $$d_{j_0}(x)=gcd_{\mathbb{Z}[x]}(f_{1j}(x), f_{2j}(x), \cdots, f_{sj}(x))$$ By assumption, we have $d_{j_0}(x)\neq 0$. Let $\overline{f_{ij}(x)}=\frac{f_{ij}(x)}{d_{j_0}(x)}$. Since $$gcd_{\mathbb{Z}[x]}(\overline{f_{1j}(x)}, \overline{f_{2j}(x)}, \cdots, \overline{f_{sj}(x)})=1$$by Lemma $\ref{2. 0. 012}$, $$d_j(x)=gcd_{\mathbb{Z}}(f_{1j}(x), f_{2j}(x), \cdots, f_{sj}(x))=
d_{j_0}(x)gcd_{\mathbb{Z}}(\overline{f_{1j}(x)},
\overline{f_{2j}(x)}, \cdots, \overline{f_{sj}(x)})\in R$$Then for $j=0, 1, \cdots, T-1$, when $n=Tm+j$, we have $$g(n)=gcd_{\mathbb{Z}}(f_1(n), f_2(n), \cdots, f_s(n))=d_j(m)$$Thus, it is easy to check that $g(x)\in R$.
\[2. 0. 015\]Let $f_1(x), f_2(x), \cdots, f_s(x)\in R$. Then we have (see Definition $\ref{2. 0. 006}$ and Corollary $\ref{2. 0. 013}$)$$gcd_{\mathbb{Z}}(f_1(x), f_2(x), \cdots,
f_s(x))=ggcd(f_1(x), f_2(x), \cdots, f_s(x))$$
Let $g(x)=gcd_{\mathbb{Z}}(f_1(x), f_2(x), \cdots, f_s(x))$. By Definition $\ref{2. 0. 5}$, $g(x)$ is a common divisor of $f_1(x),
f_2(x), \cdots, f_s(x)$. For any common divisor $p(x)\in R$ of $f_1(x), f_2(x), \cdots, f_s(x)$ we have $p(x)\mid g(x)$. Then from Definition $\ref{2. 0. 014}$(ggcd), we should only check that $g(x)$ is nonnegative, which is obvious. The proof is completed.
\[eg2\]From the theory above in this section, we can easily compute ggcd. For example, $ggcd(x^{3}+2, 3x^{2}-3x, 7x)=h(x)$ where $ h:
\mathbb{Z} \rightarrow \mathbb{Z} $ and $$h(n)=\left\{
\begin{array}{ll}
2, & \hbox{$n=2m$} \\
1, & \hbox{$n=2m+1$}
\end{array}
\right.$$
Applications
============
Application of Euclidean Division
---------------------------------
Now we will apply generalized successive Euclidean division to expand rational numbers with an integer parameter into finite simple continued fractions(see [@Number]).
\[finite\]Let $f(x), g(x)\in \mathbb{Z}[x]$ and for every $n\in
\mathbb{Z}$, $g(n)\neq 0$. Then there exists $L\in \mathbb{N}$ such that for every $n\in \mathbb{Z}$, the number of terms in expansion $h(n)=\frac{f(n)}{g(n)}$ as finite simple continued fraction is no greater than $ L$.
By generalized Euclidean division, there exist nonnegative integers $C_1, C_2$ such that $h(n)$ has uniform expansion formulas ( $n$ being divided into finite cases) as finite simple continued fractions for all but finitely many values of $n$, i.e., those integers in the interval $[-C_2,C_1]$. It follows readily that the numbers of terms for $h(n)$ are bounded.
Here is an example to illustrate Theorem $\ref{finite}$.\
When $n>4$ $$={
[ll]{} $[m-1;1, 3, m]$, &\
$[m-1;3, 1, m-1]$, &
. $$ When $n<-4$ $$={
[ll]{} $[m-1;1,2,1,-m-1]$, &\
$[m-1;4,-m]$, &
. $$
Similar with finite simple continued fraction, we can apply generalized Euclidean division on $\mathbb{Z}[x]$ to some other problems in elementary number theory, such as computing Jacobi Symbol with a parameter in $\mathbb{Z}$.
We have$$\bigg(\displaystyle\frac{4n^{2}+1}{2n+1}\bigg)=\bigg(\displaystyle\frac{2}{2n+1}\bigg)=\left\{
\begin{array}{ll}
1, & \hbox{$n=4m$} \\
-1, & \hbox{$n=4m+1$} \\
-1, & \hbox{$n=4m+2$} \\
1, & \hbox{$n=4m+3$}
\end{array}
\right.$$
Application of generalized GCD
------------------------------
Based on generalized GCD theory, we have the following applications to ideals and matrices in $\mathbb{Z}[x]$.
\[2. 0. 4\]Let $I_1, I_2$ be ideals in $\mathbb{Z}[x]$, then for every $n\in \mathbb{Z}$, $I_1(n)=I_2(n)$ as ideals in $\mathbb{Z}$ if and only if $ggcd(I_1)=ggcd(I_2)$.
\[rem1\]Consider the relation between $I_1=I_2$ and $ggcd(I_1)=ggcd(I_2)$. It is clear that $I_1=I_2$ implies $ggcd(I_1)=ggcd(I_2)$. However, the converse is not true. For example $$ggcd(2, x+1)=ggcd(4, x^{2}+1)= \left\{
\begin{array}{ll}
2, & \hbox{$x=2m+1$} \\
1, & \hbox{$x=2m$}
\end{array}
\right.$$but $(2, x+1)\neq (4, x^{2}+1)$.
\[3. 0. 5\]Let $A(x)=(a_{ij}(x))_{m\times n}, \, \, (m\leqslant n)$, where $a_{ij}(x)\in
\mathbb{Z}(x)$. For $k=1, 2, \cdots, m$, let $F_k(A(x))$ be the ideal generated by all the minors of $A(x)$ with order $k$.
1. Define generalized determinate factors $\{D_k(A(x))\}_{k=1}^{m}$ of $A(x)$ by $\{ggcd(F_k(A(x)))\}_{k=1}^{m}$;
2. Define generalized invariant factors $\{d_k(A(x))\}_{k=1}^{m}$ by $d_1(A(x))=D_1(A(x))$ and $d_k(A(x))=\{quo(D_k(A(x)), D_{k-1}(A(x)))\}(2\leq k \leq m)$ ;
3. Define the smith normal form $smith(A(x))$ of $A(x)$ of size $m\times n$ by $$\left(
\begin{array}{ccccccc}
d_1(A(x)) & 0 & 0 & 0 & 0&\cdots & 0 \\
0 & d_2(A(x)) & 0 & 0 & 0&\cdots& 0 \\
0 & 0 & \cdots & 0 & 0 &\cdots & 0 \\
0 & 0 & 0 & d_m(A(x)) & 0&\cdots & 0 \\
\end{array}
\right)_{ m \times n}$$
By the existence and uniqueness of quotient and ggcd, the above three definitions are well defined and unique for $A(x)$. Based on the classical smith normal form for integral matrices (see [@Newmann]), we have the following result.
\[3. 0. 6\]Let $A(x)=(a_{ij}(x))_{m\times n}, \, \, \, B(x)=(b_{ij}(x))_{m\times n}$, where $a_{ij}(x), b_{ij}(x)\in \mathbb{Z}(x)$. Then the following are equivalent:
1. for $k=1, 2, \cdots, m$, for all $n\in
\mathbb{Z}$, $ A(n), B(n)$ are equivalent;
2. for $k=1, 2, \cdots, m$, $D_k(A(x))=D_k(B(x))$;
3. for $k=1, 2, \cdots, m$, $d_k(A(x))=d_k(B(x))$;
4. $smith(A(x))=smith(B(x))$.
\[3. 0. 7\]By Remark $\ref{rem1}$, it is clear that for each $k=1, 2, \cdots, m$, $F_k(A(x))=F_k(B(x))$ implies for every $n\in \mathbb{Z}$, $A(n)$ and $B(n)$ are equivalent and the converse is not true. For example, let $$A(x)=\left(
\begin{array}{ccc}
x & x+2 & 0\\
x+1 & x+3 & x \\
0 & 0 & 2x\\
\end{array}
\right)= x
\left(
\begin{array}{ccc}
1 & 1 & 0\\
1& 1 & 1 \\
0 & 0 & 2\\
\end{array}
\right)+
\left(
\begin{array}{ccc}
0 & 2 & 0\\
1& 3 & 0 \\
0 & 0 & 0\\
\end{array}
\right)$$ and $$B(x)=\left(
\begin{array}{ccc}
x & x+2 & 0 \\
x+1 & x+3 & 1 \\
0& 0& 2x \\
\end{array}
\right)=
x
\left(
\begin{array}{ccc}
1 & 1 & 0\\
1& 1 & 0 \\
0 & 0 & 2\\
\end{array}
\right)+
\left(
\begin{array}{ccc}
0 & 2 & 0\\
1& 3 & 1 \\
0 & 0 & 0\\
\end{array}
\right)$$ Then $$D_1(A(x))=D_1(B(x))=1$$ $$D_2(A(x))=D_2(B(x))= \left\{ \begin{aligned}
1&\, \, \, \, x\equiv 0(mod 2)\\
2&\, \, \, \, x\equiv 1(mod 2)
\end{aligned} \right.$$ $$D_3(A(x))=D_3(B(x))=4x$$ According to Theorem $\ref{3. 0. 6}$, for all $n\in
\mathbb{Z}$, $A(n)$ and $B(n)$ are equivalent. But $$F_2(A(x))=<2, x^{2}>\neq <2, x>=F_2(B(x))$$
[00]{}
M. F. Atiyah. I. G. Macdonald, [Introduction to Commutative Algebra]{}, [Addison-Wesley Publishing Company]{}, 1969. R. P. Stanley. [Enumerative Combinatorics]{}, Vol. 1. Cambridge University Press, 1996.
K. H. Rosen. [Elementary Number Theory and Its Applications]{}, Fifth Edition, Addison-Wesley Publishing Company, 2004.
B. Mishra. [Algorithmic Algebra]{}, Springer-Veriag, 2001. I. M. Isaacs. [Algebra: A Graduate Course]{}, Wadsworth Inc. 1994.
M. Newmann. [Integral Matrices]{}, New York: Academic Press, 1972.
|
---
author:
- 'Jing Lei[^1]'
- 'Anne-Sophie Charest'
- Aleksandra Slavkovic
- Adam Smith
- Stephen Fienberg
bibliography:
- 'paper.bib'
title: |
Differentially Private Model Selection\
With Penalized and Constrained Likelihood
---
Introduction
============
Differential privacy
====================
Differentially Private Model Selection Procedures {#sec:methodology}
=================================================
Utility Analysis
================
Empirical Results
=================
Application to Real Data Sets
-----------------------------
Discussion {#sec:discussion}
==========
Appendix: Proof details {#sec:proof-detail}
=======================
[^1]: [email protected]
|
---
abstract: 'We consider a phase retrieval problem, where we want to reconstruct a $n$-dimensional vector from its phaseless scalar products with $m$ sensing vectors. We assume the sensing vectors to be independently sampled from complex normal distributions. We propose to solve this problem with the classical non-convex method of alternating projections. We show that, when $m\geq Cn$ for $C$ large enough, alternating projections succeed with high probability, provided that they are carefully initialized. We also show that there is a regime in which the stagnation points of the alternating projections method disappear, and the initialization procedure becomes useless. However, in this regime, $m$ needs to be of the order of $n^2$. Finally, we conjecture from our numerical experiments that, in the regime $m=O(n)$, there are stagnation points, but the size of their attraction basin is small if $m/n$ is large enough, so alternating projections can succeed with probability close to $1$ even with no special initialization.'
author:
- 'Irène Waldspurger [^1]'
bibliography:
- '../bib\_articles.bib'
- '../bib\_proceedings.bib'
- '../bib\_livres.bib'
- '../bib\_misc.bib'
title: Phase retrieval with random Gaussian sensing vectors by alternating projections
---
Introduction
============
The problem of reconstructing a low-rank matrix from linear observations appears under many forms in the fields of inverse problems and machine learning. An important amount of work has thus been devoted to the design of reconstruction algorithms coming with provable reconstruction guarantees. The first algorithms of this kind relied mostly on convexification techniques. They tended to have a high recovery rate, but a possibly prohibitive computational complexity. As a result, a need has emerged to prove similar guarantees for algorithms based on non-convex formulations, which are generally much faster.
In this article, we consider a subclass of low-rank recovery problems: *phase retrieval problems*. In the finite-dimensional setting, phase retrieval consists in recovering an unknown vector $x_0\in\C^n$ from $m$ phaseless linear measurements, of the form $$b_k=|\scal{a_k}{x_0}|,\quad\quad k=1,\dots,m,$$ where the *sensing vectors* $a_k\in\C^n$ are known. Phaseless measurements do not allow to distinguish $x_0$ from $ux_0$, for $u\in\C,|u|=1$, so the goal is only to recover $x_0$ *up to a global phase*. Motivations for studying these problems come in particular from optical imaging; see [@schechtman] for a recent review. Phase retrieval problems can be seen as low-rank matrix recovery problems, because knowing $|\scal{a_k}{x_0}|$ amounts to knowing $$|\scal{a_k}{x_0}|^2=\Tr(a_ka_k^*x_0x_0^*),$$ so reconstructing $x_0$ is equivalent to: $$\begin{aligned}
\label{eq:matricial_form}
\mbox{Reconstruct }X_0\in\mathcal{S}_n(\C)&\mbox{ from }\{\Tr(a_ka_k^*X_0)\}_{k=1,\dots,m}\\
&\mbox{ such that }\mathrm{rank}(X_0)=1\nonumber.\end{aligned}$$
The vector $x_0$ is uniquely determined by the $m$ phaseless measurements as soon as $m\gtrsim 4n$ [@balan]; however, reconstructing it is a priori NP-hard [@fickus]. The oldest reconstruction algorithms [@gerchberg; @fienup] were iterative: they started from a random initial guess of $x_0$, and tried to iteratively refine it by various heuristics. Although these algorithms are empirically seen to succeed in a number of cases, they can also get stuck in stagnation points, whose existence is due to the non-convexity of the problem.
To overcome these convergence problems, convexification methods have been introduced [@chai; @candes2]. These methods consider the matricial formulation , but replace the non-convex rank constraint by a more favorable convex constraint. They provably reconstruct the unknown vector $x_0$ with high probability if the sensing vectors $a_k$ are “random enough” [@candes_li; @candes_li2; @gross]. Numerical experiments show that they also perform well on more structured, non-random phase retrieval problems [@maxcut; @sun_smith].
Unfortunately, this good precision comes at a high computational cost: optimizing the $n\times n$ matrix $X_0$ is much slower that directly reconstructing the $n$-dimensional vector $x_0$. Consequently, convexification techniques are impractical when the dimension of $x_0$ exceeds a few hundred. Authors have thus recently begun to design fast non-convex algorithms, for which it is possible to establish similar reconstruction guarantees as for convexified algorithms. The methods that have been developed rely on the following two-step scheme:
1. an initialization step, that returns a point close to the solution;\[item:step1\]
2. a gradient descent (possibly with additional refinements) over a well-chosen non-convex cost function.\[item:step2\]
The intuitive reason why this scheme works is that the cost function, although globally non-convex, enjoys some good geometrical property in a neighborhood of the solution (like convexity or a weak form of it [@white]). So, if the point returned by the initialization step belongs to this neighborhood, gradient descent converges to the true solution.
A preliminary form of this scheme appears in [@netrapalli], with an alternating minimization in step \[item:step2\] instead of a gradient descent. Then, considering the cost function $$\label{eq:cost_L1}
L_1(x)=\sum_{k=1}^m\left(b_k^2-|\scal{a_k}{x}|^2\right)^2,$$ [@candes_wirtinger] proved the correctness of the two-step scheme, with high probability, in the regime $m=O(n\log n)$, for random independent Gaussian sensing vectors. In [@candes_wirtinger2; @kolte], the same result was shown in the regime $m=O(n)$ for a slightly different cost function, with additional truncation steps. In [@zhang], it was extended to the following non-smooth cost function: $$L_2(x)=\sum_{k=1}^m\left(b_k-|\scal{a_k}{x}|\right)^2.$$ Additionally, @sun_qu_wright have shown that, in the regime $m=O(n\log^3 n)$, the cost function actually has no “bad critical point”, and the initialization step is not necessary: the gradient descent in step \[item:step2\] converges to the global minimum of $L_1$, almost whatever initial point it starts from. These authors have also numerically observed that, in the regime $m=O(n)$, despite the potential presence of bad critical points, the gradient descent succeeds, with at least constant probability, starting from a random initialization.
For other low-rank recovery problems than phase retrieval, we refer for example to [@sun_luo; @ge_lee_ma] for matrix completion, to [@tu; @bhojanapalli] for the case where the measurement scheme obeys a Restricted Isometry Property, and to [@bandeira_low_rank] for $\Z_2$ synchronization problems.
In the case of phase retrieval, the most recently introduced non-convex algorithms are optimal in terms of both statistical and computational complexity, up to multiplicative constants. However, there is still a need to understand whether their theoretical reconstruction guarantees can be extended to more general classes of algorithms, that would not exactly follow the above two-step scheme, but would be closer to the algorithms that are actually used in applications. This in particular implies to answer the following two questions:
- In Step \[item:step2\], can we replace the explicit minimization of a cost function by a “less local” search, like alternating projections [@gerchberg] or Douglas-Rachford [@bauschke]?
- Is the initialization step \[item:step1\] necessary, or can Step \[item:step2\] converge to the global optimum even starting from a random initialization, at least in certain cases?
In this article, we answer the first question: we show that, in the optimal regime of $m=O(n)$ random independent Gaussian sensing vectors, replacing gradient descent with alternating projections yields exact recovery with high probability, and convergence occurs at a linear rate.
There exist absolute constants $C_1,C_2,M>0$, $\delta\in]0;1[$ such that, if $m\geq Mn$ and the sensing vectors are independently chosen according to complex normal distributions, the sequence of iterates $(z_t)_{t\in \N}$ produced by the alternating projections method satisfies $$\forall t\in\N^*,\quad\quad \inf_{\phi\in\R}||e^{i\phi}x_0-z_t||\leq \delta^t||x_0||,$$ with probability at least $$1-C_1\exp(-C_2m),$$ provided that alternating projections are correctly initialized, for example with the method described in [@candes_wirtinger2].
Alternating projections, introduced by @gerchberg, is the most ancient algorithm for phase retrieval. It is an intuitive method, whose implementation is extremely simple, and with no parameter to choose or tune; it is thus widely used. In terms of complexity, it is slower, for general measurements, than the best non-convex methods by only a logarithmic factor in the precision. For more “structured” measurements (as in all applications that we know of), it is as fast (see Paragraph \[ss:complexity\]).
We believe that the second question, about the necessity of the initialization step, is also important. In addition to being a natural theoretical question, it has practical consequences: the initialization procedure depends on the probability distribution of the sensing vectors, and, for some families of sensing vectors appearing in applications, we do not (yet) have a valid initialization procedure. We partially answer it in the case where the sensing vectors are independent and Gaussian, and reconstruction is done with alternating projections. We propose a description of when this method globally converges to the true solution, depending on the number of measurements and the initialization procedure. This description is summarized in Figure \[fig:global\_image\].
As shown in the figure, there is a regime in which the stagnation points of the alternating projections routine disappear (except possibly on a “small” set that we define), and, with high probability, alternating projections converge starting from any initialization outside the small set. This regime is $m=O(n^2)$. Our numerical experiments clearly indicate that, below this regime, there are stagnation points. It is however possible that the attraction basin of the stagnation points is small: even in the regime $m=O(n)$, we numerically see that alternating projections, starting from a random isotropic initialization[^2], succeed with probability close to $1$ despite the presence of stagnation points. We leave this assertion as a conjecture.
There exist $C_1,C_2,\gamma,M>0$, $\delta\in]0;1[$ such that, if $m\geq Mn^2$ and the sensing vectors are independently chosen according to complex normal distributions, with probability at least $$1-C_1\exp(-C_2 n),$$ the sequence of iterates $(z_t)_{t\in \N}$ produced by the alternating projections method satisfies $$\forall t\geq \gamma\log n,\quad\quad \inf_{\phi\in\R}||e^{i\phi}x_0-z_t||\leq \delta^{t-\gamma\log n}||x_0||,$$ starting from any initial point that does not belong to a small “bad set”.
Let any $\epsilon>0$ be fixed. When $m\geq Cn$, for $C>0$ large enough, alternating projections, starting from a random isotropic initialization, converge to the true solution with probability at least $1-\epsilon$.
These theorem and conjecture are the parallels for alternating projections of the results and numerical observations obtained by @sun_qu_wright for gradient descent over the cost function . The “no stagnation point” regime is much less favorable in the case of alternating projections than in the case of gradient descent: $m=O(n^2)\gg O(n\log^3 n)$. It could be due to the discontinuity of the alternating projections operator, but we have no evidence to support this fact.
0.7cm
On the side of proof techniques, there has been a lot of work on the convergence of alternating projections in non-convex settings. Transversality arguments can be shown to prove, in certain cases, local convergence guarantees (“if the initial point is sufficiently close to the correct solution, alternating projections converge to this solution”). See for example [@lewis; @drusvyatskiy]. These arguments can be used in phase retrieval, and yield local convergence results for relatively general families of sensing vectors (not necessarily random) [@noll; @chen_fannjiang]. Unfortunately, they give no control on the convergence radius of the algorithm, so the obtained results have a mainly theoretical interest.
Bounding the convergence radius requires using the statistical properties of the sensing vectors. This was first attempted in [@netrapalli], where the authors proved the global convergence of a resampled version of the alternating projections algorithm. For a non resampled version, a preliminary result was given in [@soltanolkotabi]. However, the bound on the convergence radius that underlies this result is small. As a consequence, global convergence is only proven for a suboptimal number of measurements ($m=O(n\log^2n)$), and with a complex initialization procedure.
A difficulty that we encounter is the fact that the alternating projections operator is not continuous. This difficulty also appears in the two recent articles [@zhang; @wang], where the authors consider a gradient descent over a function whose gradient is not continuous. The proof that we give for our Theorem \[thm:global\_convergence\] follows a different path as theirs (it does not use a regularity condition); the statistical tools are however similar.
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The article is organized as follows. Section \[s:setup\] precisely defines phase retrieval problems and the alternating projections algorithm. Section \[s:with\_init\] states and proves the first main result: the global convergence of alternating projections, with proper initialization, for $m=O(n)$ independent Gaussian measurements. Section \[s:without\_init\] proves the second main result: stagnation points disappear in the regime $m=O(n^2)$, making the initialization step useless. Finally, Section \[s:numerical\] presents numerical results, and conjectures that the alternating projections algorithm can succeed without special initialization in the regime $m=O(n)$, despite the presence of stagnation points. All technical lemmas are deferred to the appendices.
Notations
---------
For any $z\in\C$, $|z|$ is the modulus of $z$. We extend this notation to vectors: if $z\in\C^k$ for some $k\in\N^*$, then $|z|$ is the vector of $(\R^+)^k$ such that $$|z|_i=|z_i|,\quad\quad\forall i=1,\dots,k.$$ For any $z\in\C$, we set $E_{\phase}(z)$ to be the following subset of $\C$: $$\begin{array}{rll}
E_{\phase}(z)&=\left\{\frac{z}{|z|}\right\}&\mbox{ if }z\in\C-\{0\};\\
&=\{e^{i\phi},\phi\in\R\}&\mbox{ if }z=0.
\end{array}$$ We extend this definition to vectors $z\in\C^k$: $$E_{\phase}(z)=\prod_{i=1}^k E_{\phase}(z_i).$$ For any $z\in\C$, we define $\phase(z)$ by $$\begin{array}{rll}
\phase(z)&=\frac{z}{|z|}&\mbox{ if }z\in\C-\{0\};\\
&=1&\mbox{ if }z=0,
\end{array}$$ and extend this definition to vectors $z\in\C^k$, as for the modulus.
We denote by $\odot$ the pointwise product of vectors: for all $a,b\in\C^k$, $(a\odot b)$ is the vector of $\C^k$ such that $$(a\odot b)_i=a_ib_i,\quad\quad\forall i=1,\dots,k.$$ We define the operator norm of any matrix $A\in\C^{n_1\times n_2}$ by $$|||A||| = \sup_{v\in \C^{n_2}, ||v||=1}||Av||.$$ We denote by $A^\dag$ its Moore-Penrose pseudo-inverse. We note that $AA^\dag$ is the orthogonal projection onto $\Range(A)$.
Problem setup\[s:setup\]
========================
Phase retrieval problem
-----------------------
Les $n,m$ be positive integers. The goal of a phase retrieval problem is to reconstruct an unknown vector $x_0\in \C^n$ from $m$ measurements with a specific form.
We assume $a_1,\dots,a_m\in\C^n$ are given; they are called the *sensing vectors*. We define a matrix $A\in\C^{m\times n}$ by $$A=\begin{pmatrix}a_1^*\\\vdots\\a_m^*\end{pmatrix}.$$ This matrix is called the *measurement matrix*. The associated *phase retrieval* problem is: $$\label{eq:problem_statement}
\mbox{reconstruct }x_0\mbox{ from }b\overset{def}{=}|Ax_0|.$$ As the modulus is invariant to multiplication by unitary complex numbers, we can never hope to reconstruct $x_0$ better than *up to multiplication by a global phase*. So, instead of exactly reconstructing $x_0$, we want to reconstruct $x_1$ such that $$x_1 = e^{i\phi}x_0,\quad\quad \mbox{for some }\phi\in\R.$$
In all this article, we assume the sensing vectors to be independent realizations of centered Gaussian variables with identity covariance: $$\label{eq:def_A}
(a_{i})_j\sim\mathcal{N}\left(0,\frac{1}{2}\right)
+\mathcal{N}\left(0,\frac{1}{2}\right)i,\quad\quad
\forall 1\leq i\leq m,1\leq j\leq n.$$ The measurement matrix is in particular independent from $x_0$.
@balan and @conca have proved that, for *generic* measurement matrices $A$, Problem always has a unique solution, up to a global phase, provided that $m\geq 4n-4$. In particular, with our measurement model , the reconstruction is guaranteed to be unique, with probability $1$, when $m\geq 4n-4$.
Alternating projections
-----------------------
The alternating projections method has been introduced for phase retrieval problems by @gerchberg. It focuses on the reconstruction of $Ax_0$; if $A$ is injective, this then allows to recover $x_0$.
To reconstruct $Ax_0$, it is enough to find $z\in\C^m$ in the intersection of the following two sets.
1. $z\in \{z'\in\C^m,|z'|=b\}$;
2. $z\in\Range(A)$.
Indeed, when the solution to Problem is unique, $Ax_0$ is the only element of $\C^m$ that simultaneously satisfies these two conditions (up to a global phase).
A natural heuristic to find such a $z$ is to pick any initial guess $z_0$, then to alternatively project it on the two constraint sets. In this context, we call *projection* on a closed set $E\subset\C^m$ a function $P:\C^m\to E$ such that, for any $x\in\C^m$, $$||x-P(x)||=\inf_{e\in E}||x-e||.$$ The two sets defining constraints (1) and (2) admit projections with simple analytical expressions, which leads to the following formulas:
\[eq:gs\_image\] $$\begin{aligned}
y'_k&= b \odot \phase(y_k);& \mbox{(Projection onto set (1))}\\
y_{k+1}&= (AA^\dag) y'_k.& \mbox{(Projection onto set (2))}\end{aligned}$$
If we define $z_k$ as the unique vector such that $y_k=Az_k$, an equivalent form of these equations is: $$z_{k+1} = A^\dag(b\odot\phase(Az_k)).$$
The hope is that the sequence $(y_k)_{k\in\N}$ converges towards $Ax_0$. Unfortunately, it can get stuck in *stagnation points*. The following proposition (proven in Appendix \[s:stagnation\_points\]) characterizes these stagnation points.
\[prop:stagnation\_points\] For any $y_0$, the sequence $(y_k)_{k\in\N}$ is bounded. Any accumulation point $y_\infty$ of $(y_k)_{k\in\N}$ satisfies the following property: $$\exists u\in E_{\phase}(y_\infty),\quad\quad
(AA^\dag)(b\odot u)=y_\infty.$$ In particular, if $y_\infty$ has no zero entry, $$(AA^\dag)(b\odot \phase(y_\infty))=y_\infty.$$
Despite the relative simplicity of this characteristic property, it is extremely difficult to exactly compute the stagnation points, determine their attraction basin or avoid them when the algorithm happens to run into them.
The goal of this article is to show that, in certain settings, there are no stagnation points, or they can be avoided with a careful initialization procedure of the alternating projection routine.
Alternating projections with good initialization\[s:with\_init\]
================================================================
In this section, we prove the first of our two main results: in the regime $m=O(n)$, the method of alternating projections converges to the correct solution with high probability, if it is carefully initialized.
Local convergence of alternating projections
--------------------------------------------
This paragraph proves the key result that we will need to establish our statement. This result is a local contraction property of the alternating projections operator $x\to A^\dag(b\odot\phase(Ax))$.
\[thm:local\_convergence\] There exist $\epsilon,C_1,C_2,M>0$, and $\delta\in]0;1[$ such that, if $m\geq Mn$, then, with probability at least $$1-C_1\exp(-C_2m),$$ the following property holds: for any $x\in\C^n$ such that $$\label{eq:hyp_x}
\inf_{\phi\in\R}||e^{i\phi}x_0-x||\leq \epsilon ||x_0||,$$ we have $$\label{eq:progres_lineaire}
\inf_{\phi\in\R}||e^{i\phi}x_0-A^\dag(b\odot\phase(Ax))||\leq \delta ||x_0-x||.$$
For any $x\in\C^n$, we can write $Ax$ as $$\label{eq:Ax_orth}
Ax = \lambda_x (Ax_0) + \mu_x v^x,$$ where $\lambda_x\in\C,\mu_x\in\R^+$, and $v^x\in\Range(A)$ is a unitary vector orthogonal to $Ax_0$.
The following lemma is proven in Paragraph \[ss:diff\_phase\].
\[lem:diff\_phase\] For any $z_0,z\in\C$, $$|\phase(z_0+z)-\phase(z_0)| \leq 2. 1_{|z|\geq |z_0|/6} + \frac{6}{5}\left|\Im\left(\frac{z}{z_0}\right)\right|.$$
So, for any $x\in\C^n$, $$\begin{aligned}
|\phase(\lambda_x)(Ax_0)_i&-(b\odot\phase(Ax))_i|\\
&=\left|\phase(\lambda_x)(Ax_0)_i-|Ax_0|_i\phase((Ax)_i)\right|\\
&=\left|\phase(\lambda_x)(Ax_0)_i-|Ax_0|_i\phase(\lambda_x(Ax_0)_i+\mu_x (v^x)_i)\right|\\
&= |Ax_0|_i\left|\phase(Ax_0)_i-\phase\left((Ax_0)_i+\frac{\mu_x}{\lambda_x} (v^x)_i\right) \right|\\
&\leq 2.|Ax_0|_i1_{|\mu_x/\lambda_x||v^x|_i\geq |Ax_0|_i/6} + \frac{6}{5}\left|\Im \left(\frac{\frac{\mu_x}{\lambda_x}v^x_i}{\phase((Ax_0)_i)}\right)\right|.\end{aligned}$$ As a consequence, $$\begin{aligned}
||\phase(\lambda_x)(Ax_0)&-b\odot\phase(Ax)||\nonumber\\
&\leq \left|\left|
2.|Ax_0|\odot 1_{|\mu_x/\lambda_x||v^x|\geq |Ax_0|/6} + \frac{6}{5}\left|\Im \left(\left(\frac{\mu_x}{\lambda_x}v^x\right)\odot\overline{\phase(Ax_0)}\right)\right|\,
\right|\right|\nonumber\\
&\leq 2\left|\left|
|Ax_0|\odot 1_{6|\mu_x/\lambda_x||v^x|\geq |Ax_0|}\right|\right| + \frac{6}{5}\left|\left|\Im \left(\left(\frac{\mu_x}{\lambda_x}v^x\right)\odot\overline{\phase(Ax_0)}\right)\right|
\right|.\label{eq:error_sum}\end{aligned}$$ Two technical lemmas allow us to upper bound the terms of this sum. The first one is proved in Paragraph \[ss:first\_term\], the second one in Paragraph \[ss:second\_term\].
\[lem:first\_term\] For any $\eta>0$, there exists $C_1,C_2,M,\gamma>0$ such that the inequality $$||\,|Ax_0|\odot 1_{|v|\geq |Ax_0|}||\leq \eta ||v||$$ holds for any $v\in\Range(A)$ such that $||v||<\gamma ||Ax_0||$, with probability at least $$1-C_1\exp(-C_2m),$$ when $m\geq Mn$.
\[lem:second\_term\] For $M,C_1>0$ large enough, and $C_2>0$ small enough, when $m\geq M n$, the property $$||\Im(v\odot\overline{\phase(Ax_0)})||\leq \frac{4}{5}||v||$$ holds for any $v\in\Range(A)\cap \{Ax_0\}^\perp$, with probability at least $$1-C_1\exp(-C_2 m).$$
Let us choose $\eta>0$ such that $$12\eta + \frac{24}{25}<1.$$ We define $\gamma>0$ as in Lemma \[lem:first\_term\]. The events described in Lemmas \[lem:first\_term\] and \[lem:second\_term\] hold with probability at least $$1-2C_1\exp(-C_2m).$$ When this happens, for all $x$ such that $$\left|\frac{\mu_x}{\lambda_x}\right|< \frac{\gamma}{6} \,||Ax_0||,$$ the terms in Equation can be bounded as in the lemmas, because $$\left|\left| 6\frac{\mu_x}{\lambda_x}v^x
\right|\right|=6\left|\frac{\mu_x}{\lambda_x}\right|<\gamma||Ax_0||,$$ and $\frac{\mu_x}{\lambda_x}v^x\in \Range(A)\cap\{Ax_0\}^\perp$. So the following inequality holds: $$\begin{aligned}
\label{eq:consequence_lemmas}
||\phase(\lambda_x)(Ax_0)&-b\odot\phase(Ax)||
\leq \left(12 \eta
+\frac{24}{25} \right)\left|\frac{\mu_x}{\lambda_x}\right|.\end{aligned}$$ For any $x$ such that $\inf_{\phi\in\R}||e^{i\phi}x_0-x||\leq\epsilon||x_0||$, if we set $\epsilon^x=\inf_{\phi\in\R}\frac{||e^{i\phi}x_0-x||}{||x_0||}\leq\epsilon$, $$\inf_{\phi\in\R}||e^{i\phi}Ax_0-Ax||\leq \epsilon^x|||A|||\,||x_0||,$$ so, using Equation , $$\inf_{\phi\in\R}|e^{i\phi}-\lambda_x|^2||Ax_0||^2+|\mu_x|^2\leq (\epsilon^{x})^2|||A|||^2||x_0||^2,$$ which implies $$\begin{gathered}
|\mu_x|\leq \epsilon^x|||A|||\,||x_0||;\\
|\lambda_x| \geq 1-\epsilon^x\frac{|||A|||\,||x_0||}{||Ax_0||}.\end{gathered}$$ We can thus deduce from Equation that, on an event of probability at least $1-2C_1\exp(-C_2m)$, as soon as $\inf_{\phi\in\R}||e^{i\phi}x_0-x||\leq \epsilon||x_0||$, $$\label{eq:maj_image}
||\phase(\lambda_x)(Ax_0)-b\odot\phase(Ax)||
\leq \left(12 \eta
+\frac{24}{25} \right) \frac{\epsilon^x}{1-\epsilon^x\frac{|||A|||\,||x_0||}{||Ax_0||}}|||A|||\,||x_0||$$ if $$\label{eq:condition_gamma}
\frac{\epsilon^x}{1-\epsilon^x\frac{|||A|||\,||x_0||}{||Ax_0||}}\frac{|||A|||\,||x_0||}{||Ax_0||}
< \frac{\gamma}{6}.$$ Equation implies in particular that, if Condition holds, $$\label{eq:maj_signal}
||\phase(\lambda_x)x_0-A^\dag(b\odot\phase(Ax))||
\leq \left(12 \eta
+\frac{24}{25} \right) \frac{\epsilon^x}{1-\epsilon^x\frac{|||A|||\,||x_0||}{||Ax_0||}}|||A^\dag|||\, |||A|||\,||x_0||.$$
To conclude, it is enough to control the norms of $A$ and $A^\dag$ with the following classical result.
\[prop:davidson\] If $A$ is chosen according to Equation , then, for any $t$, with probability at least $$1-2\exp\left(-mt^2\right),$$ we have, for any $x\in\C^n$, $$\sqrt{m}\left(1-\sqrt{\frac{n}{m}}-t\right)||x||
\leq ||Ax||
\leq \sqrt{m}\left(1+\sqrt{\frac{n}{m}}+t\right)||x||.$$
From this proposition, if we choose $\delta,M,t$ such that $$\begin{gathered}
12\eta+\frac{24}{25} < \delta<1;\\
\epsilon<\min\left(\frac{1}{4},\frac{\gamma}{24},\frac{1}{2\delta}\left(\delta-12\eta-\frac{24}{25}\right)\right);\\
\frac{1+\sqrt{\frac{1}{M}}+t}{1-\sqrt{\frac{1}{M}}-t}\leq \min\left(2,\frac{(1-2\epsilon)\delta}{12\eta+\frac{24}{25}}\right).\end{gathered}$$ we have, for $m\geq Mn$, with probability at least $1-2e^{-mt^2}$, as soon as $\epsilon^x\leq\epsilon$, $$\begin{aligned}
\frac{\epsilon^x}{1-\epsilon^x\frac{|||A|||\,||x_0||}{||Ax_0||}}
\frac{|||A|||\,||x_0||}{||Ax_0||}
&\leq \frac{\epsilon}{1-\epsilon\frac{|||A|||\,||x_0||}{||Ax_0||}}
\frac{1+\sqrt{\frac{1}{M}}+t}{1-\sqrt{\frac{1}{M}}-t}\\
&\leq \frac{2\epsilon}{1-\frac{1}{4}\frac{|||A|||\,||x_0||}{||Ax_0||}}
\\
&\leq \frac{2\epsilon}{1-\frac{1}{4}\frac{1+\sqrt{\frac{1}{M}}+t}{1-\sqrt{\frac{1}{M}}-t}}\\
&\leq 4\epsilon\\
&<\frac{\gamma}{6},\end{aligned}$$ and $$\begin{aligned}
\left(12 \eta
+\frac{24}{25} \right) &\frac{\epsilon^x}{1-\epsilon^x\frac{|||A|||\,||x_0||}{||Ax_0||}}|||A^\dag|||\, |||A|||\,||x_0||\\
&\leq \left(12 \eta
+\frac{24}{25} \right) \frac{\epsilon^x}{1-2\epsilon}|||A^\dag|||\, |||A|||\,||x_0||\\
&\leq \left(12 \eta
+\frac{24}{25} \right) \frac{\epsilon^x}{1-2\epsilon}
\frac{1+\sqrt{\frac{1}{M}}+t}{1-\sqrt{\frac{1}{M}}-t}||x_0||\\
&\leq \delta \epsilon^x||x_0||.\end{aligned}$$
We now combine this with Equation : with probability at least $$1-2C_1\exp(-C_2m)-2\exp\left(-mt^2\right),$$ we have, for all $x$ such that $\inf_{\phi\in\R}||e^{i\phi}x_0-x||\leq\epsilon||x_0||$, $$||\phase(\lambda_x)x_0-A^\dag(b\odot \phase(Ax))||
\leq \delta\epsilon^x||x_0||=\delta \inf_{\phi\in\R}||e^{i\phi}x_0-x||.$$
Global convergence
------------------
In the last paragraph, we have seen that the alternating projections operator is contractive, with high probability, in an $\epsilon ||x_0||$-neighborhood of the solution $x_0$. This implies that, if the starting point of alternating projections is at distance at most $\epsilon||x_0||$ from $x_0$, alternating projections converge to $x_0$. So if we have a way to find such an initial point, we obtain a globally convergent algorithm.
Several initialization methods have been proposed that achieve the precision we need with an optimal number of measurements, that is $m=O(n)$. Let us mention the truncated spectral initialization by @candes_wirtinger2 (improving upon the slightly suboptimal spectral initializations introduced by @netrapalli and @candes_wirtinger), the null initialization by @chen_fannjiang and the method described by @gao_xu. All these methods consist in computing the largest or smallest eigenvector of $$\sum_{i=1}^m\alpha_i a_ia_i^*,$$ where the $\alpha_1,\dots,\alpha_m$ are carefully chosen coefficients, that depend only on $b$.
The method of [@candes_wirtinger2], for example, has the following guarantees.
\[thm:guarantee\_init\] Let $\epsilon>0$ be fixed.
We define $z$ as the main eigenvector of $$\frac{1}{m}\sum_{i=1}^m|a_i^*x_0|^2 a_ia_i^* 1_{|a_i^*x_0|^2\leq \frac{9}{m}\sum_{j=1}^m|a^*_ix_0|^2}.\label{eq:init_matrix}$$ There exist $C_1,C_2,M>0$ such that, with probability at least $$1-C_1\exp(-C_2 m),$$ the vector $z$ obeys $$\inf_{\phi\in\R,\lambda\in\R^*_+}||e^{i\phi}x_0-\lambda z||\leq \epsilon ||x_0||,$$ provided that $m\geq M n$.
Combining this initialization procedure with alternating projections, we get Algorithm \[alg:algo\_complet\]. As shown by the following corollary, it converges towards the correct solution, at a linear rate, with high probability, for $m=O(n)$.
**Initialization:** set $z_0$ to be the main eigenvector of the matrix in Equation .
\[cor:global\_convergence\] There exist $C_1,C_2,M>0,\delta\in]0;1[$ such that, with probability at least $$1-C_1\exp(-C_2m),$$ Algorithm \[alg:algo\_complet\] satisfies $$\label{eq:global_convergence}
\forall t\in\N^*,\quad\quad
\inf_{\phi\in\R}||e^{i\phi}x_0-z_t||\leq \delta^t ||x_0||,$$ provided that $m\geq Mn$.
Let us fix $\epsilon,\delta\in]0;1[$ as in Theorem \[thm:local\_convergence\]. Let us assume that the properties described in Theorems \[thm:local\_convergence\] and \[thm:guarantee\_init\] hold; it happens on an event of probability at least $$1-C_1\exp(-C_2m),$$ provided that $m\geq Mn$, for some constants $C_1,C_2,M>0$.
Let us prove that, on this event, Equation also holds.
We proceed by recursion. From Theorem \[thm:guarantee\_init\], there exist $\phi\in\R,\lambda\in\R^*_+$ such that $$||e^{i\phi}x_0-\lambda z_0||\leq\epsilon||x_0||.$$ So, from Theorem \[thm:local\_convergence\], applied to $x=\lambda z_0$, $$\begin{aligned}
\inf_{\phi\in\R}||e^{i\phi}x_0-z_1||&
=\inf_{\phi\in\R}||e^{i\phi}x_0-A^\dag(b\odot \phase(z_0))||\\
=\inf_{\phi\in\R}||e^{i\phi}x_0-A^\dag(b\odot \phase(\lambda z_0))||\\
&\leq \delta\inf_{\phi\in\R} ||e^{i\phi}x_0-\lambda z_0||\\
&\leq \epsilon \delta ||x_0||.\end{aligned}$$ This proves Equation for $t=1$.
The same reasoning can be reapplied to also prove the equation for $t=2,3,\dots$.
Complexity\[ss:complexity\]
---------------------------
Let $\eta>0$ be the relative precision that we want to achieve: $$\inf_{\phi\in\R}||e^{i\phi}x_0-z_T||\leq \eta||x_0||.$$ Let us compute the number of operations that Algorithm \[alg:algo\_complet\] requires to reach this precision.
The main eigenvector of the matrix defined in Equation can be computed - up to precision $\eta$ - in approximately $O(\log(1/\eta)+\log(n))$ power iterations. Each power iteration is essentially a matrix-vector multiplication, and thus requires $O(mn)$ operations.[^3] As a consequence, the complexity of the initialization is $$O(mn\left(\log(1/\eta)+\log(n)\right)).$$ Then, at each step of the `for` loop, the most costly operation is the multiplication by $A^\dag$. When performed with the conjugate gradient method, it requires $O(mn\log(1/\eta))$ operations. To reach a precision equal to $\eta$, we need to perform $O(\log(1/\eta))$ iterations of the loop. So the total complexity of Algorithm \[alg:algo\_complet\] is $$O(mn\left(\log^2(1/\eta)+\log(n)\right)).$$
Let us mention that, when $A$ has a special structure, there may exist fast algorithms for the multiplication by $A$ and the orthogonal projection onto $\Range(A)$. In the case of masked Fourier measurements considered in [@candes_li2], for example, assuming that our convergence theorem still holds, despite the non-Gaussianity of the measurements, the complexity of each of these operations reduces to $O(m\log n)$, yielding a global complexity of $$O(m\log(n)(\log(1/\eta)+\log(n))).$$ The complexity is then almost linear in the number of measurements.
Alternating projections Truncated Wirtinger flow
------------------- ----------------------------------------------------------- -----------------------------------------------------------
Unstructured case $O\left(mn\left(\log^2(1/\eta)+\log(n)\right)\right)$ $O\left(mn\left(\log(1/\eta)+\log(n)\right)\right)$
Fourier masks $O\left(m\log(n)\left(\log(1/\eta)+\log(n)\right)\right)$ $O\left(m\log(n)\left(\log(1/\eta)+\log(n)\right)\right)$
As a comparison, Truncated Wirtinger flow, which is currently the most efficient known method for phase retrieval from Gaussian measurements, has an identical complexity, up to a $\log(1/\eta)$ factor in the unstructured case (see Figure \[fig:complexity\]).
Alternating projections without good initialization\[s:without\_init\]
======================================================================
Main result
-----------
In this section, we assume that the number of measurements is quadratic in $n$ instead of linear (that is $m\geq Mn^2$, for $M$ large enough). In this setting, we show that any initialization vector $x$, unless it is almost orthogonal to the ground truth $x_0$, yields perfect recovery when provided to the alternating projection routine. This in particular proves that, in this regime, there is no stagnation point (unless possibly among the vectors almost orthogonal to $x_0$).
The convergence rate is almost as good as in the case where a good initialization is provided: after $O(\log n)$ iterations, it becomes linear.
We say that a vector $x\in\C^n$ is *not almost orthogonal* to $x_0$ if $$\mu \frac{||x_0||\,||x||}{\sqrt{n}}\leq |\scal{x_0}{x}|,$$ for some fixed constant $\mu>0$. In what follows, we assume $\mu=1$, but it is only to simplify the notations; the same result would hold for any value of $\mu$.
We remark that, in the unit sphere, the proportion (in terms of volume) of vectors that are almost orthogonal to $x_0$ goes to a constant depending on $\mu$ when $n$ goes to $+\infty$. This constant can be arbitrarily small if $\mu$ is small. As a consequence, if we choose $x\in\C^n$ according to an isotropic probability law, the probability that it is almost orthogonal to $x_0$ can be arbitrarily small.
To prove global convergence, we first need to understand what happens when we apply one iteration of the alternating projections routine to some vector $x$. We only consider vectors $x$ that are not almost orthogonal to $x_0$. We also do not consider vectors that are very close to $x_0$: these vectors are already taken care of by Theorem \[thm:local\_convergence\].
\[thm:global\_convergence\] For any $\epsilon>0$, there exist $C_1,C_2,M,\delta>0$ such that, if $m\geq Mn^2$, then, with probability at least $$1-C_1\exp(-C_2m^{1/8}),$$ the following property holds: for any $x\in\C^n$ such that $$\label{eq:global_cond}
\frac{||x_0||\,||x||}{\sqrt{n}}\leq
|\scal{x_0}{x}|\leq (1-\epsilon)||x_0||\,||x||,$$ we have $$\label{eq:global_prop}
\frac{|\scal{x_0}{A^\dag(b\odot\phase(Ax))}|}{||x_0||\,||A^\dag(b\odot\phase(Ax))||} \geq
(1+\delta)\frac{|\scal{x_0}{x}|}{||x_0||\,||x||}.$$
Before proving this theorem, let us establish its main consequence : the global convergence of alternating projections starting from any initial point that is not almost orthogonal to $x_0$. The algorithm is summarized in Algorithm \[alg:algo\_without\_init\] and global convergence is proven in Corollary \[cor:global\_convergence\_without\].
**Initialization:** set $z_0=x$.
\[cor:global\_convergence\_without\] There exist $C_1,C_2,\gamma,M>0,\Delta\in]0;1[$ such that, with probability at least $$1-C_1\exp(-C_2n),$$ Algorithm \[alg:algo\_without\_init\] satisfies: $$\label{eq:conv_rate}
\forall t\geq \gamma\log n,\quad\quad
\inf_{\phi\in\R}||e^{i\phi}x_0-z_t|| \leq \Delta^{t-\gamma\log n} ||x_0||,$$ provided that $m\geq Mn^2$.
From Theorem \[thm:local\_convergence\], there exist $C_1^{(1)},C_2^{(1)},\epsilon^{(1)},M^{(1)}>0$ such that, if $m\geq M^{(1)}n$, then, with probability at least $$1-C_1^{(1)}\exp(-C_2^{(1)}m),$$ the following property holds: any $z\in\C^n$ such that $\inf_{\phi\in\R}||e^{i\phi}x_0-z||\leq\epsilon^{(1)}||x_0||$ satisfies $$\inf_{\phi\in\R}||e^{i\phi}x_0-A^\dag(b\odot\phase(Az))||\leq\delta^{(1)}\inf_{\phi\in\R}||e^{i\phi}x_0-z||,\label{eq:local_interm}$$ for some absolute constant $\delta^{(1)}\in]0;1[$. In the following, we assume that this event is realized.
We now use Theorem \[thm:global\_convergence\], for $\epsilon={\epsilon^{(1)}}^2/2$. Let $C_1,C_2,M,\delta>0$ be defined as in this theorem. We assume that the event described in the theorem is realized, which happens with probability at least $1-C_1\exp(-C_2n)$.
We consider the sequence $(z_t)_{t\geq 0}$ defined in Algorithm \[alg:algo\_without\_init\], and distinguish two cases.
First, if the initial point $z_0=x$ is such that $$|\scal{x_0}{x}|>(1-\epsilon)||x_0||\,||x||,$$ then, setting $x'=\frac{||x_0||}{||x||}x$, $$\begin{aligned}
\inf_{\phi\in\R}||e^{i\phi}x_0-x'||&=\sqrt{||x_0||^2+||x'||^2-2|\scal{x_0}{x'}|}\\
&<||x_0||\sqrt{2\epsilon}\\
&=\epsilon^{(1)}||x_0||.\end{aligned}$$ We can thus proceed by recursion, as in the proof of Corollary \[cor:global\_convergence\], to show that: $$\label{eq:local_combined_interm}
\forall t\in\N^*,\quad\quad
\inf_{\phi\in\R}||e^{i\phi}x_0-z_t||\leq (\delta^{(1)})^t\epsilon^{(1)}||x_0||.$$ So Equation is satisfied, provided that we have chosen $\Delta\geq \delta^{(1)}$.
Second, we consider the case where the initial point $z_0=x$ is such that $$|\scal{x_0}{x}|\leq (1-\epsilon)||x_0||\,||x||.$$ Let then $\mathcal{T}$ be the smallest index $t$ such that the following inequality is not satisfied: $$\frac{||x_0||\,||z_t||}{\sqrt{n}}\leq |\scal{x_0}{z_t}|\leq
(1-\epsilon) ||x_0||\,||z_t||.\label{eq:def_mathcal_T}$$ As $z_0=x$ is not almost orthogonal to $x_0$, we must have $\mathcal{T}\geq 1$. For any $t=0,\dots,\mathcal{T}-1$, Equation of Theorem \[thm:global\_convergence\] ensures that $$\label{eq:scal_growth}
\frac{|\scal{x_0}{z_{t+1}}|}{||x_0||\,||z_{t+1}||}
\geq (1+\delta) \frac{|\scal{x_0}{z_{t}}|}{||x_0||\,||z_{t}||}.$$ In particular, $$\frac{|\scal{x_0}{z_{\mathcal{T}}}|}{||x_0||\,||z_{\mathcal{T}}||}
\geq \frac{|\scal{x_0}{z_{0}}|}{||x_0||\,||z_{0}||}\geq \frac{1}{\sqrt{n}}.$$ As Equation is not satisfied, it means that $$|\scal{x_0}{z_{\mathcal{T}}}|>(1-\epsilon)||x_0||\,||z_{\mathcal{T}}||.$$ We can now apply the same reasoning as the one that led to Equation , and get $$\forall t\geq \mathcal{T}+1,\quad\quad
\inf_{\phi\in\R}||e^{i\phi}x_0-z_t||\leq (\delta^{(1)})^{t-\mathcal{T}}\epsilon^{(1)}||x_0||.$$ This implies Equation , provided that $\mathcal{T}\leq \gamma\log n$ for some absolute constant $\gamma$. From Equation and the fact that $z_0$ is not almost orthogonal to $x_0$, $$\frac{|\scal{x_0}{z_{\mathcal{T}-1}}|}{||x_0||\,||z_{\mathcal{T}-1}||}
\geq \frac{(1+\delta)^{\mathcal{T}-1}}{\sqrt{n}}.$$ As $\mathcal{T}-1$ satisfies Equation , we must have $$\begin{gathered}
\frac{(1+\delta)^{\mathcal{T}-1}}{\sqrt{n}}\leq 1-\epsilon\leq 1;\\
\Rightarrow\quad\quad
\mathcal{T}\leq 1+\frac{\log n }{2 \log(1+\delta)}.\end{gathered}$$ And this expression can be bounded by $\gamma \log n$, for some $\gamma>0$ independent from $n$.
So we have shown that Equation holds when the events described in Theorems \[thm:local\_convergence\] and \[thm:global\_convergence\] happen. When $m\geq \max(M,M^{(1)})n^2$, this occurs with probability at least $$1-C_1^{(1)}\exp(-C_2^{(1)}m)-C_1\exp(-C_2n)
\geq 1 - (C_1+C_1^{(1)})\exp(-\min(C_2^{(1)},C_2)n).$$
Proof of Theorem \[thm:global\_convergence\]
--------------------------------------------
We will actually consider a variant of Equation , in the “image domain”, that is in $\C^m$ instead of $\C^n$. This variant is easier to analyze and, according to the following lemma (proven in Paragraph \[ss:global\_intro\]), it implies Equation .
\[lem:global\_intro\] To prove Theorem \[thm:global\_convergence\], it is enough to prove that there exist $C_1,C_2,M,\delta>0$ such that, if $m\geq Mn^2$, then, with probability at least $1-C_1\exp(-C_2m^{1/8})$, the property $$|\scal{Ax_0}{b\odot\phase(Ax)}| \geq (1+\delta) m \frac{||x_0||}{||x||}|\scal{x_0}{x}|
\label{eq:scal_augmente}$$ holds for any $x\in\C^n$ verifying Condition .
The proof of Equation is in two parts. We first prove (Lemma \[lem:net\]) that this equation holds (with high probability) for all $x$ belonging to a net with very small spacing. This part is the most technical: a direct union bound, that does not take advantage of the correlation between the vectors of the net, is not sufficient. We use a chaining argument instead. The detailed proof is in Paragraph \[ss:net\].
In a second part (Lemma \[lem:inside\_net\]), we prove that, with high probability, for any $x$ and $y$ very close, $|\scal{Ax_0}{b\odot\phase(Ax)}-\scal{Ax_0}{b\odot\phase(y)}|$ is small. This allows us to extend the inequality proven for vectors of the net to all vectors. This result is a consequence of two facts: first, the phase is a Lipschitz function outside any neighborhood of zero. Second, with high probability, for any $x$ and $y$, the vectors $Ax$ and $Ay$ have few entries that are close to zero. The detailed proof is in Paragraph \[ss:inside\_net\].
\[lem:net\] For any $n\in\N^*$, we set $$\mathcal{E}_n=\left\{x\in\C^n, ||x||=1\mbox{ and }\frac{||x_0||\,||x||}{\sqrt{n}}\leq
|\scal{x_0}{x}|\leq (1-\epsilon)||x_0||\,||x|| \right\}.$$ Let $\alpha$ be any positive number.
There exist $c,C_1,C_2,M,\delta>0$ and, for any $n\in\N^*$, a $c m^{-\alpha}$-net $\mathcal{N}_n$ of $\mathcal{E}_n$ such that, when $m\geq Mn^2$, with probability at least $$1-C_1\exp(-C_2m^{1/2}),$$ the following property holds: for any $x\in\mathcal{N}_n$, $$|\scal{Ax_0}{b\odot\phase(Ax)}| \geq (1+\delta) m\frac{||x_0||}{||x||}|\scal{x_0}{x}|.$$
\[lem:inside\_net\] For any $c>0$, there exist $C_1,C_2,C_3>0$ such that, with probability at least $$1-C_1\exp(-C_2 m^{1/8}),$$ the following property holds for any unit-normed $x,y\in\C^n$, when $m\geq 2n^2$: $$|\scal{Ax_0}{b\odot\phase(Ax)}-\scal{Ax_0}{b\odot\phase(Ay)}|
\leq C_3||x_0||^2 n m^{1/4}
\quad\mbox{if }||x-y||\leq cm^{-7/2}.$$
To conclude, we apply Lemma \[lem:net\] with $\alpha=7/2$. We define $c,C_1,C_2,M,\delta>0$, the set $\mathcal{E}_n$ and the $cm^{-7/2}$-net $\mathcal{N}_n$ as in the statement of this lemma. With probability at least $$1-C_1\exp(-C_2m^{1/2})-C_1\exp(-C_2m^{1/8}),$$ the events described in both Lemmas \[lem:net\] and \[lem:inside\_net\] happen. In this case, for any $x\in\C^n$ verifying Condition , the normalized vector $x'=x/||x||$ belongs to $\mathcal{E}_n$. As $\mathcal{N}_n$ is a $cm^{-7/2}$-net of $\mathcal{E}_n$, there exists $y\in\mathcal{N}_n$ such that $$||x'-y|| \leq cm^{-7/2}.$$ By triangular inequality, and using Lemmas \[lem:net\] and \[lem:inside\_net\], $$\begin{aligned}
\left|\scal{Ax_0}{b\odot\phase(Ax')}\right|
&\geq \left|\scal{Ax_0}{b\odot\phase(Ay)}\right|\\
&\hskip 1cm
- \left|\scal{Ax_0}{b\odot\phase(Ay)}-\scal{Ax_0}{b\odot\phase(Ax')}\right|\\
&\geq (1+\delta)m \frac{||x_0||}{||y||}|\scal{x_0}{y}|
- C_3 ||x_0||^2 nm^{1/4}\\
&= (1+\delta)m ||x_0|| \,|\scal{x_0}{y}|
- C_3 ||x_0||^2 n m^{1/4}\\
&\geq (1+\delta)m ||x_0|| \,|\scal{x_0}{x'}|
- (1+\delta)m ||x_0||^2 ||x'-y||
- C_3 ||x_0||^2 n m^{1/4}\\
&\geq (1+\delta)m ||x_0|| \,|\scal{x_0}{x'}|
- ||x_0||^2\left((1+\delta)c m^{-5/2}
+ C_3 n m^{1/4}\right).\end{aligned}$$ As $x'$ belongs to $\mathcal{E}_n$, if $m\geq Mn^2$ and $m$ is large enough, $$\begin{aligned}
||x_0||\left((1+\delta)cm^{-5/2}+C_3n m^{1/4}\right)
&\leq ||x_0||\frac{\delta m n^{-1/2}}{2}\\
&\leq \frac{\delta m}{2}|\scal{x_0}{x'}|.\end{aligned}$$ So we deduce from this and the inequality immediately before: $$\begin{aligned}
\left|\scal{Ax_0}{b\odot\phase(Ax')}\right|
&\geq \left(1+\frac{\delta}{2}\right)m ||x_0||\,|\scal{x_0}{x'}|;\\
\Rightarrow\quad\quad
\left|\scal{Ax_0}{b\odot\phase(Ax)}\right|
&\geq \left(1+\frac{\delta}{2}\right)m \frac{||x_0||}{||x||}\,|\scal{x_0}{x}|.\end{aligned}$$ By Lemma \[lem:global\_intro\], this is what we had to prove.
Numerical experiments\[s:numerical\]
====================================
In this section, we numerically validate the results obtained in Corollaries \[cor:global\_convergence\] and \[cor:global\_convergence\_without\]. We formulate a conjecture about the convergence of alternating projections with random initialization, in the regime $m=O(n)$.
The code used to generate Figures \[fig:with\_init\], \[fig:tt\_conv\_tab\] and \[fig:without\_init\] is available at <http://www-math.mit.edu/~waldspur/code/alternating_projections_code.zip>.
Alternating projections with initialization
-------------------------------------------
Our first experiment consists in a numerical validation of Corollary \[cor:global\_convergence\]: alternating projections succeed with high probability, when they start from a good initial point, in the regime where the number of measurements is linear in the problem dimension ($m=O(n)$).
We use the initialization method described in [@candes_wirtinger2], as presented in Algorithm \[alg:algo\_complet\]. We run the algorithm for various choices of $n$ and $m$, $3000$ times for each choice. This allows us to compute an empirical probability of success, for each value of $(n,m)$.
The results are presented in Figure \[fig:with\_init\]. They confirm that, when $m=Cn$, for a sufficiently large constant $C>0$, the success probability can be arbitrarily close to $1$.
Alternating projections without good initialization
---------------------------------------------------
### Disappearing of stagnation points\[sss:disappearing\]
Next, we investigate Corollary \[cor:global\_convergence\_without\]: if $m\geq Cn^2$, for $C>0$ large enough, the method of alternating projections succeeds, with high probability, starting from any initialization (that is not almost orthogonal to the true solution). In particular, there is no stagnation point, unless possibly among vectors that are *almost orthogonal* to the true solution.
To numerically validate this result, we have generated vectors $x_0$ of size $n$ and measurements matrices $A$ of size $m\times n$ for various choices of $n$ and $m$. For each $(x_0,A)$, we have randomly chosen $10000$ initializations that were not almost orthogonal to $x_0$, and we have recorded whether alternating projections, starting from these initializations, always succeeded in reconstructing $x_0$ from $|Ax_0|$. When at least one of these initializations failed, it proved that there was at least one stagnation point. Otherwise, we have considered it as a sign of absence of stagnation points.
We could thus compute, for each choice of $(n,m)$, the probability of absence of stagnation point. The result is displayed on Figure \[fig:tt\_conv\_tab\]. As foreseen by Corollary \[cor:global\_convergence\_without\], the probability becomes arbitrarily close to $1$ when $m\geq Cn^2$ for $C>0$ large enough.
The same results are presented in Figure \[fig:no\_stagnation\] under a different form. The graph on the left hand side shows, for each $n$, the number $M_n$ of measurements above which the probability that there is at least one stagnation point drops under $0.5$. The curve has a clear quadratic shape.
The plot on the right hand side represents $M_n/n^2$ as a function of $n$. It is clearly upper bounded by a constant. It also seems to be lower bounded by a positive constant (or possibly by a very slowly decaying function, like $(\log\log)^{-1}$), which indicates that the number of measurements $m=O(n^2)$ that appears in Corollary \[cor:global\_convergence\_without\] is probably optimal: when $m\ll n^2$, the probability that there are no stagnation points is small.
### Random initialization
Our last experiment consists in measuring the probability that alternating projections succeed, when started from a random initial point (sampled from the unit sphere with uniform probability).
The results are presented in Figure \[fig:without\_init\]. They lead to the following conjecture.
\[conj:convergence\_random\] Let any $\epsilon>0$ be fixed. When $m\geq Cn$, for $C>0$ large enough, alternating projections with a random isotropic initialization succeed with probability at least $1-\epsilon$.
As we have seen in Paragraph \[sss:disappearing\], in the regime $m=O(n)$, there are (attractive) stagnation points, so there are initializations for which alternating projections fail. However, it seems that these bad initializations occupy a very small volume in the space of all possible initial points. Therefore, a random initialization leads to success with high probability.
Unfortunately, proving this conjecture a priori requires to evaluate in some way the size of the attraction basin of stagnation points, which seems difficult.
Proposition \[prop:stagnation\_points\]\[s:stagnation\_points\]
===============================================================
For any $y_0$, the sequence $(y_k)_{k\in\N}$ is bounded. Any accumulation point $y_\infty$ of $(y_k)_{k\in\N}$ satisfies the following property: $$\exists u\in E_{\phase}(y_\infty),\quad\quad
(AA^\dag)(b\odot u)=y_\infty.$$ In particular, if $y_\infty$ has no zero entry, $$(AA^\dag)(b\odot \phase(y_\infty))=y_\infty.$$
The boundedness of $(y_k)_{k\in\N}$ is a consequence of the fact that $||y'_k||=||b||$ for all $k$, so $||y_{k+1}||\leq |||AA^\dag|||\,||b||$.
Let us show the second part of the statement. Let $y_\infty$ be an accumulation point of $(y_k)_{k\in\N}$, and let $\phi:\N\to\N$ be an extraction such that $$y_{\phi(n)}\to y_\infty\quad\mbox{when}\quad n\to+\infty.$$ By compacity, as $(y'_{\phi(n)})_{n\in\N}$ and $(y_{\phi(n)+1})_{n\in\N}$ are bounded sequences, we can assume, even if we have to consider replace $\phi$ by a subextraction, that they also converge. We denote by $y'_\infty$ and $y_\infty^{+1}$ their limits: $$y'_{\phi(n)}\to y'_{\infty}\quad\mbox{and}\quad y_{\phi(n)+1}\to y_\infty^{+1}\quad\mbox{when }n\to+\infty.$$
Let us define $$E_b=\{y'\in\C^m,|y'|=b\}.$$ We observe that, for any $k$, $$d(y'_{k-1},\Range(A))\geq d(y_{k},E_b)\geq d(y'_k,\Range(A)).$$ Indeed, because the operators $y\to b\odot\phase(y)$ and $y\to(AA^\dag)y$ are projections, $$\begin{array}{rcl}
d(y'_{k-1},\Range(A))=&d(y'_{k-1},y_k)&\geq d(y_k,E_b);\\
d(y_k,E_b)=&d(y_k,y'_k)&\geq d(y'_k,\Range(A)).
\end{array}$$ So the sequences $(d(y_k,E_b))_{k\in\N}$ and $(d(y'_k,\Range(A)))_{k\in\N}$ converge to the same non-negative limit, that we denote by $\delta$. In particular, $$\begin{aligned}
d(y_\infty,E_b)=\delta=d(y'_\infty,\Range(A)).\end{aligned}$$ If we pass to the limit the equalities $d(y_{\phi(n)},E_b)=||y_{\phi(n)}-y'_{\phi(n)}||$ and $d(y'_{\phi(n)},\Range(A))=||y'_{\phi(n)}-y_{\phi(n)+1}||$, we get $$||y_\infty-y'_\infty||=||y'_\infty-y_\infty^{+1}||=\delta=d(y'_\infty,\Range(A)).$$ As $\Range(A)$ is convex, the projection of $y'_\infty$ onto it is uniquely defined. This implies $$y_\infty=y_\infty^{+1},$$ and, because $\forall n,y_{\phi(n)+1}=(AA^\dag) y'_{\phi(n)}$, $$y_\infty=y_\infty^{+1}=(AA^\dag)y'_\infty.$$ To conclude, we now have to show that $y'_\infty=b\odot u$ for some $u\in E_{\phase}(y_\infty)$. We use the fact that, for all $n$, $y'_{\phi(n)}=b\odot \phase(y_{\phi(n)})$.
For any $i\in\{1,\dots,m\}$, if $(y_\infty)_i\ne 0$, $\phase$ is continuous around $(y_\infty)_i$, so $(y'_{\infty})_i= b_i\phase((y_\infty)_i)$. We then set $u_i=\phase((y_\infty)_i)$, and we have $(y'_\infty)_i=b_iu_i$.
If $(y_\infty)_i=0$, we set $u_i=\phase((y'_\infty)_i)\in E_{\phase}(0)=E_{\phase}((y_\infty)_i)$. We then have $y'_\infty=|y'_\infty|u_i=b_iu_i$.
With this definition of $u$, we have, as claimed, $y'_\infty=b\odot u$ and $u\in E_{\phase}(y_\infty)$.
Technical lemmas for Section \[s:with\_init\]
=============================================
Proof of Lemma \[lem:diff\_phase\]\[ss:diff\_phase\]
----------------------------------------------------
For any $z_0,z\in\C$, $$|\phase(z_0+z)-\phase(z_0)| \leq 2. 1_{|z|\geq |z_0|/6} + \frac{6}{5}\left|\Im\left(\frac{z}{z_0}\right)\right|.$$
The inequality holds if $z_0=0$, so we can assume $z_0\ne 0$. We remark that, in this case, $$|\phase(z_0+z)-\phase(z_0)| = |\phase(1+z/z_0)-1|.$$ It is thus enough to prove the lemma for $z_0=1$, so we make this assumption.
When $|z|\geq 1/6$, the inequality is valid. Let us now assume that $|z|<1/6$. Let $\theta\in\left]-\frac{\pi}{2};\frac{\pi}{2}\right[$ be such that $$e^{i\theta}=\phase(1+z).$$ Then $$\begin{aligned}
|\phase(1+z)-1| &=|e^{i\theta}-1|\\
&=2|\sin(\theta/2)|\\
%&=\frac{|\sin(\theta)|}{|\cos(\theta)|}\frac{|\cos(\theta)|}{|\cos(\theta/2)|}\\
&\leq |\tan\theta|\\
&=\frac{|\Im(1+z)|}{|\Re(1+z)|}\\
&\leq \frac{|\Im(z)|}{1-|z|}\\
&\leq \frac{6}{5}|\Im(z)|.\end{aligned}$$ So the inequality is also valid.
Proof of Lemma \[lem:first\_term\]\[ss:first\_term\]
----------------------------------------------------
For any $\eta>0$, there exists $C_1,C_2,M,\gamma>0$ such that the inequality $$||\,|Ax_0|\odot 1_{|v|\geq |Ax_0|}||\leq \eta ||v||$$ holds for any $v\in\Range(A)$ such that $||v||<\gamma ||Ax_0||$, with probability at least $$1-C_1\exp(-C_2m),$$ when $m\geq Mn$.
For any $S\subset\{1,\dots,m\}$, we denote by $1_S$ the vector of $\C^m$ such that $$\begin{aligned}
(1_S)_j&=1\mbox{ if }j\in S\\
&=0\mbox{ if }j\notin S.\end{aligned}$$ We use the following two lemmas, proven in Paragraphs \[sss:S\_geq\_bm\] and \[sss:S\_leq\_bm\].
\[lem:S\_geq\_bm\] Let $\beta\in]0;1/2[$ be fixed. There exist $C_1>0$ such that, with probability at least $$1-C_1\exp(-\beta^3m/e),$$ the following property holds: for any $S\subset\{1,\dots,m\}$ such that $\Card(S)\geq\beta m$, $$\label{eq:Ax01S}
||\,|Ax_0|\odot 1_S|| \geq \beta^{3/2}e^{-1/2}||Ax_0||.$$
\[lem:S\_leq\_bm\] Let $\beta\in\left]0;\frac{1}{100}\right]$ be fixed. There exist $M,C_1,C_2>0$ such that, if $m\geq M n$, then, with probability at least $$1-C_1\exp(-C_2 m),$$ the following property holds: for any $S\subset\{1,\dots,m\}$ such that $\Card (S)<\beta m$ and for any $y\in\Range(A)$, $$\label{eq:y1S}
||y\odot 1_S||\leq 10\sqrt{\beta\log(1/\beta)}||y||.$$
Let $\beta>0$ be such that $10\sqrt{\beta\log(1/\beta)}\leq \eta$. Let $M$ be as in Lemma \[lem:S\_leq\_bm\]. We set $$\gamma = \beta^{3/2}e^{-1/2}.$$ We assume that Equations and hold; from the lemmas, this occurs with probability at least $$1-C_1'\exp(-C_2'm),$$ for some constants $C_1',C_2'>0$, provided that $m\geq Mn$.
On this event, for any $v\in\Range(A)$ such that $||v||<\gamma ||Ax_0||$, if we set $S_v=\{i\mbox{ s.t. }|v_i|\geq|Ax_0|_i\}$, we have that $$\Card S_v < \beta m.$$ Indeed, if it was not the case, we would have, by Equation , $$\begin{aligned}
||v||&\geq ||v\odot 1_{S_v}||\\
&\geq ||\,|Ax_0|\odot 1_{S_v}||\\
&\geq \beta^{3/2}e^{-1/2}||Ax_0||\\
&=\gamma ||Ax_0||,\end{aligned}$$ which is in contradiction with the way we have chosen $v$.
So we can apply Equation , and we get $$\begin{aligned}
||\,|Ax_0|\odot 1_{|v|\geq |Ax_0|}||
&\leq ||v\odot 1_S||\\
&\leq 10\sqrt{\beta\log(1/\beta)}||v||\\
&\leq \eta ||v||.\end{aligned}$$
### Proof of Lemma \[lem:S\_geq\_bm\]\[sss:S\_geq\_bm\]
If we choose $C_1$ large enough, it is enough to show the property for $m$ larger than some fixed constant.
We first assume $S$ fixed, with cardinality $\Card S\geq\beta m$. We use the following lemma.
\[lem:dasgupta\] Let $k_1<k_2$ be natural numbers. Let $X\in\C^{k_2}$ be a random vector whose coordinates are independent, Gaussian, of variance $1$. Let $Y$ be the projection of $X$ onto its $k_1$ first coordinates. Then, for any $t>0$, $$\begin{aligned}
\mbox{\rm Proba}\left(\frac{||Y||}{||X||}\leq \sqrt{\frac{t k_1}{k_2}}\right)
&\leq \exp\left(k_1(1-t+\log t)\right)&\mbox{if }t<1;\\
\mbox{\rm Proba}\left(\frac{||Y||}{||X||}\geq \sqrt{\frac{t k_1}{k_2}}\right)
&\leq \exp\left(k_1(1-t+\log t)\right)&\mbox{if }t>1.\end{aligned}$$
From this lemma, for any $t\in]0;1[$, because $Ax_0$ has independent Gaussian coordinates, $$\begin{aligned}
P\left(\frac{||\,|Ax_0|\odot 1_S||}{||Ax_0||}\leq\sqrt{t\beta} \right)
\leq \exp\left(-\beta m(t-1-\ln t)\right).\end{aligned}$$ In particular, for $t=\frac{\beta^2}{e}$, $$P\left(\frac{||\,|Ax_0|\odot 1_S||}{||Ax_0||}\leq \beta^{3/2}e^{-1/2} \right)
\leq \exp\left(-\beta m\left(\frac{\beta^2}{e}-2\ln \beta\right)\right).
\label{eq:P_Ax01S}$$ As soon as $m$ is large enough, the number of subsets $S$ of $\{1,\dots,m\}$ with cardinality $\lceil \beta m\rceil$ satisfies $$\begin{aligned}
\binom{m}{\lceil\beta m\rceil}
&\leq\left(\frac{em}{\lceil \beta m\rceil}\right)^{\lceil \beta m\rceil} \nonumber\\
&\leq\exp\left(2m\beta\log\frac{1}{\beta}\right).\label{eq:maj_binom}\end{aligned}$$ (The first inequality is a classical result regarding binomial coefficients.)
We combine Equations and : Property is satisfied for any $S$ of cardinality $\lceil\beta m\rceil$ with probability at least $$1-\exp\left(-\frac{\beta^3}{e}m\right),$$ provided that $m$ is larger that some constant which depends on $\beta$.
If it is satisfied for any $S$ of cardinality $\lceil\beta m\rceil$, then it is satisfied for any $S$ of cardinality larger than $\beta m$, which implies the result.
### Proof of Lemma \[lem:S\_leq\_bm\]\[sss:S\_leq\_bm\]
We first assume $S$ to be fixed, of cardinality exactly $\lceil\beta m\rceil$.
Any vector $y\in\Range(A)$ is of the form $y=Av$, for some $v\in\C^n$. Inequality can then be rewritten as: $$\label{eq:S_leq_bm_rewritten}
||A_S v||=||\mathrm{Diag}(1_S)Av||\leq 10\sqrt{\beta\log(1/\beta)}||Av||,$$ where $A_S$, by definition, is the submatrix obtained from $A$ by extracting the rows whose indexes are in $S$.
We apply Proposition \[prop:davidson\] to $A$ and $A_S$, respectively for $t=\frac{1}{2}$ and $t=3\sqrt{\log(1/\beta)}$. It guarantees that the following properties hold: $$\begin{gathered}
\inf_{v\in\C^n}\frac{||Av||}{||v||}\geq \sqrt{m}\left(\frac{1}{2}-\sqrt{\frac{n}{m}}\right);\\
\sup_{v\in\C^n}\frac{||A_Sv||}{||v||}\leq \sqrt{\Card S}\left(1+\sqrt\frac{n}{\Card S}+3\sqrt{\log(1/\beta)}\right),\end{gathered}$$ with respective probabilities at least $$\begin{gathered}
1-2\exp\left(-\frac{m}{4}\right);\\
\mbox{and }1-2\exp\left(-9(\Card S)\log(1/\beta)\right)\geq 1-2\exp\left(-9\beta\log(1/\beta)m\right).\end{gathered}$$ Assuming $m\geq Mn$ for some $M>0$, we deduce from these inequalities that $$\begin{aligned}
\forall v\in\C^n,\quad\quad
||A_Sv||&\leq \sqrt\frac{\Card S}{m}\left(\frac{1+\sqrt\frac{n}{\Card S}+3\sqrt{\log (1/\beta)}}{\frac{1}{2}-\sqrt{\frac{n}{m}}}\right)||Av||\nonumber\\
&\leq \sqrt{\beta+\frac{1}{m}}\left(\frac{1+\sqrt{\frac{1}{\beta M}}+3\sqrt{\log(1/\beta)}}{\frac{1}{2}-\sqrt{\frac{1}{M}}}\right)||Av||,\label{eq:tmp}\end{aligned}$$ with probability at least $$1-2\exp\left(-9\beta\log(1/\beta)m\right)-2\exp\left(-\frac{m}{4}\right).$$ If we choose $M$ large enough, we can upper bound Equation by $(\epsilon+2\sqrt{\beta}(1+3\sqrt{\log(1/\beta)}))||Av||\leq (\epsilon+8\sqrt{\beta}\sqrt{\log(1/\beta)})$ for any fixed $\epsilon>0$. So this inequality implies Equation .
As in the proof of Lemma \[lem:S\_geq\_bm\], there are at most $$\exp\left(2m\beta\log\frac{1}{\beta}\right)$$ subsets of $\{1,\dots,m\}$ with cardinality $\lceil\beta m\rceil$, as soon as $m$ is large enough. As a consequence, Equation holds for any $v\in\C^n$ and $S$ of cardinality $\lceil\beta m\rceil$ with probability at least $$1-2\exp\left(-7\beta\log(1/\beta)m\right)-2\exp\left(-\left(\frac{1}{4}-2\beta\log\frac{1}{\beta}\right)m\right).$$ When $\beta\leq \frac{1}{100}$, we have $$\frac{1}{4}-2\beta\log\frac{1}{\beta}> 0,$$ so the resulting probability is larger than $$1-C_1\exp(-C_2 m),$$ for some well-chosen constants $C_1,C_2>0$.
This ends the proof. Indeed, if Equation holds for any set of cardinality $\lceil\beta m\rceil$, it also holds for any set of cardinality $\Card S<\beta m$, because $||A_{S'} v||\leq ||A_S v||$ whenever $S'\subset S$. This implies Equation .
Proof of Lemma \[lem:second\_term\]\[ss:second\_term\]
------------------------------------------------------
For $M,C_1>0$ large enough, and $C_2>0$ small enough, when $m\geq M n$, the property $$\label{eq:second_term}
||\Im(v\odot\overline{\phase(Ax_0)})||\leq \frac{4}{5}||v||$$ holds for any $v\in\Range(A)\cap \{Ax_0\}^\perp$, with probability at least $$1-C_1\exp(-C_2 m).$$
If we multiply $x_0$ by a positive real number, we can assume $||x_0||=1$. Moreover, as the law of $A$ is invariant under right multiplication by a unitary matrix, we can assume that $$x_0=\left(\begin{smallmatrix}1\\0\\\vdots\\0\end{smallmatrix}\right).$$ Then, if we write $A_1$ the first column of $A$, and $A_{2:n}$ the submatrix of $A$ obtained by removing this first column, $$\label{eq:range_perp}
\Range(A)\cap\{Ax_0\}^\perp
=\left\{w-\frac{\scal{w}{A_1}}{||A_1||^2}A_1,
w\in\Range(A_{2:n})
\right\}.$$ We first observe that $$\sup_{w\in\Range(A_{2:n})-\{0\}}\frac{|\scal{w}{A_1}|}{||w||}$$ is the norm of the orthogonal projection of $A_1$ onto $\Range(A_{2:n})$. The $(n-1)$-dimensional subspace $\Range(A_{2:n})$ has an isotropic distribution in $\C^m$, and is independent of $A_1$. Thus, from Lemma \[lem:dasgupta\] coming from [@dasgupta], for any $t>1$, $$\sup_{w\in\Range(A_{2:n})-\{0\}}\frac{|\scal{w}{A_1}|}{||w||\,||A_1||}< \sqrt{\frac{t(n-1)}{m}},$$ with probability at least $$1-\exp\left(-(n-1)(t-1-\ln t)\right).$$ We take $t=\frac{m}{n-1}(0.04)^2$ (which is larger than $1$ when $m\geq Mn$ with $M>0$ large enough), and it implies that $$\label{eq:second_005}
\sup_{w\in\Range(A_{2:n})-\{0\}}\frac{|\scal{w}{A_1}|}{||w||\,||A_1||}< 0.04$$ with probability at least $$1-\exp(-c_2m)$$ for some constant $c_2>0$, provided that $m\geq Mn$ with $M$ large enough.
Second, as $A_{2:n}$ is a random matrix of size $m\times(n-1)$, whose entries are independent and distributed according to the law $\mathcal{N}(0,1/2)+\mathcal{N}(0,1/2)i$, we deduce from Proposition \[prop:davidson\] applied with $t=0.01$ that, with probability at least $$1-2\exp\left(-10^{-4}m\right),$$ we have, for any $x\in\C^{n-1}$, $$\label{eq:norm_C}
||A_{2:n}x|| \geq \sqrt{m}\left(1-\sqrt{\frac{(n-1)}{m}}-0.01\right)||x||\geq 0.98\sqrt{m}||x||,$$ provided that $m\geq 10000n$.
We now set $$C = \mathrm{Diag}(\overline{\phase(A_1)})A_{2:n}.$$ The matrix $\left(\begin{matrix}\Im C&\Re C\end{matrix}\right)$ has size $m\times(2(n-1))$; its entries are independent and distributed according to the law $\mathcal{N}(0,1/2)$. So by [@davidson Thm II.13] (applied with $t=0.01$), with probability at least $$1-\exp(-5.10^{-5}m),$$ we have, for any $x\in\R^{2(n-1)}$, $$\label{eq:norm_ImReC}
\left|\left|\left(\begin{matrix}\Im C&\Re C\end{matrix}\right)x
\right|\right|\leq \sqrt{\frac{m}{2}}\left(1+\sqrt\frac{2(n-1)}{m}+0.01\right)||x||
\leq 1.02\sqrt\frac{m}{2}||x||,$$ provided that $m\geq 20000n$.
When Equations and are simultaneously valid, any $w=A_{2:n}w'$ belonging to $\Range(A_{2:n})$ satisfies: $$\begin{aligned}
\left|\left|\Im(w\odot\overline{\phase(Ax_0)})\right|\right|
&=\left|\left|\Im(Cw')\right|\right|\nonumber\\
&=\left|\left|\begin{pmatrix}\Im C&\Re C\end{pmatrix}\begin{pmatrix}
\Re w'\\\Im w'\end{pmatrix}
\right|\right|\nonumber\\
&\leq 1.02\sqrt\frac{m}{2}\left|\left|\begin{pmatrix}
\Re w'\\\Im w'\end{pmatrix}
\right|\right|\nonumber\\
&=1.02\sqrt\frac{m}{2}||w'||\nonumber\\
&\leq \frac{1.02}{0.98\sqrt{2}}||A_{2:n}w'||\nonumber\\
&= \frac{1.02}{0.98\sqrt{2}}||w||\nonumber\\
&\leq 0.75 ||w||.\label{eq:norms_combined}\end{aligned}$$
We now conclude. Equations , and hold simultaneously with probability at least $$1-C_1\exp(-C_2 m)$$ for any $C_1$ large enough and $C_2$ small enough, provided that $m\geq Mn$ with $M$ large enough. Let us show that, on this event, Equation also holds. Any $v\in\Range(A)\cap\{Ax_0\}^\perp$, from Equality , can be written as $$v=w-\frac{\scal{w}{A_1}}{||A_1||^2}A_1,$$ for some $w\in\Range(A_{2:n})$. Using Equation , then Equation , we get: $$\begin{aligned}
\left|\left|\Im(v\odot\overline{\phase(Ax_0)})\right|\right|
&\leq\left|\left|\Im(w\odot\overline{\phase(Ax_0)})\right|\right|
+\left|\left|\frac{\scal{w}{A_1}}{||A_1||^2}A_1\right|\right|\\
&\leq\left|\left|\Im(w\odot\overline{\phase(Ax_0)})\right|\right|
+0.04 ||w||\\
&\leq 0.79 ||w||.\end{aligned}$$ But then, by Equation again, $$||v||^2=||w||^2-\frac{\scal{w}{A_1}^2}{||A_1||^2}\geq (1-(0.04)^2)||w||^2.$$ So $$\begin{aligned}
\left|\left|\Im(v\odot\overline{\phase(Ax_0)})\right|\right|
&\leq 0.79 ||w||\\
&\leq \frac{0.79}{\sqrt{1-(0.04)^2}}||v||\\
&\leq \frac{4}{5}||v||.\end{aligned}$$
Technical lemmas for Section \[s:without\_init\]
================================================
Proof of Lemma \[lem:global\_intro\]\[ss:global\_intro\]
--------------------------------------------------------
To prove Theorem \[thm:global\_convergence\], it is enough to prove that there exist $C_1,C_2,M,\delta>0$ such that, if $m\geq Mn^2$, then, with probability at least $1-C_1\exp(-C_2m^{1/8})$, the property $$|\scal{Ax_0}{b\odot\phase(Ax)}| \geq (1+\delta) m \frac{||x_0||}{||x||}|\scal{x_0}{x}|
\tag{\ref{eq:scal_augmente}}$$ holds for any $x\in\C^n$ verifying Condition .
Let us define $\lambda_1(A)\geq\dots\geq\lambda_n(A)$ to be the $n$ singular values of $A$. From Proposition \[prop:davidson\], setting $t=\delta'/\sqrt{n}$ for $\delta'$ small enough, if $M$ is high enough, we have with probability larger than $1-C_1\exp(-C_2m/n)\geq 1-C_1\exp(-C_2m^{1/2})$, $$\begin{gathered}
\frac{\lambda_1^2(A)}{\lambda_n^2(A)}-1\leq \frac{\delta}{3}\frac{1}{\sqrt{n}},\\
\mbox{and }\lambda_1(A)\lambda_n(A)\leq \left(\frac{1+\delta}{1+2\delta/3}\right)m,\end{gathered}$$ when $m\geq Mn^2$.
In this case, we have in particular, for any $x$ satisfying Equation , $$\frac{\lambda_1^2(A)}{\lambda_n^2(A)}-1
\leq \frac{\delta}{3}\frac{|\scal{x_0}{x}|}{||x_0||\,||x||}.$$ For any $x$, $$\begin{aligned}
|\scal{Ax_0}{b\odot\phase(Ax)}|
&=|\scal{Ax_0}{(AA^\dag)(b\odot\phase(Ax))}|\\
&=|\scal{(A^*A)x_0}{A^\dag(b\odot\phase(Ax))}|\\
&\leq \lambda_n^2(A)|\scal{x_0}{A^\dag(b\odot\phase(Ax))}|\\
&\hskip 2cm+ |\scal{(A^*A-\lambda_n^2(A)\Id)x_0}{A^\dag(b\odot\phase(Ax))}|\\
&\leq \lambda_n^2(A)|\scal{x_0}{A^\dag(b\odot\phase(Ax))}|\\
&\quad\quad+(\lambda_1^2(A)-\lambda_n^2(A))||x_0||\,||A^\dag(b\odot\phase(Ax))||.\end{aligned}$$ So when $x$ satisfies Equations and , $$\begin{aligned}
\frac{|\scal{x_0}{A^\dag(b\odot\phase(Ax))}|}{||x_0||\,||A^\dag(b\odot\phase(Ax))||}
&\geq\frac{1}{\lambda_n^2(A)}\frac{|\scal{Ax_0}{b\odot\phase(Ax)}|}{||x_0||\,||A^\dag(b\odot\phase(Ax))||}
- \left(\frac{\lambda_1^2(A)}{\lambda_n^2(A)}-1\right)\\
&\geq\frac{1}{\lambda_n^2(A)}\frac{|\scal{Ax_0}{b\odot\phase(Ax)}|}{||x_0||\,||A^\dag(b\odot\phase(Ax))||}
- \frac{\delta}{3}\frac{|\scal{x_0}{x}|}{||x_0||\,||x||}\\
&\geq(1+\delta)\frac{m}{\lambda_n^2(A)}\frac{|\scal{x_0}{x}|}{||x||\,||A^\dag(b\odot\phase(Ax))||}
- \frac{\delta}{3}\frac{|\scal{x_0}{x}|}{||x_0||\,||x||}\\
&\geq(1+\delta)\frac{m}{\lambda_n(A)}\frac{|\scal{x_0}{x}|}{||x||\,||b||}
- \frac{\delta}{3}\frac{|\scal{x_0}{x}|}{||x_0||\,||x||}\\
&=(1+\delta)\frac{m}{\lambda_n(A)}\frac{|\scal{x_0}{x}|}{||x||\,||Ax_0||}
- \frac{\delta}{3}\frac{|\scal{x_0}{x}|}{||x_0||\,||x||}\\
&\geq (1+\delta)\frac{m}{\lambda_1(A)\lambda_n(A)}\frac{|\scal{x_0}{x}|}{||x||\,||x_0||}
- \frac{\delta}{3}\frac{|\scal{x_0}{x}|}{||x_0||\,||x||}\\
&\geq \left(1+\frac{\delta}{3}\right)\frac{|\scal{x_0}{x}|}{||x_0||\,||x||}.\end{aligned}$$ So Equation is also satisfied (although for a smaller value of $\delta$).
Proof of Lemma \[lem:net\]\[ss:net\]
------------------------------------
For any $n\in\N^*$, we set $$\mathcal{E}_n=\left\{x\in\C^n, ||x||=1\mbox{ and }\frac{||x_0||\,||x||}{\sqrt{n}}\leq
|\scal{x_0}{x}|\leq (1-\epsilon)||x_0||\,||x|| \right\}.$$ Let $\alpha$ be any positive number.
There exist $c,C_1,C_2,M,\delta>0$ and, for any $n\in\N^*$, a $c m^{-\alpha}$-net $\mathcal{N}_n$ of $\mathcal{E}_n$ such that, when $m\geq Mn^2$, with probability at least $$1-C_1\exp(-C_2m^{1/2}),$$ the following property holds: for any $x\in\mathcal{N}_n$, $$|\scal{Ax_0}{b\odot\phase(Ax)}| \geq (1+\delta) m\frac{||x_0||}{||x||}|\scal{x_0}{x}|.$$
For any $n\in\N^*$, $k\in\N$, let $\mathcal{M}_n^k$ be a $2^{-k}$-net of $\mathcal{E}_n$. As $\mathcal{E}_n$ is a closed subset of the complex unit sphere of dimension $n$, we can construct $\mathcal{M}_n^k$ as $$\mathcal{M}_n^k=\{P_{\mathcal{E}_n}(y),y\in\mathcal{V}_n^k\},$$ where $\mathcal{V}_n^k$ is a $2^{-(k+1)}$-net of the unit sphere, and, for any $y$, $P_{\mathcal{E}_n}(y)$ is a point in $\mathcal{E}_n$ whose distance to $y$ is minimal. From [@vershynin Lemma 5.2], this implies that we can choose $\mathcal{M}_n^k$ such that $$\Card\mathcal{M}_n^k \leq \left(1+\frac{2}{2^{-(k+1)}}\right)^{2n}
\leq 2^{2n(k+3)}.\label{eq:Card_Mnk}$$
For any $x\in\C^n$, we set $$F(x)=\E\left(\scal{Ax_0}{b\odot\phase(Ax)}\right)$$ (where the expectation denotes the expectation over $A$ with $x_0$ and $x$ fixed).
The main difficulty consists in showing that $\scal{Ax_0}{b\odot\phase(Ax)}$ is close to its expectation for all $x\in\mathcal{M}_n^K$, with $K\in\N^*$ relatively large. This is what the following lemma does; it is proved in Paragraph \[sss:ecart\_net\].
\[lem:ecart\_net\] For any $\eta,\mathcal{A}>0$, there exist $c,C_1,C_2,M>0$ such that, when $m\geq Mn^2$, for any $k\in\N$ such that $k\leq \mathcal{A} \log m-c$, with probability at least $$1-C_1\exp(-C_2 m^{1/2}),$$ the following property holds: for any $x\in\mathcal{M}_n^k,y\in\mathcal{M}_n^{k+1}$ such that $||x-y||\leq 2^{-(k-1)}$, $$|\left(\scal{Ax_0}{b\odot\phase(Ax)}-F(x)\right)-
\left(\scal{Ax_0}{b\odot\phase(Ay)}-F(y)\right)| \leq \frac{\eta}{(k+1)^2}\frac{m}{\sqrt{n}} ||x_0||^2 .$$ In the case $k=0$, we additionally have, with the same probability: for all $x\in\mathcal{M}_n^0$, $$|\left(\scal{Ax_0}{b\odot\phase(Ax)}-F(x)\right)| \leq \eta\frac{m}{\sqrt{n}}
||x_0||^2.$$
Let $\eta,\mathcal{A}>0$ be temporarily fixed. We set $K=\lceil \mathcal{A}\log m-c\rceil$. The event described in the previous lemma holds for all $k\leq K-1$ with probability at least $1-K C_1\exp(-C_2 m^{1/2})$.
For any $x\in\mathcal{M}_n^K$, there exists a sequence $(y_0,y_1,\dots,y_{K-1},y_K)$ such that $$\begin{gathered}
y_K=x;\\
\forall k\leq K, y_k\in \mathcal{M}_n^k;\\
\forall k\leq K-1,||y_k-y_{k+1}||\leq 2^{-k}.\end{gathered}$$ So when the event of Lemma \[lem:ecart\_net\] holds, we have, for any $x\in\mathcal{M}_n^K$, $$\begin{aligned}
&\left|\scal{Ax_0}{b\odot\phase(Ax)}-F(x)\right|\\
&\quad\leq \left|\scal{Ax_0}{b\odot\phase(Ay_0)}-F(y_0)\right|\\
&\quad\quad + \sum_{k=0}^{K-1}\left|\left(\scal{Ax_0}{b\odot\phase(Ay_k)}-F(y_k)\right)
-\left(\scal{Ax_0}{b\odot\phase(Ay_{k+1})}-F(y_{k+1})\right)
\right|\\
&\quad\leq \eta \frac{m}{\sqrt{n}}\left(1+\sum_{k=0}^{K-1}\frac{1}{(k+1)^2}\right)||x_0||^2\\
&\quad \leq\eta\left(1+\frac{\pi^2}{6}\right)\frac{m}{\sqrt{n}}||x_0||^2.\end{aligned}$$
To conclude, we only have to evaluate $F$. This is done by the following lemma, proven in Paragraph \[sss:F\].
\[lem:F\] There exist $\delta>0$ such that, for any $x\in\mathcal{E}_n$, $$|F(x)|\geq (1+\delta)m\frac{||x_0||}{||x||}|\scal{x_0}{x}|.$$
We combine this lemma and the equation before the lemma: with probability at least $1-KC_1\exp(-C_2m^{1/2})$, for any $x\in\mathcal{M}_n^K$, $$\begin{aligned}
|\scal{Ax_0}{b\odot\phase(Ax)}|
&\geq |F(x)| - \eta\left(1+\frac{\pi^2}{6}\right)\frac{m}{\sqrt{n}}||x_0||^2\\
&\geq (1+\delta)m\frac{||x_0||}{||x||}|\scal{x_0}{x}| - \eta\left(1+\frac{\pi^2}{6}\right)\frac{m}{\sqrt{n}}||x_0||^2\\
&\geq \left(1+\delta-\eta\left(1+\frac{\pi^2}{6}\right)\right)m\frac{||x_0||}{||x||}|\scal{x_0}{x}|.\end{aligned}$$ For the last inequality, we have used the fact that $x\in\mathcal{E}_n$, so $|\scal{x_0}{x}|\geq ||x_0||\,||x||/\sqrt{n}$.
We can choose $\eta>0$ sufficiently small so that $1+\delta-\eta\left(1+\frac{\pi^2}{6}\right)>1+\frac{\delta}{2}$. We fix $\mathcal{A}$ to be any real number larger than $\alpha/\log 2$. Then, from the definition of $K$, $$2^{-K}\leq 2^{-\mathcal{A}\log m+c}= 2^c m^{-\mathcal{A}\log 2}\leq 2^cm^{-\alpha}.$$ As $K\leq \mathcal{A}\log m-c+1$, we can upper bound $1-KC_1\exp(-C_2m^{1/2})$ by $1-C_1'\exp(-C_2'm^{1/2})$, for $C'_1,C'_2>0$ well-chosen. If we summarize, we get that, with probability at least $1-C_1'\exp(-C_2'm^{1/2})$, $$\begin{aligned}
\forall x\in\mathcal{M}_n^K,\quad\quad
|\scal{Ax_0}{b\odot\phase(Ax)}|
\geq \left(1+\frac{\delta}{2}\right)m\frac{||x_0||}{||x||}|\scal{x_0}{x}|,\end{aligned}$$ and $\mathcal{M}_n^K$ is a $2^cm^{-\alpha}$-net of $\mathcal{E}_n$. The lemma is proved.
### Proof of Lemma \[lem:ecart\_net\]\[sss:ecart\_net\]
For any $\eta,\mathcal{A}>0$, there exist $c,C_1,C_2,M>0$ such that, when $m\geq Mn^2$, for any $k\in\N$ such that $k\leq \mathcal{A} \log m-c$, with probability at least $$1-C_1\exp(-C_2 m^{1/2}),$$ the following property holds: for any $x\in\mathcal{M}_n^k,y\in\mathcal{M}_n^{k+1}$ such that $||x-y||\leq 2^{-(k-1)}$, $$|\left(\scal{Ax_0}{b\odot\phase(Ax)}-F(x)\right)-
\left(\scal{Ax_0}{b\odot\phase(Ay)}-F(y)\right)| \leq \frac{\eta}{(k+1)^2}\frac{m}{\sqrt{n}}||x_0||^2 .$$ In the case $k=0$, we additionally have, with the same probability: for all $x\in\mathcal{M}_n^0$, $$|\left(\scal{Ax_0}{b\odot\phase(Ax)}-F(x)\right)| \leq \eta\frac{m}{\sqrt{n}}||x_0||^2.$$
We only prove the first part of the lemma. The proof of the second one follows the same principle.
As our expressions are all homogeneous in $x_0$, we can assume that $||x_0||=1$.
For any $j=1,\dots,m$, let us denote by $a_j^*$ the $j$-th line of $A$. We have $$\scal{Ax_0}{b\odot\phase(Ax)}=
\sum_{j=1}^m|a_j^*x_0|^2\phase(a_j^*x)\phase(\overline{a_j^*x_0}).$$ As all the $a_j^*$ are identically distributed, $$\forall j,\quad\quad
\E\left(|a_j^*x_0|^2\phase(a_j^*x)\phase(\overline{a_j^*x_0})\right)
=\frac{1}{m}\E\scal{Ax_0}{b\odot\phase(Ax)}=\frac{1}{m}F(x).$$ So for any fixed $x,y$, we have $$\begin{aligned}
\left(\scal{Ax_0}{b\odot\phase(Ax)}-F(x)\right)&-
\left(\scal{Ax_0}{b\odot\phase(Ay)}-F(y)\right)\nonumber\\
&=\sum_{j=1}^m \left(|a_j^*x_0|^2Z_j-\E(|a_j^*x_0|^2Z_j)\right),
\label{eq:sum_Ajx0_Z}\end{aligned}$$ with $$Z_j=\phase(a_j^*x)\phase(\overline{a_j^*x_0})-\phase(a_j^*y)\phase(\overline{a_j^*x_0}).$$ Were there no terms “$|a_j^*x_0|^2$” in Equation , we could apply Bennett’s concentration inequality: the random variables $Z_j$ are bounded by $2$ in modulus, and, as we are going to see, their variance is small if $x$ and $y$ are close. Bennett’s inequality would then guarantee that the term in Equation is small with high probability. Unfortunately, the $|a_j^*x_0|^2$ are not almost surely bounded, so we cannot directly apply Bennett’s inequality.
To overcome this problem, we first condition over $Ax_0$. When conditioned over $Ax_0$, the random variables $|a_j^*x_0|^2 Z_j$ are almost surely bounded; we will prove that they still have a small variance. We still cannot directly apply Bennett’s inequality, because the bounds depend on $j$, but we can adapt its proof, and get a concentration inequality for the following sum: $$\sum_{j=1}^m|a_j^*x_0|^2Z_j - |a_j^*x_0|^2\E(Z_j|Ax_0).$$ After that, we will also need to derive a concentration inequality for $$\sum_{j=1}^m|a_j^*x_0|^2\E(Z_j|Ax_0)-\E(|a_j^*x_0|^2Z_j),$$ but it will be easier.
The first step is to control the distribution of the $|a_j^*x_0|$. The idea is that there are a few indexes $j$ for which $|a_j^*x_0|$ is large, but these are sufficiently rare so that the sum $\sum_j|a_j^*x_0|^2Z_j$, when conditioned over $Ax_0$, essentially behaves as if all random variables were bounded by the same constant.
The proof of the following lemma is in Paragraph \[sss:dist\_Ajx0\].
\[lem:dist\_Ajx0\] For some constants $C_1,C_2>0$, the following event happens with probability at least $1-C_1e^{-C_2 \sqrt{m}}$: for any $s\in\{1,\dots,\lfloor m^{1/4}\rfloor\}$, $$\begin{aligned}
\Card\left\{j\in\{1,\dots,m\},|a_j^*x_0|\geq s\right\}&\leq \frac{m}{s^2} \max(m^{-1/2},e^{-s^2/2})\end{aligned}$$ and $$\begin{aligned}
\Card\left\{j\in\{1,\dots,m\},|a_j^*x_0|> m^{1/4}\right\}&=0.\end{aligned}$$
Let us denote by $\mathcal{E}_0$ the event described in the previous lemma: $$\begin{aligned}
\mathcal{E}_0=\Big(
\forall s\in\{1,\dots,\lfloor m^{1/4}\rfloor\}, &\Card\left\{j\in\{1,\dots,m\},|a_j^*x_0|\geq s\right\} \leq \frac{m}{s^2} \max(m^{-1/2},e^{-s^2/2});\nonumber\\
\mbox{and }&\Card\left\{j\in\{1,\dots,m\},|a_j^*x_0|> m^{1/4}\right\}=0
\Big).\label{eq:def_E0}\end{aligned}$$ The second step is to get an upper bound on the variance of the $Z_j$, conditioned by $Ax_0$. The proof of the following lemma is in Paragraph \[sss:var\_bound\].
\[lem:var\_bound\] There exists a constant $C>0$ depending only on $\epsilon$ such that, for any fixed unit-normed $x,y$ such that $$\label{eq:var_cond_xy}
|\scal{x_0}{x}|\leq (1-\epsilon)||x_0||\,||x||\quad\mbox{and}\quad
|\scal{x_0}{y}|\leq (1-\epsilon)||x_0||\,||y||,$$ we have, for any $j$, $$\Var(Z_j|Ax_0)\leq C\left(1+\frac{|a_j^*x_0|^2}{||x_0||^2}\right)
||x-y||^2\log\left(4||x-y||^{-1}\right).$$
From the previous lemma, we deduce that, if $x\in\mathcal{M}_n^k,y\in\mathcal{M}_n^{k+1}$ are fixed and satisfy $||x-y||\leq 2^{-(k-1)}$, we have $$\begin{gathered}
\Var(\Re Z_j|Ax_0)\leq \Var(Z_j|Ax_0)\leq C'\left(1+|a_j^*x_0|^2\right) \gamma^{-2k},\label{eq:var_bound}\\
\Var(\Im Z_j|Ax_0)\leq \Var(Z_j|Ax_0)\leq C'\left(1+|a_j^*x_0|^2\right) \gamma^{-2k}
\nonumber.\end{gathered}$$ where $\gamma$ can be any real number in $]1;2[$, and $C'>0$ is a large enough constant (depending on $\gamma$).
To follow the proof of Bennett’s inequality, we now have to upper bound, for suitable values of $\lambda>0$, $$\begin{aligned}
&\E\left(\exp\left(\lambda \sum_{j=1}^m\Re\left(|a_j^*x_0|^2Z_j-|a_j^*x_0|^2\E(Z_j|Ax_0)\right)\right)\Bigg| Ax_0\right)\\
&\hskip 7cm
=\prod_{j=1}^m \E(e^{\lambda \Re (|a_j^*x_0|^2 Z_j-|a_j^*x_0|^2 \E(Z_j|Ax_0))}\Big| Ax_0).\end{aligned}$$ We use here the fact that, even when conditioned on $Ax_0$, the $Z_j$ are independent random variables.
The upper bound relies on the following lemma, proven in Paragraph \[sss:esp\_exp\_bound\].
\[lem:esp\_exp\_bound\] Let $Z$ be any real random variable such that $|Z|\leq 2$ with probability $1$. If we set $\sigma^2=\Var(Z)$, then, for any $\lambda\in\R^+$, $$\E\left(e^{\lambda (Z-\E(Z))}\right) \leq 1+\frac{\sigma^2}{16}\left(e^{4\lambda}-1-4\lambda\right).$$
From Equation and the previous lemma, for any $\lambda\geq 0$, $$\begin{aligned}
\E&\Big(e^{\lambda |a_j^*x_0|^2\Re(Z_j-\E(Z_j|Ax_0))}\Big|Ax_0\Big)\nonumber\\
&\leq 1+ \frac{C'(1+|a_j^*x_0|^2)\gamma^{-2k}}{16}\left(e^{4|a_j^*x_0|^2\lambda}-1-4|a_j^*x_0|^2\lambda\right).\nonumber\end{aligned}$$ So we can upper bound $$\begin{aligned}
&\E\left(\exp\left(\lambda \sum_{j=1}^m\Re\left(|a_j^*x_0|^2Z_j-|a_j^*x_0|^2\E(Z_j|Ax_0)\right)\right)\Bigg| Ax_0\right)
\nonumber\\
&\hskip 1cm
\leq \exp\left(\sum_{j=1}^m\log\left(1+ \frac{C'(1+|a_j^*x_0|^2)\gamma^{-2k}}{16}\left(e^{4|a_j^*x_0|^2\lambda}-1-4|a_j^*x_0|^2\lambda\right)\right) \right).
\label{eq:exp_bound}\end{aligned}$$ On the event $\mathcal{E}_0$ defined in Equation , we can simplify the sum inside the exponential. Specifically, if we define the function $$\begin{aligned}
P:s\in\R^+\to \frac{m}{\max(s,1)^2} \max(m^{-1/2},e^{-s^2/2}),\end{aligned}$$ we have that, on the event $\mathcal{E}_0$, for any non-decreasing function $f:\R^+\to\R^+$, $$\begin{aligned}
\sum_{j=1}^m f(|a_j^*x_0|)
&\leq \sum_{s=1}^{+\infty} f(s)\Big(\Card\{j,|a_j^*x_0|\geq s-1\}-\Card\{j,|a_j^*x_0|\geq s\}\Big)\\
&=\sum_{s=0}^{+\infty}(f(s+1)-f(s))\Card\{j,|a_j^*x_0|\geq s\}+mf(0)\\
&\leq \sum_{s=1}^{\lfloor m^{1/4}\rfloor}(f(s+1)-f(s))P(s)+P(0)f(0)\\
&\leq \sum_{s=1}^{\lfloor m^{1/4}\rfloor} f(s)\left(P(s-1)-P(s)\right)+f(\lfloor m^{1/4}\rfloor+1)P(\lfloor m^{1/4}\rfloor)\\
&= \sum_{s= 1}^{\lfloor m^{1/4}\rfloor}f(s)\int_{s}^{s+1} (-P'(t-1))dt+f(\lfloor m^{1/4}\rfloor+1)P(\lfloor m^{1/4}\rfloor)\\
&\leq \int_1^{ m^{1/4}+1}f(t)(-P'(t-1))dt+f(\lfloor m^{1/4}\rfloor+1)P(\lfloor m^{1/4}\rfloor).\end{aligned}$$ By a direct computation, we see that, if $C>0$ is properly chosen, we can bound: $$\begin{aligned}
-P'(s-1)&\leq C\frac{m}{s^2}e^{-s^2/4}\mbox{ if }s\leq \sqrt{\log m}+1,\\
&\leq C\frac{m^{1/2}}{s^3}\mbox{ if }\sqrt{\log m}+1<s\leq m^{1/4}+1;\\
P(\lfloor m^{1/4}\rfloor)&\leq C.\end{aligned}$$ So $$\begin{aligned}
\frac{1}{m}\sum_{j=1}^m f(|a_j^*x_0|)
\leq C\left(\int_1^{\sqrt{\log m}+1}\frac{f(t)}{t^2}e^{-t^2/4}dt+
m^{-1/2}\int_{\sqrt{\log m}+1}^{m^{1/4}+1}\frac{f(t)}{t^3}dt\right)+\frac{C}{m}f(m^{1/4}+1).\end{aligned}$$ We plug this inequality into Equation . For any $\lambda\geq 0$, we set $$f_\lambda(x)=\log\left(1+\frac{C'(1+x^2)\gamma^{-2k}}{16}(e^{4\lambda x^2}-1-4\lambda x^2)\right),$$ and, on the event $\mathcal{E}_0$, we have: $$\begin{aligned}
&\E\left(\exp\left(\lambda \sum_{j=1}^m\Re\left(|a_j^*x_0|^2Z_j-|a_j^*x_0|^2\E(Z_j|Ax_0)\right)\right)\Bigg| Ax_0\right)
\nonumber\\
&\hskip 1cm\leq \exp\left(Cm\left(\int_1^{\sqrt{\log m}+1}\frac{f_{\lambda}(t)}{t^2}e^{-t^2/4}dt+
m^{-1/2}\int_{\sqrt{\log m}+1}^{m^{1/4}+1}\frac{f_{\lambda}(t)}{t^3}dt
+\frac{1}{m}f(m^{1/4}+1)\right)\right).\label{eq:bound_exp_int}\end{aligned}$$ We upper bound the sum of the integrals, using standard analysis techniques. The detailed proof is in Paragraph \[sss:eval\_lambda\_small\].
\[lem:eval\_lambda\_small\] There exists a constant $\tilde C>0$ depending only on $\gamma$ and $\epsilon>0$ such that, for any $\lambda\in]0;\frac{1}{40}[$, $$\int_1^{\sqrt{\log m}+1}\frac{f_{\lambda}(t)}{t^2}e^{-t^2/4}dt+
m^{-1/2}\int_{\sqrt{\log m}+1}^{m^{1/4}+1}\frac{f_{\lambda}(t)}{t^3}dt+
\frac{1}{m}f_{\lambda}(m^{1/4}+1)
\leq \tilde C \gamma^{-2k}\lambda^2,$$ provided that
$$\begin{gathered}
\Big(\log(\max(1,\gamma^k/\lambda))+1\Big)\left(\frac{\gamma^k}{\lambda}\right)^{4/3}\leq m^{1/2};\label{eq:eval_cond1}\\
\frac{m^{1/2}\lambda \gamma^{-2k}}{1+\log m} \geq 1.\label{eq:eval_cond2}\end{gathered}$$
We apply this result with $$\lambda = \frac{\eta \gamma^{2k}}{8C\tilde C (k+1)^2m^{1/4}},$$ where $\eta>0$ is the fixed constant chosen in the statement of Lemma \[lem:ecart\_net\], $C$ is the constant of Equation and $\tilde C$ is the one of Lemma \[lem:eval\_lambda\_small\]. We consider only the values of $k\in\N$ such that $$\label{eq:cond_k_1}
\gamma^{2k}< \frac{C\tilde C}{5 \eta}m^{1/4},$$ which in particular ensures that $$\lambda< \frac{1}{40}.$$ With this definition, Conditions and are satisfied. Indeed, as $\gamma>1$, $$\begin{gathered}
\frac{\gamma^k}{\lambda}= \frac{8C\tilde C(k+1)^2m^{1/4}}{\eta \gamma^k}=O(m^{1/4});\\
\Rightarrow\quad\quad
\left(\log(\max(1,\gamma^k/\lambda))+1\right)\left(\frac{\gamma^k}{\lambda}\right)^{4/3} = O(m^{1/3}\log m) \leq m^{1/2},\end{gathered}$$ if $m$ is large enough. For the second condition, because of Equation , $$\begin{aligned}
\frac{m^{1/2}\lambda\gamma^{-2k}}{1+\log m}
&=\frac{m^{1/2}}{1+\log m} \frac{\eta}{8C\tilde C(k+1)^2m^{1/4}}\\
&\geq \frac{m^{1/4}}{1+\log m}\frac{\eta}{8CC'\left(1+\log(C\tilde Cm^{1/4}/(5\eta))/(2\log(\gamma))\right)^2}\\
&\geq c\frac{m^{1/4}}{(1+\log m)^3}\\
&\geq 1,\end{aligned}$$ if $m$ is large enough. (In the second inequality, $c>0$ is a positive constant.)
As the two conditions are satisfied, we can combine Lemma \[lem:eval\_lambda\_small\] and Equation . We get that, on the event $\mathcal{E}_0$, $$\begin{aligned}
\E\left(\exp\left(\lambda \sum_{j=1}^m\Re\left(|a_j^*x_0|^2Z_j-|a_j^*x_0|^2\E(Z_j|Ax_0)\right)\right)\Bigg| Ax_0\right)
&\leq
\exp\left(C\tilde C m \gamma^{-2k}\lambda^2\right)\\
&=\exp\left( \frac{\eta^2 \gamma^{2k} m^{1/2}}{64 C\tilde C(k+1)^4} \right).\end{aligned}$$ So, by Markov’s inequality, on the event $\mathcal{E}_0$, if $m\geq n^2$, $$\begin{aligned}
P&\left(\sum_{j=1}^m\Re \left(|a_j^*x_0|^2Z_j-|a_j^*x_0|^2\E(Z_j|Ax_0)\right)\geq \frac{\eta m}{4(k+1)^2\sqrt{n}} \Bigg| Ax_0 \right)\\
&\leq P\left(\sum_{j=1}^m\Re \left(|a_j^*x_0|^2Z_j-|a_j^*x_0|^2\E(Z_j|Ax_0)\right)\geq \frac{\eta m^{3/4}}{4(k+1)^2} \Bigg| Ax_0 \right)\\
&\leq \exp\left( \frac{\eta^2 \gamma^{2k} m^{1/2}}{64C\tilde C(k+1)^4 } \right)\exp\left(-\frac{\lambda\eta m^{3/4}}{4(k+1)^2 }\right)\\
&= \exp\left(-\frac{\eta^2\gamma^{2k} m^{1/2}}{64C\tilde C(k+1)^4}\right).\end{aligned}$$ We integrate over $Ax_0$, and obtain $$\begin{aligned}
P&\left(\mathcal{E}_0\cap\left\{\sum_{j=1}^m\Re \left(|a_j^*x_0|^2Z_j-|a_j^*x_0|^2\E(Z_j|Ax_0)\right)\geq \frac{\eta m}{4(k+1)^2\sqrt{n}} \right\} \right)\\
&\leq \exp\left(-\frac{\eta^2\gamma^{2k} m^{1/2}}{16C\tilde C(k+1)^4}\right).\end{aligned}$$ We can apply the same reasoning to $-\Re(Z_j),\Im(Z_j)$ and $-\Im(Z_j)$. This yields: $$\begin{aligned}
P&\left(\mathcal{E}_0\cap\left\{\left|\sum_{j=1}^m\left(|a_j^*x_0|^2Z_j-|a_j^*x_0|^2\E(Z_j|Ax_0)\right)\right|\geq \frac{\eta m}{2(k+1)^2\sqrt{n}} \right\} \right)\nonumber\\
&\leq 4\exp\left(-\frac{\eta^2\gamma^{2k} m^{1/2}}{16C\tilde C(k+1)^4}\right).
\label{eq:concentration_part1}\end{aligned}$$ Now that we have a bound for $\sum_{j=1}^m\left(|a_j^*x_0|^2Z_j-|a_j^*x_0|^2\E(Z_j|Ax_0)\right)$, we remember that we also have to bound $$\sum_{j=1}^m\left(|a_j^*x_0|^2\E(Z_j|Ax_0)-\E(|a_j^*x_0|^2Z_j)\right).$$ We remark that, for all $j$, $\E(Z_j|Ax_0)=\E(Z_j|a_j^*x_0)$, so that the random variables $|a_j^*x_0|^2\E(Z_j|Ax_0)$, for $j=1,\dots,m$, are independent and identically distributed.
We begin with the following lemma, proven in Paragraph \[sss:maj\_esp\].
\[lem:maj\_esp\] There exist a constant $C>0$ depending only on $\epsilon$ such that, for any fixed unit-normed $x,y$ such that $$|\scal{x_0}{x}|\leq (1-\epsilon)||x_0||\,||x||\quad\mbox{and}\quad
|\scal{x_0}{y}|\leq (1-\epsilon)||x_0||\,||y||,$$ and any $j=1,\dots,m$, $$|\E(Z_j|a_j^*x_0)| \leq C \min\left(1,||x-y||\left(1+\frac{|a_j^*x_0|}{||x_0||}\right)\right).$$
To simplify the expressions, we still assume that $||x_0||=1$. If $||x-y||\leq 2^{-(k-1)}$, the previous lemma guarantees that, for any $j$, $$\label{eq:maj_esp_gamma}
|\E(Z_j|a_j^* x_0)|\leq 2C \min\left(1,\gamma^{-k}(1+|a_j^*x_0|)\right),$$ where $\gamma$ is still our real number in $]1;2[$.
This inequality allows us to upper bound $\E\left(e^{\lambda\left(|a_j^*x_0|^2\E(Z_j|Ax_0)-\E(|a_j^*x_0|^2Z_j)\right)}\right)$, for $\lambda$ small enough. The next lemma is proved in Paragraph \[sss:esp\_exp\_bound2\].
\[lem:esp\_exp\_bound2\] There exist constants $c,C'>0$, that depend only on $\gamma$ and $\epsilon$, such that, for any $\lambda\in[-c;c]$, $$\begin{aligned}
\log\left(\E\left(e^{\lambda\Re\left(|a_j^*x_0|^2\E(Z_j|Ax_0)-\E(|a_j^*x_0|^2Z_j)\right)}\right) \right)
&\leq C' \lambda^2 \gamma^{-2k},\\
\mbox{and }
\log\left(\E\left(e^{\lambda\Im\left(|a_j^*x_0|^2\E(Z_j|Ax_0)-\E(|a_j^*x_0|^2Z_j)\right)}\right) \right)
&\leq C' \lambda^2 \gamma^{-2k}.\end{aligned}$$
So by Markov’s inequality, taking $$\lambda=\frac{\eta\gamma^{2k}}{8C'(k+1)^2m^{1/4}},$$ for $k$ such that $$\label{eq:cond_k_2}
\gamma^{2k}\leq \frac{8cC'}{\eta}m^{1/4},$$ we have, when $m\geq n^2$, $$\begin{aligned}
P&\left(\Re\left(\sum_{j=1}^m\left(|a_j^*x_0|^2\E(Z_j|Ax_0)-\E(|a_j^*x_0|^2Z_j)\right)\right)\geq \frac{\eta m}{4(k+1)^2\sqrt{n}} \right)\\
&\leq P\left(\Re\left(\sum_{j=1}^m\left(|a_j^*x_0|^2\E(Z_j|Ax_0)-\E(|a_j^*x_0|^2Z_j)\right)\right)\geq \frac{\eta m^{3/4}}{4(k+1)^2} \right)\\
&\leq \E\left(\exp\left(\lambda\Re\left(\sum_{j=1}^m\left(|a_j^*x_0|^2\E(Z_j|Ax_0)-\E(|a_j^*x_0|^2Z_j)\right)\right)\right)\right)
\exp\left(-\frac{\lambda\eta m^{3/4}}{4(k+1)^2}\right)\\
&=\exp\left(m \log\left(\E\left(e^{\lambda\Re\left(|a_j^*x_0|^2\E(Z_j|Ax_0)-\E(|a_j^*x_0|^2Z_j)\right)}\right) \right)- \frac{\lambda\eta m^{3/4}}{4(k+1)^2} \right)\\
&\leq\exp\left(m C'\lambda^2\gamma^{-2k}- \frac{\lambda\eta m^{3/4}}{4(k+1)^2} \right)\\
&=\exp\left(-\frac{m^{1/2}\eta^2\gamma^{2k}}{64 C'(k+1)^4}\right).\end{aligned}$$ The same inequality holds if we replace $\Re$ by $-\Re,\Im$ or $-\Im$, so we obtain: $$\begin{aligned}
P&\left(\left|\sum_{j=1}^m\left(|a_j^*x_0|^2\E(Z_j|Ax_0)-\E(|a_j^*x_0|^2Z_j)\right)\right|\geq \frac{\eta m}{2(k+1)^2\sqrt{n}} \right)
\leq 4\exp\left(-\frac{m^{1/2}\eta^2\gamma^{2k}}{64 C'(k+1)^4}\right).\end{aligned}$$ We are close to the end. The previous equation, combined with Equation yields, by triangular inequality, that for any fixed $x\in\mathcal{M}_n^k,y\in\mathcal{M}_n^{k+1}$ such that $||x-y||\leq 2^{-(k-1)}$, $$\begin{aligned}
P&\left(\mathcal{E}_0\cap\left\{\left|\sum_{j=1}^m\left(|a_j^*x_0|^2Z_j-\E(|a_j^*x_0|^2Z_j)\right)\right|
\geq \frac{\eta m}{(k+1)^2\sqrt{n}} \right\} \right)\\
&\leq 8\exp\left(-\mathcal{C}\frac{\gamma^{2k}}{(k+1)^4}m^{1/2}\right),\end{aligned}$$ where $\mathcal{C}$ is a constant that depends only on $\eta,\epsilon$ and $\gamma$. We recall that $Z_j$ depends on $x$ and $y$, although it does not appear in the notation.
From Equation , $$\Card\mathcal{M}_n^k\leq 2^{2n(k+3)}\quad\mbox{and}\quad
\Card\mathcal{M}_n^{k+1}\leq 2^{2n(k+4)}.$$ The number of possible pairs $(x,y)\in\mathcal{M}_n^k\times\mathcal{M}_n^{k+1}$ is then bounded by $$2^{2n(2k+7)}\leq e^{10n(k+1)},$$ and by union bound, $$\begin{aligned}
P&\left(\mathcal{E}_0\cap\left\{\exists x,y\in\mathcal{M}_n^k\times\mathcal{M}_n^{k+1}, \left|\sum_{j=1}^m\left(|a_j^*x_0|^2Z_j-\E(|a_j^*x_0|^2Z_j)\right)\right|
\geq \frac{\eta m}{(k+1)^2\sqrt{n}} \right\} \right)\\
&\leq 8\exp\left(-\mathcal{C}\frac{\gamma^{2k}}{(k+1)^4}m^{1/2}+10 n(k+1)\right).\end{aligned}$$ From Lemma \[lem:dist\_Ajx0\], the probability of $\mathcal{E}_0$ is at least $1-C_1 e^{-C_2m^{1/2}}$ for some constants $C_1,C_2>0$, so $$\begin{aligned}
P&\left(\forall x,y\in\mathcal{M}_n^k\times\mathcal{M}_n^{k+1}, \left|\sum_{j=1}^m\left(|a_j^*x_0|^2Z_j-\E(|a_j^*x_0|^2Z_j)\right)\right|
< \frac{\eta m}{(k+1)^2\sqrt{n}} \right)\\
&\geq 1- 8\exp\left(-\mathcal{C}\frac{\gamma^{2k}}{(k+1)^4}m^{1/2}+10 n(k+1)\right)-C_1\exp(-C_2m^{1/2}).\end{aligned}$$ There exists a constant $\mathcal{C'}$ depending only on $\gamma$ such that $\gamma^{2k}\geq \mathcal{C}'(k+1)^5$ for any $k\in\N$. If we assume that $m\geq Mn^2$ for some $M>0$, we have $$\begin{aligned}
P&\left(\forall x,y\in\mathcal{M}_n^k\times\mathcal{M}_n^{k+1}, \left|\sum_{j=1}^m\left(|a_j^*x_0|^2Z_j-\E(|a_j^*x_0|^2Z_j)\right)\right|
< \frac{\eta m}{(k+1)^2\sqrt{n}} \right)\\
&\geq 1- 8\exp\left(-m^{1/2}(k+1) (\mathcal{C}\mathcal{C}'-10 M^{-1/2})\right)-C_1\exp(-C_2m^{1/2})\\
&\geq 1- 8\exp\left(- (\mathcal{C}\mathcal{C}'-10 M^{-1/2})m^{1/2}\right)-C_1\exp(-C_2m^{1/2}).\end{aligned}$$ When $M>0$ is large enough, this can be lower bounded by $1-C_1\exp(-C_2m^{1/2})$, where the constants $C_1,C_2>0$ depend on $\eta,\epsilon$ and $\gamma$ but not on $k,m$ or $n$.
We recall Equations and : the reasoning holds only for the values of $k$ such that $$\gamma^{2k}< \alpha m^{1/4},$$ where, again, $\alpha>0$ is a constant that depends only on $\eta,\epsilon$ and $\gamma$. This means that, if we have chosen $\gamma\in]1;2[$ sufficiently close to $1$, it holds for any $k$ satisfying $$k < \mathcal{A} \ln m - c,$$ where $c\in\R$ is a constant that does not depend on $n$ or $m$.
### Proof of Lemma \[lem:F\]\[sss:F\]
There exist $\delta>0$ such that, for any $x\in\mathcal{E}_n$, $$|F(x)|\geq (1+\delta)m\frac{||x_0||}{||x||}|\scal{x_0}{x}|.$$
We write $$x=\alpha x_0 + \beta x',$$ with $\alpha,\beta\in\C$ and $x'\in \C^n$ such that $\scal{x_0}{x'}=0$ and $||x'||=1$.
$$\begin{aligned}
F(x)&=\E(\scal{Ax_0}{b\odot\phase(Ax)})\\
&=\sum_{j=1}^m\E\left(\overline{(Ax_0)_j}|(Ax_0)_j|\phase((Ax)_j)\right)\\
&=m \E\left(\overline{(Ax_0)_1}|(Ax_0)_1|\phase((Ax)_1)\right)\\
&=m \E\left(\overline{(Ax_0)_1}|(Ax_0)_1|\phase(\alpha (Ax_0)_1 + \beta (Ax')_1)\right)\\
&=m ||x_0||^2 \phase(\alpha) \E\left(\frac{\overline{(Ax_0)_1}}{||x_0||}\frac{|(Ax_0)_1|}{||x_0||}\phase\left(\frac{(Ax_0)_1}{||x_0||} + \frac{\beta}{\alpha||x_0||} (Ax')_1\right)\right)\\
&=m||x_0||^2\phase(\alpha)\E\left(\overline{Z_1}|Z_1|\phase\left(Z_1+ \frac{|\beta|}{|\alpha|\,||x_0||} Z_2 \right)\right).\end{aligned}$$
where $Z_1=\frac{(Ax_0)_1}{||x_0||}$ and $Z_2=\phase(\beta/\alpha)(Ax')_1$ are independent complex Gaussian variables with variance $1$.
The expectation cannot be analytically computed, but it can be lower bounded by a simple function. The following lemma is proven in Paragraph \[sss:min\_f\].
\[lem:min\_f\] For any $t\in\R^+$, we set $$f(t)=\E\left(\overline{Z_1}|Z_1|\phase\left(Z_1+t Z_2\right)\right).$$ The function $f$ is real-valued. For any $\gamma>0$, there exist $\delta>0$ such that $$\forall t\in[\gamma;+\infty[,\quad\quad
f(t)\geq \frac{1+\delta}{\sqrt{1+t^2}}.$$
As $x$ belongs to $\mathcal{E}_n$, we have: $$\begin{aligned}
\frac{|\beta|}{|\alpha|\,||x_0||}
&=\frac{\sqrt{||x||^2-|\alpha|^2||x_0||^2}}{|\alpha|\,||x_0||}\\
&=\sqrt{\frac{1}{|\alpha|^2||x_0||^2}-1}\\
&=\sqrt{\frac{||x_0||^2}{|\scal{x_0}{x}|^2}-1}\\
&\geq \sqrt{\frac{1}{(1-\epsilon)^2}-1}.\end{aligned}$$ Consequently, we can apply the lemma with $\gamma = \sqrt{\frac{1}{(1-\epsilon)^2}-1}$. It implies that, for some $\delta>0$ that depends only on $\epsilon$, $$\begin{aligned}
|F(x)|&\geq m||x_0||^2(1+\delta)\frac{1}{\sqrt{1+\left(\frac{|\beta|}{|\alpha|\,||x_0||}\right)^2}}\\
&= m||x_0||^2(1+\delta)|\alpha|\,||x_0||\\
&=(1+\delta)m \frac{||x_0||}{||x||} |\scal{x_0}{x}|.\end{aligned}$$
### Proof of Lemma \[lem:dist\_Ajx0\]\[sss:dist\_Ajx0\]
For some constants $C_1,C_2>0$, the following event happens with probability at least $1-C_1e^{-C_2 \sqrt{m}}$: for any $s\in\{1,\dots,\lfloor m^{1/4}\rfloor\}$, $$\begin{aligned}
\Card\left\{j\in\{1,\dots,m\},|a_j^*x_0|\geq s\right\}&\leq \frac{m}{s^2} \max(m^{-1/2},e^{-s^2/2})\end{aligned}$$ and $$\begin{aligned}
\Card\left\{j\in\{1,\dots,m\},|a_j^*x_0|> m^{1/4}\right\}&=0.\end{aligned}$$
We recall that $A_1x_0,\dots,A_mx_0$ are independent complex Gaussian random variables with variance $||x_0||^2=1$. In particular, for any $s\in\N$, $$\begin{aligned}
P(|a_j^*x_0|\geq s)&=e^{-s^2};\\
\E(1_{|a_j^*x_0|\geq s})&=e^{-s^2};\\
\Var(1_{|a_j^*x_0|\geq s})&\leq e^{-s^2}.\end{aligned}$$
We first consider the values of $s$ belonging to $\{1,\dots,\lfloor\sqrt{\log m}\rfloor\}$. For any of these $s$, by Bennett’s inequality, if we denote by $h$ the function $h:x\in\R^+\to (1+x)\log(1+x)-x$, $$\begin{aligned}
&P\left(\Card\left\{j\in\{1,\dots,m\},|a_j^*x_0|\geq s\right\}\geq \frac{m}{s^2}e^{-s^2/2} \right)\\
=&P\left(\sum_{j=1}^m \left(1_{|a_j^*x_0|\geq s}-\E\left(1_{|a_j^*x_0|\geq s}\right)\right)
\geq m\left(\frac{e^{-s^2/2}}{s^2}-e^{-s^2}\right) \right)\\
\leq&\exp\left(-m e^{-s^2}h\left(\frac{e^{s^2/2}}{s^2}-1\right)\right)\\
=&\exp\left(-m\frac{e^{-s^2/2}}{s^2}\left(\frac{s^2}{2}-2\log(s)-1+s^2 e^{-s^2/2}\right)\right)\\
\leq&\exp\left(-c_1m e^{-s^2/2}\right),\end{aligned}$$ for some absolute constant $c_1>0$. As $s\leq \sqrt{\log m}$, this yields: $$\begin{aligned}
P\left(\Card\left\{j\in\{1,\dots,m\},|a_j^*x_0|\geq s\right\}\geq \frac{m}{s^2}e^{-s^2/2} \right)\leq
\exp(-c_1m^{1/2}).\end{aligned}$$ Second, we consider the values of $s$ in $\{\left\lfloor\sqrt{\log m}\right\rfloor+1,\dots,\lfloor m^{1/4}+1\rfloor\}$. $$\begin{aligned}
&P\left(\Card\left\{j\in\{1,\dots,m\},|a_j^*x_0|\geq s\right\}\geq \frac{m^{1/2}}{s^2} \right)\\
=&P\left(\sum_{j=1}^m \left(1_{|a_j^*x_0|\geq s}-\E\left(1_{|a_j^*x_0|\geq s}\right)\right)
\geq m\left(\frac{m^{-1/2}}{s^2}-e^{-s^2}\right) \right)\\
\leq&\exp\left(-me^{-s^2}h\left(\frac{m^{-1/2}}{s^2}e^{s^2}-1\right)\right)\\
=&\exp\left(- m^{1/2}\left(1-\frac{\log m}{2s^2}-\frac{2\log s}{s^2}-\frac{1}{s^2}+m^{1/2}e^{-s^2}
\right)\right)\\
\overset{(a)}{\leq}&\exp\left(-m^{1/2}\left(1-\frac{1}{2}-\frac{\log (\log m)}{\log m}-\frac{1}{\log m}
\right)\right)\\
\leq&\exp\left(-\frac{m^{1/2}}{4}\right).\end{aligned}$$ as soon as $m$ is large enough. For (a), we have used the inequality $s\geq\sqrt{\log m}$.
To conclude, we observe that, if $$\Card\left\{j\in\{1,\dots,m\},|a_j^*x_0|\geq s\right\}\leq \frac{m^{1/2}}{s^2}$$ for $s=\lfloor m^{1/4}+1\rfloor>m^{1/4}$, we must have $$\Card\left\{j\in\{1,\dots,m\},|a_j^*x_0| > m^{1/4} \right\}=0.$$ So we see that the desired event holds, for $m$ large enough, with probability at least $$\begin{aligned}
1- \sqrt{\log m}e^{-c_1\sqrt{m}}-m^{1/4}e^{-\sqrt{m}/4},\end{aligned}$$ which can be bounded by $1-C_1e^{-C_2\sqrt{m}}$ for $C_1,C_2>0$ well-chosen.
### Proof of Lemma \[lem:var\_bound\]\[sss:var\_bound\]
There exists a constant $C>0$ depending only on $\epsilon$ such that, for any fixed unit-normed $x,y$ such that $$\label{eq:var_cond_xy}
|\scal{x_0}{x}|\leq (1-\epsilon)||x_0||\,||x||\quad\mbox{and}\quad
|\scal{x_0}{y}|\leq (1-\epsilon)||x_0||\,||y||,$$ we have, for any $j$, $$\Var(Z_j|Ax_0)\leq C\left(1+\frac{|a_j^*x_0|^2}{||x_0||^2}\right)
||x-y||^2\log\left(4||x-y||^{-1}\right).$$
By the definition of $Z_j$, it suffices to prove $$\label{eq:var_reformulation}
\Var(\phase(a_j^*x)-\phase(a_j^*y)|Ax_0)\leq C\left(1+\frac{|a_j^*x_0|^2}{||x_0||^2}\right) ||x-y||^2\log\left(4||x-y||^{-1}\right).$$ We write $$x = \alpha_xx_0+x'\mbox{ and }y=\alpha_y x_0+\beta x'+y'',$$ where $\alpha_x,\alpha_y,\beta$ are complex numbers and $x',y''\in\C^n$ satisfy $\scal{x'}{x_0}=\scal{y''}{x_0}=\scal{x'}{y''}=0$. Because of Equation , and because $x,y$ are unit-normed,
$$\begin{gathered}
||x'||\geq \sqrt{\epsilon(2-\epsilon)}\geq \sqrt{\epsilon};\label{eq:var_x_prime}\\
|\beta-1|=\frac{|\scal{y-x}{x'}|}{||x'||^2}\leq \frac{1}{\sqrt{\epsilon}}||y-x||;\label{eq:var_beta}\\
||\alpha_xx_0-\beta\alpha_yx_0||=\frac{|\scal{x-y}{x_0}|}{||x_0||}\leq ||x-y||;\label{eq:var_alpha}\\
||y''||=\frac{|\scal{y-x}{y''}|}{||y''||}\leq ||x-y||.\label{eq:var_y_seconde}\end{gathered}$$
As $|Z_j|$ is bounded (by $2$), the desired inequality is true for $||x-y||\geq \sqrt{\epsilon}/2$, provided that $C$ is large enough, so we can assume $||x-y||<\sqrt{\epsilon}/2$, which in particular guarantees that $|\beta|>1/2$.
As $$\begin{aligned}
\Var(\phase(a_j^*x)-\phase(a_j^*y)|Ax_0)&\leq
\E\left(\left|\phase(a_j^*x)-\phase(a_j^*y)\right|^2\Big|Ax_0\right)\\
&=2\left(1-\Re\left(\E\left(\phase(\overline{a_j^*x})\phase(a_j^*y)|Ax_0\right)\right)\right),\end{aligned}$$ we only need, in order to prove Equation , to show that, for some constant $C>0$, $$\begin{aligned}
\label{eq:var_reformulation2}
1-\Re\left(\E\left(\phase(\overline{a_j^*x})\phase(a_j^*y)|Ax_0\right)\right)
\leq C\left(1+\frac{|a_j^*x_0|^2}{||x_0||^2}\right)||x-y||^2\log(4||x-y||^{-1}).\end{aligned}$$
We have $$\begin{aligned}
\phase(a_j^*x)&=\phase\left(\frac{a_j^*x'}{||x'||}+\frac{\alpha_x}{||x'||}a_j^*x_0\right);\\
\phase(a_j^*y)&=\phase\left(\frac{a_j^*x'}{||x'||}+\frac{\alpha_y}{\beta||x'||}a_j^*x_0+\frac{1}{\beta||x'||}a_j^*y'' \right)\phase(\beta),\end{aligned}$$ and $\frac{a_j^*x'}{||x'||}$ is a complex Gaussian random variable with variance $1$, independent from $Ax_0$ and $a_j^*y''$. So $$\begin{aligned}
&\quad 1-\Re\left(\E\left(\phase(\overline{a_j^*x})\phase(a_j^*y)|Ax_0,a_j^*y''\right)\right)\\
&=1-\frac{1}{\pi}\Re\left(\phase(\beta)\int_\C \phase\left(\overline{z+\frac{\alpha_x}{||x'||}a_j^*x_0}\right)\phase\left(z+\frac{\alpha_y}{\beta||x'||}a_j^*x_0+\frac{1}{\beta||x'||}a_j^*y'' \right)e^{-|z|^2}d^2z\right).\end{aligned}$$ We upper bound this quantity with the following proposition, proven in Paragraph \[sss:controle\_G\].
\[prop:controle\_G\] Let us define the function $$\begin{array}{rccc}
G:&\C^2&\to&\C\\
&(a,b)&\to&1-\frac{1}{\pi}\Re\int_\C\phase(\overline{z+a})\phase(z+b)e^{-|z|^2}d^2z.
\end{array}$$ For some constant $c_1>0$, the following inequalities are true: $$\begin{aligned}
\forall a,b\in\C,\quad\quad
|\Re G(a,b)|&\leq c_1 |a-b|^2\max\left(1,\log\left(|a-b|^{-1}\right)\right),\\
|\Im G(a,b)|&\leq c_1 |a-b|.\end{aligned}$$
So $$\begin{aligned}
&\quad 1-\Re\left(\E\left(\phase(\overline{a_j^*x})\phase(a_j^*y)|Ax_0,a_j^*y''\right)\right)\\
&=1-\Re(\phase(\beta))+\Re(\phase(\beta))\Re G\left(\frac{\alpha_x}{||x'||}a_j^*x_0,\frac{\alpha_y}{\beta||x'||}a_j^*x_0+\frac{1}{\beta||x'||}a_j^*y''\right)\\
&\hskip 2cm-\Im(\phase(\beta))\Im G\left(\frac{\alpha_x}{||x'||}a_j^*x_0,\frac{\alpha_y}{\beta||x'||}a_j^*x_0+\frac{1}{\beta||x'||}a_j^*y''\right)\\
&\leq c_1\left|\frac{\alpha_y-\beta\alpha_x}{\beta||x'||}a_j^*x_0+\frac{1}{\beta||x'||}a_j^*y'' \right|^2\max\left(1,\log\left(\left|\frac{\alpha_y-\beta\alpha_x}{\beta||x'||}a_j^*x_0+\frac{1}{\beta||x'||}a_j^*y''\right|^{-1}\right)\right)\\
&\hskip 2cm+|1-\Re(\phase\beta)| + |\Im(\phase(\beta))|\left|\frac{\alpha_y-\beta\alpha_x}{\beta||x'||}a_j^*x_0+\frac{1}{\beta||x'||}a_j^*y'' \right|\\
&\leq c_1\left(\left|\frac{\alpha_y-\beta\alpha_x}{\beta||x'||}a_j^*x_0\right|+\left|\frac{1}{\beta||x'||}a_j^*y'' \right|\right)^2\max\left(1,\log\left(\left|\frac{\alpha_y-\beta\alpha_x}{\beta||x'||}a_j^*x_0\right|+\left|\frac{1}{\beta||x'||}a_j^*y''\right|\right)^{-1}\right)\\
&\hskip 2cm+2|1-\beta|^2 + |\beta-1|\left(\left|\frac{\alpha_y-\beta\alpha_x}{\beta||x'||}a_j^*x_0\right|+\left|\frac{1}{\beta||x'||}a_j^*y'' \right|\right)\\
&\leq 2c_1\left|\frac{\alpha_y-\beta\alpha_x}{\beta||x'||}a_j^*x_0\right|^2\max\left(1,\log\left(\left|\frac{\alpha_y-\beta\alpha_x}{\beta||x'||}a_j^*x_0\right|^{-1}\right)\right)\\
&\hskip 2cm + 2c_1\left|\frac{1}{\beta||x'||}a_j^*y'' \right|^2\max\left(1,\log\left(\left|\frac{1}{\beta||x'||}a_j^*y''\right|^{-1}\right)\right)\\
&\hskip 2cm+2|1-\beta|^2 + |\beta-1|\left(\left|\frac{\alpha_y-\beta\alpha_x}{\beta||x'||}a_j^*x_0\right|+\left|\frac{1}{\beta||x'||}a_j^*y'' \right|\right)\\
&\overset{(*)}{\leq} 2c_1\left|\frac{\alpha_y-\beta\alpha_x}{\beta||x'||}\right|^2\max(||x_0||,|a_j^*x_0|)^2 \max\left(1,\log\left(\left|\frac{\alpha_y-\beta\alpha_x}{\beta||x'||}.\max(||x_0||,|a_j^*x_0|)\right|^{-1}\right)\right)\\
&\hskip 2cm + 2c_1\left|\frac{1}{\beta||x'||}a_j^*y'' \right|^2\max\left(1,\log\left(\left|\frac{1}{\beta||x'||}a_j^*y''\right|^{-1}\right)\right)\\
&\hskip 2cm+2|1-\beta|^2 + |\beta-1|\left(\frac{\left|\alpha_y-\beta\alpha_x\right|\,||x_0||}{\beta||x'||}\frac{|a_j^*x_0|}{||x_0||}+\left|\frac{1}{\beta||x'||}a_j^*y'' \right|\right)\\
&\leq 2c_1\left|\frac{\alpha_y-\beta\alpha_x}{\beta||x'||}\right|^2\max(||x_0||,|a_j^*x_0|)^2 \max\left(1,\log\left(\left|\frac{\alpha_y-\beta\alpha_x}{\beta||x'||}||x_0||\right|^{-1}\right)\right)\\
&\hskip 2cm + 2c_1\left|\frac{1}{\beta||x'||}a_j^*y'' \right|^2\max\left(1,\log\left(\left|\frac{1}{\beta||x'||}a_j^*y''\right|^{-1}\right)\right)\\
&\hskip 2cm+2\frac{||x-y||^2}{\epsilon} + \frac{||x-y||}{\sqrt{\epsilon}}\left(2\frac{||x-y||}{\sqrt{\epsilon}}\frac{|a_j^*x_0|}{||x_0||}+\frac{2}{\sqrt{\epsilon}}\left|a_j^*y'' \right|\right)\\
&\leq c_2 ||x-y||^2\left(1+\frac{|a_j^*x_0|}{||x_0||}\right)^2\max(1,\log||x-y||^{-1})+ c_2|a_j^*y''|^2\max(1,\log|a_j^*y''|^{-1})\\
&\hskip 2cm + c_2||x-y||\,|a_j^*y''|.\end{aligned}$$ For $(*)$, we have used the fact that $t\to t^2\max(1,\log(1/t))$ is non-decreasing. For the last two lines, we have used this same fact and Equations , and .
The random variable $a_j^*y''$ is complex and Gaussian, has variance $||y''||^2$ and is independent from $Ax_0$, so, taking the expectation over $a_j^*y''$ then using Equation , we get: $$\begin{aligned}
&\quad 1-\Re\left(\E\left(\phase(\overline{a_j^*x})\phase(a_j^*y)|Ax_0\right)\right)\\
&\leq c_2 ||x-y||^2\left(1+\frac{|a_j^*x_0|}{||x_0||}\right)^2\max(1,\log||x-y||^{-1})+ c_3||y''||^2\max(1,\log||y''||^{-1})\\
&\hskip 2cm + c_3||y''||\,||x-y||\\
&\leq c_4 ||x-y||^2\left(1+\frac{|a_j^*x_0|}{||x_0||}\right)^2\max(1,\log||x-y||^{-1}).\end{aligned}$$ As $||x-y||\leq 2$ (because $x$ and $y$ are unit-normed), this implies Equation and concludes.
### Proof of Lemma \[lem:esp\_exp\_bound\]\[sss:esp\_exp\_bound\]
Let $Z$ be any real random variable such that $|Z|\leq 2$ with probability $1$. If we set $\sigma^2=\Var(Z)$, then, for any $\lambda\in\R^+$, $$\E\left(e^{\lambda (Z-\E(Z))}\right) \leq 1+\frac{\sigma^2}{16}\left(e^{4\lambda}-1-4\lambda\right).$$
Let us define $Z'=Z-\E(Z)$. We have $|Z'|\leq 4$ with probability $1$, $\E(Z')=0$ and $\E(Z'^2)=\sigma^2$. Then, $$\begin{aligned}
\E\left(e^{\lambda Z'}\right)
&=\E\left(1+\lambda Z' +\sum_{k\geq 2}\frac{\lambda^kZ'^k}{k!}\right)\\
&\leq 1 + \sum_{k\geq 2}\E\left(\frac{\lambda^k Z'^2 4^{k-2}}{k!}\right)\\
&=1+\frac{\sigma^2}{16}\left(e^{4\lambda}-1-4\lambda\right).\end{aligned}$$
### Proof of Lemma \[lem:eval\_lambda\_small\]\[sss:eval\_lambda\_small\]
There exists a constant $\tilde C>0$ depending only on $\gamma$ and $\epsilon>0$ such that, for any $\lambda\in]0;\frac{1}{40}[$, $$\int_1^{\sqrt{\log m}+1}\frac{f_{\lambda}(t)}{t^2}e^{-t^2/4}dt+
m^{-1/2}\int_{\sqrt{\log m}+1}^{m^{1/4}+1}\frac{f_{\lambda}(t)}{t^3}dt+
\frac{1}{m}f_{\lambda}(m^{1/4}+1)
\leq \tilde C \gamma^{-2k}\lambda^2,$$ provided that
$$\begin{gathered}
\Big(\log(\max(1,\gamma^k/\lambda))+1\Big)\left(\frac{\gamma^k}{\lambda}\right)^{4/3}\leq m^{1/2};\tag{\ref{eq:eval_cond1}}\\
\frac{m^{1/2}\lambda \gamma^{-2k}}{1+\log m} \geq 1.\tag{\ref{eq:eval_cond2}}\end{gathered}$$
As we only consider the function $f_\lambda$ on $]1;+\infty[$, we can upper bound it by the slightly simpler expression $$\tilde f_\lambda(x)=\log\left(1+\frac{C'\gamma^{-2k}}{8}x^2(e^{4\lambda x^2}-1-4\lambda x^2)\right).$$ Let $X_0$ be the (unique) positive number such that $$\begin{aligned}
\frac{C'\gamma^{-2k}}{8}X_0^2(e^{4\lambda X_0^2}-1-4\lambda X_0^2)&=1.\nonumber\\
\iff\hskip 1cm \lambda X_0^2(e^{4\lambda X_0^2}-1-4\lambda X_0^2)&=\frac{8}{C'}\lambda\gamma^{2k}.\label{eq:eval_def_x0}\end{aligned}$$ The function $\tilde f_\lambda$ satisfies the following inequalities: $$\begin{gathered}
\forall x\in\R^+,\quad\quad
\tilde f_\lambda(x)\leq \frac{C'\gamma^{-2k}}{8}x^2(e^{4\lambda x^2}-1-4\lambda x^2);\end{gathered}$$ $$\begin{aligned}
\forall x\geq X_0,\quad\quad
\tilde f_\lambda(x)&\leq \log\left(\frac{C'\gamma^{-2k}}{8}x^2(e^{4\lambda x^2}-1-4\lambda x^2)\right)+\log 2\\
&\leq \log\left(\frac{C'\gamma^{-2k}}{8}x^2e^{4\lambda x^2}\right)+\log 2\\
&\leq \log\left(\frac{C'}{4}\right)+2\log x + 4\lambda x^2.\end{aligned}$$ In particular, if $X_0\leq 2m^{1/4}$, $$\begin{aligned}
\frac{1}{m}f_{\lambda}(m^{1/4}+1)
&\leq \frac{1}{m}f_{\lambda}(2m^{1/4})\\
&\leq \frac{1}{m}\left(\log\left(\frac{C'}{4}\right)+2\log(2m^{1/4})+16\lambda m^{1/2}\right)\\
&\leq D\left(\frac{\log m+\lambda m^{1/2}}{m}\right)\\
&\overset{\eqref{eq:eval_cond2}}{\leq} \frac{2D\lambda}{m^{1/2}}\\
&\overset{\eqref{eq:eval_cond2}}{\leq} 2D\lambda^2 \gamma^{-2k},\end{aligned}$$ and if $X_0>2m^{1/4}$, from the definition of $X_0$, we see that $$\begin{aligned}
\frac{1}{m}f_{\lambda}(m^{1/4}+1)
&\leq \frac{1}{m}\log(1+\frac{C'\gamma^{-2k}}{8}X_0^2(e^{4\lambda X_0^2}-1-4\lambda X_0^2))\\
&\leq \frac{\log 2}{m}\\
&\leq (\log 2)\frac{\lambda^2\gamma^{-2k}}{(m^{1/2}\lambda\gamma^{-2k})^2}\\
&\overset{\eqref{eq:eval_cond2}}{\leq} (\log 2)\lambda^2 \gamma^{-2k}.\end{aligned}$$ So $\frac{1}{m}f_{\lambda}(m^{1/4}+1)$ is bounded by $\tilde C\gamma^{-2k}\lambda^2$ and we only have to show the same bound for the integral terms.
Using the inequalities we have established over $\tilde f_{\lambda}$, $$\begin{aligned}
\int_1^{\sqrt{\log m}+1}&\frac{f_{\lambda}(t)}{t^2}e^{-t^2/4}dt+
m^{-1/2}\int_{\sqrt{\log m}+1}^{m^{1/4}+1}\frac{f_{\lambda}(t)}{t^3}dt
\nonumber\\
&\leq \frac{C'\gamma^{-2k}}{8}\int_0^{+\infty} (e^{4\lambda t^2}-1-4\lambda t^2)e^{-t^2/4} dt\label{eq:eval_term1}\\
&\quad +m^{-1/2}\frac{C'\gamma^{-2k}}{8}\int_{\sqrt{\log m}+1}^{\max(X_0,\sqrt{\log m}+1)}\frac{1}{t}(e^{4\lambda t^2}-1-4\lambda t^2)dt\label{eq:eval_term2}\\
&\quad +m^{-1/2}\int_{\min(m^{1/4}+1,\max(X_0,\sqrt{\log m}+1))}^{m^{1/4}+1}\frac{1}{t^3}\left(\log\left(\frac{C'}{4}\right)+ 2\log t + 4\lambda t^2\right)dt\label{eq:eval_term3}.
%&=\frac{C'\gamma^{-2k}}{8}\sqrt{5\pi}\left(\frac{1}{1-20\lambda}-1-10\lambda\right) + \dots\end{aligned}$$ We separately study each of the three right-side terms.
For Term , we can do an exact computation, taking into account the fact that $\lambda\leq 1/40$: $$\begin{aligned}
\mbox{\eqref{eq:eval_term1}}
&=\frac{C'\sqrt{\pi}}{8}\gamma^{-2k} \left(\frac{1}{\sqrt{1-16\lambda}}-1-8\lambda\right)\\
&\leq \frac{C''\sqrt{\pi}}{8}\gamma^{-2k}\lambda^2.\end{aligned}$$ For Term , if $X_0<\sqrt{\log m}+1$, then it is zero. Otherwise, $X_0\geq \sqrt{\log m}+1$ and $$\begin{aligned}
\mbox{\eqref{eq:eval_term2}}
&\leq m^{-1/2}\frac{C'\gamma^{-2k}}{8}\int_0^{X_0}\frac{1}{t}(e^{4\lambda t^2}-1-4\lambda t^2)dt\nonumber\\
&=m^{-1/2}\frac{C'\gamma^{-2k}}{8}\int_0^{2\sqrt{\lambda}X_0}\frac{1}{t}(e^{t^2}-1-t^2)dt.\label{eq:eval_term2_upper_bound}\end{aligned}$$ When $\frac{8}{C'}\lambda\gamma^{2k}\leq 1$, we check from the definition of $X_0$ (Equation ) that $\lambda X_0^2\leq 1$, so $2\sqrt{\lambda}X_0\leq 2$ and $$\begin{aligned}
\mbox{\eqref{eq:eval_term2}}
&\leq m^{-1/2}\frac{C''\gamma^{-2k}}{8}\int_0^{2\sqrt{\lambda}X_0}t^3 dt.\\
&= m^{-1/2} \frac{C''\gamma^{-2k}}{2} \lambda^2X_0^4.\end{aligned}$$ From Equation again, we see that, as $\lambda X_0^2\leq 1$, $$\begin{aligned}
\frac{8}{C'}\lambda\gamma^{2k}&\geq C'''(\lambda X_0^2)^3;\nonumber\\
\Rightarrow\hskip 1cm
\left(\frac{8}{C'C'''}\right)^{2/3}\frac{\gamma^{4k/3}}{\lambda^{4/3}}&\geq X_0^4.
\label{eq:eval_x0_small_equiv}\end{aligned}$$ From Condition , we know that $\left(\frac{\gamma^k}{\lambda}\right)^{4/3}\leq m^{1/2}$, so $$\begin{aligned}
\mbox{\eqref{eq:eval_term2}}
&\leq m^{-1/2}C''''\gamma^{-2k}\lambda^2\left(\frac{\gamma^{k}}{\lambda}\right)^{4/3} \leq C'''' \gamma^{-2k}\lambda^2.\end{aligned}$$ On the other hand, when $\frac{8}{C'}\lambda\gamma^{2k}> 1$, $2\sqrt{\lambda}X_0$ is bounded away from zero. We evaluate the integral in Equation by parts: $$\begin{aligned}
\mbox{\eqref{eq:eval_term2}}
&\leq m^{-1/2}\frac{C'\gamma^{-2k}}{8}\int_0^{2\sqrt{\lambda}X_0}\frac{1}{t}(e^{t^2}-1-t^2)dt\\
&\leq m^{-1/2} C''\gamma^{-2k}\frac{1}{\lambda X_0^2} e^{4\lambda X_0^2}.\end{aligned}$$ From Equation , we can compute that, when $2\sqrt{\lambda}X_0$ is bounded away from zero, $$\frac{1}{X_0^2} e^{4\lambda X_0^2} \leq C'''\frac{\lambda^2\gamma^{2k}}{\left(1+\log\left(8\lambda\gamma^{2k}/C'\right)\right)^2},$$ which yields, together with Condition : $$\begin{aligned}
\mbox{\eqref{eq:eval_term2}}
&\leq \frac{m^{-1/2}C''C''' \lambda}{\left(1+\log\left(8\lambda\gamma^{2k}/C'\right)\right)^2}
\leq m^{-1/2}C''C'''\lambda
\leq C''C''' \lambda^2\gamma^{-2k}.\end{aligned}$$ Finally, we consider the last term. When $X_0\geq m^{1/4}+1$, it is zero, so we only have to consider the case where $X_0< m^{1/4}+1$. $$\begin{aligned}
\mbox{\eqref{eq:eval_term3}}
&\leq m^{-1/2}C''\int_{\max(X_0,1)}^{m^{1/4}+1}\frac{1+\log t+\lambda t^2}{t^3}dt\nonumber\\
&= m^{-1/2}C''\left[-\frac{3+2\log t}{4t^2} +\lambda\log t \right]_{\max(X_0,1)}^{m^{1/4}+1}\nonumber\\
&\leq m^{-1/2}C''\left( \frac{1+\log(\max(1,X_0))}{X_0^2} + \lambda \log\left(\frac{m^{1/4}+1}{\max(X_0,1)}\right)\right)\nonumber\\
&\leq m^{-1/2}C''\left( \frac{1+\log(\max(1,X_0))}{X_0^2} + \lambda \log\left(m^{1/4}+1\right)\right).\label{eq:eval_term3_simple}\end{aligned}$$ The second part of Equation can be upper bounded as desired, thanks to Condition : $$\begin{aligned}
m^{-1/2}C'' \lambda \log(m^{1/4}+1)
&\leq m^{-1/2}C''\lambda(1+\log m)\nonumber\\
&\leq C'' \lambda^2\gamma^{-2k}.\label{eq:eval_term3_1}\end{aligned}$$ For the first part, let us distinguish the cases $\frac{8}{C'}\lambda\gamma^{2k}\leq 1$ and $\frac{8}{C'}\lambda\gamma^{2k}>1$.
In the case where $\frac{8}{C'}\lambda\gamma^{2k}\leq 1$, we see (in a similar way as in Equation ) that $$c_1 \frac{\gamma^{k/3}}{\lambda^{1/3}}\leq X_0\leq c_2 \frac{\gamma^{k/3}}{\lambda^{1/3}},$$ so $$\begin{aligned}
m^{-1/2}\left(\frac{1+\log(\max(1,X_0))}{X_0^2}\right)
&\leq m^{-1/2}C'''\left(1+\log(\max(1,\gamma^k/\lambda))\right)\frac{\lambda^{2/3}}{\gamma^{2k/3}}\nonumber\\
&= m^{-1/2}C'''\left(1+\log(\max(1,\gamma^k/\lambda))\right)\lambda^2\gamma^{-2k} \left(\frac{\gamma^k}{\lambda}\right)^{4/3}\nonumber\\
&\leq C''' \lambda^2 \gamma^{-2k}.\label{eq:eval_term3_2}\end{aligned}$$ For the last equality, we have used Condition .
In the case where $\frac{8}{C'}\lambda\gamma^{2k}>1$, as we have already seen, $\sqrt{\lambda}X_0$ is bounded away from $0$, so, for some constant $C'''>0$, $$X_0\geq C''' \lambda^{-1/2},$$ which implies $$\begin{aligned}
m^{-1/2}\left(\frac{1+\log(\max(1,X_0))}{X_0^2}\right)
&\leq C'''' m^{-1/2}\lambda \left(1+\log(\max(1,\lambda^{-1/2}))\right)\nonumber\\
&\leq m^{-1/2}C'''' \lambda \left(1+\log\left(\max\left(1,\frac{1}{\lambda\gamma^{-2k}}\right)\right)\right).\label{eq:eval_term3_final}\end{aligned}$$ From Condition , we know that $$\begin{gathered}
\lambda \gamma^{-2k}\geq m^{-1/2}(1+\log m);\\
\Rightarrow\quad\quad
\frac{1+\log\left(\max\left(1,\frac{1}{\lambda\gamma^{-2k}}\right)\right)}{\lambda\gamma^{-2k}}\leq m^{1/2} \frac{1+\log\left(\frac{m^{1/2}}{1+\log m}\right)}{1+\log m}\leq m^{1/2}.\end{gathered}$$ We plug this into Equation and get $$\label{eq:eval_term3_3}
m^{-1/2}\left(\frac{1+\log(\max(1,X_0))}{X_0^2}\right)
\leq C'''' \lambda^2\gamma^{-2k}.$$ Finally, we combine Equations , and . With Equation , they show that $$\mbox{\eqref{eq:eval_term3}}\leq \mathcal{C}\lambda^2\gamma^{-2k},$$ for some constant $\mathcal{C}>0$.
### Proof of Lemma \[lem:maj\_esp\]\[sss:maj\_esp\]
There exist a constant $C>0$ depending only on $\epsilon$ such that, for any fixed unit-normed $x,y$ such that $$|\scal{x_0}{x}|\leq (1-\epsilon)||x_0||\,||x||\quad\mbox{and}\quad
|\scal{x_0}{y}|\leq (1-\epsilon)||x_0||\,||y||,$$ and any $j=1,\dots,m$, $$|\E(Z_j|a_j^*x_0)| \leq C \min\left(1,||x-y||\left(1+\frac{|a_j^*x_0|}{||x_0||}\right)\right).$$
As $Z_j=\phase(a_j^*x)\phase(\overline{a_j^*x_0})-\phase(a_j^*y)\phase(\overline{a_j^*x_0})$, $$|\E(Z_j|a_j^*x_0)| = |\E(\phase(a_j^* x)|a_j^*x_0)-\E(\phase(a_j^*y)|a_j^*x_0)|.$$ As in the proof of Lemma \[lem:var\_bound\], we write $$x=\alpha_x x_0+x'\mbox{ and }y=\alpha_yx_0+\beta x'+y'',$$ where $\alpha_x,\alpha_y,\beta$ are complex numbers and $x',y''\in\C^n$ satisfy $\scal{x'}{x_0}=\scal{y''}{x_0}=\scal{x'}{y''}=0$. We recall Equations to : $$\begin{gathered}
||x'||\geq \sqrt{\epsilon(2-\epsilon)}\geq \sqrt{\epsilon};\tag{\ref{eq:var_x_prime}}\\
|\beta-1|=\frac{|\scal{y-x}{x'}|}{||x'||^2}\leq \frac{1}{\sqrt{\epsilon}}||y-x||;\tag{\ref{eq:var_beta}}
\\
||\alpha_xx_0-\alpha_yx_0||=\frac{|\scal{x-y}{x_0}|}{||x_0||}\leq ||x-y||;\tag{\ref{eq:var_alpha}}
\\
||y''||=\frac{|\scal{y-x}{y''}|}{||y''||}\leq ||x-y||.\tag{\ref{eq:var_y_seconde}}\end{gathered}$$ The variable $Z_j$ is bounded in modulus by $2$, so the desired inequality holds for $||x-y||\geq\sqrt{\epsilon}/2$ if we choose $C\geq 4/\sqrt{\epsilon}$. In what follows, we assume that $||x-y||<\sqrt{\epsilon}/2$, which notably guarantees that $|\beta|>1/2$.
The random variables $a_j^*x_0,a_j^*x'$ and $a_j^*y''$ are independent complex Gaussians, with respective variances $||x_0||^2,||x'||^2,||y''||^2$. Thus, $$\begin{aligned}
\E(\phase(a_j^*x)|a_j^*x_0)
&=\E\left(\phase\left(\frac{\alpha_x}{||x'||}a_j^*x_0 + \frac{a_j^* x'}{||x'||}\right)\Bigg| a_j^*x_0\right)\nonumber\\
&=\frac{1}{\pi}\int_\C \phase\left(\frac{\alpha_x}{||x'||}a_j^*x_0 + z \right)e^{-{|z|^2}}d^2z,\label{eq:int_phase_x}\end{aligned}$$ and similarly, $$\begin{aligned}
\E(\phase(a_j^*y)|a_j^*x_0,a_j^*y'')
&=\frac{\phase(\beta)}{\pi}\int_\C \phase\left(\frac{\alpha_y}{\beta ||x'||}a_j^*x_0 +\frac{a_j^* y''}{\beta ||x'||} + z \right)e^{-{|z|^2}}d^2z.
\label{eq:int_phase_y}\end{aligned}$$ The function $$a\in\C\to
\frac{1}{\pi}\int_\C\phase(a+z)e^{-{|z|^2}}d^2z
=\frac{1}{\pi}\int_\C\phase(z)e^{-{|z-a|^2}}d^2z$$ is Lipschitz (as can be seen by derivation under the integral sign). If we denote by $D>0$ the Lipschitz constant, Equations and imply that $$\begin{aligned}
&\left|\E(\phase(a_j^*x)|a_j^*x_0)-\overline{\phase(\beta)}\E(\phase(a_j^*y)|a_j^*x_0,a_j^*y'')\right|\\
&\quad\quad \leq D\left|\left|\frac{\alpha_x}{||x'||}a_j^*x_0-\left(\frac{\alpha_y}{\beta ||x'||}a_j^*x_0 +\frac{a_j^* y''}{\beta ||x'||}\right)\right|\right|\\
&\quad\quad\leq D \left(||x-y|| \frac{|a_j^*x_0|}{||x_0||}
\left(\frac{1}{\sqrt{\epsilon}}+\frac{2}{\epsilon}\right)
+\frac{2}{\sqrt{\epsilon}} |a_j^* y''|\right)\\
&\quad\quad\leq D ||x-y|| \left( \frac{|a_j^*x_0|}{||x_0||}
\left(\frac{1}{\sqrt{\epsilon}}+\frac{2}{\epsilon}\right)
+\frac{2}{\sqrt{\epsilon}} \frac{|a_j^* y''|}{||y''||}\right).\end{aligned}$$ For the last two inequalities, we have used Equations to . We finally take the expectation over $a_j^* y''$; by triangular inequality, $$\begin{aligned}
&\left|\E(\phase(a_j^*x)|a_j^*x_0)-\overline{\phase(\beta)}\E(\phase(a_j^*y)|a_j^*x_0)\right|\\
&\quad\quad
\leq D ||x-y|| \left( \frac{|a_j^*x_0|}{||x_0||}
\left(\frac{1}{\sqrt{\epsilon}}+\frac{2}{\epsilon}\right)
+\sqrt{\frac{\pi}{\epsilon}} \right)\\
&\quad\quad\leq C ||x-y||\left(1+\frac{|a_j^*x_0|}{||x_0||}\right),\end{aligned}$$ when $C>0$ is large enough. Additionally, $$\begin{aligned}
&\left|\E(\phase(a_j^*y)|a_j^*x_0)-\overline{\phase(\beta)}\E(\phase(a_j^*y)|a_j^*x_0)\right|\\
&\quad\quad \leq |1-\beta|\\
&\quad\quad \leq 2\frac{|1-\beta|}{|\beta|}\\
&\quad\quad \leq \frac{4}{\sqrt{\epsilon}}||y-x||.\end{aligned}$$ So by triangular inequality, $$\begin{aligned}
&\left|\E(\phase(a_j^*x)|a_j^*x_0)-\E(\phase(a_j^*y)|a_j^*x_0)\right|\\
&\quad\quad\leq C' ||x-y||\left(1+\frac{|a_j^*x_0|}{||x_0||}\right),\end{aligned}$$
We also have $$|\E(Z_j|a_j^*x_0)|\leq C,$$ for any constant $C\geq 2$, so $$|\E(Z_j|a_j^*x_0)|\leq C\min\left(1,||x-y||\left(1+\frac{|a_j^*x_0|}{||x_0||}\right)\right)$$ when $C>0$ is large enough.
### Proof of Lemma \[lem:esp\_exp\_bound2\]\[sss:esp\_exp\_bound2\]
There exist constants $c,C'>0$, that depend only on $\gamma$ and $\epsilon$, such that, for any $\lambda\in[-c;c]$, $$\begin{aligned}
\log\left(\E\left(e^{\lambda\Re\left(|a_j^*x_0|^2\E(Z_j|Ax_0)-\E(|a_j^*x_0|^2Z_j)\right)}\right) \right)
&\leq C' \lambda^2 \gamma^{-2k},\\
\mbox{and }
\log\left(\E\left(e^{\lambda\Im\left(|a_j^*x_0|^2\E(Z_j|Ax_0)-\E(|a_j^*x_0|^2Z_j)\right)}\right) \right)
&\leq C' \lambda^2 \gamma^{-2k}.\end{aligned}$$
We only prove the first inequality; the proof of the second one is identical. We assume that $\lambda$ is positive; the same reasoning holds with minor modifications when $\lambda$ is negative.
To simplify the notations, we set $$\mathcal{Z}_j=|a_j^*x_0|^2\E(Z_j|a_j^*x_0)
-\E(|a_j^*x_0|^2Z_j).$$ We recall from Equation that $$|a_j^*x_0|^2 |\E(Z_j|a_j^*x_0)|\leq 2C |a_j^*x_0|^2 \min(1,\gamma^{-k}(1+|a_j^*x_0|)).$$ As a consequence, because $a_j^*x_0$ is a complex Gaussian random variable with variance $||x_0||^2=1$, $$\begin{aligned}
|\E(|a_j^*x_0|^2Z_j)|
&=|\E\left(|a_j^*x_0|^2 \E(Z_j|a_j^*x_0)\right)|\\
&\leq 2C \E\left( |a_j^*x_0|^2 \min(1,\gamma^{-k}(1+|a_j^*x_0|))\right)\\
&\leq 2C\gamma^{-k}\E(|a_j^*x_0|^2(1+|a_j^*x_0|))\\
&=2C\left(1+\frac{3}{4}\sqrt{\pi}\right)\gamma^{-k}.\end{aligned}$$ Combining this with Equation , we see that there exists a constant $C''>0$ such that $$\begin{aligned}
|\mathcal{Z}_j|=
\Big||a_j^*x_0|^2\E(Z_j|a_j^*x_0)
-\E(|a_j^*x_0|^2Z_j)\Big| \leq C''(1+|a_j^*x_0|)^2\min(1,\gamma^{-k}(1+|a_j^*x_0|)).\end{aligned}$$ Let us note that, because $\E(\mathcal{Z}_j)=0$, $$\begin{aligned}
\log(\E(e^{\lambda\Re(\mathcal{Z}_j)}))
&\leq \E(e^{\lambda\Re(\mathcal{Z}_j)})-1\\
&=\E(e^{\lambda\Re(\mathcal{Z}_j)}-\lambda\Re(\mathcal{Z}_j)-1).\end{aligned}$$ The function $f:x\to e^{\lambda x}-\lambda x-1$ is non-decreasing over $\R^+$, and satisfies $f(x)\leq f(|x|)$ for any $x\in\R$. Hence, $$\begin{aligned}
&\log(\E(e^{\lambda\Re(\mathcal{Z}_j)}))\nonumber\\
\leq \E&\left(
e^{\lambda C''(1+|a_j^*x_0|)^2\min(1,\gamma^{-k}(1+|a_j^*x_0|))}
-\lambda C''(1+|a_j^*x_0|)^2\min(1,\gamma^{-k}(1+|a_j^*x_0|))
-1\right)\nonumber\\
&=\frac{1}{\pi}\int_\C \left(e^{\lambda C''(1+|z|)^2\min(1,\gamma^{-k}(1+|z|))}-\lambda C''(1+|z|)^2\min(1,\gamma^{-k}(1+|z|))-1\right)e^{-|z|^2}d^2z\nonumber\\
&=2\int_0^{+\infty} \left(e^{\lambda C''(1+r)^2\min(1,\gamma^{-k}(1+r))}-\lambda C''(1+r)^2\min(1,\gamma^{-k}(1+r))-1\right)re^{-r^2}dr\nonumber\\
&=2\int_1^{+\infty} \left(e^{\lambda C''r^2\min(1,\gamma^{-k}r)}-\lambda C''r^2\min(1,\gamma^{-k}r)-1\right)(r-1)e^{-(r-1)^2}dr\nonumber\\
&\leq C'''\int_0^{+\infty}\left(e^{\lambda C''r^2\min(1,\gamma^{-k}r)}-\lambda C''r^2\min(1,\gamma^{-k}r)-1\right)e^{-r^2/2}dr\nonumber\\
&= C'''\int_0^{\gamma^k}\left(e^{\lambda C''r^3\gamma^{-k}}-\lambda C''r^3\gamma^{-k}-1\right)e^{-r^2/2}dr\label{eq:esp_term1}\\
&\hskip 2cm
+ C'''\int_{\gamma^k}^{+\infty}\left(e^{\lambda C''r^2}-\lambda C''r^2-1\right)e^{-r^2/2}dr.\label{eq:esp_term2}\end{aligned}$$ We need to show that both components and are upper bounded by $C'\lambda^2\gamma^{-2k}$ for some constant $C'>0$ sufficiently large, provided that $|\lambda|\leq c$ for some constant $c>0$.
For Term , we use the fact that, when $r\leq C''^{-1/3}\gamma^{k/3}\lambda^{-1/3}$, $$\begin{gathered}
\lambda C''r^3\gamma^{-k}\leq 1;\\
\Rightarrow\quad
e^{\lambda C''r^3\gamma^{-k}}-\lambda C''r^3\gamma^{-k}-1
\leq (\lambda C''r^3\gamma^{-k})^2.\end{gathered}$$ It yields: $$\begin{aligned}
\mbox{\eqref{eq:esp_term1}}
&\leq C'''\int_0^{\min(\gamma^k,C''^{-1/3}\gamma^{k/3}\lambda^{-1/3})} (\lambda C''r^3\gamma^{-k})^2e^{-r^2/2}dr\nonumber\\
&\hskip 2cm+
C'''\int_{\min(\gamma^k,C''^{-1/3}\gamma^{k/3}\lambda^{-1/3})}^{\gamma^k}e^{\lambda C''r^3\gamma^{-k}}e^{-r^2/2}dr\nonumber\\
&\leq C''' C''^2\lambda^2\gamma^{-2k}\int_0^{+\infty}r^6e^{-r^2/2}dr
+
C'''\int_{\min(\gamma^k,C''^{-1/3}\gamma^{k/3}\lambda^{-1/3})}^{\gamma^k}e^{\lambda C''r^3\gamma^{-k}}e^{-r^2/2}dr.\label{eq:esp_term1_tmp}\end{aligned}$$ For the second term of this sum, if we assume that $$\lambda<\frac{1}{4C''},$$ we have $$\begin{aligned}
\int_{\min(\gamma^k,C''^{-1/3}\gamma^{k/3}\lambda^{-1/3})}^{\gamma^k}e^{\lambda C''r^3\gamma^{-k}}e^{-r^2/2}dr&= \int_{\min(\gamma^k,C''^{-1/3}\gamma^{k/3}\lambda^{-1/3})}^{\gamma^k}e^{r^2\left(\lambda C''r\gamma^{-k}-\frac{1}{2}\right)}dr\\
&\leq \int_{\min(\gamma^k,C''^{-1/3}\gamma^{k/3}\lambda^{-1/3})}^{\gamma^k}e^{r^2\left(\lambda C''-\frac{1}{2}\right)}dr\\
&\leq \int_{\min(\gamma^k,C''^{-1/3}\gamma^{k/3}\lambda^{-1/3})}^{\gamma^k}e^{-r^2/4}dr\\
&\leq \int_{C''^{-1/3}\gamma^{k/3}\lambda^{-1/3}}^{+\infty}e^{-r^2/4}dr\\
&\leq C''' \frac{e^{-\left(C''^{-1/3}\gamma^{k/3}\lambda^{-1/3}\right)^2/4}}{C''^{-1/3}\gamma^{k/3}\lambda^{-1/3}}\\
&\leq C'''' \lambda^2\gamma^{-2k}.\end{aligned}$$ For the last inequality, we have used the fact that there exists a constant $D>0$ such that $e^{-x}\leq D x^{-5/2}$, for all $x>0$.
Plugging this into Equation , we get $$\label{eq:esp_term1_final}
\mbox{\eqref{eq:esp_term1}}\leq C'\lambda^2\gamma^{-2k}.$$ For Term , still under the assumption $\lambda<1/(4C'')$, $$\begin{aligned}
\mbox{\eqref{eq:esp_term2}}&\leq
C''' \int_{\gamma^k}^{\max(\gamma^k,(\lambda C'')^{-1/2})}(\lambda C''r^2)^2 e^{-r^2/2}dr
+C''' \int_{\max(\gamma^k,(\lambda C'')^{-1/2})}^{+\infty}e^{\lambda C''r^2}e^{-r^2/2}dr\nonumber\\
&\leq C'''C''^2 \lambda^2\int_{\gamma^k}^{+\infty}r^4e^{-r^2/2}dr
+C'''\int_{\max(\gamma^k,(\lambda C'')^{-1/2})}^{+\infty}e^{-r^2/4}dr\nonumber\\
&\leq C''''\left(\lambda^2 \gamma^{3k}e^{-\gamma^{2k}/2}+\min\left(\frac{e^{-\gamma^{2k}/4}}{\gamma^k},\sqrt{\lambda C''}e^{-1/(4\lambda C'')} \right)\right)\nonumber\\
&\overset{(*)}{\leq} \tilde C(\lambda^2 \gamma^{-2k}+\min(\gamma^{-4k},\lambda^4))\nonumber\\
&\leq 2\tilde C\lambda^2\gamma^{-2k}.\label{eq:esp_term2_final}\end{aligned}$$ For Inequality $(*)$, we have used the existence of a constant $D$ such that, for all $k$, $\gamma^{3k}e^{-\gamma^{2k}/2}\leq D\gamma^{-2k}$ and, for all $\lambda$ staying in a bounded interval, $\sqrt{\lambda}e^{-1/(4\lambda C'')}\leq D\lambda^4$.
Equations and , combined with Equation , show that, when $\lambda<1/(4C'')$, $$\log(\E(e^{\lambda\Re(\mathcal{Z}_j)}))
\leq C' \lambda^2\gamma^{-2k},$$ for some constant $C'>0$ that depends only upon $\gamma$.
### Proof of Lemma \[lem:min\_f\]\[sss:min\_f\]
For any $t\in\R^+$, we set $$f(t)=\E\left(\overline{Z_1}|Z_1|\phase\left(Z_1+t Z_2\right)\right).$$ The function $f$ is real-valued. For any $\gamma>0$, there exist $\delta>0$ such that $$\forall t\in[\gamma;+\infty[,\quad\quad
f(t)\geq \frac{1+\delta}{\sqrt{1+t^2}}.$$
As $(\overline{Z}_1,\overline{Z}_2)$ has the same distribution as $(Z_1,Z_2)$, $$\forall t\in\R^+,\quad\quad
f(t) = \E(Z_1|Z_1|\phase(\overline{Z}_1+t\overline{Z}_2))=\overline{f(t)},$$ so $f(t)$ is a real number, for any $t\geq 0$.
Let us now show the second part of the result. We have $$\begin{aligned}
f(t)
&=\frac{1}{\pi^2}\int_{\C^2}\overline{z_1}|z_1|\phase(z_1+tz_2)e^{-|z_1|^2}e^{-|z_2|^2}d^2z_1d^2z_2\\
&=\frac{1}{\pi^2}\int_{\C^2}\overline{y_1}|y_1|\phase(y_2)e^{-|y_1|^2}e^{-|y_2-y_1/t|^2}d^2y_1d^2y_2\\
&=\frac{1}{\pi^2}\int_{\C^2}\overline{y_1}|y_1|\phase(y_2)e^{-|y_1|^2}e^{-|y_2|^2}
\left(\sum_{k\geq 0}\frac{1}{k!}\left(y_2\overline{y_1}/t+y_1\overline{y_2}/t-|y_1|^2/t^2\right)^k\right)
d^2y_1d^2y_2\\
&=\frac{1}{\pi^2}\sum_k
\sum_{k_1+k_2\leq k}\frac{(-1)^{k-(k_1+k_2)}}{k_1!k_2!(k-k_1-k_2)!}
\frac{1}{t^{2k-(k_1+k_2)}}\times\\
&\hskip 4cm
\int_{\C^2}y_1^{k-k_1} \overline{y_1}^{k-k_2+1}|y_1|y_2^{k_1}\overline{y_2}^{k_2}\phase(y_2)e^{-|y_1|^2}e^{-|y_2|^2}
d^2y_1d^2y_2\\
&\overset{(*)}{=}\frac{1}{\pi^2}\sum_k
\sum_{2k_1+1\leq k}\frac{(-1)^{k-2k_1-1}}{k_1!(k_1+1)!(k-2k_1-1)!}\frac{1}{t^{2k-2k_1-1}}
\int_{\C^2}|y_1|^{2(k-k_1)+1}|y_2|^{2k_1+1}e^{-|y_1|^2}e^{-|y_2|^2}
d^2y_1d^2y_2\\
&\overset{(**)}{=}\sum_{l}\frac{1}{t^{2l+1}}
\left(\frac{1}{\pi}\int_{\C^2}|y|^{2l+3}e^{-|y|^2}d^2y\right)\sum_{k_1\leq l}
\frac{(-1)^{k_1+l}}{k_1!(k_1+1)!(l-k_1)!}
\left(\frac{1}{\pi}\int_{\C^2}|y|^{2k_1+1}e^{-|y|^2}d^2y\right)\\
&\overset{(***)}{=}\sum_{l}\frac{\pi}{t^{2l+1}}
(l+1)(l+2)\binom{2(l+2)}{l+2}
\sum_{k_1\leq l}
(-1)^{k_1+l}\binom{2(k_1+1)}{k_1+1}
\binom{l}{k_1}
2^{-2(l+k_1+3)}
.\end{aligned}$$ Equality $(*)$ is true because the integral is zero if $k_2\ne k_1+1$, as can be seen with a change of variable $y_1\to uy_1$ for $u$ a complex number of modulus $1$. Equality $(**)$ is obtained by setting $l=k-k_1-1$. Equality $(***)$ is a consequence of the following inequality, valid for all odd $K$: $$\frac{1}{\pi}\int_\C |y|^Ke^{-|y|^2}d^2y=\sqrt{\pi}2^{-K}\frac{K!}{\left(\frac{K-1}{2}\right)!}.$$
This reasoning is valid only for $t$ large enough; for small values of $t$, the series may not converge. We see that, in order for all the involved series to be absolutely convergent, it is enough that the following one is absolutely convergent: $$\begin{aligned}
&\sum_k
\sum_{k_1+k_2\leq k}\frac{1}{k_1!k_2!(k-k_1-k_2)!}
\frac{1}{t^{2k-(k_1+k_2)}}\times\\
&\hskip 3cm
\int_{\C^2}\left|y_1^{k-k_1} \overline{y_1}^{k-k_2+1}|y_1|y_2^{k_1}\overline{y_2}^{k_2}\phase(y_2)e^{-|y_1|^2}e^{-|y_2|^2}\right|
d^2y_1d^2y_2.\end{aligned}$$ When $t\geq 2$, for example, this series can be upper bounded by $$\begin{aligned}
&\sum_k
\sum_{k_1+k_2\leq k}\frac{1}{k_1!k_2!(k-k_1-k_2)!}
\frac{1}{t^{2k-(k_1+k_2)}}
\int_{\C^2}|y_1|^{2k-(k_1+k_2)+2}|y_2|^{k_1+k_2}e^{-|y_1|^2}e^{-|y_2|^2}
d^2y_1d^2y_2\\
&=\int_\C\left(\sum_k \frac{1}{k!}|y_1|^2
\left(\frac{|y_1||y_2|}{t}+\frac{|y_1||y_2|}{t}+\frac{|y_1|^2}{t^2}\right)^k
e^{-|y_1|^2}e^{-|y_2|^2}\right)
d^2y_1d^2y_2\\
&=\int_\C |y_1|^2\exp\left(-|y_1|^2-|y_2|^2+2\frac{|y_1||y_2|}{t}+\frac{|y_1|^2}{t^2}
\right)d^2y_1d^2y_2\\
&\leq \int_\C |y_1|^2\exp\left(-\left(1-\frac{1}{t}-\frac{1}{t^2}\right)|y_1|^2-\left(1-\frac{1}{t}\right)|y_2|^2
\right)d^2y_1d^2y_2\\
&\leq \int_\C |y_1|^2\exp\left(-\frac{1}{4}|y_1|^2-\frac{1}{2}|y_2|^2
\right)d^2y_1d^2y_2<+\infty.\end{aligned}$$ So the series converge.
For any $l\in\N,k_1\in\{0,\dots,l\}$, we set $$\begin{gathered}
c_{l,k_1}=\binom{2(k_1+1)}{k_1+1}
\binom{l}{k_1}
2^{-2(l+k_1+3)};\\
C_l=(l+1)(l+2)\binom{2(l+2)}{l+2}\sum_{k_1\leq l}(-1)^{k_1+l}c_{l,k_1}.\end{gathered}$$ The series $\sum_{k_1\leq l}(-1)^{k_1+l}c_{l,k_1}$ is alternating, and we can check that $$\begin{aligned}
\max_{k_1\leq l}|c_{l,k_1}|=|c_{l,[l/2]}|.\end{aligned}$$ This allows us to see that $$\begin{aligned}
\left|\sum_{k_1\leq l}(-1)^{k_1+l}c_{l,k_1}\right|
&\leq \max_{k_1\leq l}|c_{l,k_1}|\\
&=c_{l,[l/2]}\\
&\leq \frac{1}{8\pi}\frac{1}{l2^l},\end{aligned}$$ We do not derive the second inequality in full detail: the principle is to compute the upper limit of the sequence $(c_{l,[l/2]}l2^l)_{l\in\N}$ with Sterling’s formula, then to study the variations of this sequence, to show that it is bounded by its upper limit. Hence, using this inequality and the fact that, for any $s$, $\binom{2s}{s}\leq 2^{2s}/\sqrt{\pi s}$, we see that $$|C_l|\leq \frac{l+1}{l}.\sqrt{\frac{l+2}{\pi}}\frac{2^{l+1}}{\pi}.$$ So for any $l\geq 3$, $$\begin{aligned}
|C_l|&\leq \frac{l2^{l+1}}{\pi^{3/2}}.\end{aligned}$$ We explicitly compute $C_0,C_1,C_2$: $$C_0=\frac{3}{8};\quad\quad
C_1=-\frac{15}{64};\quad\quad
C_2=\frac{105}{512}.$$ Hence, combining the previous results, for any $t\geq 2$, $$\begin{aligned}
f(t)&=\pi \sum_{l\geq 0}\frac{C_l}{t^{2l+1}}\\
&\geq \pi \sum_{l=0}^2\frac{C_l}{t^{2l+1}}-
-\frac{1}{\sqrt{\pi}}\sum_{l=3}^{+\infty}\frac{l2^{l+1}}{t^{2l+1}}\\
&=\pi\left(\frac{3}{8}\frac{1}{t}-\frac{15}{64}\frac{1}{t^3}+\frac{105}{512}\frac{1}{t^5}\right)
-\frac{16}{\sqrt{\pi}t^5}\frac{3t^2-4}{t^4-4}.\end{aligned}$$ From here, we can easily verify with a computer that, for any $t>2.5$, $$\label{eq:t_large}
f(t)>\frac{1.05}{\sqrt{1+t^2}}.$$
Let us now show that $f(t)>(1+t^2)^{-1/2}$ for any $t\in]0;2.5]$. If we set $$Y_1 = \frac{-tZ_1+Z_2}{\sqrt{1+t^2}}\quad\mbox{and}\quad
Y_2 = \frac{Z_1+tZ_2}{\sqrt{1+t^2}},$$ we see that $Y_1$ and $Y_2$ are independent Gaussian random variables, with variance $1$, and that $$f(t)=\frac{1}{1+t^2}\E\left((\overline{Y_2-tY_1})|Y_2-tY_1|\phase(Y_2)
\right).$$ We set $$g(t)=\E\left((\overline{Y_2-tY_1})|Y_2-tY_1|\phase(Y_2)\right).$$ A straight computation yields $$\begin{gathered}
g'(t)=\E\left(
\left(-\frac{3}{2}\overline{Y}_1|Y_2-tY_1|-\frac{1}{2}Y_1\frac{(\overline{Y_2-tY_1})^2}{|Y_2-tY_1|}\right)\phase(Y_2)
\right);
\label{eq:g_prime}\\
g''(t)=\E\left(
\left(\frac{3}{2}|Y_1|^2\phase(\overline{Y_2-tY_1})
+\frac{3}{4}\overline{Y}_1^2\phase(Y_2-tY_1)\right.\right.\nonumber\\
\left.\left.\hskip 6cm-\frac{1}{4} Y_1^2\phase(\overline{Y_2-tY_1})^3
\right)
\phase(Y_2)\right).\label{eq:g_seconde}\end{gathered}$$ For any $u,t>0$, we see by triangular inequality that $$\begin{aligned}
&\left|\phase(Y_2-tY_1)-\phase(Y_2-uY_1)\right|\\
&\quad\quad\leq
\left|\frac{Y_2-tY_1}{|Y_2-tY_1|}-\frac{Y_2-uY_1}{|Y_2-tY_1|}\right|
+\left|\frac{Y_2-uY_1}{|Y_2-tY_1|}-\frac{Y_2-uY_1}{|Y_2-uY_1|}\right|\\
&\quad\quad\leq 2|t-u|\frac{|Y_1|}{|Y_2-tY_1|},\end{aligned}$$ which also implies $$\begin{aligned}
\left|\phase(Y_2-tY_1)^3-\phase(Y_2-uY_1)^3\right|\leq 6|t-u|\frac{|Y_1|}{|Y_2-tY_1|}.\end{aligned}$$ Plugging this into Equation : $$\begin{aligned}
|g''(t)-g''(u)|
&\leq 6|t-u| \E\left(\frac{|Y_1|^3}{|Y_2-tY_1|}\right)\\
&\leq 6|t-u| \E\left(\frac{|Y_1|^3}{|Y_2|}\right)\\
&=\frac{9}{2}\pi |t-u|.\end{aligned}$$ We deduce from here that, for any $u,t$ such that $0\leq u\leq t$, $$\begin{aligned}
g(t)&=g(u)+(t-u)g'(u)+\frac{(t-u)^2}{2}g''(u)+\int_u^t(t-s)(g''(s)-g''(u))ds\\
&\geq g(u)+(t-u)g'(u)+\frac{(t-u)^2}{2}g''(u)-\frac{9}{2}\pi\int_u^t(t-s)(s-u)ds\\
&= g(u)+(t-u)g'(u)+\frac{(t-u)^2}{2}g''(u)-\frac{9}{2}\pi \frac{(t-u)^3}{6}.\end{aligned}$$ In $u=0$, Equations and allow us to compute $g'(0)$ and $g''(0)$: we have $g'(0)=0$ and $g''(0)=\frac{3}{2}$. Thus, from the last equation, for any $t\geq 0$, $$g(t)\geq 1 + \frac{3}{4}t^2-\frac{3}{4}\pi t^3,$$ which allows us to verify (with a computer) that, for any $t\in]0;0.1]$, $$f(t)=\frac{g(t)}{1+t^2}\geq \frac{1+\frac{3}{4}t^2-\frac{3}{4}\pi t^3}{1+t^2}>\frac{1}{\sqrt{1+t^2}}.$$ We can apply the same reasoning to values of $u$ that are different from $0$. Equations and do not appear to have a simple analytic expression when $u\ne 0$. They can however be computed with a computer. We do so for $u=0.1,0.2,0.3,0.4,\dots,2.4$, and successively show that the previous inequality also holds on the intervals $[0.1;0.2],[0.2,0.3],\dots,[2.7,2.5]$.
We have thus proven that $f(t)>(1+t^2)^{-1/2}$ for any $t\in]0;2.5]$. By compacity (as $f$ is continuous), it means that there exists $\delta>0$ such that $$\forall t\in[\gamma;2.5],\quad\quad
f(t)\geq \frac{1+\delta}{\sqrt{1+t^2}}.$$ Together with Equation , this implies the lemma.
### Proof of Proposition \[prop:controle\_G\]\[sss:controle\_G\]
Let us define the function $$\begin{array}{rccc}
G:&\C^2&\to&\C\\
&(a,b)&\to&1-\frac{1}{\pi}\Re\int_\C\phase(\overline{z+a})\phase(z+b)e^{-|z|^2}d^2z.
\end{array}$$ For some constant $c_1>0$, the following inequalities are true: $$\begin{aligned}
\forall a,b\in\C,\quad\quad
|\Re G(a,b)|&\leq c_1 |a-b|^2\max\left(1,\log\left(|a-b|^{-1}\right)\right),\\
|\Im G(a,b)|&\leq c_1 |a-b|.\end{aligned}$$
$$\begin{aligned}
|\Re G(a,b)|&=\frac{1}{\pi}\left|\Re\int_\C\left(1-\phase(\overline{z+a})\phase(z+b)\right)e^{-|z|^2}d^2z \right|\\
&=\frac{1}{\pi}\left|\Re\int_\C\left(1-\phase\left(1+\frac{b-a}{z+a}\right)\right)e^{-|z|^2}d^2z \right|\\
&\leq \frac{1}{\pi}\left|\Re\int_{|z+a|>2|b-a|} \left(1-\phase\left(1+\frac{b-a}{z+a}\right)\right)e^{-|z|^2}d^2z \right|\\
&\quad\quad+\frac{2}{\pi}\left|\int_{|z+a|\leq 2|b-a|} e^{-|z|^2}d^2z \right|\\
&\overset{(a)}{\leq} \frac{c_2}{\pi}\left|\int_{|z+a|>2|b-a|} \left|\frac{b-a}{z+a}\right|^2 e^{-|z|^2}d^2z \right|
+\frac{2}{\pi}\left|\int_{|z+a|\leq 2|b-a|} 1d^2z \right|\\
&\leq \frac{c_2}{\pi}\left|\int_{1\geq |z+a|>2|b-a|} \left|\frac{b-a}{z+a}\right|^2 e^{-|z|^2}d^2z \right|
+ \frac{c_2}{\pi}\left|\int_{|z+a|>1} \left|\frac{b-a}{z+a}\right|^2 e^{-|z|^2}d^2z \right|
+8|b-a|^2\\
& \leq \frac{c_2}{\pi}\left|\int_{1\geq |z+a|>2|b-a|} \left|\frac{b-a}{z+a}\right|^2d^2z \right|
+ \frac{c_2}{\pi}|b-a|^2\left|\int_{\C} e^{-|z|^2}d^2z \right|
+8|b-a|^2\\
& \leq c_1 |b-a|^2\max\left(1,\log\left(|b-a|^{-1}\right)\right).\end{aligned}$$
Inequality (a) comes from the fact that $z\in\C\to \Re(1-\phase(1+z))\in\R$ is a $\mathcal{C}^\infty$ function on $\{z\in\C,|z|<1/2\}$, and its derivative in $0$ is $0$ (because the function reaches a local minimum at this point). So by compacity, there exists a constant $c_2>0$ such that, for any $z$ verifying $|z|<1/2$, $$\left|\Re(1-\phase(1+z))\right|\leq c_2 |z|^2.$$
The proof of the second inequality is identical, except that we bound $\left|\Im\left(1-\phase\left(1+\frac{b-a}{z-a}\right)\right)\right|$ by $c_2\left|\frac{b-a}{z+a}\right|$ on the set $\{z,|z+a|>2|b-a|\}$.
Proof of Lemma \[lem:inside\_net\]\[ss:inside\_net\]
----------------------------------------------------
For any $c>0$, there exist $C_1,C_2,C_3>0$ such that, with probability at least $$1-C_1\exp(-C_2 m^{1/8}),$$ the following property holds for any unit-normed $x,y\in\C^n$, when $m\geq 2n^2$: $$|\scal{Ax_0}{b\odot\phase(Ax)}-\scal{Ax_0}{b\odot\phase(Ay)}|
\leq C_3||x_0||^2 nm^{1/4}
\quad\mbox{if }||x-y||\leq cm^{-7/2}.$$
We write $$\begin{aligned}
&|\scal{Ax_0}{b\odot\phase(Ax)}-\scal{Ax_0}{b\odot\phase(Ay)}|\\
&\quad =\left| \sum_{i=1}^m\overline{(Ax_0)_i}|(Ax_0)_i|\left(\phase((Ax)_i)-\phase((Ay)_i)\right)\right|\\
&\quad\leq \sum_{i=1}^m |(Ax_0)_i|^2\left|\phase((Ax)_i)-\phase((Ay)_i)\right|\\
&\quad\leq 2\sum_{i=1}^m |(Ax_0)_i|^2\min\left(1,\frac{\left|(Ax)_i-(Ay)_i\right|}{|(Ax)_i|}\right)\\
&\quad\leq 2\,\sum_{\mathclap{|(Ax)_i|\leq 1/m^2}} |(Ax_0)_i|^2
+2\,\sum_{\mathclap{|(Ax)_i|> 1/m^2}} |(Ax_0)_i|^2\frac{\left|(Ax)_i-(Ay)_i\right|}{|(Ax)_i|}\\
&\quad\leq 2\,\sum_{\mathclap{|(Ax)_i|\leq 1/m^2}} |(Ax_0)_i|^2
+2 m^2 |||A|||^3\, ||x-y|| \,||x_0||^2.\end{aligned}$$
From Proposition \[prop:davidson\], if $m\geq 2n^2\geq 2n$, $|||A|||\leq 3\sqrt{m}$ with probability at least $$1-2\exp(-m).$$ On this event, we can deduce from the previous inequality that, for any $x,y$ such that $||x-y||\leq cm^{-7/2}$, $$\begin{aligned}
&|\scal{Ax_0}{b\odot\phase(Ax)}-\scal{Ax_0}{b\odot\phase(Ay)}|\\
&\quad \leq 2\,\sum_{\mathclap{|(Ax)_i|\leq 1/m^2}} |(Ax_0)_i|^2
+ 54 m^{7/2}||x-y||\,||x_0||^2\\
&\quad \leq 2\,\sum_{\mathclap{|(Ax)_i|\leq 1/m^2}} |(Ax_0)_i|^2
+ 54 c ||x_0||^2.\end{aligned}$$
To upper bound the first term of the right-hand side, we use two auxiliary lemmas, proven in Paragraphs \[sss:Card\_Ix\] and \[sss:A\_I\].
\[lem:Card\_Ix\] For any unit-normed $x\in\C^n$, we define $I_x=\left\{i\in\{1,\dots,m\},|(Ax)_i|\leq \frac{1}{m^2}\right\}$. There exist $C_1,C_2>0$ such that, when $m\geq n^2$, the event $$\Big( \forall x, \Card I_x< nm^{1/8}\Big)$$ has probability at least $$1-C_1\exp(-C_2m^{1/8}).$$
\[lem:A\_I\] There exist $C>0$ such that, with probability at least $$1-\exp(-nm^{1/4}),$$ for any $I\subset\{1,\dots,m\}$ such that $\Card I \leq nm^{1/8}$, $$\sum_{i\in I}|(Ax_0)_i|^2 \leq C ||x_0||^2 nm^{1/4}.$$
We combine these lemmas with the last inequality. This proves that, with probability at least $$1-C_1\exp(-C_2m^{1/8}),$$ (for some constants $C_1,C_2>0$ possibly different from the ones introduced in Lemma \[lem:Card\_Ix\]), $$\begin{aligned}
|\scal{Ax_0}{b\odot\phase(Ax)}-\scal{Ax_0}{b\odot\phase(Ay)}|
&\leq ||x_0||^2 \left(2Cnm^{1/4} +54c \right),\\
&\leq C_3||x_0||^2 nm^{1/4},\end{aligned}$$ for all $x,y$ verifying $||x-y||\leq cm^{-7/2}$.
### Proof of Lemma \[lem:Card\_Ix\]\[sss:Card\_Ix\]
For any unit-normed $x\in\C^n$, we define $I_x=\left\{i\in\{1,\dots,m\},|(Ax)_i|\leq \frac{1}{m^2}\right\}$. There exist $C_1,C_2>0$ such that, when $m\geq n^2$, the event $$\Big( \forall x, \Card I_x< nm^{1/8}\Big)$$ has probability at least $$1-C_1\exp(-C_2m^{1/8}).$$
Let $\mathcal{M}\geq 1$ be temporarily fixed.
For any $n,m$, let $\mathcal{N}_{n,m}$ be a $\frac{1}{\mathcal{M}m^2}$-net of the unit sphere of $\C^n$. From [@vershynin Lemma 5.2], there is one of cardinality at most $$\left(1+4\mathcal{M} m^2\right)^{2n}\leq (5\mathcal{M}m^2)^{2n}.$$ We define two events: $$\begin{gathered}
\mathcal{E}_1=\left\{\forall x\in\mathcal{N}_{n,m},\Card\left\{i,|(Ax)_i|\leq \frac{2}{m^2}\right\}<nm^{1/8} \right\};\\
\mathcal{E}_2=\{\forall i\in\{1,\dots,m\},||a_i^*||\leq \mathcal{M} \}.\end{gathered}$$ (We recall that $a_i^*$ is the $i$-th line of $A$.)
On the intersection of these two elements, we have $\Card I_x<nm^{1/8}$ for any unit-normed $x\in\C^n$. Indeed, for any such $x$, there exists $x'\in\mathcal{N}_{n,m}$ such that $||x-x'||\leq 1/(\mathcal{M}m^2)$. For any $i\in I_x$, $$\begin{aligned}
|(Ax')_i| &\leq |(Ax)_i| + |a_i^*(x-x')|\\
&\leq \frac{1}{m^2} + ||a_i^*||\,||x-x'||\\
&\leq \frac{2}{m^2}.\end{aligned}$$ As a consequence, $I_x\subset\left\{i,|(Ax')_i|\leq \frac{2}{m^2}\right\}$, whose cardinality is strictly less than $nm^{1/8}$ because we are on event $\mathcal{E}_1$.
Let us find lower bounds on the probabilities of $\mathcal{E}_1$ and $\mathcal{E}_2$.
For any $x\in\mathcal{N}_{n,m}$, for any $i=1,\dots,m$, $$P\left(|(Ax)_i|\leq\frac{2}{m^2}\right)= 1-e^{-\frac{4}{m^4}}\leq \frac{4}{m^4},$$ because $(Ax)_i$ is a complex Gaussian random variable with variance $1$. So by Hoeffding’s inequality, for $x$ fixed, $$\begin{aligned}
P\left(\Card\left\{i,|(Ax)_i|\leq \frac{2}{m^2}\right\}\geq nm^{1/8} \right)
&=P\left(\sum_{i=1}^m 1_{|(Ax)_i|\leq 2/m^2}\geq nm^{1/8}\right)\\
&\leq P\left(\sum_{i=1}^m 1_{|(Ax)_i|\leq 2/m^2}\geq m\E\left(1_{|(Ax)_1|\leq 2/m^2}\right) + \left(nm^{1/8}-\frac{4}{m^3}\right)\right)\\
&\leq \exp\left(-\frac{4}{m^3}h\left(\frac{m^{3+1/8}n}{4}-1\right)\right),\end{aligned}$$ where $h$ is the function $t\to (1+t)\log(1+t)-t$.
We simplify: $$\begin{aligned}
P\left(\Card\left\{i,|(Ax)_i|\leq \frac{2}{m^2}\right\}\geq 2n \right)
&\leq \exp\left(-nm^{1/8} \log(m^{3+1/8}n/4)+nm^{1/8}-\frac{4}{m^3}\right)\\
&\leq \exp\left(-nm^{1/8} \left(\log(m^{3+1/8}n)-3\right) \right).\end{aligned}$$ Finally, as the cardinality of $\mathcal{N}_{n,m}$ is at most $(5\mathcal{M}m^2)^{2n}$, $$\begin{aligned}
P(\mathcal{E}_1)
&\geq 1 - (5\mathcal{M}m^2)^{2n}e^{-nm^{1/8} \left(\log(m^{3+1/8}n)-3\right)}\nonumber\\
&=1 - \exp\left(-nm^{1/8} \left(\log(m^{3+1/8}n)-3\right) + 2n\log(5\mathcal{M}m^2))\right).\label{eq:PE1}\end{aligned}$$
Let us now consider $\mathcal{E}_2$. For any $i$, $a_i^*$ is a random vector with $n$ independent random complex Gaussian coordinates, of variance $1$. Gaussian measure concentration results (see for example [@barvinok Proposition 2.2]) imply that, for any $\delta>0$, $$\begin{aligned}
P\left(||a_i^*||> \sqrt{n+\delta} \right)
&\leq \left(1+\frac{\delta}{n}\right)^ne^{-\delta}.\end{aligned}$$ For $\delta = \mathcal{M}^2-n$, we get $$\begin{aligned}
P\left(||a_i^*||> \mathcal{M} \right)
&\leq \left(\frac{\mathcal{M}^2}{n}\right)^n e^{-\left(\mathcal{M}^2-n\right)}\\
&\leq 3 \mathcal{M}^{2n}e^{-\mathcal{M}^2}.\end{aligned}$$ As a consequence, $$\label{eq:PE2}
P(\mathcal{E}_2)\geq 1- 3m\mathcal{M}^{2n}e^{-\mathcal{M}^2}.$$ We can take, for example, $\mathcal{M}=\sqrt{m}$. We evaluate Equations and for this value of $\mathcal{M}$ and get, when $m\geq n^2$, $$P(\mathcal{E}_1\cap\mathcal{E}_2)
\geq 1 - C_1e^{-C_2 m^{1/8}}.$$
### Proof of Lemma \[lem:A\_I\]\[sss:A\_I\]
There exist $C>0$ such that, with probability at least $$1-\exp(-nm^{1/4}),$$ for any $I\subset\{1,\dots,m\}$ such that $\Card I \leq nm^{1/8}$, $$\sum_{i\in I}|(Ax_0)_i|^2 \leq C ||x_0||^2 n m^{1/4}.$$
By homogeneity, we can assume $||x_0||=1$.
The random variables $(Ax_0)_1,\dots,(Ax_0)_m$ are independent and (complex) Gaussian with variance $1$. Hence, by Bernstein’s inequality for subexponential variables, there exist a constant $c>0$ such that, for any $t>0$, and for any fixed $I\subset\{1,\dots,m\}$, $$P\left(\sum_{i\in I}|(Ax_0)_i|^2 \geq \Card I +t \right)
\leq \exp\left(-c\min\left(t,\frac{t^2}{\Card I}\right)\right).$$ In particular, if $\Card I = nm^{1/8}$, $$P\left(\sum_{i\in I}|(Ax_0)_i|^2 \geq nm^{1/8} + \frac{2}{c}nm^{1/4} \right)
\leq \exp\left(-2nm^{1/4}\min\left(1,\frac{2}{c}m^{1/8}\right)\right).$$ So as soon as $m$ is large enough, $$P\left(\sum_{i\in I}|(Ax_0)_i|^2 \geq \frac{3}{c}nm^{1/4} \right)
\leq \exp(-2n m^{1/8}).$$ There are less than $m^{nm^{1/8}}=e^{nm^{1/8}\log m}$ subsets of $\{1,\dots,m\}$ with cardinality $nm^{1/8}$, so $$P\left(\exists I\mbox{ s.t. }\Card I\leq nm^{1/8},\sum_{i\in I}|(Ax_0)_i|^2 \geq \frac{3}{c}n m^{1/4} \right)
\leq \exp(-n m^{1/4}).$$
[^1]: MIT Institute for Data, Systems and Society; e-mail address: `[email protected]`.
[^2]: By “isotropic”, we mean that the law of the initial vector is invariant under linear unitary transformations.
[^3]: These matrix-vector multiplications can be computed without forming the whole matrix (which would require $O(mn^2)$ operations), because this matrix factorizes as $$\frac{1}{m}A^* \mathrm{Diag}(|Ax_0|^2\odot I) A,$$ where $I\in\R^m$ is such that $\forall i\leq m,I_i=1_{|A_ix_0|^2\leq\frac{9}{m}\sum_{j=1}^m|A_ix_0|^2}$.
|
ITEP/TH-07/15
ABSTRACT
[Reshetikhin-Turaev (a.k.a. Chern-Simons) TQFT is a functor that associates vector spaces to two-dimensional genus $g$ surfaces and linear operators to automorphisms of surfaces. The purpose of this paper is to demonstrate that there exists a Macdonald $q,t$-deformation – refinement – of these operators that preserves the defining relations of the mapping class groups beyond genus 1. For this we explicitly construct the refined TQFT representation of the genus 2 mapping class group in the case of rank one TQFT. This is a direct generalization of the original genus 1 construction of arXiv:1105.5117, opening a question if it extends to any genus. Our construction is built upon a $q,t$-deformation of the square of $q$-6j symbol of $U_q(sl_2)$, which we define using the Macdonald version of Fourier duality. This allows to compute the refined Jones polynomial for arbitrary knots in genus 2. In contrast with genus 1, the refined Jones polynomial in genus 2 does not appear to agree with the Poincare polynomial of the triply graded HOMFLY knot homology. ]{}
**
Introduction {#introduction .unnumbered}
============
Do Chern-Simons TQFT representations of mapping class groups of surfaces have non-trivial deformations? In the case of a torus, it is known [@Kirillov; @AS] that the answer is positive. Since these representations ultimately determine the TQFT knot invariants, as explained in [@AS], this implies existence of a deformation – often called refinement – of the HOMFLY polynomials of torus knots. [@AS] observed that refined torus knot invariants agree with the homological knot invariants – namely, the superpolynomials of [@superpoly], the Poincare polynomials for the triply graded knot homology – for all torus knots (colored by symmetric or antisymmetric representations). This observation was especially interesting since homological invariants of knots [@Khovanov; @KhovanovRozansky; @KhovanovRozanskyStrings] are generally computationally harder [@BarNatan] than TQFT invariants, so the observation of [@AS] led to an alternative, more accessible, way to study torus knot homology.
A natural question is how far-going the deformation of TQFT representations, described in [@Kirillov; @AS], actually is. There is an ongoing debate in the mathematics and physics community whether it can be extended beyond genus 1, or not. There are arguments both for and against such extension. In this paper, we hope to give convincing evidence that the deformation exists in genus 2, and is related to Macdonald polynomials as directly as in genus 1. This raises a question if this deformation can be similarly carried over in genus 3 and higher, possibly resulting in a full-scale Chern-Simons-Macdonald TQFT. The algebraic approach that we choose in this paper seems to be well-suited to answer this question, and we plan to continue investigating this problem in genus 3 and higher.
The genus 2 construction that we suggest shares all features of the genus 1 construction of [@AS], except one. What appears to break down is the striking close relation to homological Poincare polynomials. This was to be expected in the light of the conjecture of [@AS] that the refined TQFT computes an index on knot homologies, which accidentally happens to coincide with the Poincare polynomial for simple enough knots and representations. Already in genus 1, if one replaces torus knots by torus links with more than one connected component, or if one replaces the symmetric coloring representations with arbitrary Young diagrams, the literal equality between refined and Poincare polynomials no longer holds true. What happens is that, when one looks at the refined TQFT in increasing generality, signs inevitably start appearing in the coefficients of refined TQFT invariants. This could not happen for the actual Poincare polynomial, but is totally expectable from an index, which is, after all, an Euler characteristic w.r.t. one of the gradings of knot homology.
It appears that generalization to genus 2 knots makes the situation generic enough so that the coincidence with the Poincare polynomial is almost never reached – unless the knot is actually a torus knot and the above-discussed conditions on the coloring representations are met. It is enough to look at the simplest examples of twist knots ($4_1,6_1,8_1,\ldots$) in s. 6 below to see that they are very different from those of [@superpoly; @superpoly2]. This, unfortunately, seems to imply that there is little to compare to on the knot homology side. One can only check agreement with Jones polynomials and topological invariance (the latter is a non-trivial check, that we report for a number of interesting knots below). At the same time, the very existence of refined Chern-Simons in genus 2 suggests existence of extra grading(s) in knot homology, in addition to those already known. If/once these extra gradings are defined, the index conjecture of [@AS] for refined Chern-Simons invariants could be checked.
We expect our results to agree with the doubly affine Hecke algebra (DAHA) approach to deformed knot invariants [@DAHA0; @DAHA1; @DAHA2]. While DAHA computations for most genus 2 knots are not available yet, some of them can be computed using a generalization of DAHA described in [@CherednikPi]: for example, we were able to confirm the matching for the $6_1$ knot [@CherednikLetter]. Our results suggest that spherical DAHA (a.k.a. the elliptic Hall algebra) has a genus 2 analog, generated by knot operators of refined Chern-Simons theory on a genus 2 surface. This will be studied elsewhere.
####
From the Reshetikhin-Turaev algebraic viewpoint on TQFT [@Reshetikhin; @ReshetikhinKirillov; @ReshetikhinTuraev; @TuraevBlueBook] based on representation theory of the quantum group $U_q(sl_2)$, the present paper relies upon a curious fact: while the q-6j symbols of $U_q(sl_2)$ (and associated fundamental identities such as the pentagon and Yang-Baxter equations) do not seem to admit nice Macdonald deformations, *their squares do*:
$$\begin{aligned}
\left\{\left\{\begin{array}{ccc} j_{12} & j_{13} & j_{23} \\ j_{34} & j_{24} & j_{14} \end{array} \right\}\right\}_{q,t} = \left\{\begin{array}{ccc} j_{12} & j_{13} & j_{23} \\ j_{34} & j_{24} & j_{14} \end{array} \right\}_q^2 \ + \ O(q - t)\end{aligned}$$
\
Since the q-6j symbols enter the genus 2 representations only in the squared form, this is enough for the purposes of present paper. The object in the l.h.s. of this equality is an interesting new quantity, which we define and discuss in certain detail in this paper. It is an intriguing question what exactly is the representation theory meaning of this deformation. This observation can also have important consequences for the full refined TQFT, if it exists: it suggests that refined Chern-Simons theory is less local, than usual Chern-Simons theory, since some quantities (the squares of q-6j symbols) that used to be broken up into elementary constituents (the individual q-6j symbols) no longer do so.
The same idea echoes in a different (though related) TQFT, the Turaev-Viro [@TuraevViro] a.k.a. BF theory, where the q-6j symbol is an elementary building block of a 3-manifold invariant – a local weight associated to a single tetrahedron of an arbitrary triangulation. The square of the q-6j symbol is then the weight associated with the simplest triangulation of a 3-sphere into two tetrahedra. The fact that the weight of the whole triangulation admits a deformation, but the local weight of a single tetrahedron does not, might suggest a non-local Macdonald deformation of Turaev-Viro theory. This interesting possibility also needs to be investigated.
** ![The $g+1$ Dehn twists around the A-cycles.[]{data-label="Atwists"}](A-twist.eps "fig:"){width="35.00000%"}
![The $g$ Dehn twists around the B-cycles.[]{data-label="Btwists"}](B-twist.eps){width="30.00000%"}
TQFT representations of mapping class groups
============================================
It is well-known [@Humphries] that the mapping class group of a genus $g$ closed oriented two-dimensional surface is generated by $2g+1$ Dehn twists along the $A$- and $B$-cycles, shown on Fig.\[Atwists\] and Fig.\[Btwists\], resp., that satisfy algebraic relations [@Wajnryb]. These relations can be divided into three types: the degree 2 and 3 relations of a braid group,
$$\begin{aligned}
A_n A_m = A_m A_n, \ \ \ \forall \ n,m\end{aligned}$$
$$\begin{aligned}
B_n B_m = B_m B_n, \ \ \ \forall \ n,m\end{aligned}$$
$$\begin{aligned}
A_n B_m = B_m A_n, \ \ \ \forall \ n,m \ \ \mbox{such that} \ \ {\mathit i}( A_n, B_m ) = 0\end{aligned}$$
$$\begin{aligned}
A_n B_m A_n = B_m A_n B_m, \ \ \ \forall \ \ n,m \ \ \mbox{such that} \ \ {\mathit i}( A_n, B_m ) = 1\end{aligned}$$
\
where $i$ is the intersection form, and more exotic higher degree relations, that reflect the difference between mapping class groups and braid groups. One could say that a mapping class group is a braid group with additional higher degree relations. We do not write these additional relations here in full generality, one can easily find them in [@Wajnryb]. In the case of genus $g=2$ these additional relations become especially simple and we present a complete set of them in eq. (\[RelationsExtra\]).
**
![Basis vectors in the TQFT vector space, associated to a genus $g$ surface.[]{data-label="basis-g"}](basis.eps){width="60.00000%"}
Rank one, level $K$ Chern-Simons [@Witten; @ReshetikhinTuraev; @TuraevBlueBook] TQFT [@Atiyah] is a functor that associates to that surface a vector space, spanned by vectors labeled as on Fig.\[basis-g\]., where $j_1, \ldots, j_{g+1}$ and $j^\pm_1, \ldots, j^\pm_{g-2}$ are integers in $0, \ldots, K$ such that whenever a triple $(j, j^{\prime}, j^{\prime \prime})$ meets at a vertex, they satisfy the so-called admissibility condition
$$\begin{aligned}
| j^{\prime} - j^{\prime \prime}| \leq j \leq j^{\prime} + j^{\prime \prime}, \ \ \ j + j^{\prime} + j^{\prime \prime} = \mbox{ even number } \leq 2K\end{aligned}$$
\
It also associates linear maps to bordisms [@Atiyah]; in particular, this implies that the mapping class group of every surface is represented on its vector space by linear operators. To completely describe these representations, it suffices to describe the matrix elements of the generators $A_1, \ldots, A_{g+1}; B_1, \ldots, B_g$. This can be done using any formalism for Chern-Simons TQFT: either by representation theory of the quantum group $U_q(sl_2)$ [@Reshetikhin; @ReshetikhinKirillov; @ReshetikhinTuraev; @TuraevBlueBook] or equivalently by skein theory [@Masbaum1; @Masbaum2; @Roberts; @ACampo].
Unrefined TQFT representations for $g = 1,2$
============================================
In this paper, we focus specifically on the cases of $g = 1$ and $g = 2$. This is enough to demonstrate that TQFT representations admit Macdonald deformations beyond the torus case. In these cases the matrix elements of the TQFT representation can be actually expressed in a simple closed form, which is straightforward to prove using either of the methods of [@ReshetikhinTuraev; @TuraevBlueBook] or [@Masbaum1; @Masbaum2; @Roberts; @ACampo]. This form is suggestive of Macdonald deformations. We first describe this closed form, and then give a Macdonald deformation of it.
![Basis vectors in the TQFT vector space, cases $g = 1$ and $g = 2$.[]{data-label="basis-12"}](basis1.eps "fig:"){width="15.00000%"} ![Basis vectors in the TQFT vector space, cases $g = 1$ and $g = 2$.[]{data-label="basis-12"}](basis2.eps "fig:"){width="20.00000%"}
#### For genus 1,
the basis vectors are labeled by a single integer, $0 \leq j \leq K$, as on Fig.\[basis-12\]. Let us denote that basis vector $| j \rangle$. There are two generators, $A$ and $B$, the representations of which are given [@Witten] by the following formulas: for $q = e^{\frac{2 \pi i}{K + 2}}$,
$$\begin{aligned}
\langle i | \ A \ | j \rangle = q^{\ j^2/4 + j/2} \ \delta_{i j}\end{aligned}$$
$$\begin{aligned}
\langle i | \ A B A \ | j \rangle = \dfrac{1}{\sqrt{2K+4}} \ \big[(i+1)(j+1)\big], \ \ \ [x] \equiv \dfrac{q^{x/2} - q^{-x/2}}{q^{1/2} - q^{-1/2}}
\label{UnrefinedS}\end{aligned}$$
\
There is a single relation, $ABA = BAB$. The elements $S = ABA$ and $T = A^{-1}$ are often called the modular $S$- and $T$-matrices, because the relations they satisfy closely resemble the $SL(2,{\mathbb Z})$ relations: $S^4 = 1$ and $(ST)^3 = {\rm const} \cdot 1$, with the only difference being an unimportant constant that can be removed by rescaling $T$.
#### For genus 2,
the basis vectors are labeled by triples of integers, $0 \leq j_1,j_2,j_3 \leq K$, as on Fig.\[basis-12\], satisfying an admissibility condition. Let us denote that basis vector $| j_1,j_2,j_3 \rangle$. There are five generators, $A_1,A_2,A_3$ and $B_1,B_2$, with representations
$$\begin{aligned}
\langle i_1,i_2,i_3 | \ A_{n} \ | j_1,j_2,j_3 \rangle = q^{ \ j_n^2/4 + j_n/2} \ \delta_{i_1 j_1} \ \delta_{i_2 j_2} \ \delta_{i_3 j_3}, \ \ \ n = 1,2,3\end{aligned}$$
[ $$\begin{aligned}
\langle i_1,i_2,i_3 | \ B_1 \ | j_1,j_2,j_3 \rangle = \delta_{i_3 j_3} \ [i_1 + 1][i_2 + 1] \ \sum\limits_{s = 0}^{K} \ q^{-s^2/4-s/2} \ [s+1] \ \left\{\begin{array}{ccc} i_3 & i_2 & i_1 \\ s & j_1 & j_2 \end{array} \right\}_q^2\end{aligned}$$ $$\begin{aligned}
\langle i_1,i_2,i_3 | \ B_2 \ | j_1,j_2,j_3 \rangle = \delta_{i_1 j_1} \ [i_2 + 1][i_3 + 1] \ \sum\limits_{s = 0}^{K} \ q^{-s^2/4-s/2} \ [s+1] \ \left\{\begin{array}{ccc} i_3 & i_2 & i_1 \\ j_2 & j_3 & s \end{array} \right\}_q^2\end{aligned}$$]{} where the quantity in brackets is the q-6j symbol of the Hopf algebra $U_q(sl_2)$, [@ReshetikhinKirillov]:
[ $$\begin{aligned}
\left\{\begin{array}{ccc} j_{12} & j_{13} & j_{23} \\ j_{34} & j_{24} & j_{14} \end{array} \right\}_q = \sum\limits_{z} \ \dfrac{ (-1)^z [z+1]! }{ [J_{1} - z]! [J_2 - z]! [J_3 - z]! } \ \prod\limits_{1 \leq a < b < c \leq 4} \dfrac{\Delta\big( j_{ab}, j_{ac}, j_{bc} \big)}{ [z - j_{ab}/2 - j_{ac}/2 - j_{bc}/2 ]! }
\label{KRformula}\end{aligned}$$ $$\begin{aligned}
[x]! \equiv [1][2] \ldots [x]\end{aligned}$$ $$\begin{aligned}
2J_1 = j_{12} + j_{34} + j_{13} + j_{24}, \ 2J_2 = j_{12} + j_{34} + j_{23} + j_{14}, \ 2J_3 = j_{13} + j_{24} + j_{23} + j_{14}\end{aligned}$$ $$\begin{aligned}
\Delta_{ijk} = {\cal N}_{ijk} \ \left( \dfrac{[i/2 + j/2 - k/2]![i/2 - j/2 + k/2]![-i/2 + j/2 + k/2]!}{[i/2 + j/2 + k/2 + 1]!} \right)^{\frac{1}{2}}\end{aligned}$$]{}\
The coefficients ${\cal N}_{ijk}$ are often called Verlinde coefficients, and in this case[^1] are very simple: they take values 1 and 0 depending if the triple $(i,j,k)$ is admissible or not, resp. The generators satisfy the defining relations of the braid group,
$$\begin{aligned}
A_1 B_1 A_1 \propto B_1 A_1 B_1, \ A_2 B_1 A_2 \propto B_1 A_2 B_1\end{aligned}$$
$$\begin{aligned}
A_2 B_2 A_2 \propto B_2 A_2 B_2, \ A_3 B_2 A_3 \propto B_2 A_3 B_2\end{aligned}$$
$$\begin{aligned}
A_1 A_2 \propto A_2 A_1, \ \ \ A_1 B_2 \propto B_2 A_1, \ \ \ A_1 A_3 \propto A_3 A_1\end{aligned}$$
$$\begin{aligned}
B_1 B_2 \propto B_2 B_1, \ \ \ B_1 A_3 \propto A_3 B_1, \ \ \ A_2 A_3 \propto A_3 A_2\end{aligned}$$
\
and a few more exotic relations, which we now write explicitly [@Wajnryb]
$$\begin{aligned}
\nonumber ( A_1 B_1 A_2 )^4 \propto A_3^2, \ \ \ I^6 \propto 1, \ \ \ H^2 \propto 1 \\
\nonumber \\
H A_n \propto A_n H, \ \ \ H B_n \propto B_n H, \ \ \ \ \ \ \forall n
\label{RelationsExtra}\end{aligned}$$
\
with notations $I = A_1 B_1 A_2 B_2 A_3$ and $H = A_3 B_2 A_2 B_1 A_1 A_1 B_1 A_2 B_2 A_3$. Here, $\propto$ means that the matrices are equal up to a scalar multiple: this implies that TQFT representation is only projective, as it is well known to be the case in general [@Masbaum2].
Refined TQFT representation
===========================
There is an expectation that Chern-Simons TQFT representations of mapping class groups admit a one-parameter deformation, which is characterized, in particular, by deforming the $sl_N$ characters a.k.a. the Schur symmetric polynomials
[ $$\begin{aligned}
& \chi_{1}(x_1, \ldots, x_N) = \sum\limits_{i} x_i, \\
\\
& \chi_{2}(x_1, \ldots, x_N) = \sum\limits_{i} x_i^2 + \sum\limits_{i < j} x_i x_j, \\
\\
& \chi_{3}(x_1, \ldots, x_N) = \sum\limits_{i} x_i^3 + \sum\limits_{i < j} x_i^2 x_j + \sum\limits_{i < j < k} x_i x_j x_k, \ \ \ldots\end{aligned}$$]{}\
into the Macdonald polynomials [@Macdonald]:
[ $$\begin{aligned}
& M_{1}(x_1, \ldots, x_N) = \sum\limits_{i} x_i, \\
\\
& M_{2}(x_1, \ldots, x_N) = \sum\limits_{i} x_i^2 + \dfrac{(1-q^2)(1-t)}{(1-q)(1-qt)} \sum\limits_{i < j} x_i x_j, \\
\\
& M_{3}(x_1, \ldots, x_N) = \sum\limits_{i} x_i^3 + \dfrac{(1-q^3)(1-t)}{(1-q)(1-q^2t)} \sum\limits_{i < j} x_i^2 x_j + \dfrac{(1-q^2)(1-q^3)(1-t)^2}{(1-q)^2(1-qt)(1-q^2t)} \sum\limits_{i < j < k} x_i x_j x_k , \ \ \ldots\end{aligned}$$]{}\
Macdonald polynomials depend on two parameters $q$ and $t$, where $t = q^{\beta}$ and $\beta \in {\mathbb C}^{\star}$ is the deformation parameter, so that $\beta = 1$ is the undeformed point. These polynomials are especially simple in the case of rank one, i.e. $N = 2$ eigenvalues:
$$\begin{aligned}
\chi_{j}(x_1, x_2) = \dfrac{x_1^{j+1} - x_2^{j+1}}{x_1 - x_2}\end{aligned}$$
\
and, similarly,
$$\begin{aligned}
M_{j}(x_1, x_2) = \sum\limits_{l = 0}^{j} \ x_1^{j - l} x_2^l \ \prod\limits_{i = 0}^{l-1} \frac{[j - i]}{[j - i + \beta - 1]} \frac{[i + \beta]}{[i + 1]}\end{aligned}$$
\
**For genus 1**, a deformation of the TQFT representation has been described in [@Kirillov; @AS]. Let us briefly review it here, concentrating on the rank one, i.e. $N = 2$. The vector space of the refined TQFT remains the same, but the matrix elements of the generators $S$ and $T$ (or equivalently $A$ and $B$) deform,
$$\begin{aligned}
\langle i | \ T \ | j \rangle \equiv T_i \delta_{ij} \ = \ q^{-j^2/4} t^{-j/2} \ \delta_{i j}\end{aligned}$$
$$\begin{aligned}
\langle i | \ S \ | j \rangle \equiv S_{ij} \ = \ S_{00} \ q^{-ij/2} \ g_{i}^{-1} \ M_{i}\big( t^{\frac{1}{2}}, t^{\frac{-1}{2}} \big) M_{j}\big( t^{\frac{1}{2}} q^i, t^{\frac{-1}{2}} \big)\end{aligned}$$
\
where now $q = e^{\frac{2 \pi i}{K + 2 \beta}}, t = q^{\beta} = e^{\frac{2 \pi \beta i}{K + 2 \beta}}$ (or, equivalently, $t = e^{\frac{2 \pi i}{N}} q^{-K/N}$) and
$$\begin{aligned}
g_i = \prod\limits_{m = 0}^{i-1} \dfrac{[i - m] [m + 2 \beta]}{[i - m + \beta - 1][m + \beta + 1]}\end{aligned}$$
\
is the quadratic norm of the Macdonald polynomials under a natural orthogonality condition [@AS]. These refined operators satisfy the same relations, as the original ones,
$$\begin{aligned}
S^2 = 1, \ \ \ (ST)^3 = \mbox{ central }\end{aligned}$$
\
**For genus 2**, following the same path, we assume that the vector space is undeformed and the basis vectors are still labeled by triples of integers, $0 \leq j_1,j_2,j_3 \leq K$ satisfying an admissibility condition. We suggest the following formulas for the deformed representations of the five generators, $A_1,A_2,A_3$ and $B_1,B_2$:
$$\begin{aligned}
\langle i_1,i_2,i_3 | \ A_{\alpha} \ | j_1,j_2,j_3 \rangle = T_{j_{\alpha}}^{-1} \ \delta_{i_1 j_1} \ \delta_{i_2 j_2} \ \delta_{i_3 j_3}, \ \ \ \alpha = 1,2,3
\label{RefA}\end{aligned}$$
[ $$\begin{aligned}
\langle i_1,i_2,i_3 | \ B_1 \ | j_1,j_2,j_3 \rangle = \delta_{i_3 j_3} \ \dfrac{ \dim_{q,t}(i_1) \dim_{q,t}(i_2) }{ {\cal N}_{i_1 i_2 i_3} } \ \sum\limits_{s = 0}^{K} \ T_s \ \dim_{q,t}(s) \ \left\{\left\{\begin{array}{ccc} i_3 & i_2 & i_1 \\ s & j_1 & j_2 \end{array} \right\}\right\}_{q,t}
\label{RefB1}\end{aligned}$$ $$\begin{aligned}
\langle i_1,i_2,i_3 | \ B_2 \ | j_1,j_2,j_3 \rangle = \delta_{i_1 j_1} \ \dfrac{ \dim_{q,t}(i_2) \dim_{q,t}(i_3) }{ {\cal N}_{i_1 i_2 i_3} } \ \sum\limits_{s = 0}^{K} \ T_s \ \dim_{q,t}(s) \ \left\{\left\{\begin{array}{ccc} i_3 & i_2 & i_1 \\ j_2 & j_3 & s \end{array} \right\}\right\}_{q,t}
\label{RefB2}\end{aligned}$$]{} The logic behind this suggestion is simple: each part of the original formula is replaced by its Macdonald counterpart. E.g. $\dim_{q,t}(i) \ = \ S_{0 i} / S_{0 0} \ = \ M_{i}\big( t^{\frac{1}{2}}, t^{\frac{-1}{2}} \big) $ is a $q,t$-deformation of the quantum dimension $[i+1]$ of the $i$-th representation of $U_q(sl_2)$, ${\cal N}_{ijk}$ is a $q,t$-deformation of the Verlinde coefficients[^2] discussed in [@AS],
\
and the quantity in brackets is the deformation of the square (!) of the q-6j symbol,
$$\begin{aligned}
\left\{\left\{\begin{array}{ccc} j_{12} & j_{13} & j_{23} \\ j_{34} & j_{24} & j_{14} \end{array} \right\}\right\}_{q,t} = \left\{\begin{array}{ccc} j_{12} & j_{13} & j_{23} \\ j_{34} & j_{24} & j_{14} \end{array} \right\}_q^2 \ + \ O(q - t)\end{aligned}$$
\
that we define and describe in the next section. This is the main new ingredient, not present/seen in genus 1, and the central algebraic quantity of the present paper.
#### Conjecture I.
Operators (\[RefA\]), (\[RefB1\]), (\[RefB2\]) satisfy
$$\begin{aligned}
A_1 B_1 A_1 \propto B_1 A_1 B_1, \ A_2 B_1 A_2 \propto B_1 A_2 B_1\end{aligned}$$
$$\begin{aligned}
A_2 B_2 A_2 \propto B_2 A_2 B_2, \ A_3 B_2 A_3 \propto B_2 A_3 B_2\end{aligned}$$
$$\begin{aligned}
A_1 A_2 \propto A_2 A_1, \ \ \ A_1 B_2 \propto B_2 A_1, \ \ \ A_1 A_3 \propto A_3 A_1\end{aligned}$$
$$\begin{aligned}
B_1 B_2 \propto B_2 B_1, \ \ \ B_1 A_3 \propto A_3 B_1, \ \ \ A_2 A_3 \propto A_3 A_2\end{aligned}$$
$$\begin{aligned}
( A_1 B_1 A_2 )^4 \propto A_3^2, \ \ \ I^6 \propto 1, \ \ \ H^2 \propto 1\end{aligned}$$
$$\begin{aligned}
H A_n \propto A_n H, \ \ \ H B_n \propto B_n H, \ \ \ \ \ \ \forall n\end{aligned}$$
\
with notations $I = A_1 B_1 A_2 B_2 A_3$ and $H = A_3 B_2 A_2 B_1 A_1 A_1 B_1 A_2 B_2 A_3$ and, again, $\propto$ is used to stress that the representation is projective. While we cannot yet prove the conjecture in full generality, for any given $K > 0$ it is straightforward to prove by computing the matrices and checking the relations directly. We completed this verification for $1 \leq K \leq 8$; the following is the example of $K = 2$.
**
#### Example: K=2.
The basis of the TQFT vector space consists of 10 vectors
$$\begin{aligned}
|0, 0, 0\rangle, |1, 1, 0\rangle, |2, 2, 0\rangle, |1, 0, 1\rangle, |0, 1, 1\rangle, |2, 1, 1\rangle, |1, 2, 1\rangle, |2, 0, 2\rangle, |1, 1, 2\rangle, |0, 2, 2\rangle\end{aligned}$$
\
The generators are represented by $10 \times 10$ matrices: $B_1$ is represented by
{width="60.00000%"}
$B_2$ is represented by
{width="60.00000%"}
and $A_1,A_2,A_3$ are trivial. It is straightforward to check that all the relations of the mapping class group are satisfied. Here $Q = e^{\frac{\pi i}{2 \beta + 2}} = \sqrt{q}$ is a refinement parameter, that reduces to the standard TQFT value at $\beta = 1$, that is, $Q = e^{\frac{\pi i}{4}}$, $q = t = e^{\frac{\pi i}{2}}$. It is equally straightforward to produce such matrices for any $K$.
The deformation of the q-6j symbol squared
==========================================
It appears that the square of the $q$-6j symbol admits a $q,t$-deformation. The definition of this object is the following: it is the unique solution[^3] to the linear system of equations that we suggest to call the *Macdonald duality* equation:
[ $$\begin{aligned}
\left\{\left\{\begin{array}{ccc} j_{12} & j_{13} & j_{23} \\ j_{34} & j_{24} & j_{14} \end{array} \right\}\right\}_{q,t} \ = \ \sum\limits_{ i_{12},i_{13},i_{23},i_{14},i_{24},i_{34} = 0 }^{K} \ \prod\limits_{a < b} S_{j_{ab}, i_{ab}} \ \left\{\left\{\begin{array}{ccc} i_{34} & i_{24} & i_{14} \\ i_{12} & i_{13} & i_{23} \end{array} \right\}\right\}_{q,t}
\label{duality}\end{aligned}$$]{}\
that has the same symmetries (24 permutations) and zeroes (if any of the 4 triples are non-admissible) as the standard q-6j symbol. The representation-theory meaning of this quantity, covariant under Macdonald duality, remains to be seen. The equation is the Macdonald analog of the well-known *Fourier duality* of the square of the q-6j symbol, originally found in the Regge quantum gravity literature [@Duality1; @Duality2]:
[ $$\begin{aligned}
\left\{\begin{array}{ccc} j_{12} & j_{13} & j_{23} \\ j_{34} & j_{24} & j_{14} \end{array} \right\}_q^2 \ = \ \sum\limits_{ i_{12},i_{13},i_{23},i_{14},i_{24},i_{34} = 0 }^{K} \ \prod\limits_{a < b} \ S^{(q=t)}_{j_{ab}, i_{ab}} \ \left\{\begin{array}{ccc} i_{34} & i_{24} & i_{14} \\ i_{12} & i_{13} & i_{23} \end{array} \right\}_{q}^2\end{aligned}$$]{}\
The reason for this name is that the explicit form (\[UnrefinedS\]) of the unrefined $S$-matrix looks like a (discrete or difference) Fourier transform. The fact that the refined S-matrix is an analog and a generalization of the Fourier transform, in particular that it is self-dual ($S^2 = 1$), has been discussed in detail in [@CherednikFourier]. By solving Macdonald duality, we can compute any desired number of examples. A first few are as follows:
$$\begin{aligned}
\left\{\left\{\begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & 0 \end{array} \right\}\right\}_{q,t} \equiv 1\end{aligned}$$
$$\begin{aligned}
\left\{\left\{\begin{array}{ccc} 0 & 0 & 0 \\ 1 & 1 & 1 \end{array} \right\}\right\}_{q,t} = \dfrac{t^{1/2}(1 - qt)}{(1 + t)^2 (1 - q)}\end{aligned}$$
$$\begin{aligned}
\left\{\left\{\begin{array}{ccc} 0 & 1 & 1 \\ 2 & 1 & 1 \end{array} \right\}\right\}_{q,t} = \dfrac{t(1 - qt)^2}{(1-q)^2(1+t)^4}\end{aligned}$$
$$\begin{aligned}
\left\{\left\{\begin{array}{ccc} 0 & 2 & 2 \\ 2 & 2 & 2 \end{array} \right\}\right\}_{q,t} = \dfrac{t^{2}(1 - q^2t)^2(1-qt)^4(1-q^2t^2)(1-t)}{(1-q)^3(1+t)^4(1-q^3t)(1-qt^2)^4}\end{aligned}$$
$$\begin{aligned}
\left\{\left\{\begin{array}{ccc} 1 & 1 & 2 \\ 2 & 2 & 1 \end{array} \right\}\right\}_{q,t} = \dfrac{t^{3/2}(1-t)(1-qt)^3(1-q^2t^2)}{(1+t)^5(1-q)^3(1-qt^2)^2}\end{aligned}$$
$$\begin{aligned}
\nonumber & \left\{\left\{\begin{array}{ccc} 2 & 2 & 2 \\ 2 & 2 & 2 \end{array} \right\}\right\}_{q,t} = \dfrac{t^2(1+qt)(1-t)(1-q^2t)^3(1-qt)^6}{(1+t)^5(1-q)^4(1-q^3t)^3(1-qt^2)^5} \times \emph{} \\
\nonumber \\
& \emph{} \big( \ 1-2 t+q t-q t^2+3 q^2 t-2 q^2 t^2-2 q^3 t+3 q^3 t^2-q^4 t+q^4 t^2-2 q^5 t^2+q^5 t^3 \ \big)\end{aligned}$$
\
and so on. One can see that the non-vanishing quantities with one 0-index are
$$\begin{aligned}
\left\{\left\{\begin{array}{ccc} 0 & n & n \\ v & u & u \end{array} \right\}\right\}_{q,t} = \dfrac{ {\cal N}_{n,u,v} }{ \dim_{q,t}(n) \dim_{q,t}(u) }
\label{0index}\end{aligned}$$
\
and the non-vanishing quantities with one 1-index are
[$$\begin{aligned}
\left\{ \left\{ \begin{array}{ccc} 1 & n & n+1 \\ v & u+1 & u \end{array} \right\} \right\}_{q,t} = \dfrac{{\cal N}_{n+1,u-1,v}}{ \dim_{q,t}(n+1) \dim_{q,t}(u+1) } \ \dfrac{\left[ \frac{n+u+v}{2} + 1 + \beta \right] \ \left[ \frac{n+u-v}{2} + 1 \right] \ \big[ u + 2 \beta \big] \ \big[ n + 2 \beta \big] }{ \big[ u + \beta \big] \ \big[ u + \beta + 1 \big] \ \big[ n + \beta \big] \ \big[ n + \beta + 1 \big] }\end{aligned}$$]{}\
[$$\begin{aligned}
\left\{ \left\{ \begin{array}{ccc} 1 & n & n+1 \\ v & u-1 & u \end{array} \right\} \right\}_{q,t} = - \dfrac{{\cal N}_{n+1,u+1,v}}{ \dim_{q,t}(n+1) \dim_{q,t}(u-1) } \ \dfrac{\left[ \frac{u+v-n}{2} + 1 + \beta \right] \ \left[ \frac{u-v-n}{2} + 1 \right] \ \big[ n + 2 \beta \big] }{ \big[ u + 2 \beta - 1 \big] \ \big[ n + \beta \big] \ \big[ n + \beta + 1 \big]}\end{aligned}$$]{}\
generalizing the well-known specializations of q-6j symbols. The quantities with indices $2,3,4,\ldots$ can be described with equally explicit formulas, see s. 7.
####
Eq. (\[0index\]) explains, among other things, consistency with the genus 1 case: indeed, genus 2 TQFT has a subsector $i_1 = j_1 = 0$ which looks precisely like a torus, and the B-twist in that subsector reproduces the torus B-twist as expected:
$$\begin{aligned}
\sum\limits_{s = 0}^{K} \ T_s \ \dim_{q,t}(s) \ \left\{\left\{\begin{array}{ccc} i & i & 0 \\ j & j & s \end{array} \right\}\right\}_{q,t} \ \propto \ \sum\limits_{s = 0}^{K} \ T_s \ \dim_{q,t}(s) \ {\cal N}_{i j s} = (T S T)_{i j}\end{aligned}$$
\
Here we used the well-known formula for $TST$ in terms of ${\cal N}$ [@AS].
![Heegaard splitting: gluing an $S^3$ from two genus 2 handlebodies.[]{data-label="Heegaard"}](Heegaard.eps){width="100.00000%"}
Knot invariants
===============
As explained in [@AS], to compute the TQFT knot invariants, in addition to the representation of the mapping class group one also needs the *knot operators* ${\cal O}_j(K)$, that the TQFT functor associates to the bordisms inserting $K$ colored by representation $j$. We define these operators below. The $j$-colored knot invariant of $K$ is
$$\begin{aligned}
{\cal Z}_j( K ) = \langle 0, 0, 0 | \ I {\cal O}_j(K) \ | 0, 0, 0 \rangle\end{aligned}$$
\
This represents the geometric operation of gluing an $S^3$ from two genus $2$ handlebodies. One first takes a vector $| 0, 0, 0 \rangle$ – the state corresponding to an empty handlebody – then acts on it by the knot operator to insert a knot into it, and finally takes a scalar product with another vector $I | 0, 0, 0 \rangle$ to glue in the second handlebody. Note that the boudaries of the handlebodies are not glued identically, as this would not result in an $S^3$; instead, they are glued with the help of an specific transformation $I = A_1 B_1 A_2 B_2 A_3$ in analogy with the torus case. This way to obtain $S^3$ is called Heegaard splitting [@Heegaard], see Fig.\[Heegaard\].
####
Based on the computations below, and on the relation to mapping class groups, we propose the following topological invariance conjecture:
#### Conjecture II.
${\cal Z}_j( K )$ is an invariant of knots modulo framing, i.e. for any two topologically equivalent knots $K$ and $K^{\prime}$ in genus 2, ${\cal Z}_j( K ) = T_j^{\alpha} {\cal Z}_j( K^{\prime} )$ for some $\alpha$.
####
Note that, by construction, at $q = t$ the refined Chern-Simons invariant ${\cal Z}_j( K )$ coincides with the Jones polynomial of $K$. If Conjecture II is true, for $q \neq t$ the invariant ${\cal Z}_j( K )$ is a new, potentially stronger, knot invariant, that can be called the refined Jones polynomial. It remains to be seen to which extent the refined Jones polynomial is stronger in distinguishing knots: one very interesting check, for example, would be to compute it for any pair of mutants [@mutants] – inequivalent knots, indistinguishable by the ordinary Jones polynomial. This remains to be done.
$
{\color{blue}{{\cal O}^{(1)}_j}} = \begin{array}{c} \includegraphics[width=0.2\textwidth]{O2.eps} \end{array} \ \ \
{\color{blue}{{\cal O}^{(2)}_j}} = \begin{array}{c} \includegraphics[width=0.2\textwidth]{O12.eps} \end{array} \ \ \
{\color{blue}{{\cal O}^{(3)}_j}} = \begin{array}{c} \includegraphics[width=0.2\textwidth]{O1.eps} \end{array}
$
Let us start with simple knot operators, representing unknots that wind around the first handle (the 3-unknot) the second (the 1-unknot) or both (the 2-unknot). Their matrix elements, computed, say, using the methods of [@Masbaum1], have a form
[ $$\begin{aligned}
\langle i_1,i_2,i_3 | \ {\cal O}^{(1)}_j \ | j_1,j_2,j_3 \rangle = \delta_{i_1 j_1} \ [i_2 + 1] [i_3 + 1] \ \left\{\begin{array}{ccc} i_3 & i_2 & i_1 \\ j_2 & j_3 & j \end{array} \right\}_q^2\end{aligned}$$ $$\begin{aligned}
\langle i_1,i_2,i_3 | \ {\cal O}^{(2)}_j \ | j_1,j_2,j_3 \rangle = \delta_{i_2 j_2} \ [i_1 + 1] [i_3 + 1] \ \left\{\begin{array}{ccc} i_3 & i_2 & i_1 \\ j_1 & j & j_3 \end{array} \right\}_q^2\end{aligned}$$ $$\begin{aligned}
\langle i_1,i_2,i_3 | \ {\cal O}^{(3)}_j \ | j_1,j_2,j_3 \rangle = \delta_{i_3 j_3} \ [i_1 + 1] [i_2 + 1] \ \left\{\begin{array}{ccc} i_3 & i_2 & i_1 \\ j & j_1 & j_2 \end{array} \right\}_q^2\end{aligned}$$]{}\
We propose the following Macdonald deformation of these formulas:
[ $$\begin{aligned}
\langle i_1,i_2,i_3 | \ {\cal O}^{(1)}_j \ | j_1,j_2,j_3 \rangle = \delta_{i_1 j_1} \ g_j \ \dfrac{ \dim_{q,t}(i_2) \dim_{q,t}(i_3) }{ {\cal N}_{i_1 i_2 i_3} } \ \left\{\left\{\begin{array}{ccc} i_3 & i_2 & i_1 \\ j_2 & j_3 & j \end{array} \right\}\right\}_{q,t}\end{aligned}$$ $$\begin{aligned}
\langle i_1,i_2,i_3 | \ {\cal O}^{(2)}_j \ | j_1,j_2,j_3 \rangle = \delta_{i_2 j_2} \ g_j \ \dfrac{ \dim_{q,t}(i_1) \dim_{q,t}(i_3) }{ {\cal N}_{i_1 i_2 i_3} } \ \left\{\left\{\begin{array}{ccc} i_3 & i_2 & i_1 \\ j_1 & j & j_3 \end{array} \right\}\right\}_{q,t}\end{aligned}$$ $$\begin{aligned}
\langle i_1,i_2,i_3 | \ {\cal O}^{(3)}_j \ | j_1,j_2,j_3 \rangle = \delta_{i_3 j_3} \ g_j \ \dfrac{ \dim_{q,t}(i_1) \dim_{q,t}(i_2) }{ {\cal N}_{i_1 i_2 i_3} } \ \left\{\left\{\begin{array}{ccc} i_3 & i_2 & i_1 \\ j & j_1 & j_2 \end{array} \right\}\right\}_{q,t}\end{aligned}$$]{}\
Once an unknot is inserted, one can use the action of the mapping class group to wind it into something non-trivial. Fig.\[figureeight\] illustrates how this is done, starting from a 2-unknot, then doing transformations $A_1^{-1} B_1$ and $A_3^{-1} B_2$ to wind it around the handles, and finally $A_2^{-1}$ to complete the knot. What one obtains is a figure eight knot, a.k.a. $4_1$. This gives an explicit formula for the knot operator, that inserts the $4_1$ knot, colored by representation $j$:
$$\begin{aligned}
{\cal O}_j( 4_1 ) = U \ {\cal O}^{(2)}_j \ U^{-1}, \ \ \ U = A_2^{-1} A_1^{-1} B_1 A_3^{-1} B_2\end{aligned}$$
\
More generally, quite a large family of genus 2 pretzel knots can be obtained by further acting on the figure eight knot by the three $A$-twist operators:
![Using the genus 2 automorphisms to wind an unknot into a figure eight.[]{data-label="figureeight"}](O12twisted.eps){width="100.00000%"}
$$\begin{aligned}
{\cal O}_j\Big( \ \mbox{Pretzel}_{n_1 n_2 n_3} \ \Big) = U \ {\cal O}^{(2)}_j \ U^{-1}, \ \ \ U = A_1^{-m_1} A_2^{-m_2} A_3^{-m_3} \ A_1^{-1} B_1 A_3^{-1} B_2\end{aligned}$$
\
where $(n_1, n_2, n_3) = (2 m_1 + 1, 2 m_2, 2 m_3 + 1)$. This 3-parametric family includes many quite non-trivial knots, and will be the main playground in the present paper. Using level $K$ refined TQFT representations, we straightforwardly find
$$\begin{aligned}
\dfrac{ {\cal Z}_1( \ \mbox{Pretzel}_{1 2 1} = 4_1 \ ) }{ {\cal Z}_1( \ \mbox{Pretzel}_{1 0 1} = \bigcirc \ ) } = q^{-1} + q^{K/2 - 1} - q^{K - 2} + q^{K - 1} - q^{3K/2 - 2}\end{aligned}$$
\
Using here the substitution $t = - q^{-K/2}$, we obtain
$$\begin{aligned}
\dfrac{ {\cal Z}_1( \ \mbox{Pretzel}_{1 2 1} = 4_1 \ ) }{ {\cal Z}_1( \ \bigcirc \ ) } = t^{-3} q^{-2} \ \big( 1 - t + t q - t^2 q + t^3 q \big)\end{aligned}$$
\
This is the same procedure that has been used in [@AS], only now in genus 2 setting. It is straightforward and quite fast: more examples are provided in the next section.
Refined Chern-Simons Invariants of Pretzel knots
================================================
In this section we provide more examples of refined knot invariants in genus 2. With the full machinery of the mapping class group at hand, one can compute refined Chern-Simons invariants for any knot in genus 2. For illustration, we present a detailed exposition for the Pretzel knots.
####
Note that 3-index Pretzel knots possess both cyclic $(n_1,n_2,n_3) \simeq (n_2,n_3,n_1)$ and reversal $(n_1,n_2,n_3) \simeq (n_3,n_2,n_1)$ symmetries, hence, they are completely symmetric. All knot invariants are normalized by the invariant of the unknot, and further normalized to be a polynomial in non-negative powers of $q^{\geq 0} t^{\geq 0}$, starting with 1.
####
Let us start by gradually increasing $n$’s in the small positive area, keeping the color $j = 1$. This gives a bunch of simple knots from the Rolfsen table:
$$\begin{aligned}
\begin{array}{c|c|ccc}
(n_1,n_2,n_3) & \mbox{ Knot } K & \mbox{ Normalized Invariant } {\cal Z}_1(K) / {\cal Z}_1( \bigcirc )
\\ \hline & &
\\
(1,2,1) & 4_1 & \begin{array}{ll} 1-t+t q-t^2 q+t^3 q \\ \emph{} \end{array}
\\ \hline & &
\\
(1,4,1) & 6_1 & \begin{array}{ll} 1-t+t q-2 t^2 q+t^2 q^2+t^3 q-t^3 q^2+t^4 q^2 \\ \emph{} \end{array}
\\ \hline & &
\\
(1,2,3), (3,2,1) & 6_2 & \begin{array}{ll} 1 - t + 2 t q - 2 t^2 q + t^3 q + t^2 q^2 - 2 t^3 q^2 + t^4 q^2 \\ \emph{} \end{array}
\\ \hline & &
\\
(1,6,1) & 8_1 &
\begin{array}{ll} 1-t+t q-2 t^2 q+t^2 q^2+t^3 q \\
- 2 t^3 q^2+t^3 q^3+t^4 q^2-t^4 q^3+t^5 q^3 \\ \emph{} \end{array}
\\ \hline & &
\\
(3,4,1), (1,4,3) & 8_4 & \begin{array}{ll} 1-t+2 t q-3 t^2 q+2 t^2 q^2+t^3 q \\ - 3 t^3 q^2+t^3 q^3+2 t^4 q^2-2 t^4 q^3+t^5 q^3 \\ \emph{} \end{array}
\\ \hline & &
\\
(1,2,5), (5,2,1) & 8_2 & \begin{array}{ll} 1-t+2 t q-2 t^2 q+2 t^2 q^2+t^3 q \\ -3 t^3 q^2+t^3 q^3+t^4 q^2-2 t^4 q^3+t^5 q^3 \\ \emph{} \end{array}
\\ \hline & &
\\
(3,2,3) & 8_5 & \begin{array}{ll} 1-t+3 t q-3 t^2 q+2 t^2 q^2+t^3 q \\ -4 t^3 q^2+t^3 q^3+2 t^4 q^2-2 t^4 q^3+t^5 q^3 \\ \emph{} \end{array}
\end{array}\end{aligned}$$
Note that some knots have two different genus 2 realizations, differing by a permutation of $n_1$ and $n_3$. The fact that the answers match provides a simple check of topological invariance of the refined TQFT. Continuing to 10 crossings,
$$\begin{aligned}
\begin{array}{c|c|ccc}
(n_1,n_2,n_3) & \mbox{ Knot } K & \mbox{ Normalized Invariant } {\cal Z}_1(K) / {\cal Z}_1( \bigcirc )
\\ \hline & &
\\
(1,8,1) & 10_1 & \begin{array}{ll} 1-t+t q-2 t^2 q+t^2 q^2 +t^3 q \\ -2 t^3 q^2+t^3 q^3 + t^4 q^2-2 t^4 q^3+ \\ t^4 q^4+t^5 q^3-t^5 q^4+t^6 q^4 \\ \emph{} \end{array}
\\ \hline & &
\\
(3,6,1), (1,6,3) & 10_4 & \begin{array}{ll} 1-t+2 t q-3 t^2 q+2 t^2 q^2+t^3 q \\ -4 t^3 q^2 +2 t^3 q^3+2 t^4 q^2-3 t^4 q^3 + \\ t^4 q^4 + 2 t^5 q^3-2 t^5 q^4+t^6 q^4 \\ \emph{} \end{array}
\\ \hline & &
\\
(1,4,5), (5,4,1) & 10_8 & \begin{array}{ll} 1 - t + 2 t q - 3 t^2 q + 3 t^2 q^2 + \\ t^3 q - 4 t^3 q^2 + 2 t^3 q^3 + 2 t^4 q^2 - 4 t^4 q^3 + \\ t^4 q^4+2 t^5 q^3-2 t^5 q^4+t^6 q^4 \\ \emph{} \end{array}
\\ \hline & &
\\
(3,4,3) & 10_{61} & \begin{array}{ll} 1-t+3 t q-4 t^2 q+3 t^2 q^2+t^3 q-\\5 t^3 q^2+2 t^3 q^3+3 t^4 q^2-4 t^4 q^3+\\t^4 q^4+2 t^5 q^3-2 t^5 q^4+t^6 q^4 \\ \emph{} \end{array}
\\ \hline & &
\\
(1,2,7), (7,2,1) & 10_2 & \begin{array}{ll} 1-t+2 t q-2 t^2 q+2 t^2 q^2+t^3 q \\ -3 t^3 q^2 + 2 t^3 q^3+t^4 q^2-3 t^4 q^3+ \\ t^4 q^4+t^5 q^3-2 t^5 q^4+t^6 q^4 \\ \emph{} \end{array}
\\ \hline & &
\\
(3,2,5), (5,2,3) & 10_{46} & \begin{array}{ll} 1-t+3 t q-3 t^2 q+3 t^2 q^2+t^3 q \\ -5 t^3 q^2+2 t^3 q^3+2 t^4 q^2-4 t^4 q^3+ \\t^4 q^4+2 t^5 q^3-2 t^5 q^4+t^6 q^4 \\ \emph{} \end{array}
\end{array}\end{aligned}$$
{width="50.00000%"}
####
Another interesting series of examples, allowing to further test topological invariance, is obtained by allowing some of the indices $n_1,n_2,n_3$ to be negative or zero:
$$\begin{aligned}
\begin{array}{c|c|ccc}
(n_1,n_2,n_3) & \mbox{ Knot } K & \mbox{ Normalized Invariant } {\cal Z}_1(K) / {\cal Z}_1( \bigcirc )
\\ \hline & &
\\
(-1,0,3) & 3_1 & 1 + tq - t^2 q
\\ \hline & &
\\
(-3,0,-3) & 3_1 \ \# \ 3_1 & (1 + tq - t^2 q)^2
\\ \hline & &
\\
(3,0,-3) & 3_1 \ \# \ \overline{3_1} & (1 + tq - t^2 q)(1 - t - t^2 q)
\\ \hline & &
\\
(1,-2, 1) & 3_1 \simeq T_{2,3} & 1 + tq - t^2 q
\\ \hline & &
\\
(1,-2, 3) & 5_1 \simeq T_{2,5} & 1 + tq - t^2 q + t^2 q^2 - t^3 q^2
\\ \hline & &
\\
(1,-2, 5) & 7_1 \simeq T_{2,7} & 1 + tq - t^2 q + t^2 q^2 - t^3 q^2 + t^3 q^3 - t^4 q^3
\\ \hline & &
\\
(3,-2, 3) & 8_{19} \simeq T_{3,4} & 1+t q+t q^2-t^2 q-t^3 q^2
\\ \hline & &
\\
(3,-2, 5) & 10_{124} \simeq T_{3,5} & 1 + t q + t q^2 - t^2 q + t^2 q^3 - t^3 q^2 - t^3 q^3
\end{array}\end{aligned}$$
Two of the knots are composite – $(-3,0,-3)$ a.k.a the *Granny knot*, and $(3,0,-3)$ a.k.a. the *square knot* – they are connected sums (denoted $\#$) of trefoils. The refined invariants of these knots factorize, suggesting this is the general behaviour w.r.t. the connected sum operation. The others give alternative genus 2 realizations of torus knots, incluing the most complicated $(3,-2,3) = 8_{19} \simeq T_{3,4}$ and $(3,-2,5) = 10_{124} \simeq T_{3,5}$. The answers, that we obtain here with a genuinely genus 2 computation, match the corresponding results of the genus 1 computations of [@AS]. This provides a non-trivial check of topological invariance of the refined TQFT.
As explained in [@noroots], in refined Chern-Simons theory one can expect to unify all of the above examples into a single *evolution* formula a-la [@evolution], making the dependence on the winding numbers $m_1,m_2,m_3$ fully explicit[^4]. Let us briefly review here the argument of [@noroots]. First, by definition, the knot invariant is given by
$$\begin{aligned}
{\cal Z}_j\big( \ \mbox{Pretzel}_{n_1,n_2,n_3} \ \big) = \langle 0, 0, 0 | \ I \ A_1^{-m_1} \ A_2^{-m_2} \ A_3^{-m_3} \ {\cal O}_j(\bigcirc) \ | 0, 0, 0 \rangle\end{aligned}$$
\
Second, this formula can be expanded as a sum over intermediate states,
$$\begin{aligned}
{\cal Z}_j\big( \ \mbox{Pretzel}_{n_1,n_2,n_3} \ \big) = \sum\limits_{k_1,k_2,k_3} \ \Gamma^{(k_1,k_2,k_3)}_j \ T_{k_1}^{m_1} \ T_{k_2}^{m_2} \ T_{k_3}^{m_3}\end{aligned}$$
\
where we used the fact that the $A$-twists are diagonal, and denoted
$$\begin{aligned}
\Gamma^{(k_1,k_2,k_3)}_j = \langle 0, 0, 0 | \ I \ | k_1,k_2,k_3 \rangle \cdot \langle k_1,k_2,k_3 \ | \ {\cal O}_j(\bigcirc) \ | 0, 0, 0 \rangle\end{aligned}$$
\
Finally – and this was the main point of [@noroots] – knot operators in refined Chern-Simons theory are highly sparse. Even though *a priori* the sum in the above formula goes over all $k_1,k_2,k_3$ in the admissible set, the matrix elements of knot operators, namely, $\Gamma^{(k_1,k_2,k_3)}_j$, are nonzero only for a few values of $k$, which are actually independent on $K$ at all. For example, in the fundamental case $j = 1$ these are
$$\begin{aligned}
\Gamma^{(2,2,0)}_1 = \Gamma^{(0,2,2)}_1 = \dfrac{t^2q-1}{tq(1-t)} \ \Gamma^{(0,0,0)}_1\end{aligned}$$
$$\begin{aligned}
\Gamma^{(2,0,2)}_1 = \dfrac{(1-t^2q)(1-q)}{t q^2 (1-t)^2} \ \Gamma^{(0,0,0)}_1, \ \ \ \Gamma^{(2,2,2)}_1 = \dfrac{(1-t^2q)(1+tq)}{t^2q^2(1-t)} \ \Gamma^{(0,0,0)}_1\end{aligned}$$
\
and all the other $\Gamma$’s vanish. This implies that
[ $$\begin{aligned}
\nonumber & \dfrac{ {\cal Z}_1\big( \ \mbox{Pretzel}_{n_1,n_2,n_3} \ \big) }{ {\cal Z}_1\big( \ \mbox{Pretzel}_{1,0,1} \ \big) } = \dfrac{q^2t^2(1-t)^2}{(1-qt)^2} \ \left( 1 + \dfrac{t^2q-1}{tq(1-t)} \ \Big[ \ (qt)^{-m_1-m_2} + (qt)^{-m_2-m_3} \ \Big] + \right. \\ &
\left. + \dfrac{(1-t^2q)(1-q)}{t q^2 (1-t)^2} \ (qt)^{-m_1-m_3} + \dfrac{(1-t^2q)(1+tq)}{t^2q^2(1-t)} \ (qt)^{-m_1-m_2-m_3} \right)\end{aligned}$$]{}\
One can check that this, indeed, reproduces all the examples above. This seems to be a deformation of the formula of [@pretzels], and it would be interesting to understand how to generalize this to Pretzel knots in higher genus, along the lines of [@pretzels].
The algebra of knot operators
=============================
If one inserts one and the same knot several times, it is natural to expect that the result can be expressed as a linear combination of single insertions, summed over various colors. This implies that knot operators form an algebra. In the usual Chern-Simons TQFT it was very simple, and looked like
$$\begin{aligned}
q = t: \ \ \ {\cal O}_i \big( \bigcirc \big) {\cal O}_j \big( \bigcirc \big) = \mathop{\sum\limits_{0 \leq k \leq K}}_{(i,j,k) \mbox{ admiss. }} {\cal O}_k \big( \bigcirc \big)\end{aligned}$$
\
The refined knot operators, that we constructed above, enjoy a similar algebra:
$$\begin{aligned}
{\cal O}_i \big( \bigcirc \big) {\cal O}_j \big( \bigcirc \big) = g_i \ g_j \ \sum\limits_{0 \leq k \leq K} \ {\cal N}_{i j k} \ {\cal O}_k \big( \bigcirc \big)\end{aligned}$$
\
One can think of this as a recursion relation, expressing knot operators with higher colors through the knot operators through lower colors. Solving it order by order, one finds completely explicit formulas
$$\begin{aligned}
& {\cal O}_0 = \big( {\cal O}_1 \big)^0 \ \equiv \ 1 \\
& \nonumber \\
& {\cal O}_1 = \big( {\cal O}_1 \big)^1 \\
& \nonumber \\
& {\cal O}_2 = \big( {\cal O}_1 \big)^2 - \dfrac{(1-q)(1+t)}{1-q t} \ \big( {\cal O}_1 \big)^0 \\
& \nonumber \\
& {\cal O}_3 = \big( {\cal O}_1 \big)^3 - \dfrac{(1-q)(2qt+q+t+2)}{1-q^2 t} \ \big( {\cal O}_1 \big)^1 \\
& \nonumber \ldots\end{aligned}$$
\
expressing everything in terms of ${\cal O}_1$. It is not only easy to solve order by order, a general solution is not hard either, because the exact same algebra is satisfied by the Macdonald polynomials (this is one of the alternative definitions of ${\cal N}$, see [@AS]):
$$\begin{aligned}
M_i\big( x, x^{-1} \big) M_j \big( x, x^{-1} \big) = g_i \ g_j \ \sum\limits_{0 \leq k \leq K} \ {\cal N}_{i j k} \ M_k \big( x , x^{-1} \big)\end{aligned}$$
\
This implies that knot operators ${\cal O}_j$ are recovered from the simplest knot operator ${\cal O}_1$ in the same way [^5] as Macdonald polynomials $M_j(x,x^{-1})$ are recovered from the simplest Macdonald polynomial $M_1(x,x^{-1}) = x + x^{-1}$. The easiest way to do this recovery is to first express Macdonald polynomials through the Schur polynomials,
$$\begin{aligned}
M_j\big( x, x^{-1} \big) = \sum\limits_{l = 0}^{[j/2]} \ \dfrac{q^{l} [j - 2 l + 1]}{[j - l + 1]} \ \prod\limits_{m = 0}^{l - 1} \dfrac{[j - l + 1 + m] [m + \beta - 1]}{[m + 1][j + \beta - 1 - m]} \ \chi_{j-2l}(x, x^{-1})\end{aligned}$$
\
and then express the Schur polynomials through the desired basis – powers of $x+x^{-1}$:
$$\begin{aligned}
\chi_{j-2l}\big( x, x^{-1} \big) = \sum\limits_{p = 0}^{[j/2]-l} \ \dfrac{(-1)^p (j - 2l - p)!}{p!(j - 2l - 2p)!} \ (x+x^{-1})^{j-2l-2p}\end{aligned}$$
\
Note, that the last formula is written in terms of the usual, not q-deformed, factorials. Putting these two together and replacing $x + x^{-1} \mapsto {\cal O}_1$, we find an explicit formula for all knot operators, colored by arbitrary representations $j$:
[ $$\begin{aligned}
\boxed{ \ \ \ {\cal O}_j = \sum\limits_{l = 0}^{[j/2]} \ \dfrac{q^{l} [j - 2 l + 1]}{[j - l + 1]} \ \prod\limits_{m = 0}^{l - 1} \dfrac{[j - l + 1 + m] [m + \beta - 1]}{[m + 1][j + \beta - 1 - m]} \ \sum\limits_{p = 0}^{[j/2]-l} \dfrac{(-1)^p(j - 2l - p)!}{p!(j - 2l - 2p)!} \ \big( {\cal O}_1 \big)^{j-2l-2p} \ \ \ }
\label{General1}\end{aligned}$$]{}\
Note that this formula is completely general and applies to refined Chern-Simons TQFT in any genus, if it exists. The only external input, required by this formula, is the knowledge of the fundamental knot operator ${\cal O}_1$. Fortunately, we possess the duality definition eq. (\[duality\]), which allows us to directly compute the genus-2 $\ {\cal O}_1$:
[$$\begin{aligned}
\langle n+1, u+1, v | \ {\cal O}^{(1)}_1 \ | n, u, v \rangle = \dfrac{\left[ \frac{n+u+v}{2} + 1 + \beta \right] \ \left[ \frac{n+u-v}{2} + 1 \right] \ \big[ u + 2 \beta \big] \ \big[ n + 2 \beta \big] }{ \big[ u + \beta \big] \ \big[ u + \beta + 1 \big] \ \big[ n + \beta \big] \ \big[ n + \beta + 1 \big] }
\label{General2}\end{aligned}$$ $$\begin{aligned}
\langle n+1, u-1, v | \ {\cal O}^{(1)}_1 \ | n, u, v \rangle = - \dfrac{\left[ \frac{u+v-n}{2} + 1 + \beta \right] \ \left[ \frac{u-v-n}{2} + 1 \right] \ \big[ n + 2 \beta \big] }{ \big[ u + 2 \beta - 1 \big] \ \big[ n + \beta \big] \ \big[ n + \beta + 1 \big] }
\label{General3}\end{aligned}$$]{}\
and all the other matrix elements vanish. Together, eqs. (\[General1\]),(\[General2\]),(\[General3\]) give an explicit formula for all the $q,t$-deformed squares of $q$-6j symbols.
Discussion
==========
#### $\bullet$ Distinguishing mutants.
One of the most straightforward and interesting applications of refined Chern-Simons theory could be distinguishing mutants [@mutants] – knots that cannot be distinguished by the usual Jones polynomials, or generally by HOMFLY polynomials colored by highest weights of symmetric or antisymmetric representations. Unfortunately, the knots that we have computed so far (the 3-index pretzels) do not have any non-trivial mutants among them. However, there might be such among the non-pretzel knots in genus 2, and it would be very interesting to check if refined Jones polynomials distinguish them or not. Another obvious possibility is to go to genus 3, where there exist non-trivial pretzel mutants.
#### $\bullet$ Higher genus.
The construction of present paper relies upon the Macdonald duality equation, that constrains the matrix elements of knot operators in genus 2. This duality equation is a deformation of the known Fourier duality equation for the squares of q-6j symbols. Following the same steps as we do in higher genus, one inevitably discovers that the matrix elements of knot operators are no longer degree 2 contractions of the q-6j symbols, but rather degree 4 contractions. This degree does not grow: for generic $g$ it stays degree 4. Genus 2 is a distinguished case from this point of view. To obtain a refined q,t-deformation of these degree 4 contractions, one can expect to look for degree 4 generalizations of Fourier/Macdonald duality; this remains to be done.
#### $\bullet$ Higher rank.
The main problem with generalization to higher rank is the fact that basis vectors in the vector space, associated to a surface of genus 2 (or higher), is no longer a decorated knot: it is a decorated trivalent graph. For a TQFT of type $A_n$, the decoration will include, as a part, assigning multiplicities of tensor products of representations to the trivalent intersections. It is not completely clear how this will affect the central identity of the present construction – the Macdonald duality. In addition, introducing and handling multiplicities is simply very hard technically.
The usual solution to this problem is to only consider knots colored by the highest weights of symmetric or antisymmetric representations. This, however, does not seem to be possible within the mapping class group approach, since we do not choose which decorated graphs to include into the definition of the basis – this is forced on us by the values of $N$ and $K$. The right methods and language to generalize to higher rank remain to be found.
#### $\bullet$ Higher genus DAHA’s.
As discussed before in [@GorskyNegut], knot operators in refined Chern-Simons theory on a torus generate an algebra which is isomorphic to the spherical DAHA, also known as the elliptic Hall algebra. Our results seem to suggest that a similar algebra exists in genus 2, generated by all knot insertion operators along all possible knots. In principle, using the formulas of present paper it should be possible to learn quite a lot about this algebra.
#### $\bullet$ Refined Chern-Simons as a two-parameter quantization.
It is known that knot operators in ordinary Chern-Simons theory provide a quantization of the Poisson algebra of functions on the moduli space of flat connections on the surface. The parameter $q$ plays the role of a quantum parameter, with $q \rightarrow 1$ being the classical limit, where the Poisson algebra is recovered. The fact that there exists a Macdonald q,t-deformation, with two independent “quantum” parameters, suggests that there exist two independent Poisson brackets for functions on the moduli space of flat connections. It would be interesting to make this and other statements about the “classical” limit of refined Chern-Simons theory more precise.
#### $\bullet$ Elliptic quantum groups.
One place where q,t-6j symbols with two deformation parameters appear in mathematical physics are the elliptic quantum groups, such as $U_{q,t}(sl_2)$ [@Felder]. However, these q,t-6j symbols also typically contain a third “spectral” or “dynamical” parameter, and satisfy a dynamical Yang-Baxter equation. The relation between elliptic quantum groups and refined Chern-Simons theory, if any, should involve a way to eliminate the spectral parameter.
#### $\bullet$ Topological string theory.
Given a 3-manifold $M$, is known [@CSstring] that the partition function of topological string theory on $T^{\star} M$ agrees with the partition function of Chern-Simons theory on $M$. As explained in [@AS], there is a refined version of this relation. Namely, for Seifert 3-manifolds $M$ (in particular, for $S^3$) Chern-Simons partition function can be refined, with the refinement following entirely from the action of the genus 1 mapping class group. The resulting partition function agrees with the partition function of the refined topological string on $T^{\star} M$ [@AS]. One can think of this relation as an alternative way to compute the refined topological string partition function on backgrounds of the form $T^{\star} M$. The results of present paper imply an extension of the class of manifolds that can be accessed this way, from Seifert to more general ones, constructed with the genus 2 mapping class group.
Acknowledgements {#acknowledgements .unnumbered}
================
We are indebted to G.Masbaum for enlightening explanations of the higher genus TQFT representations of mapping class groups. We are grateful to M.Aganagic, I.Cherednik, E.Gorsky, R.Kashaev, A.Morozov, N.Reshetikhin and C.Vafa for fruitful discussions. The work of S.A. was partly supported by the grants RFBR 15-01-04217 and 15-51-50034-YaF. The work of Sh.Sh. was partly supported by the grants RFBR 15-01-05990, 15-31-20832-Mol-a-ved and NSh-1500.2014.2.
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[^1]: We will see later that they deform non-trivially in the refined case. See also [@AS].
[^2]: Note, that coefficients ${\cal N}_{ijk}$ were trivial in the usual TQFT – either 1 or 0, depending on whether a triple is admissible or not – but in the refined setting they are not even integers anymore, but rational functions of $q$ and $t$, and the full formula (\[Verlinde\]) has to be used to describe them.
[^3]: We have verified that the solution exists and is unique for $1 \leq K \leq 8$.
[^4]: As a reminder, $(n_1, n_2, n_3) = (2 m_1 + 1, 2 m_2, 2 m_3 + 1)$.
[^5]: In the torus case this fact was pointed out in [@GorskyNegut]. One can see that it is a very general fact.
|
---
abstract: 'Vapor-liquid-solid (VLS) route and its variants are routinely used for scalable synthesis of semiconducting nanowires yet the fundamental processes remain unknown. Here, we employ atomic-scale computations based on model potentials to study the stability and growth of gold-catalyzed silicon nanowires (SiNWs). Equilibrium studies uncover segregation at the solid-like surface of the catalyst particle, a liquid AuSi droplet, and a silicon-rich droplet-nanowire interface enveloped by heterogeneous truncating facets. Supersaturation of the droplets leads to rapid 1D growth on the truncating facets and much slower nucleation-controlled 2D growth on the main facet. Surface diffusion is suppressed and the excess Si flux occurs through the droplet bulk which, together with the Si-rich interface and contact line, lowers the nucleation barrier on the main facet. The ensuing step flow is modified by Au diffusion away from the step edges. Our study highlights key interfacial characteristics for morphological and compositional control of semiconducting nanowire arrays.'
author:
- Hailong Wang
- 'Luis A. Zepeda-Ruiz'
- 'George H. Gilmer'
- Moneesh Upmanyu
title: 'Atomistics of vapor-liquid-solid nanowire growth'
---
Introduction {#introduction .unnumbered}
============
A characteristic feature of the VLS route for nanowire synthesis is the presence of a catalyst particle, usually a droplet, that mediates the mass transfer from the vapor phase precursor to the growing nanowire[@nw:WagnerEllis:1964; @nw:GivargizovChernov:1973; @nw:WuLieber:2004; @nw:SchmidtGosele:2005; @nw:SchmidtGosele:2009; @nw:Ross:2010]. The technique was demonstrated successfully in the 1960s in the context of crystalline whisker growth[@nw:WagnerEllis:1964], and has become increasingly relevant of late as it offers direct control over nanowire composition and morphology, aspects of critical importance in several nanowire-based applications[@nw:Lieber:1998; @nw:Lieber:2001; @nw:Lieber:2003; @nano:BarkerArias:2002; @nw:SamuelsonBjork:2004; @nw:RoperVoorhees:2007; @nw:RossTersoffReuter:2005; @nw:SchmidtGosele:2007; @Thompson2009; @nw:SchwalbachVoorhees:2009; @nw:MadrasDrucker:2009; @nw:IrreraPecoraPriolo:2009]. The salient features can be readily seen in the schematic illustration in Fig. \[fig:figure1\]a (inset). A low melting eutectic droplet is supersaturated following breakdown of the precursor gas on its surface. Past a critical point, crystallization at the droplet-nanowire interface results in tip growth. The droplet then acts as a conduit for 1D crystal growth with diameters that are set by its dimensions.
The atomic-scale structure, energetics and dynamics of the alloyed particle directly influence the morphology and composition of the as-grown nanowires, yet the prevailing ideas continue to be based on continuum frameworks[@nw:Givargizov:1975; @nw:DubrovskiNickolai:2004; @nw:SchmidtGosele:2007; @nw:RoperVoorhees:2007; @nw:SchwarzTersoff:2009; @nw:DubrovskiiGlas:2009; @nw:SchwarzTersoff:2011; @nw:SchwarzTersoff:2011b]. As an example, gold-catalyzed growth of silicon nanowires is a well-studied system yet the morphology and composition of the particle-nanowire interface is unknown. Basic crystal growth parameters such as the size dependent equilibrium composition $X_{Si}^{\ast}$ above which the droplet rejects the excess silicon onto the nanowire remain unknown; continuum models models ignore the effect of segregation and related interfacial phenomena. Such microscopic features modify the droplet morphology through the balance of interfacial tension at the contact line which in turn dynamically regulates the chemical potential difference between the two phases, i.e. the driving force for growth. Likewise, the atomic-scale processes that sustain the growth of the nanowire remain unknown. The steady-state is limited by reduction of the precursor gas on the droplet surface[@nw:KodambakaRoss:2006] and the 2D nucleation-limited step flow at the droplet-nanowire interface[@nw:; @KimRoss:2008; @nw:KimRoss:2009]. Understanding the interplay with the dynamics that feeds the growth of the nuclei is crucial for compositional control of the as-grown nanowire[@nw:PereaLauhon:2006; @nw:PereaLauhon:2009; @nw:SchwalbachVoorhees:2009; @nw:HemesathVoorheesLauhon:2012] and requires an atomistic understanding of the near-equilibrium particle behavior.
{width="2\columnwidth"}
We focus exclusively on a small diameter $2R=10$nm, $\langle111\rangle$ silicon nanowire growing isothermally above the eutectic temperature at $T=873$K. Both experiments and [*ab-initio*]{} computational techniques are handicapped in accessing the spatio-temporal scales associated with the energetics and 3D dynamics during growth[@nw:LeeHwang:2010; @nw:HaxhimaliAstaHoyt:2009; @nw:RyuCai:2011]. We capture these processes in their full complexity by employing atomistic simulations based on an angular embedded-atom-method (AEAM) framework for describing the hybrid metallic-covalent interactions[@nw:DongareZhigilei:2009]. The computed surfaces of equilibrated isolated droplets are ordered and Si-rich, yet decorated by a submonolayer of Au. The surface structure leads to sluggish atomic mobilities such that most of the excess Si flux during growth is directed through the droplet bulk. Studies on droplet-capped SiNWs show that the solid-liquid interface is again Si-rich and the nanowire sidewalls are decorated by Au monolayers. Of note is the interface morphology composed of a main facet that truncates into smaller staircase-like, stepped facets at the contact line. The latter grow rapidly in a 1D fashion while crystallization on the main facet proceeds via nucleation of 2D islands which then grow via step flow. Interestingly, the motion of the steps is limited by Au diffusion into the droplets and/or the nanowire sidewalls. Studies with varying levels of droplet supersaturation yield the nucleation barrier which is lower than that for homogeneously nucleation on a planar Si(111)-AuSi interface, highlighting the combined role of segregation, atomic diffusivities, and interface morphology in regulating nanowire growth.
Results {#results .unnumbered}
=======
Size corrected liquidus composition
-----------------------------------
We first quantify the equilibrium state of the alloy system by extracting the liquidus composition, $X_{Si}^\ast(2R, T=873$K). We capture the effect of the solid-liquid interface and the contact line by extracting the equilibrium state of a $2R=10$nm diameter, $\langle111\rangle$ SiNW capped by a droplet at each end. The geometry allows the nanowire to act as a source/sink as the droplets adjust to the size-and morphology-dependent liquidus composition. The starting configuration consists of a nanowire with a hexagonal cross-section bounded by $\{112\}$-like sidewall facets (Methods). In particular, we have performed separate computations for SiNWs with $\{110\}$ and {112} sidewall facets. The choices are motivated by past experiments that have shown that the sidewalls can be a mixture of $\{$112$\}$ and $\{110\}$ facets[@nw:RossTersoffReuter:2005; @nw:DavidGentile:2008]. Figure \[fig:figure1\]b shows the atomic configuration of the equilibrated SiNW system, in this particular case bounded by $\{110\}$ facets. We immediately see the formation of an Au submonolayer on the sidewalls, consistent with prior experimental observations[@nw:HannonRossTromp:2006]. The cross-section maintains its six-fold symmetry for both $\{$110$\}$ or $\{$112$\}$ family of facets implying that the two surfaces are energetically quite close; we do not observe their co-existence[@nw:RossTersoffReuter:2005; @nw:DavidGentile:2008]. This is not surprising as the thermodynamic shape is inherently size dependent[@gbe:Herring:1951] and the diameters of the synthesized SiNWs are much larger. The $\{112\}$ facet is unstable for Au coverages typically observed during nanowire growth[@nw:RossTersoffReuter:2005], yet we do not observe axial serrations due to the small nanowire lengths. We also cannot rule out the effects of surface stresses at these small sizes[@nw:LiangUpmanyuHuang:2005] as they can change the relative stability between $\{$110$\}$ or $\{$112$\}$ family of facets. The average composition within the interior of the capping droplets is $X_{Si}^\ast (2R)=0.46$ and is indicated on the phase diagram in Fig. \[fig:figure1\]a. The size-corrected supersaturation with respect to the bulk liquidus composition is $\Delta_X=X_{Si}^\ast(2R) - X_{Si}^\ast\approx1\%$ and the equivalent undercooling $\Delta_T\approx 18$K.
Surface segregation, ordering and kinetics
------------------------------------------
The composition of the capping droplets is non-uniform and we decouple the surface effects by first studying isolated droplets at $X_{Si}=X_{Si}^\ast (2R)$. Atomic configuration of a midsection through an equilibrated droplet is shown in Fig. \[fig:figure2\]a. The non-uniform composition arises mainly due to surface segregation. Details are extracted using normalized density profiles $\langle\rho({\bf r})/\bar{\rho}\rangle$ plotted as a function of the distance $r$ from the droplet center of mass (Fig. \[fig:figure2\]b). The overall profile (dark solid line) has a well-defined peak at the surface suggestive of a solid-like surface structure. The multiple alternating peaks in the Au and Si profiles indicate a Si-rich subsurface decorated by a submonolayer of Au. While the Si segregation is consistent with experiments on eutectic AuSi thin films[@tsf:ShpyrkoPershan:2006], the Au submonolayer has not been reported earlier.
![: (a) Midsection through an atomic plot of a 32,000-atom AuSi system equilibrated at the model liquidus composition $X_{Si}^{\ast}=0.46$. (b and c) Radial variation of (b) the ensemble averaged and normalized density profile ${\rho}(r)/\bar{\rho}$ for the combined system (solid dark line) and for Au and Si, as indicated, and (c) ensemble averaged diffusivities of Au and Si, $D_{Au/Si}(r)$ extracted from equilibrium MD simulations of the droplets. The dotted lines in (c) correspond to the bulk diffusivities extracted from simulations on an equilibrated bulk alloy system (see Methods). \[fig:figure2\] ](Fig2NcomLowRes.pdf){width="\columnwidth"}
We further isolate the effects of surface curvature via segregation studies on thin AuSi films and the results are summarized in Supplementary Figure S1. At the eutectic point, $X_{Si}\approx0.31$ and $T=590$K, we see ripples in the density profiles that extend to $\sim$7 layers and decay rapidly into the bulk. The density variations are larger than those usually seen on liquid surfaces and suggest a compositionally ordered pre-frozen surface qualitatively similar to the surface ordering seen in eutectic AuSi thin films[@tsf:ShpyrkoPershan:2006]. The ordering decreases with increasing $X_{Si}$ and temperature, and the extracted density profiles are similar to those observed in the liquid droplets, i.e. the surface curvature has little effect. The surfaces are always decorated by a Au submonolayer. It is expectedly less ordered compared to monolayers on crystalline Si surfaces, and this feature likely makes it difficult to be resolved by X-ray based surface characterization techniques[@tsf:ShpyrkoPershan:2006]. The discrepancy can certainly be an artifact of the EAM framework used to describe the metallic bonds[@intpot:WebbGrest:1986]; it underestimates the surface tension of liquid Au by 20% and can therefore artificially favor the formation of the Au submonolayer. Nonetheless, it is natural to expect that the stability of Au monolayers on crystalline Si surfaces[@tsf:Lelay:1981] also extends to partially crystalline Si surfaces. As confirmation, we still see the Au decoration in segregations studies with modified Au-Au interactions using a charge gradient approach fit to the surface tension of liquid Au[@intpot:WebbGrest:1986] (Supplementary Figure S2 and Supplementary Methods). The Au coverage is slightly reduced in extent yet the qualitative trends remain unchanged. The effect of surface structure extends to the diffusive kinetics as well. We quantify the interplay by monitoring the atomic mean-square displacements (MSD) that yield the atomic diffusivities $D_{Au/Si}$ (Supplementary Figure S3). The radial variation is plotted in Fig. \[fig:figure2\]c (Methods). For both Au and Si, the surface diffusivity is less than half of the bulk value and further highlights the dramatic effect of the surface structure.
Equilibrated nanowire-particle system
-------------------------------------
The substrate is explicitly included by mating an equilibrated droplet-SiNW system onto a dehydrogenated Si(111) surface (Methods). Figure \[fig:figure3\]a and Supplementary Figure S4 show the atomic configurations of the nanowire system with {110} and {112} sidewall facets, respectively. A stable Au submonolayer is evident on the nanowire sidewalls (right, Fig. \[fig:figure3\]a). The surface segregation is similar to that on isolated droplets (Fig. \[fig:figure2\]b), i.e. the substrate has a minor effect. There are differences in the structure of the Au submonolayer for the two classes of sidewall facets. We see Si segregation at the droplet-nanowire interface within liquid layers adjacent to the interface and parallel to $($111$)$ planes. Ripples in the in-plane density profiles within the droplet extracted as a function of distance from the interface $z$ extend to a thickness of $\approx5\,{\rm \AA}$ (Fig. \[fig:figure3\]b). The sharp Si and Au peaks adjacent to the interface at $z\approx1.5\,{\rm \AA}$ (arrow) indicate a Si-rich layer with some in-plane order. The remainder of the segregated layer is mostly liquid characterized by a smaller and broader Si-peak at $z\approx2.5\,{\rm \AA}$ (arrow), followed by a Au-rich region ($z\approx4\,{\rm \AA}$) which ultimately settles into the bulk composition. The compositional distribution for the entire nanowire-droplet system is summarized schematically in Fig. \[fig:figure3\]c.
{width="2\columnwidth"}
Surface plot of the solid nanowire sans the the liquid atoms yields insight into the interface morphology (middle panel, Fig. \[fig:figure3\]a). Clearly, it is not flat. The $\langle111\rangle$ facet is dominant yet it is truncated along the periphery next to the contact line. These truncated regions are 2-4 atomic diameters wide and their length varies non-uniformly along the contact line. Multiple midsections reveal a stepped structure yet they are not smoothly curved; rather they correspond to two distinct orientations inclined at $\beta\approx32^\circ$ and $\beta\approx50^\circ$ to the $\langle111\rangle$ sidewall facet (vertical), which we identify to be the $\{113\}$ and {120} family of planes. Supplementary Figure S4 reveals a similar morphology for SiNWs with $\langle112\rangle$ sidewall facets with the exception that the truncating facets consist of {113} and {111} family of planes. The six facets alternate between these two inclinations along the contact line, as indicated in the surface plot and the midsections (Fig. \[fig:figure3\]a, right). The facets are also observed in the doubly-capped nanowire system shown in Fig. \[fig:figure1\]b. The six edge corners where truncating facets intersect are inherently rough as they have to absorb the differences in the facet inclinations, as illustrated schematically. The roughening can also be seen in the surface plot in that the step morphology kinks at the corner edges. The droplet shape is therefore asymmetric since the angles $\alpha$ and $\beta$ that follow from the Young’s balance along the contact line are shaped by the facet energetics and geometry. The asymmetry is also evident in the two midsections: $\alpha\approx11^\circ$ at the $\{113\}$ truncating facet and changes to $\alpha\approx15^\circ$ at the $\{120\}$ truncating facet.
The presence of nanometer-wide stable truncating facets is consistent with recent reports in several semiconducting nanowire systems, including the Au-Si system[@nw:OhChisholmRuhle:2010; @nw:WenTersoffRoss:2011; @nw:GamalskiHofmann:2011]. Limitations of the model system notwithstanding, such a complex 3D morphology composed of multiple families of truncating facets is similar to that recently reported in \[0001\] sapphire nanowires[@nw:OhChisholmRuhle:2010], suggesting that this is perhaps a general feature during VLS growth. For the specific case of Si, it is interesting that the $\{113\}$ and $\{111\}$ facets routinely decorate the [sidewalls]{} during the complex sawtooth faceting in much larger diameter $\langle111\rangle$ SiNWs [@nw:RossTersoffReuter:2005; @nw:DavidGentile:2008], suggesting that they perhaps nucleate first at the solid-liquid interface.
Nanowire growth kinetics
------------------------
The heterogeneous nature of interface facets has important ramifications for nanowire growth which we now study computationally. We simplify the catalysis by directly abstracting individual silicon atoms onto the droplet surface to a prescribed level of supersaturation, and employ classical molecular dynamics to study the non-equilibrium response of the entire droplet-SiNW-substrate system (Supplementary Methods). Figure \[fig:figure4\] shows the structural evolution at and around the interface as well as the in-plane density profile $\rho_{\parallel}(z)$ for a droplet with initial composition $X_{Si}=0.48$ ($\Delta_T\approx36$K). The initial growth stage for $t<0.4$ns is marked by layer-by-layer crystallization on the truncating facets. The [*circumferential*]{} growth is likely aided by their small widths and the proximity to the Si-rich contact line (circled, Fig. \[fig:figure4\]a). We see 1D lengthening of kinks associated with the stepped structure of these facets. Interestingly, the growth occurs preferentially away from the corner edges implying that they favor dissolution. For details, see Supplementary Movie 1. The extent of truncation changes due to their growth, consistent with the oscillatory growth wherein the facet widths decrease between bilayer additions on the main facet[@nw:OhChisholmRuhle:2010; @nw:WenTersoffRoss:2011; @nw:GamalskiHofmann:2011]. We also see smaller clusters of 2-3 Si atoms that continually form and dissolve on the main facet yet the density profile normal to the interface remains unchanged (bottom row), i.e. the crystallization is limited to the truncating facets.
{width="1.75\columnwidth"}
The $t=0.8$ns configuration plotted in Fig. \[fig:figure4\]b shows that, barring the corner edges, the growth has advanced by almost a full layer on the truncating facets. The droplet is still supersaturated and it now drives the nucleation on the main facet. It is worth pointing out that structurally the growth on the truncating facets and on the main facet is essentially independent, since a 2D island reaching the edge of one facet has no nearest neighbors with an island on the adjacent facet. They require the influence of the much weaker next nearest-neighbor interactions to extend across. We observe early stages of classical 2D layer-by-layer growth wherein several nuclei form and attempt to overcome the nucleation barrier. Observe that the nuclei form slightly away from the center of the main facet and their location is correlated with the corner edges. The disorder together with the enhanced Si-segregation at the contact line likely aids the nucleation on the main facet. The diffusion along the surface is indeed slower yet the bulk liquid can easily provide a pathway. The precursor cluster that eventually becomes supercritical is circled in Fig. \[fig:figure4\]b. Figures \[fig:figure4\]c-\[fig:figure4\]e reveal the key features of the subsequent bilayer growth on the main facet, i.e. it remains layer-by-layer and dominated by a single growing nucleus.
The density profiles evolve accordingly. The intensity of the two silicon peaks increases at the expense of the Au peak. A sharp peak in the adjacent liquid layer indicates build-up of Si while the third layer becomes Au-rich (arrow in Fig. \[fig:figure4\]d). Evidently, Au diffuses rapidly away from the interface. At $t=2$ns, the first layer has not completely formed yet the step flow has slowed down due to the low supersaturation left in the droplet, $X_{Si}=0.463$. Note that there is negligible growth at the edge corners (circled, Fig. \[fig:figure4\]e). We see Au and liquid Si within the mostly crystallized Si layer although the liquid Si atoms exhibit in-plane order. Eventually, some of the Au atoms diffuse away into the liquid droplet and aid further crystallization (not shown).
We quantify these observations by monitoring the crystallized volume at the interface $V(t)$ and the droplet composition $X_{Si}(t)$ (Fig. \[fig:figure4\]g). Crystallization kinetics on the truncating and main facets are plotted separately. Initially for $t<1$ns the increase in $V(t)$ is entirely due to crystallization on the truncating facets. A linear fit yields a growth rate of $v\approx10.5$cm/s. The growth is instantaneous indicative of a small nucleation barrier. While the barrier can be artificially reduced due to the larger supersaturations in the computations, note that the Si-rich contact line is an easy source for Si adatoms. Also, as in the experiments, the facet widths rarely exceed a few nanometers such that the critical nuclei sizes are likely larger than the facet widths[@sold:RohrerMullins:2001]. The circumferential 1D growth is also aided by the stepped structure of these facets and we quantify this effect by extracting Si adatom enthalpies $H_{ad}$ on the $\{111\}$ and the $\{113\}$ family planes (Methods). Not surprisingly, the close-packed $\{111\}$ facets have a significantly higher enthalpy than the stepped $\{113\}$ facets, $\Delta H_{ad}\approx2.0$eV/adatom. The adatom diffusivity on the $\{113\}$ facets is the suppressed since the atom makes fewer diffusion hops before another atom crystallizes in close proximity. The adatoms can readily form clusters and therefore lower the nucleation barrier, which in turn can induce rapid 1D growth. The role of such structural effects in introducing kinetic anisotropies during faceted crystal growth is well-established[@cg:ZepedaRuizGilmer:2006].
At $t\approx1.0$ns, the growth rate on the truncating facets decreases, either due to site-saturation or the onset of the oscillatory growth mode. The growth on the main facet, non-existent so far, dramatically increases. This is indicative of the growth of a supercritical nuclei and is again consistent with our qualitative observations (Fig. \[fig:figure4\]c-\[fig:figure4\]e). In order to ascertain if the growth is layer-by-layer or kinetically rough, we monitor the size evolution of the visibly identifiable nuclei on the main facet (Fig. \[fig:figure4\]h). We see a dramatic increase in the size of one of the nuclei while the other nuclei fluctuate about a relatively smaller size ($<35$ atoms). At longer times $t\approx10$ns, one of these smaller nuclei shrinks while the other two persist without any appreciable increase in size. Clearly, one 2D island dominates the layer growth and the encompassing steps are the main contributions to surface roughening. Moreover, the step orientations reflect the symmetry of the growth direction while that is not the case for the fluctuating subcritical islands. The size evolution shows that even though the driving force is larger than that in experiments, kinetic roughening or the increase in step density resulting from growth on the main facet is non-existent[@sold:JacksonGilmer:1976]. The persistence of the subcritical nuclei can signify that the supersaturation is on the borderline between single nucleus and polynuclear growth; then at lower supersaturations we expect the growth of the nanowires to be universally by the single nucleus mechanism. Following nucleation, the growth is steady-state as the volume increases and composition decreases. A linear fit in this regime yields the overall growth velocity, $v\approx6.3$cm/s. The growth rate also includes contributions from the truncating facets. Subtracting the former from the overall growth rate yields the step velocity on the main facet, $v_s\approx3$cm/s. Figure \[fig:figure4\]f is a schematic summary of the morphological features implicated in the overall growth.
Effect of supersaturation
-------------------------
In order to make contact with experimental scales, we quantify the nucleation barrier on the main facet by performing computations with decreasing initial supersaturation. Representative results for $X_{Si}=0.463$ ($\Delta_T\approx5.4$K) are shown in Fig. \[fig:figure5\]. The truncating facets again grow rapidly and circumferentially (Fig. \[fig:figure5\]a). The growth is instantaneous yet slower, and the growth on the main facet is non-existent (Fig. \[fig:figure5\]b). The crystallization is again preferentially at the centers of the truncating facets, presumably due to a 1D interplay between crystallization at the facet center and melting at the rough corner edges (Supplementary Movie 2). At $t\approx3.0$ns, we see transient crystallization of subcritical nuclei on the main facet yet there is no supersaturation left for their growth. Then, $X_{Si}=0.463$ represents a critical initial composition below which there is no nucleation on the main facet, i.e. nanowire growth can only occur at larger supersaturations.
{width="1.5\columnwidth"}
Increasing the initial composition in the range $X_{Si}\ge0.485$ ($\Delta_T\ge45$K) decreases the dead-zone that precedes the nucleation on the main facet. Multiple stable islands contribute to the growth of a single layer such that the step density increases and there is a gradual transition to a kinetically roughened regime wherein the second bilayer begins to crystallize before the complete growth of the first bilayer. This is evident in configurations for $X_{Si}=0.485$ shown in Supplementary Figure S5. Several Au atoms remain confined within the first layer. They likely retard the step flow and that triggers the onset of a 3D growth mode. Midsections reveal additional Si diffusion down the sidewalls that further aids the absorption of the supersaturation (inset).
Figure \[fig:figure6\]a shows the driving force dependence of the overall growth velocities for varying supersaturations. The velocities are linear fits to the initial crystallization rates. In the small driving force limit $X_{Si}\le0.48$ ($\Delta_T<36$K), the velocity increases rapidly. Results of similar computations on a planar Si(111)-AuSi interface are also plotted for comparison (see Methods). Crystallization at the planar interface entails homogeneous nucleation and is always slower at comparable driving forces, suggesting a higher nucleation barrier. Separate plots for the contributions of the truncating and main facets are plotted in Fig. \[fig:figure6\]b. The velocity of the truncating facets increases linearly with supersaturation and indicates that its growth is not nucleation-controlled, at least for undercoolings as low as $\Delta_T=5.4$K. The average kinetic coefficient is $\approx3.0$mm/(sK) and is of the order of kinetic coefficients in binary alloys[@sold:HoytAstaKarma:2002].
The crystallization rate on the main facet increases non-linearly with the supersaturation signifying classical nucleation-controlled growth. It is possible that at much smaller driving forces the growth is linear as the finite facet size requires equilibrium between melting and nucleation[@nw:HaxhimaliAstaHoyt:2009], although it is not clear if this applies to a fully faceted interface. Accordingly, we fit the variation to a functional form $v=v_0\exp(-\Delta G/k_BT)$, where the exponential factor is the probability that a spontaneous fluctuation will result in a critical nucleus, and $v_0$ is the 2D growth rate of the supercritical nucleus that is almost independent of the supersaturation. The nucleation barrier varies as $\Delta G \propto 1/\Delta_X$ and the exponential fit to the extracted velocities yields a value $\Delta G=4$meV/$\Delta_X$ at $T=873$K. A similar fit to the planar interface data yields a barrier that is $63\%$ larger and is consistent with estimates based on nucleation of a 2D island (Supplementary Discussion 1). At larger driving forces characterized by kinetically rough 3D growth, the velocity increases almost linearly with supersaturation. The plot also shows results of computations on undersaturated droplets ($X_{Si}<X_{Si}^\ast$). At small driving forces, the nucleation and growth dynamics associated with the melting kinetics is completely reversed; the truncating facets melt first as the energy of formation of a vacancy on these facets is lower, and eventually the main facet melts preferentially from the corner edges. The layer-by-layer melting velocities are smaller compared to those during growth due to the different morphological features implicated in the two processes, which by itself is important as the growth is a steady-state between growth and melting kinetics[@sold:RohrerMullins:2001; @nw:HaxhimaliAstaHoyt:2009].
{width="2\columnwidth"}
Mechanistic insight during growth
---------------------------------
{width="1.5\columnwidth"}
Maps of atomic displacements over prescribed time intervals corroborate the observed trends. Figure \[fig:figure7\]a-d shows a $t=0.5-1.0$ns map for select Si and Au atoms within the $X_{Si}=0.48$ computation. The displacements correspond to newly crystallized Si atoms (Si-map) and Au atoms initially at the interface (Au-map). The 2D step flow results in linear decrease of the droplet composition from $X_{Si}=0.474$ to $X_{Si}=0.469$ (Fig. \[fig:figure4\]g). The Si crystallization is mostly from adjacent bilayers and driven by displacement chains. Surprisingly, we also see long-range fast diffusion from as far as the droplet surface (Fig. \[fig:figure7\]a). The flux is mostly through the amorphous bulk of the droplet and we see negligible surface diffusion (Fig. \[fig:figure7\]b). The step flow is aided by Au diffusion into the droplet (Figs. \[fig:figure7\]c) and also along the interface directed away from the moving steps (Fig. \[fig:figure7\]d). We see evidence for both short-range atomic hops into adjacent liquid bilayer and long-range diffusion towards the droplet surface. The diffusion is markedly along the bulk and also towards the nanowire sidewalls; the latter represents a direct link between nanowire growth and Au decoration on the sidewalls.
Discussion {#discussion .unnumbered}
==========
The atomic-scale structure and dynamics during VLS growth of SiNWs is clearly sensitive to segregation at the surface and the droplet-nanowire interface. The robust surface layer is expected to be stable under reactor conditions where the droplet is exposed to the precursor gas atmosphere. The Au submonolayer together with the inhibited surface diffusion renders the droplet surface in immediate contact with the vapor sufficiently Au-rich to drive the precursor breakdown over large pressure and temperature ranges. The heterogeneously faceted interface morphology impacts all aspects of the growth process[@nw:WenTersoffRoss:2011] and requires reconsideration of current growth models. The truncating facets induce variations in the Si segregation and droplet morphology along the contact line. The relative disordered corner edges impact interface nucleation and growth. They likely serve as efficient sources of steps for rapid crystallization on the truncating facets, analogous to the screw dislocation-based spiral crystal growth[@tsf:BurtonCabreraFrank:1951]. At much lower supersaturations, the dynamics may involve a competition between nucleation at the facet centers and melting at the edge corners. Nevertheless, the barrier is clearly much smaller than that on the main facet. The latter is clearly the rate limiting event. The growth of the truncating facets also decreases the extent of these ordered regions, and they can facilitate mass transfer from the Si-rich contact line onto the main facet and thereby lower the barrier compared to an equivalently supersaturated planar Si-AuSi interface; the computations show that this is indeed the case. The extracted nucleation barrier $\Delta G$ allows us to establish nucleation- and catalysis-controlled regimes. Under typical growth conditions, the atomic incorporation rate is of the order of a single Si atom per millisecond (Supplementary Discussion 2). Equating this incorporation rate to the nucleation controlled growth rate, the extracted barrier $\Delta G=4$meV$/\Delta_X$ yields a supersaturation of $\Delta_X^c\approx0.002$ for isothermal nanowire growth at $T=873$K, or a critical undercooling of $\Delta_T^c\approx3.6$K. The growth is then nucleation-controlled for $\Delta_T<\Delta_T^c$ and catalysis-controlled otherwise. Solidification from melts typically occurs at undercooling of less than a Kelvin[@book:Woodruff:1973], and if that is the case during VLS growth of nanowires, it must nucleation-controlled. We should emphasize that since the barrier is expected to be nanowire size dependent, the interplay between nucleation and catalysis can be quantitatively different at larger sizes and is a focus of future studies.
An interesting coda is the transition to kinetically rough growth mode at larger supersaturations is intriguing as the step flow is increasingly limited by Au-diffusion away from the step-edges. The inter-island Au atoms are stable over the MD time scales ($<10$ns), and although we cannot rule out their slow diffusion out of the nanowire eventually, there is growing evidence of catalyst particle trapping within the as-grown nanowires[@nw:MoutanabbirSeidman:2013]. Detailed understanding of this regime opens up the possibility of doping nanowires to non-equilibrium compositions [*during*]{} their synthesis[@nw:SchwalbachVoorhees:2009], a scenario worthy of detailed explorations by itself.
Methods
=======
[**Model system**]{}: We use classical inter-atomic potentials to model the eutectic AuSi system. Hydrogenation effects are ignored due to hydrogen desorption and Au-surface passivation at these elevated temperatures. The potentials can be a limitation as they represent approximations to the electronic degrees of freedom. Although the AEAM model system is fit to several experimental and first-principles data and is based on established frameworks for both pure Au and Si, its ability to make quantitative predictions, notably interface properties, may be limited. For example, the pure Au potential underestimates the liquid free energies[@intpot:WebbGrest:1986] while the pure Si potential is limited in describing surface reconstructions on crystal facets[@intpot:LiBroughton:1988]. Nevertheless, validation studies presented here show that the predictions of the model potential are in agreement with several alloy properties. The AEAM framework reproduces the primary feature of the binary phase diagram, i.e. a low melting eutectic. The noteworthy deviations are an underestimation of the eutectic temperature $T_E$ ($590$K compared to $636$K) and overestimation of the eutectic composition $X_{Si}^{E}$ ($31\%$ compared to $19\%$). We have also performed contact angle studies of eutectic and hypereutectic AuSi droplets on Si(111) surface using semi-grandcanonical Monte-Carlo (SGMC) simulations (Supplementary Methods). The final configuration of a $2R=10$nm size droplet on a Si(111) surface at the growth temperature is shown in Supplementary Figure S6. The contact angle $\theta=143^\circ$ is close to the experimental value[@tsf:ResselHomma:2003], $\theta=136^\circ$.
[**Equilibrium liquidus concentration for $2R=10\,$nm SiNW**]{}: The initial structure consists of a pristine $2R=10\,$nm $\langle111\rangle$ SiNW with $\{110\}$ or $\{112\}$ family of lateral facets capped by two AuSi particles. The Si(111)-AuSi interfaces are initially flat. The SiNW length is $\approx8$nm, large enough such that the interfaces do not interact. The atomic structure of the particles is generated randomly at a density corresponding to liquid AuSi and the initial composition is hypoeutectic, $X_{Si}=0.25$. The entire system consists of $\sim64000$ atoms and is relaxed using SGMC computations. A local order parameter is employed to demarcate the liquid and solid Si atoms[@sold:ButaAstaHoyt:2008]. The order parameter is an extension of the pure Si case based on the local 3D structure around each Si atom. Additional validation of the surface segregation is done using charge-gradient corrections to the Au-Au interactions that correctly reproduces the liquid Au surface tension (Supplementary Methods).
[**Equilibrium studies on isolated AuSi particles and thin films**]{}: Density profiles of the equilibrated droplets are averaged over bilayer thick spherical shells around the center of mass of the droplet, and the ensemble average is over 1000 different atomic configurations. In order to quantify the effect of droplet surface curvature, the density profiles are also extracted for thin films. We employ a thin film of thickness $H=10\,$nm, free at both surfaces and periodic in the film plane. SGMC simulations are used to equilibrate the initially liquid film. Supplementary Figure S1 shows the equilibrated configuration for a 27000-atom simulation and the corresponding normalized density profiles as a function of a distance from the surface $h-H$ extracted at the bulk eutectic composition and temperature, $X_{Si}=0.31$ and $T_E=590$K. Layering of both Au and Si is clearly visible and a Au monolayer decorates the Si-rich surface. Supplementary Figure S1 also shows the density profiles for thin films at the growth temperature selected for this study ($T=873$K) and for size-corrected liquidus point, ($T=873$K, $X_{Si}=0.46$). Comparison with the profile at the eutectic point clearly shows a decrease in the thickness of the solid surface layer with increasing temperature and Si supersaturation.
[**Au and Si diffusivities within the droplet**]{}: MD simulations are performed on the equilibrated droplets to extract the ensemble-averaged MSD of both Au and Si (Supplementary Methods). A net simulation time of a few tens of picoseconds is sufficient to extract statistically meaningful data. For each atomic species at an initial distance $r$ from the mass center of the particle, the MSD increases linearly with time and the slope yields the diffusivity$D(r)$. For atoms in the bulk and on the surface, the slope is different. The MSDs for Au and Si atoms initially at the surface of the $2R=10$nm Au$_{54}$Si$_{46}$ droplet are plotted in Supplementary Figure S3. The computations on bulk eutectic AuSi alloy yield a diffusivity of $D_{Au}=1.54\times10^{-9}$m$^2$/s at $873$K; the value is in excellent agreement with past experiments on Au diffusivity in eutectic AuSi alloys[@diff:BrusonGerl:1982], $D_{Au}\approx1.7\times10^{-9}$m$^2$/s.
[**Equilibrium state of the droplet-nanowire-substrate system**]{}: The initial configuration is generated by taking one half of the double droplet configuration and placing it on a ($16.0\,{\rm nm}\times15.4\,{\rm nm}$) Si(111) substrate. The dimensions of the substrate are the same as in the contact angle studies. The bottom bilayer is fixed. The exposed area is decorated by a Au monolayer with a structure similar to the classical $\sqrt{3}\times\sqrt{3}\,\uppercase{R}\,30^\circ$ monolayer structure[@tsf:NagaoHenzler:1998], and the entire system is equilibrated using canonical Monte-Carlo simulations followed by MD simulations.
[**Nanowire growth kinetics**]{}: The catalytic incorporation of individual Si atoms takes place over much larger time-scales, and we expect the interface growth to be orders of magnitude faster. Therefore, there is no additional Si flux imposed on the droplet during the course of the MD computations and we investigate the effect of an initially prescribed driving force that evolves in a manner consistent with nucleation-limited growth. Although the supersaturation is arbitrary and likely larger compared to the critical supersaturation in experiments, a systematic variation of the driving force allows us to extract the key features associated with the near-equilibrium behavior of the nanowire during its growth. The supersaturation for growth is generated by attempting to place Si atoms on the surface within an MD simulation and then accepting the move if an Au atom is within a nearest-neighbor distance. Alternate procedure wherein the droplet is supersaturated by directly introducing atoms within an isolated droplet have negligible effect on the ensuing dynamics; the segregation at the surface and interface is fairly robust. The MD simulations employed are same as those for diffusivity calculations.
Si(111)-AuSu interface kinetics and step energetics {#si111-ausu-interface-kinetics-and-step-energetics .unnumbered}
---------------------------------------------------
The simulation cell contains of a crystalline Si(111) substrate (8640 atoms) abutting a AuSi alloy (34560 atoms). The dimensions of simulation cell is $14.14 a_0\times14.69 a_0\times 25.97a_0$, where $a_0$ is the lattice parameter at $873$K. The initial AuSi alloy with the desired concentration is randomly structured and MD simulations are performed at $873$K. Periodic boundary conditions are applied along the in-plane directions with free surfaces normal to the interface. The Si(111) substrate is initially fixed and the AuSi alloy is relaxed for $0.1$ns. The planar nucleation/melting simulations are carried by fixing the bottom bilayer of the substrate. The crystalline and liquid Si atoms are identified using the local order parameter described earlier. The simulation cell for extracting the step enthalpies is similar; it $19.80 a_0\times19.60a_0\times10.39 a_0$ in size and consists of a Si(111) slab (16128 atoms) and a AuSi alloy slab (16128 atoms). The composition of the initial AuSi alloy is $X_{Si}=0.46$ and it is relaxed using isobaric, isothermal MD simulations (Supplementary Methods). The simulation cell is fully periodic so that the cell consists of Si(111)-AuSi interfaces. The dimension normal to the free surface adjusts to maintain zero pressure while two in-plane dimensions $(1\bar{1}0)$ and $(11\bar{2})$ are fixed. Following relaxation ($0.3$ns), the total potential energy $U_1$ is averaged over $0.1$ns. Single bilayer steps are created on both surfaces of Si slab while preserving the atoms in the Si(111) crystal and liquid AuSi. The steps are oriented along the index $(112)$ direction. The total potential potential energy is recorded as $U_2$ and the difference $\sigma=(U_2-U_1)/L=0.16$eV/nm, where $L$ is the total step length.
[**Si adatom energetics on truncating and main facets**]{}: Canonical MD simulations are performed at temperature $T=1$K. Each computational cell consists of an unreconstructed crystal Si slab, free at both both surfaces and periodic in the slab plane. The dimensions of simulation cells are \[$7.07 \times 9.38 a_0 \times 6.63 a_0$\] and \[$7.07 a_0 \times 7.35 a_0 \times 5.20 a_0$\] for $(113)$ and $(111)$ surfaces and contains 3520 and 2160 atoms, respectively. The slabs were relaxed for $10$ps. A single Si atom is placed within the interaction range on each of the surfaces and relaxed for another 10 ps. All possible positions with maximum nearest neighbors were simulated to obtain the lowest potential energy. The maximum potential energy difference before and after addition of the Si adatom is recorded as the Si adatom enthalpy $H_{ad}$. The extracted enthalpies are $-5.4$eV/atom and $-3.5$eV/atom for the $(113)$ and $(111)$ surfaces, respectively. Note that the stepped structure of unreconstructed $(113)$ surfaces can be interpreted as $(111)$ steps separated by single atomic rows of $(001)$ terraces. The $\{120\}$ facets are also stepped and we therefore expect similar trends.
[**Acknowledgements**]{}: We thank A. Dongare and L. Zheiglei for providing us with an early version of the AEAM interatomic potential, and Alain Karma and Albert Davydov for helpful discussions. The computations were performed on [*st*]{}AMP supercomputing resources at Northeastern University. The study is supported by National Science Foundation DMR CMMT Program (1106214). Part of this work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344 (LAZ and GHG).
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Supplementary Information {#supplementary-information .unnumbered}
=========================
Supplementary Figures {#supplementary-figures .unnumbered}
=====================
![(top left) Atomic configuration and the corresponding normalized density profiles as a function of distance from the surface ($h-H$) in an AuSi thin film at the eutectic point for the model system. (a) The normalized density profiles at (a) eutectic point $X_{Si}^\ast(590\,{\rm K})=0.31$, (b) elevated temperature $T=873$K, and (c) elevated temperature and supersaturation, $X_{Si}(873\,{\rm K})=0.46$. \[fig:Supp2\] ](FigSupp2LowRes.pdf){width="0.6\columnwidth"}
![\[fig:Supp5\] a) Surface tension $\gamma$ versus $-\beta$ for AEAM Au at 1400K. $\beta=-0.025$ is a fit to the experimental value. (b) Segregation profiles for the AEAM Au/Si alloy thin film, before and after charge gradient correction.](FigSupp-CGLowRes.pdf){width="\columnwidth"}
![Temporal evolution of the ensemble-averaged mean square displacement of surface and bulk atoms in equilibrium MD simulation of an equilibrated $2R=10$nm isolated Au$_{53}$Si$_{46}$ particle. \[fig:Supp3\] ](FigSupp3LowRes.pdf){width="0.55\columnwidth"}
![Equilibrium morphology of a $2R=10$nm nanowire with {112} sidewall facets. (left inset) Schematic showing the relationship with 110 sidewall facets. (left) Atomic configuration of the complete nanowire-droplet-substrate system. (middle inset) (right) A mid-section slice along one of the $\langle$$112$$\rangle$ directions. Dashed lines in the figure indicate the truncating facets. The SiNW structure is the same as in the double-capped simulations, i.e. it has a hexagonal cross-section with {110} facets. The nanowire stem is $3.5$nm in length. \[fig:Supp4\] ](112ResultsLowRes.pdf){width="0.8\columnwidth"}
![Same as in Figure 5 in the main text, but for $X_{Si}=0.485$ ($\Delta_T\approx45$K). The $t=4$ns configuration is shown with and without liquid Si (red) atoms. We see multiple growing nuclei on the main facet indicating that they are supercritical. The nuclei form preferentially near the corner edges where the main facet meets two truncating facets.[]{data-label="fig:Supp7"}](FigSupp485LowRes.pdf){width="\columnwidth"}
![Atomic-configuration of an equilbrated eutectic droplet Au$_{33}$Si$_{67}$ on a Si(111) surface. The color scheme is similar to that in the main text with the exception that we make no distinction between liquid and solid Si atoms. The inset shows the morphology of eutectic AuSi droplets on Si(111) reported in prior experiments [@tsf:ResselHomma:2003]. The image is reproduced with the authors’ permissions. \[fig:Supp0\] ](FigSupp0LowRes.pdf){width="0.7\columnwidth"}
Supplementary Methods {#supplementary-methods .unnumbered}
=====================
Atomic-scale simulations {#atomic-scale-simulations .unnumbered}
------------------------
The semi-grand canonical Monte-Carlo simulations employed to relax the system at the growth temperature consist of translational and Au$\leftrightarrow$Si exchange (transmutation) moves [@book:AllenTildesley:1989; @gbseg:Seidman:2002]. The ratio between translational and exchange moves is fixed at $10:1$. The acceptance ratio for translation moves is constrained to $50\%$ by adjusting the magnitude of the translations. The equilibration is performed until the energy, pair distribution functions and density profiles converge. In some cases, canonical (NVT) Monte-Carlo computations are performed by turning off the transmutation moves.
The MD simulations are performed by integrating the Newton’s equations of motion using velocity-Verlet scheme. The time step is $1$fs and the trajectories for the canonical ensemble are generated using a Nosé-Hoover to fix the temperature at $873$K [@book:AllenTildesley:1989]. Isobaric isothermal simulations (NPT) are performed by using a Parinello-Rahman barostat that allows the cell volume to adjust to the imposed press. The fictitious mass of the barostat is fixed at XX.
Contact angle of $2R=10\,$nm droplet on Si(111) surface {#contact-angle-of-2r10nm-droplet-on-si111-surface .unnumbered}
-------------------------------------------------------
A $2R=10$nm size spherical droplet consisting of $\sim32000$ atoms is constructed by assigning random coordinates about positions corresponding to average Au-Si equilibrium distance based on the model angular EAM potential. One half of the droplet is then placed on an atomically flat Si(111) substrate (thickness 8 bilayers). The substrate dimensions are as mentioned in the main text. It is periodic along the in-plane directions. The bottom layer is fixed to prevent the entire system from translating. The exposed regions are decorated with a $(\sqrt{3}\times\sqrt{3})\uppercase{R}\,30^\circ$ Au monolayer and then entire system is then equilibrated to the growth temperature using SGMC minimization followed by NVT MD. Upon equilibration, we see some evidence for disorder within the monolayer. Systematic thin film studies indicate that for our model system, the monolayer undergoes a surface transition at and around $900^\circ$K. A similar transition has been reported in experiments [@tsf:NagaoHenzler:1998], albeit at $1000^\circ$K. The difference between the experimental and computational contact angles is around $7^\circ$. We attribute this mainly to the differences in the manner in which the droplets attain the eutectic composition. In experiments, the silicon from the substrate melts into the initially Au-rich droplet to arrive at the equilibrium composition, resulting in a the crater like morphology of the substrate. On the other hand, the droplet in the computations is already at liquidus composition before it stabilizes on the substrate. The substrate is almost flat and therefore increases the measured contact angle with respect to the substrate surface plane.
Charge gradient corrections for AEAM {#charge-gradient-corrections-for-aeam .unnumbered}
------------------------------------
The computational cell for the MD simulations is chosen to be an FCC crystal slab with dimensions $10\,a_0\times10\,a_0$ in the periodic surface plane, and a thickness of $8a_0$. The slabs are melted and equilibrated for $1$ns; the effect of temperature was studied by equilibrating the system for $250$ps. Data runs $1$ns in duration are carried out for each temperature $T$. The mechanical definition is used to extract the surface tension [@intseg:YuStroud:1997], $$\tag{S1}
\gamma=\frac{1}{2}\int_{-\infty}^{\infty}[\sigma_{xx}+\sigma_{yy}-2\sigma_{zz}]\le\frac{1}{S}\sum_{i}\{\frac{p_x^2+p_y^2-2p_z^2}{2
m_i}+\sum_{i{\neq}j}\frac{f_xr_x+f_yr_y-2f_zr_z}{4}\}>\,.$$ Here the $\sigma_{\alpha\beta}$ are components of the surface stress tensor, $\alpha, \beta=1, 2, 3$ along the $x$, $y$ and $z$ directions in a Cartesian coordinate system. $S$ is the surface area, $m_i$ is the mass of atom $i$, $p_\alpha$ is the ${\alpha}$th momentum component of atom $i$, $r_\alpha$ is the ${\alpha}$th component of the distance atoms $i$ and $j$, $f_\alpha$ is the ${\alpha}$th component force component of atom $i$, and both the summations over $i$ and $j$ run over all the atoms in the systems.
The Au-Au interactions employed as part of the AEAM framework used in this study are of the form suggested in Ref. [@tsf:Zhou:2001]. For pure liquid Au at $T=1400$K, the extracted surface tension is $0.9$J/m compared to the experimental value of $1.11$J/m. The difference is significant and can affect the segregation qualitatively. To test this, we employ a charge-gradient correction [@intpot:WebbGrest:1986] to fit the Au-Au interactions to the experimental value of the liquid/vapor surface tension; the Au-Si and the Si-Si interactions remain unchanged.
In this framework, the potential energy $U_i$ of atom $i$ is, $$\begin{aligned}
& U_i=\sum_i\left\{F(\eta_i)+\sum_{j\neq{i}}\frac{1}{2}\Phi(r)\right\}\nonumber\\
& \eta_i=\rho_i+\beta|\nabla\rho_i|^2,\quad \rho_i=\sum_{j\neq{i}}f(r)\,.
\tag{S2}\end{aligned}$$ Here $F$ is the usual embedding potential energy, $\Phi$ is the pair potential energy, $\rho_i$ the local charge density at atom $i$, and $\eta_i$ is the local charge density of atom $i$ with gradient corrections, and $r$ is the distance between atoms $i$ and $j$. The constant $\beta$ emerges as the main fitting parameter for the liquid surface tension. Notice that the formulation in Ref. [@intpot:WebbGrest:1986] introduces another fitting parameter $c$ which is set to zero here. Then, the $\alpha$th component of force $m_ia_i^\alpha$ on atom $i$ becomes, $$\begin{aligned}
\tag{S3}
m_ia_i^\alpha=&-\sum_{j{\neq}i}\left(\frac{1}{2}\frac{\partial\Phi}{\partial{r}}+\frac{\partial{F}}{\partial{r}}\right)\frac{r_{\alpha}}{r}\nonumber\\
&-2\beta\frac{\partial{F}}{\partial{\eta_i}}
\left\{\sum_{j{\neq}i}\frac{\partial{f}}{\partial{r}}\frac{r_\alpha}{r}\sum_{j{\neq}i}\frac{\partial{f}}{\partial{r}}\frac{1}{r}+
\sum_{j{\neq}i}\frac{\partial{f}}{\partial{r}}\frac{r_\beta}{r}
\sum_{j{\neq}i}\frac{{r_\alpha}{r_\beta}}{r^2}(\frac{\partial^2{f}}{\partial{r^2}}-\frac{\partial{f}}{\partial{r}}\frac{1}{r})\right\}\,,\nonumber\end{aligned}$$ where the first term is the standard EAM-based force and the second term is the charge gradient correction contribution. Using this formulation, we have re-calculated the surface tension of pure liquid Au $\gamma$ as a function of the fitting parameter $\beta$ at 1400K. The results are shown in Supplementary Figure S2. The surface tension increases monotonically with the magnitude of $\beta$ with the value $\beta=-0.025$ in excellent agreement with the experimental value. Simple tests are first performed to quantify the effect of this modified potential on the bulk properties of Au at $1400$K using MD simulations. For $\beta=-0.025$, the cohesive energy is $-3.626$eV/atom and the lattice constant is $a_0=4.197A$. As comparison, for $\beta=0$ which reduces to the original AEAM potential, the corresponding values are $-3.624$eV/atom and $a_0=4.206$[Å]{}, respectively.
This modified Au-Au potential within the AEAM formulation is employed to recalculate the density profiles for AuSi thin films. The surface segregation in AuSi film with $33\%$ Si for charge gradient correction $\beta=0.025$, which is also qualitatively same the result without charge gradient correction. The ensemble-averaged segregation profiles are show in Fig. \[fig:Supp5\]B. Again, we see strong subsurface Si segregation which is decorated by a submonolayer of Au. The quantitative differences are minimal, notably that the peak associated with Au submonolayer decoration on the surface is slightly decreased.
Supplementary Discussion 1 {#supplementary-discussion-1 .unnumbered}
==========================
We estimate the nucleation barrier on the planar Si(111)-AuSi interface by assuming that the nuclei is a circular 2D island. The barrier for homogeneous nucleation on the planar Si(111) surface is roughly $\Delta G \sim \sigma^2h/(\rho\,\mu_{sl})$, where $\sigma$ is the step energy per step step (bilayer) height $h$, $\rho$ is the density of solid silicon, and $\mu_{sl}$ is the free energy difference between solid and liquid phases. Our extracted values of the step enthalpies on Si(111) vary between $\sigma h\approx0.16$eV/nm. $\mu_{sl}$ for the small supersaturations in the AuSi model system is $\sim100\,\Delta_X$kJ/mol, approximated as the enthalpy of mixing predicted by the model AuSi system \[34\]. Using these values yields a barrier $\Delta G\approx20$meV$/\Delta_X$ that is of the same order as the extracted value of $\Delta G\approx7$meV$/\Delta_X$. The self-consistent comparison is strong evidence that the nucleation barrier during nanowire growth is lowered due to the multi-faceted morphology of the growth front.
Supplementary Discussion 2 {#supplementary-discussion-2 .unnumbered}
==========================
The Si incorporation rate follows from the volume growth rate due to the flux made available by the surface catalysis. It scales as $k_{vl}\,p\,A_s$, where $p$ is the precursor gas partial pressure, $k_{vl}$ is normalized catalyst rate constant per unit partial pressure, and $A_s$ is the exposed surface area of the droplet. The rate constant reported in recent experiments on growth of Si nanocrystals within an AuSi droplet exposed to a silane flux is $k_{vl}\sim10^3\,\, {\rm nm\, s}^{-1}\,{\rm Torr}^{-1}$ \[26\]. Approximating the droplet surface area to that of a hemisphere of radius $5$nm, under typical processing conditions ($p\approx10^{-5}$ Torr) yields the incorporation rate assumed in the text. As a simple check, the axial nanowire growth rate then is of the order of a nanometer per minute which is consistent with growth rate observed in in-situ experiments.
The transition from flux-controlled to nucleation-controlled growth occurs when the flux due to catalysis equals the crystallization rate on the main facet, $$\tag{S4}
v=v_0\exp\left(-\frac{\Delta G/\Delta_x}{k_BT}\right)\nonumber$$ From Fig 6b in the main text, we have $v_0=0.58$m/s and $\Delta G/k_BT=0.055$. For Si volume flux of the order of $I_{Si}\approx1$atom/ms, the axial nanowire growth rate is, $$\tag{S5}
v=\frac{I_{Si}}{\rho\pi R^2}\nonumber,$$ where $\rho=50$ atom/nm$^3$ is the density of crystallized Si and $R=5$nm is the nanowire radius in our simulation. Equating Eqs. S1 and S2, we get $\Delta_x=0.002$.
[1]{}
B. Ressel, K. C. Prince, S. Heun, and Y. Homma. Wetting of [Si]{} surfaces by [Au-Si]{} liquid alloys. , 93(7):3886–3892, 2003.
M. P. Allen and D. J. Tildesley. . Oxford University Press, 1989.
D. Seidman. Sub-nanoscale studies of segregation at grain boundaries: [S]{}imulations and experiments. , 32:235–269, 2002.
T. Nagao, S. Hasegawa, K. Tsuchie, S. Ino, C. Voges, G. Klos, H. Pfnür, and M. Henzler. Structural phase transitions of $\mathrm{Si}(111)-(\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3})r30\ifmmode^\circ\else\textdegree\fi{}-\mathrm{Au}$: Phase transitions in domain-wall configurations. , 57(16):10100–10109, Apr 1998.
W. Yu and D. Stroud. Molecular-dynamics study of surface segregation in liquid semiconductor alloys. , 56(19):12243–12249, 1997.
X. W. Zhou, H. N. G. Wadley, R. A. Johnson, D. J. Larson, N. Tabat, A. Cerezo, A. K. [Petford-Long]{}, G. D. W. Smith, P. H. Clifton, R. L. Martens, and T. F. Kelly. Atomic scale structure of sputtered metal multilayers. , 49(11):4005–4015, 2001.
E. B. [Webb III]{} and G. S. Grest. Liquid/vapor surface tension of metals: [E]{}mbedded atom method with charge gradient corrections. , 86(10):2066–2069, 2001.
|
---
address: |
$^{\ast}$Computational Linguistics & $^{\dagger}$IWR\
Heidelberg University, Germany\
{beilharz,karimova,riezler}@cl.uni-heidelberg.de,\
[email protected]
bibliography:
- 'references.bib'
title: |
LibriVoxDeEn: A Corpus for German-to-English Speech Translation\
and Speech Recognition
---
Introduction
============
Direct speech translation has recently been shown to be feasible using a single sequence-to-sequence neural model, trained on parallel data consisting of source audio, source text and target text. The crucial advantage of such end-to-end approaches is the avoidance of error propagation as in a pipeline approaches of speech recognition and text translation. While cascaded approaches have an advantage in that they can straightforwardly use large independent datasets for speech recognition and text translation, clever sharing of sub-networks via multi-task learning and two-stage modeling [@WeissETAL:17; @AnastasopoulosChiang:18; @SperberETAL:19] has closed the performance gap between end-to-end and pipeline approaches. However, end-to-end neural speech translation is very data hungry while available datatsets must be considered large if they exceed 100 hours of audio. For example, the widely used Fisher and Call-home Spanish-English corpus [@PostETAL:13] comprises 162 hours of audio and $138,819$ parallel sentences. Larger corpora for end-to-end speech translation have only recently become available for speech translation from English sources. For example, 236 hours of audio and $131,395$ parallel sentences are available for English-French speech translation based on audio books [@KocabiyikogluETAL:18; @BerardETAL:18]. For speech translation of English TED talks, 400-500 hours of audio aligned to around $250,000$ parallel sentences depending on the language pair have been provided for eight target languages by . Pure speech recognition data are available in amounts of $1,000$ hours of read English speech and their transcriptions in the [LibriSpeech]{} corpus provided by .
When it comes to German sources, the situation regarding corpora for end-to-end speech translation as well as for speech recognition is dire. To our knowledge, the largest freely available corpora for German-English speech translation comprise triples for 37 hours of German audio, German transcription, and English translation [@StuekerETAL:12]. Pure speech recognition data are available from 36 hours [@Radeck-ArnethETAL:15] to around 200 hours [@BaumannETAL:18].
We present a corpus of sentence-aligned triples of German audio, German text, and English translation, based on German audio books. The corpus consists of over 100 hours of audio material aligned to over 50k parallel sentences. Our approach mirrors that of in that we start from freely available audio books. The fact that the audio data is read speech keeps the number of disfluencies low. Furthermore, we use state-of-the art tools for audio-text and text-text alignment, and show in a manual evaluation that the speech alignment quality is in general very high, while the sentence alignment quality is comparable to widely used corpora such as that of and can be adjusted by cutoffs on the automatic alignment score. To our knowledge, the presented corpus is to data the largest resource for end-to-end speech translation for German.
Overview
========
In the following, we will give an overview over our corpus creation methodology. More details will be given in the following sections.
1. Creation of German corpus (see Section )
- Data download
- Download German audio books from *LibriVox* web platform [^1]
- Collect corresponding text files by crawling public domain web pages[^2]
- Audio preprocessing
- Manual filtering of audio pre- and postfixes
- Text preprocessing
- Noise removal, e.g. special symbols, advertisements, hyperlinks
- Sentence segmentation using *spaCy*[^3]
- Speech-to-text alignments
- Manual chapter segmentation of audio files
- Audio-to-text alignments using forced aligner *aeneas*[^4]
- Split audio according to obtained timestamps using *SoX*[^5]
2. Creation of German-English Speech Translation Corpus (see Sections and )
- Download English translations for German texts
- Text preprocessing (same procedure as for German texts)
- Bilingual text-to-text alignments
- Manual text-to-text alignments of chapters
- Dictionary creation using parallel DE-EN *WikiMatrix*[^6] corpus [@SchwenkETAL:19]
- German-English sentence alignments using *hunalign* [@VargaETAL:05]
- Data filtering based on *hunalign* alignment scores
Source Corpus Creation {#source_corpus}
======================
Data Collection
---------------
We acquired pairs of German books and their corresponding audio files starting from *LibriVox*, an open source platform for people to publish their audio recordings of them reading books which are available open source on the platform *Project Gutenberg*. Source data were gathered in a semi-automatic way: The URL links were collected manually by using queries containing metadata descriptions to find German books with LibriVox audio and possible German transcripts. These were later automatically scraped using *BeautifulSoup4*[^7] and *Scrapy*[^8], and saved for further processing and cleaning. Public domain web pages crawled include <https://gutenberg.spiegel.de>, <http://www.zeno.org>, and <https://archive.org>.
Data Preprocessing
------------------
We processed the audio data in a semi-automatic manner which included manual splitting and alignment of audio files into chapters, while also saving timestamps for start and end of chapters. We removed boilerplate intros and outros and as well as noise at the beginning and end of the recordings. Preprocessing the text included removal of several items, including special symbols like \*, advertisements, hyperlinks in \[\], <>, empty lines, quotes, - preceding sentences, indentations, and noisy OCR output.
German sentence segmentation was done using *spaCy* based on a medium sized German corpus[^9] that contains the TIGER corpus[^10] and the WikiNER dataset [^11] dataset. Furthermore we added rules to adjust the segmenting behavior for direct speech and for semicolon-separated sentences.
Text-to-Speech Alignment
------------------------
To align sentences to onsets and endings of corresponding audio segments we made use of *aeneas* – a tool for an automatic synchronization of text and audio. In contrast to most forced aligners, *aeneas* does not use automatic speech recognition (ASR) to compare an obtained transcript with the original text. Instead, it works in the opposite direction by using dynamic time warping to align the mel-frequency cepstral coefficients extracted from the real audio to the audio representation synthesized from the text, thus aligning the text file to a time interval in the real audio.
Furthermore, we used the maps pointing to the beginning and the end of each text row in the audio file produced with *SoX* to split the audio into sentence level chunks. The timestamps were also used to filter boilerplate information about the book, author, speaker at the beginning and end of the audio file.
Statistics on the resulting corpus are given in Table \[tab:source\].
\#books \#chapters \#sentences \#hours \#words
--------- ------------ ------------- --------- -----------
86 1,556 419,449 **547** 4,082,479
\#books \#chapters \#sentences \#hours \#words
--------- ------------ --------------- --------- ----------------
19 365 \[DE\] 53.168 **133** \[DE\] 898.676
\[EN\] 50.883 \[EN\] 989.768
\#books \#chapters \#sentences \#hours \#words
--------- ------------ --------------- --------- ----------------
19 365 \[DE\] 50.427 **110** \[DE\] 860.369
\[EN\] 50.883 \[EN\] 948.565
Target Corpus Creation {#target_corpus}
======================
Data Collection and Preprocessing
---------------------------------
In collecting and preprocessing the English texts we followed the same procedure as for the source language corpus, i.e., we manually created queries containing metadata descriptions of English books (e.g. author names) corresponding to German books which then were scraped. The *spaCy* model for sentence segmentation used a large English web corpus[^12]. See Section for more information.
Text-to-Text Alignment
----------------------
To produce text-to-text alignments we used *hunalign* with a custom dictionary of parallel sentences, generated from the *WikiMatrix* corpus. Using this additional dictionary improved our alignment scores. Furthermore we availed ourselves of a realign option enabling to save a dictionary generated in a first pass and profiting from it in a second pass. The final dictionary we used for the alignments consisted of a combination of entries of our corpora as well as the parallel corpus *WikiMatrix*. For further completeness we reversed the arguments in *hunalign* to not only obtain German to English alignments, but also English to German. These tables were merged to build the union by dropping duplicate entries and keeping those with a higher confidence score, while also appending alignments that may only have been produced when aligning in a specific direction.
Statistics on the resulting text alignments are given in Table \[tab:pre\].
Data Filtering and Corpus Structure {#corpus_filtering}
===================================
Corpus Filtering
----------------
A last step in our corpus creation procedure consisted out filtering out empty and incomplete alignments, i.e., alignments that did not consist of a DE-EN sentence pair. This was achieved by dropping all entries with a *hunalign* score of -0.3 or below. Table \[tab:post\] shows the resulting corpus after this filtering step. Moreover, many-to-many alignments by *hunalign* were re-segmented to source-audio sentence level for German, while keeping the merged English sentence to provide a complete audio lookup. The corresponding English sentences were duplicated and tagged with `<MERGE>` to mark that the German sentence was involved into a many-to-many alignment.
The size of our final cleaned and filtered corpus is thus comparable to the cleaned Augmented LibriSpeech corpus that has been used in speech translation experiments by .
Statistics on the resulting filtered text alignments are given in Table \[tab:post\].
Corpus Structure
----------------
Our corpus is structured in following folders:
- - contains German text files for each book
- - contains English text files for each book
- - alignment maps produced by *aeneas*
- sentence level audio files
- - text2speech, a lookup table for speech alignments
- text2text, a lookup table for text-to-text alignments
Further information about the corpus and a download link can be found here: <https://www.cl.uni-heidelberg.de/statnlpgroup/librivoxdeen/>.
**Bin** ***hunalign* confidence (avg)** **audio-text alignment (max 3)** **text-text alignment (max 5)**
------------- --------------------------------- ---------------------------------- ---------------------------------
Low 0.17 2.73 3.43
Moderate 0.59 2.65 3.63
High 1.06 2.71 4.35
**Average** **0.61** **2.69** **3.80**
Corpus Evaluation
=================
Human Evaluation
----------------
For a manual evaluation of our dataset, we split the corpus into three bins according to ranges $(-0.3,0.3]$, $(0.3,0.8]$ and $(0.8,\infty)$ of the *hunalign* confidence score (see Table \[tab:bins\]).
**Low** **Moderate** **High**
--------- --------------------- -------------------- -----------
**Bin** $-0.3 < x \leq 0.3$ $0.3 < x \leq 0.8$ $0.8 < x$
: Bins of text alignment quality according to *hunalign* confidence score[]{data-label="tab:bins"}
The evaluation of the text alignment quality was conducted according to the 5-point scale used in :
1. Wrong alignment
2. Partial alignment with slightly compositional translational equivalence
3. Partial alignment with compositional translation and additional or missing information
4. Correct alignment with compositional translation and few additional or missing information
5. Correct alignment and fully compositional translation
The evaluation of the audio-text alignment quality was conducted according to the following 3-point scale:
1. Wrong alignment
2. Partial alignment, some words or sentences may be missing
3. Correct alignment, allowing non-spoken syllables at start or end.
The evaluation experiment was performed by two annotators who each rated 30 items from each bin, where 10 items were the same for both annotators in order to calculate inter-annotator reliability.
Evaluation Results
------------------
Table \[tab:res\] shows the results of our manual evaluation. The audio-text alignment was rated as in general as high quality. The text-text alignment rating increases corresponding to increasing *hunalign* confidence score which shows that the latter can be safely used to find a threshold for corpus filtering. Overall, the audio-text and text-text alignment scores are very similar to those reported by .
The inter-annotator agreement between two raters was measured by Krippendorff’s $\alpha$-reliability score [@Krippendorff:13] for ordinal ratings. The inter-annotator reliability for text-to-text alignment quality ratings scored 0.77, while for audio-text alignment quality ratings it scored 1.00.
Examples
--------
In the following, we present selected examples for text-text alignments for each bin. A closer inspection reveals properties and shortcomings of *hunalign* scores which are based on a combination of dictionary-based alignments and sentence-length information.
Shorter sentence pairs are in general aligned correctly, irrespective of the score (compare examples with score $0.30$. $0.78$ and $1.57$, $2.44$ below). Longer sentences can include exact matches of longer substrings, however, they are scored based on a bag-of-words overlap (see the examples with scores $0.41$ and $0.84$ below).
- - Schigolch Yes, yes; und mir träumte von einem Stück Christmas Pudding.
- She only does that to revive old memories. LULU.
- - Und hätten dreißigtausend Helfer sich ersehn.
- And feardefying Folker shall our companion be; He shall bear our banner; better none than he.
- - Kakambo verlor nie den Kopf.
- Cacambo never lost his head.
- - Es befindet sich gar keine junge Dame an Bord, versetzte der Proviantmeister.
- He is a tall gentleman, quiet, and not very talkative, and has with him a young lady — There is no young lady on board, interrupted the AROUND THE WORLD IN EIGPITY DAYS. purser..
- - Ottilie, getragen durch das Gefühl ihrer Unschuld, auf dem Wege zu dem erwünschtesten Glück, lebt nur für Eduard.
- Ottilie, led by the sense of her own innocence along the road to the happiness for which she longed, only lived for Edward.
- - Was ist geschehen? fragte er.
- What has happened ? he asked.
- - Es sind nun drei Monate verflossen, daß wir Charleston auf dem Chancellor verlassen, und zwanzig Tage, die wir schon auf dem Flosse, von der Gnade der Winde und Strömungen abhängig, verbracht haben!
- JANUARY st to th.More than three months had elapsed since we left Charleston in the Chancellor, and for no less than twenty days had we now been borne along on our raft at the mercy of the wind and waves.
- - Charlotte stieg weiter, und Ottilie trug das Kind.
- Charlotte went on up the cliff, and Ottilie carried the child.
- - Fin de siecle, murmelte Lord Henry.
- Fin de siecle, murmured Lord Henry.
Conclusion
==========
We presented a corpus of aligned triples of German audio, German text, and English translations for speech translation from German to English. The audio data in our corpus are read speech, based on German audio books, ensuring a low amount of speech disfluencies. The audio-text alignment and text-to-text sentence alignment was done with state-of-the-art alignment tools and checked to be of high quality in a manual evaluation. The audio-text alignment was generally rated very high. The text-text sentence alignment quality is comparable to widely used corpora such as that of . A cutoff on a sentence alignment quality score allows to filter the text alignments further for speech translation, resulting in a clean corpus of $50,427$ German-English sentence pairs aligned to 110 hours of German speech. A larger version of the corpus, comprising 133 hours of German speech and high-quality alignments to German transcriptions is available for speech recognition.
Acknowledgments
===============
The research reported in this paper was supported in part by the German research foundation (DFG) under grant RI-2221/4-1.
Bibliographical References
==========================
[^1]: <https://librivox.org>
[^2]: https://gutenberg.spiegel.de,http://www.zeno.org, https://archive.org
[^3]: <https://spacy.io/>
[^4]: <https://github.com/readbeyond/aeneas>
[^5]: <http://sox.sourceforge.net/>
[^6]: <https://ai.facebook.com/blog/wikimatrix/>
[^7]: <https://www.crummy.com/software/BeautifulSoup/bs4/doc/>
[^8]: <https://scrapy.org/>
[^9]: <https://spacy.io/models/de#de_core_news_md>
[^10]: [https://www.ims.uni-stuttgart.de/forschung/\
ressourcen/korpora/tiger.html](https://www.ims.uni-stuttgart.de/forschung/\ressourcen/korpora/tiger.html)
[^11]: <https://dx.doi.org/10.1016/j.artint.2012.03.006>
[^12]: <https://spacy.io/models/en#en_core_web_lg>
|
---
author:
- |
Jie Liao$^{1}$, Yeping Li$^{2}$[^1],\
$^1$[*Department of Mathematics, East China University of Science and Technology, Shanghai 200237, P. R. China.*]{}\
$^2$[*Department of Mathematics, East China University of Science and Technology, Shanghai 200237, P. R. China.*]{}
title: 'Global existence and $L^{p}$ convergence rates of planar waves for three-dimensional bipolar Euler-Poisson systems'
---
[**Abstract:**]{} In the paper, we consider a multi-dimensional bipolar hydrodynamic model from semiconductor devices and plasmas. This system takes the form of Euler-Poisson with electric field and frictional damping added to the momentum equations. We show the global existence and $L^{p}$ convergence rates of planar diffusion waves for multi-dimensional bipolar Euler-Poisson systems when the initial data are near the planar diffusive waves. A frequency decomposition and approximate Green function based on delicate energy method are used to get the optimal decay rates of the planar diffusion waves. To our knowledge, the $L^p(p\in[2,+\infty])$-convergence rate of planar waves improves the previous results about the $L^2$-convergence rates.\
[**Key words:**]{} Bipolar Euler-Poisson system, planar wave, approximate Green function, smooth solution, energy estimates.\
[**AMS subject classifications:**]{} 35M20, 35Q35, 76W05.\
Introduction.
=============
In this paper, we consider the following bipolar Euler-poisson system (hydrodynamic model) in three space dimension: $$\label{l1}
\left\{
\begin{array}{lcr}
\partial_{t} \rho^{+}+ \mathrm{div}(\rho^{+} u^{+})=0,\vspace{2.5mm}\\
\partial_{t} (\rho^{+} u^{+}_{i})+\mathrm{div}(\rho^{+} u^{+}_{i}u^{+})+\partial_{x_{i}} P(\rho^{+})=-\rho^{+} u^{+}_{i} +\rho^{+}\partial_{x_{i}}\phi ,~~1\leq i\leq 3, \vspace{2.5mm}\\
\partial_{t} \rho^{-}+ \mathrm{div}(\rho^{-} u^{-})=0,\vspace{2.5mm}\\
\partial_{t} (\rho^{-} u^{-}_{i})+\mathrm{div}(\rho^{-} u^{-}_{i}u^{-})+\partial_{x_{i}} P(\rho^{-})=-\rho^{-} u^{-}_{i} - \rho^{-}\partial_{x_{i}}\phi ,~~1\leq i\leq 3, \vspace{2.5mm}\\
\Delta \phi = \rho^{+} -
\rho^{-},\lim_{|x|\rightarrow\infty}|\nabla\phi|=0,
\end{array}\right.$$ with initial data $$\label{l2}
(\rho^{\pm}, u^{\pm})(x,0)=(\rho_0^{\pm}(x), u_0^{\pm}(x)),$$ where $\rho^{\pm}$ are the two particles’s densities, $\rho^{\pm}u^{\pm}= (\rho^{\pm}u^{\pm}_{1} , \rho^{\pm}u^{\pm}_{2},
\rho^{\pm}u^{\pm}_{3})$ are current densities, $\phi$ is the electrostatic potential, and $P(\rho^{\pm})$ are pressures. As usual, we assume the pressure $P(\rho)$ be smooth function in a neighborhood of a constant state $\rho^*$ with $P^{\prime}(\rho)>0$. The bipolar Euler-Poisson equations are generally used in the description of charged particle fluids, for example, electrons and holes in semiconductor devices, positively and negatively charged ions in a plasma. This model takes an important role in the fields of applied and computational mathematics, and we can see more details in [@J; @MRS; @SM] etc..
Due to their physical importance, mathematical complexity and wide range of applications, many efforts were made for the multi-dimensional bipolar hydrodynamic equations from semiconductors or plasmas. Li [@L2] showed existence and some limit analysis of stationary solutions for the multi-dimensional bipolar Euler-Poisson system. Ali and Jüngel [@AJ], Li and Zhang [@LZ] and Peng and Xu [@PX] studied the global smooth solutions of the Cauchy problem for multidimensional bipolar hydrodynamic models in the Sobolev space $H^l(\mathbb{R}^d)(l>1+\frac d2)$ and in the Besov space, respectively. Ju [@J1] discussed the global existence of smooth solutions to the initial boundary value problem for the three-dimensional bipolar Euler-Poisson system. Li and Yang [@LY] and Wu and Wang [@WW] showed global existence and $L^2$ decay rate of the smooth solutions to the three dimensional bipolar Euler-Poisson systems when the initial data are small perturbation of the constant stationary solution. Huang, Mei and Wang [@HMW] showed large time behavior of solution to $n$-dimensional bipolar hydrodynamic model for semiconductors when the initial data are near to the planar diffusion waves. Ali and Chen [@AC] studied the zero-electron-mass limit in the Euler-Poisson system for both well- and ill-prepared initial data. Lattanzio [@L] and Li [@L1] investigated the relaxation limit of the multi-dimensional bipolar isentropic Euler-Poisson model for semiconductors, respectively. Ju, etc. [@JLLJ] discussed the quasi-neutral limit of the two-fluid multi-dimensional Euler-Poisson system. Moreover, it is worth to mentioning that there are a lot of reference about the one-dimensional bipolar Euler-Poisson equation, and the interesting reader can refer to [@DMRS; @GM; @GHL; @HL; @HMWY; @HZ1; @HZ2; @N1; @T; @ZH; @ZL] and the reference therein. In particular, motivation by [@GM; @[H-L]], Gasser, Hsiao and Li [@GHL] found that the frictional damping is the key to the nonlinear diffusive phenomena of hyperbolic waves, and investigated the diffusion wave phenomena of smooth “small" solutions for the one-dimensional bipolar hydrodynamic model. Huang and Li [@HL] also studied the large-time behavior and quasi-neutral limit of $L^\infty$ solution of the one-dimensional Euler-Poisson equations for large initial data with vacuum. That is, they showed that the weak entropy solution of the one-dimensional bipolar Euler-Poisson system converges to the nonlinear diffusion waves. Then Huang, Mei and Wang [@HMW] showed the planar diffusive wave stability to $n(n\geq2)$-dimensional bipolar hydrodynamic model for semiconductors, and obtained the optimal $L^2$ and $L^\infty$ decay rates. In this paper, we are going to reconsider global existence of the smooth solution for the multi-dimensional bipolar Euler-Poisson systems, in particular, we try to establish the $L^{p}(p\in[2,+\infty])$ convergence rates of planar waves.
In the following discussion, we assume that the initial data are a small perturbation of the diffusion profile constructed later with small wave strength. Let the initial data $\rho_0^{\pm}(x)$ be strictly positive and satisfy $$\lim_{x_1\rightarrow \pm\infty}\rho_0^{\pm}(x)=\rho_\pm, $$ where $ \rho_\pm>0$ are two far field constants with $\rho_-\ne
\rho_+$. Similar as the consideration of planar diffusion waves of damped Euler equations in [@[W-Y3]; @lwy09], to define the multi-dimensional planar diffusion wave, we first consider the one dimensional diffusion equation $$\label{p5}
\partial_t w= P(w)_{x_1x_1},$$ which can be derived from the bipolar Euler-Poisson equations with the relaxation terms in one dimensional case by imposing the Darcy’s law, cf. [@GHL; @GM]. Then a multi-dimensional diffusion wave $w(x, t)$ is a one dimensional profile in multi-dimensional space. That is, $w(x,t)=W(x_1/\sqrt{1+t})$ is a self-similar solution of the equation (\[p5\]) connecting two end states $\rho_\pm$ at $x_1=\pm\infty$. Denote $\zeta=\frac{x_1}{\sqrt{t+1}}$, then $W(\zeta)$ satisfies $$-\frac 12 \zeta\partial_\zeta
W=\partial_\zeta(P'(W(\zeta))\partial_\zeta W).$$
For simplicity, let the initial velocity $u_0^{\pm}(x)$ vanish as $x_1\rightarrow \pm\infty$, that is, $$\lim_{x_1\rightarrow \pm\infty}u^{\pm}_0(x)=0,$$ which implies that there is no mass flux coming in from $x_1=\pm\infty$. This assumption could be removed in a technical way similar to the argument for one dimensional problem because the momentum at $x_1=\pm\infty$ decays exponentially induced by the linear relaxation terms.
We now recall the bipolar Euler-Poisson systems (\[l1\]) in one space dimension: $$\label{l3}
\left\{
\begin{array}{lcr}
\partial_{t} \rho^{+}+ \partial_{x_{1}}(\rho^{+} u_{1}^{+})=0,\vspace{2.5mm}\\
\partial_{t} (\rho^{+} u^{+}_{1})+ \partial_{x_{1}}(\rho^{+} u^{+}_{1}u^{+}_{1})+\partial_{x_{1}} P(\rho^{+})=-\rho^{+} u^{+}_{1} +\rho^{+}E , \vspace{2.5mm}\\
\partial_{t} \rho^{-}+ \partial_{x_{1}}(\p^{-} u^{-}_{1})=0,\vspace{2.5mm}\\
\partial_{t} (\p^{-} u^{-}_{1})+\partial_{x_{1}}(\p^{-} u^{-}_{1}u^{-}_{1})+\partial_{x_{1}} P(\p^{-})=-\p^{-} u^{-}_{1} - \p^{-}E , \vspace{2.5mm}\\
\partial_{x_{1}}E = \p^{+} - \p^{-},\lim_{x_1\rightarrow-\infty}E(x_1,t)=0.
\end{array}\right.$$ Denote the solution of (\[l3\]) by $ (\pp^{\pm}, \u_1^{\pm},\tilde
{E})(x_1,t)$. When $$\lim\limits_{x_1\rightarrow\pm\infty} \pp^{\pm}(x_1, 0)=\p_{\pm},\ \
\lim\limits_{x_1\rightarrow\pm\infty} \tilde{u}^{\pm}_1(x_1, 0)=0,$$ the time-asymptotic behavior of $(\pp^+,\u_1^+,\pp^-,
\u_1^-)(x_1,t)$ has been studied in [@GHL], which is shown to be a nonlinear diffusion profile governed by Darcy’s law. Roughly speaking, the solution $\pp^{\pm}(x_1, t)$ converge to a same diffusion wave $W(x_1/\sqrt{1+t})$ up to a constant shift in $x_1$. Note that more detailed assumptions on the initial data of the one-dimensional problem will be specified in Theorem 2.1.
In this paper, we will generalize this time asymptotic behavior towards a planar diffusion wave to three-dimensional case and establish the related $L^p~(2\leq p\leq\infty)$ convergence rates.
As in the consideration of planar diffusion waves of damped Euler equation in [@[W-Y3]; @lwy09] and of the bipolar Euler-Poisson system in [@HMW], we do not directly compare the solution of the problem (\[l1\]) with the diffusion wave $W(x_1/\sqrt{1+t})$, instead, we will compare it with the solution of one dimensional problem (\[l3\]). For this, without loss of generality, let us first assume the initial density $\pp^{\pm}(x_1, 0)$ in (\[l3\]) satisfy $$\label{a1}
\int^{+\infty}_{-\infty}(\pp^{\pm}(x_1 ,0)-W(x_1))dx_1=0.$$ For the multi-dimensional problem, the shift function $\delta_0(x')$ , where we used the notation $x'=(x_{2},x_{3})$, can be chosen as in [@[W-Y3]; @lwy09] such that the initial density function satisfies $$\int^{+\infty}_{-\infty}(\p^{\pm}(x,
0)-W(x_1+\delta_0^{\pm}(x')))dx_1=0.$$ Note that $\delta_0^{\pm}(x')$ is then uniquely determined by $$\delta_0^{\pm}(x') = \frac{1}{\rho_+ - \rho_-}\int_{-\infty}^{\infty}
(\rho^{\pm}(x,0)-W(x_1))dx_1,$$ for $\rho_-\ne \rho_+$. Moreover, we assume that basically the shift is uniform in directions other than $x_1$ at infinity, that is, $$\lim_{|x'|\rightarrow+\infty}\frac{1}{\rho_+ -
\rho_-}\int^{+\infty}_{-\infty}(\p^{\pm}(x,
0)-W(x_1))dx_1=\delta_*^{\pm},$$ Note that this assumption simplifies the problem and it remains unsolved for general perturbation when this assumption fails. An immediate consequence of this assumption is that $$\lim_{|x'|\rightarrow+\infty}\delta_0^{\pm}(x')=\delta_*^{\pm}.$$ And for simplicity, we assume $\delta_*^{\pm}=\delta_*$ be same constants. With these notations, the main purpose here is to show that the solutions $(\p^{\pm}, u^{\pm})$ of (\[l1\]) converge to $(\bar{\p}^{\pm},
\bar{u}^{\pm})$ with certain time decay rates, where $$\label{l4}
\left\{
\begin{array}{lcr}
\bar{\p}^{\pm}(x,t)=\pp^{\pm} (x_1+\delta(x', t), t),\vspace{2.5mm}\\
\bar{u}^{\pm}(x,t)=(\tilde{u}^{\pm}_1(x_1+\delta(x', t), t), 0 ,0),\vspace{2.5mm}\\
\bar E (x,t)=(\tilde E(x_1+\delta(x', t), t), 0 ,0) ,\vspace{2.5mm}\\
\delta(x', t)=\delta_*+e^{- t}(\delta_0(x')-\delta_*),
\end{array}
\right.$$ in which $\pp^{\pm},\tilde{u}^{\pm}_1$ and $\tilde E$ are solution of (\[l3\]). In the following discussion, we will also assume that the shift generated by the initial data satisfies $$\label{l11}
|\partial_{x'}^\beta(\delta_0^{\pm}(x')-\delta_*) |\le
C(1+|x'|^2)^{-N},$$ for any multi-index $\beta$ and any positive integers $N$. Here, $C$ is a constant depending only on $\beta$. This assumption implies that the shift $\delta_0(x')$ decays to $\delta_*$ almost exponentially. Again, this assumption can be reduced to the constraint on the initial perturbation. More precise construction of the background planar diffusion wave $(\bar{\rho}^{\pm},
\bar{u}^{\pm})(x,t)$ and its properties will be given in Lemma 2.3 below.
Throughout this paper, we denote any generic constant by $C$. The usual Sobolev space is denoted by $W^{s,p}({\R}^n)$, $s\in {\bf
Z_+}$, $p\in
[1,\infty]$ with the norm $$\|f\|_{W^{s,p}} :=\sum^s_{|{\alpha}|=0}\|\partial^{\alpha}f\|_{L^p},$$ where $\P^{\alpha}$ used for $\P^{\alpha}_x$ without confusion. In particular, $W^{s,2}({\R}^n)=H^s({\R}^n)$. Set $$\label{l5}
\begin{array}{lcr}
V^{\pm}(x,t)=\p^{\pm}(x,t)-\bar{\p}^{\pm}(x,t), \vspace{2.5mm}\\
U^{\pm}(x,t)=(u_1^{\pm}(x,t)-\bar{u}^{\pm}_1(x,t), u_2^{\pm}(x,t), u_3^{\pm}(x,t))
, \vspace{2.5mm}\\
\nabla \varphi = \nabla \phi -\bar E ~(~\rm{Note}~\mathrm{div}(\nabla \phi -\bar E )=0), \vspace{2.5mm}\\
K(x,t) = V^{+}(x,t) - V^{-}(x,t),
\end{array}$$ and also denote $$\label{l6}
\nu^{\pm} (x,0)=\int^{x_1}_{-\infty}V^{\pm}(x_1,x', 0)dx_1, ~~~
\nu^{\pm}_t(x,0)=\int^{x_1}_{-\infty}V^{\pm}_t(x_1, x', 0)dx_1.$$ Note that the time derivative on the initial data can be defined by the compatibility of the initial data through the equation (\[l7\]).
Now we state the main results in this paper. Note that we consider only the spatial dimension $n=3$ in this paper. However, we will still use notation $n$ in the below theorem for convenience to extend our result to other high dimensional cases, since other higher dimensional cases can be considered similarly.
Let $(\bar{\rho}^{\pm}, \bar{u}^{\pm})(x,t)$ in (\[l4\]) be planar diffusion waves with a shift $\delta(x', t)$ constructed above. (See more precisely its properties in Lemma 2.3.) For $k\geq 4$, assume that the initial data $(\rho^{\pm}, u^{\pm})(x,0)$ satisfy the smallness assumption $$\begin{array}{rl}
|\p_+-\p_- |+ \|(\nu^{\pm}, \nu^{\pm}_t)(\cdot, 0) \|_{L^2\cap
L^1}+\|(\p^{\pm}-\bar{\p}^{\pm})(\cdot,0) \|_{L^1} +
\|(\p^{\pm}-\bar{\p}^{\pm}, u^{\pm}-\bar{u}^{\pm})(\cdot, 0)
\|_{H^k} \leq \epsilon_0,
\end{array}$$where $\epsilon_0>0$ is a sufficiently small constant. Then\
(i) (Global existence) There exist unique global classical solution $(\rho^{\pm}, u^{\pm}, \nabla\phi)$ to the system (\[l1\])-(\[l2\]) that $$V^{\pm}(t,x), U^{\pm}(t,x)\in C([0,\infty),H^{k}({\R}^n))\cap C^1((0,\infty), H^{k-1}({\R}^n)), \hspace{2mm}
\nabla\phi \in W^{k,6}({\R}^n).$$ (ii) ($L^{p}$ convergence) Moreover, for $|\r|\leq k-2$, $p\in[2, \infty]$, we have $$\begin{array}{rl}
\|\P^\r_xV^{\pm} \|_{L^p} \leq &C \epsilon_0 (1+t)^{-\frac{n}{2}(1-\frac{1}{p})-\frac{|\r|+1}{2}}, \vspace{2.5mm}\\
\|\P^\r_xU^{\pm} \|_{L^p} \leq
&C \epsilon_0 (1+t)^{-\frac{n}{2}(1-\frac{1}{p})-\frac{|\r|+2}{2}}.
\end{array}$$ (iii) (Estimates on $\varphi$ and $K=V^{+}-V^{-}$) For $|\gamma|\leq k-2$, $$\|\P^\gamma_x K \|_{L^2} \leq C \epsilon_0 (1+t)^{-{5\over 4}n-2- {|\gamma|\over 2}}
, \hspace{3mm}
\|\P^\gamma_x \nabla\varphi \|_{L^6} \leq C \epsilon_0 (1+t)^{-{5\over 4}n-2- {|\gamma|\over 2}} ,$$ and for $|\gamma|= k-1$, $$\|\P^\gamma_x K \|_{L^2} \leq C \epsilon_0 (1+t)^{-{5\over 4}n-1- {k\over 2}}
, \hspace{3mm}
\|\P^\gamma_x \nabla\varphi \|_{L^6} \leq C \epsilon_0 (1+t)^{-{5\over 4}n-1- {k\over 2}} .$$
As noted in [@lwy09], in general, if the shift of the profile is not exactly captured, the decay rates for $V^{\pm}$ and $U^{\pm}$ should be $\frac 12$ lower than the one given in the above theorem even in one space dimensional case. Here, the reason that the above decay estimate holds is that the shift due to the initial perturbation introduced above so that when we apply the Green function, the term corresponding to the initial data yields an extra $(1+t)^{-\frac 12}$ decay after taking the anti-derivative of the initial perturbation. Moreover, under the condition (\[l11\]) on the initial shift, even though the anti-derivative of the perturbation can not be defined for all time as the shift function is not precisely defined, we know that $\delta_*$ is exactly the final shift when $t$ tends to infinity of the profile because the initial perturbation will spread out eventually.
We note here that the $L^{2}$ decay rates of $K$ is higher than that of $V^{\pm}$. However, we can only get the decay estimates on derivatives of $K$ up to $(k-1)$-th order.
Compared with [@LY; @WW], our initial data are the small perturbation of the planar waves, instead of the constant states. In the meanwhile, here we can show the $L^p(p\in[2,+\infty])$ convergence rates of the planar waves of the three-dimensional bipolar Euler-Poisson equations. This improved the results in [@HMW]. Moreover, here we only consider the case that the far fields of two particles’ velocity in the $x_1$-direction are same, see (\[a1\]), namely, the switch-off case. However, we believe that the same results also hold for the switch-on case. Indeed, using the gap function with exponential decay in [@HMW], we can show the similar results for the switch-on case.
The outline of the proof of the main theorems is as follows. First, we notice that the equations for $V^{\pm}$ are coupled by $\nabla\varphi$, which is expressed by nonlocal Riesz potential $\nabla \varphi = \nabla \Delta^{-1} K$, with $K=V^+-V^-$. So we need to have some good estimates on $K$ before the estimate of $V^{+}$ and $V^{-}$. Luckily we note that $K$ satisfies the damping “Klein-Gordon" type equations with an addition good term to perform the energy estimate. The estimates of $K$ and $\nabla\varphi$ are given in Section 2, where the algebraic decay rates of $K$ in the $L^2$-norm are derived by some delicate energy methods, which will be used to obtain the $L^p(p\in[2,+\infty])$-convergence rates of the solutions in the subsequent. Next, we use the frequency decomposition based energy method introduced in [@lwy09], which combines the approximation Green function and energy method, to prove global existence and $L^{p}$ convergence results, see (i) and (ii) in Theorem 1.1. This method captures the low frequency component in the approximate Green function and avoids the singularity in the high frequency component. That is, we firstly show the precise algebraic decay estimate of $V^{\pm}$ in the low frequency component, which dominant the decay of the perturbation, and then obtain the better decay rates of the high frequency component of $V^{\pm}$ in the $L^2$-norm by energy methods. For the high frequency component, one has an additional $\rm
Poincar\acute{e}$-type inequality to close the energy estimate. Note that the lack of $\rm Poincar\acute{e}$ inequality in the whole space is usually the essential difficulty in the energy estimate which is in contrast to the problem in a torus. This in some sense illustrates the essence of the Green function on the decay rate related to the frequency.
The rest of the paper is arranged as follows. In Section 2, we will reformulate the system around a planar diffusion wave defined in (\[p5\]) and then state some known properties of this background diffusion wave. In Section 3, we will study the energy estimate of $K=V^{+}-V^{-}$ and prove part (iii) in Theorem 1.1. The frequency decomposition based energy method will be carried out in Section 4 and 5, where in Section 4, we will study the approximate Green function and then the main $L^p$ estimates on the low frequency component of $V^{\pm}$, and in Section 5, we will study the $L^2$ energy estimates on the high frequency component of $V^{\pm}$. Finally, we will complete the proof to part (i) and (ii) of Theorem 1.1 in Section 6.
Preliminaries.
==============
In this section, we will first derive the equations for the perturbation functions $V^{\pm}$ and $U^{\pm}$ defined in (\[l5\]). Then we will recall some results on the background diffusion waves.
Reduced system.
---------------
We first derive the system for the perturbation of the nonlinear planar diffusion wave. Then, from (\[l1\]) and (\[l3\]), we have the equations for $V^{\pm}$ that $$\label{l7}
V^{\pm}_{t}+(\bar{\p}^{\pm}+V^{\pm}){\rm div}U ^{\pm}
= R_{\rho^{\pm}}-(U^{\pm}\cdot\nabla)(\bar{\rho}^{\pm}+V^{\pm})
-V^{\pm}(\bar{u}_1^{\pm})_{x_1}-\bar{u}_1^{\pm}V^{\pm}_{x_1},$$ where $$R_\rho^{\pm}= [-\bar{\p^{\pm}}(x, t)\delta^{\pm}_t(x', t)]_{x_1}.$$ Similarly, the equations for $U_1^{\pm}$ are $$(U_1^{\pm})_t+(\bar{\p}^{\pm}+V^{\pm})^{-1}[P(\bar{\p}^{\pm}+V^{\pm})-P(\bar{\p}^{\pm})]_{x_1}+
U_1^{\pm} =
\frac{P(\bar{\p}^{\pm})_{x_1}V^{\pm}}{\bar{\p}^{\pm}(\bar{\p}^{\pm}+V^{\pm})}
+R_{u_1^{\pm}}-R_1^{\pm} \pm \partial_{x_{1}} \varphi,$$ and for $i= 2, 3$, $$(U_i^{\pm})_t+(\bar{\p}^{\pm}+V^{\pm})^{-1}[P(\bar{\p}^{\pm}+V^{\pm})-P(\bar{\p}^{\pm})]_{x_i}+
U_i^{\pm}
=\frac{P(\bar{\p}^{\pm})_{x_i}V^{\pm}}{\bar{\p}^{\pm}(\bar{\p}^{\pm}+V^{\pm})}-
(\bar{\p}^{\pm})^{-1}P(\bar{\p}^{\pm})_{x_i}-R_i^{\pm} \pm
\partial_{x_{i}} \varphi,$$ where $$\begin{array}{rl}
R_{u_1^{\pm}}=&-[\tilde{u}_1^{\pm}(x_1+\delta(x', t), t)\delta_t(x', t)]_{x_1},\vspace{2.5mm}\\
R_1^{\pm}=&U^{\pm}\cdot\nabla(\bar{u}_1^{\pm}+U_1^{\pm})+\bar{u}_1^{\pm}(U_1^{\pm})_{x_1},\vspace{2.5mm}\\
R_i^{\pm}=&U^{\pm}\cdot\nabla U_i^{\pm}+\bar{u}_1^{\pm}(U_i^{\pm})_{x_1},~ 2\leq i\leq n.
\end{array}$$ The equation for $\varphi$ is simply $$\Delta \varphi = V^{+} - V^{-} =K,$$ it is directly that the perturbed electric field $\nabla \varphi$ can be expressed by the Riesz potential as a nonlocal term $$\label{l12}
\nabla \varphi = \nabla \Delta^{-1} K.$$ Then the system for the perturbation $(V^{\pm}, U^{\pm}, \varphi)$ can be summarized as $$\label{l13}
\left\{
\begin{array}{lcr}
V^{\pm}_{t}+(\bar{\p}^{\pm}+V^{\pm}){\rm div}U^{\pm}=Q^{\pm},\vspace{2.5mm}\\
(U_i^{\pm})_t+(\bar{\p}^{\pm}+V^{\pm})^{-1}({\mathcal P}(V^{\pm},
\bar{\p}^{\pm})V^{\pm})_{x_i}+ U_i^{\pm}=H^{\pm}_i \pm
\partial_{x_{i}}\varphi, \quad 1\le i\le 3,
\end{array}
\right.$$ where ${\mathcal P}(V^{\pm}, \bar{\p}^{\pm})=\dis{\int^1_0 }P^\prime(\bar{\p}^{\pm}+\theta V^{\pm})d\theta$, and $$\begin{array}{rl}
Q^{\pm}=&R_{\rho^{\pm}}-(U^{\pm}\cdot\nabla)(\bar{\rho}^{\pm}+V^{\pm})-V^{\pm}(\bar{u}^{\pm}_1)_{x_1}
-(\bar{u}^{\pm}_1)V^{\pm}_{x_1},\vspace{2.5mm}\\
H_1^{\pm}=&R_{u^{\pm}_1}+\frac{P(\bar{\p}^{\pm})_{x_1}V^{\pm}}{\bar{\p}^{\pm}(\bar{\p}^{\pm}+V^{\pm})}-R_1^{\pm},\vspace{2.5mm}\\
H_i^{\pm}=&-\frac{P(\bar{\p}^{\pm})_{x_i}}{\bar{\p}^{\pm}} +\frac{P(\bar{\p}^{\pm})_{x_i}
V^{\pm}}{\bar{\p}^{\pm}(\bar{\p}^{\pm}+V^{\pm})}-R_i^{\pm}, ~~2\le i\le 3.
\end{array}$$ Moreover, we can deduce the equation for $V^{\pm}(x,t)$ from (\[l13\]) as $$\label{l14}
V^{\pm}_{tt}-\triangle[{\mathcal P}(V^{\pm}, \bar{\p}^{\pm})V^{\pm}]+ V^{\pm}_{t}=\tilde{Q}(V^{\pm}, U^{\pm},\bar{\p}^{\pm}, \bar{u}^{\pm}_1)
\mp \mathrm{div}[( \bar{\p}^{\pm}+V^{\pm} )\nabla\varphi],$$ where $$\begin{array}{rl}
&\tilde{Q}(V^{\pm},U^{\pm}, \bar{\p}^{\pm}, \bar{u}^{\pm}) \vspace{2.5mm}\\
= &[(R_{\rho^{\pm}})_t+R_{\rho^{\pm}} ]-(1
+\partial_t)(V^{\pm}\bar{u}^{\pm}_1)_{x_1}
-\mathrm{div}[(\bar{\p}^{\pm}+V^{\pm})_tU^{\pm}]-\mathrm{div}[(\bar{\p}^{\pm}+V^{\pm})H^{\pm}],
\end{array}$$ with $H^{\pm}=(H_1^{\pm}, \cdots, H_3^{\pm})$. By linearizing (\[l14\]) around $\bar{\p}$, we have $$\label{l15}
\begin{array}{rl}
&V^{\pm}_{tt}-\triangle(a^{\pm}(x,t)V^{\pm})+ V^{\pm}_{t}\vspace{2.5mm}\\
=&\tilde{Q}(V^{\pm}, U^{\pm},\bar{\p}^{\pm}, \bar{u}^{\pm}_1)
+\triangle ({\mathcal P}_1(\bar{\p}, V)V^2) \mp \mathrm{div}[(
\bar{\p}^{\pm}+V^{\pm} )\nabla\varphi]
\vspace{2.5mm}\\
=: &F^{\pm} \mp \mathrm{div}[( \bar{\p}^{\pm}+V^{\pm}
)\nabla\varphi],
\end{array}$$ where $a^{\pm}(x,t)=P^\prime(\bar{\p}^{\pm})$ and $${\mathcal P}_1(\bar{\p}^{\pm}, V^{\pm})=\int^1_0 (\int^{\theta_1}_0
P^{\prime\prime}(\bar{\p}^{\pm}+\theta_2V^{\pm})d\theta_2 ) d\theta_1.$$ Since $$(R_{\rho^{\pm}})_t+R_{\rho^{\pm}}= (-\bar{\p}^{\pm}_t(x,
t)\delta_t(x^\prime, t) )_{x_1},$$ direct calculation shows that $F^{\pm}=F(V^{\pm}, U^{\pm}, \bar{\p}^{\pm},\bar{u}^{\pm})$ is in divergence form, that is, $$\label{l18}
F^{\pm}=\sum (F^{\pm,i})_{x_i}+\sum (F^{\pm,ij})_{x_ix_j},$$ where, without confusion, we omit the $\pm$ sign, $$\begin{array}{rl}
F^1=&-\bar{\p}_t\delta_t-(\bar{\rho}\bar{u}_1\delta_t)_{x_1}, \ \ \ F^i=-P(\bar{\p})_{x_i}, \,\, 2\le i\le n, \vspace{2.5mm}\\
F^{11}=& \bar{\p}(2\bar{u}_1U_1+U^2_1)+V(\bar{u}_1+
U_1)^2+{\mathcal P}_1(\bar{\p}, V)V^2,\vspace{2.5mm}\\
F^{1i}=&F^{i1}=2 [(\bar{\p}+V)(\bar{u}_1+U_1)U_i ],\,\, 2\le i\le n, \vspace{2.5mm}\\
F^{ij}=&(\bar{\p}+V)U_iU_j+\delta_{ij}{\mathcal P}_1(\bar{\p},
V)V^2,\,\, 2\le \ i,j\le n.
\end{array}$$ Here $\delta_{ij}$ is the Kronecker symbol. On the other hand, by linearizing (\[l13\])$_2$ around $\bar{u}^{\pm}$, we have $$\label{l16}
U^{\pm}_t + (\bar{\p}^{\pm})^{-1}\nabla(a^{\pm}(x,t) V^{\pm}) + U^{\pm}=\bar{H}^{\pm} \pm \nabla\varphi ,$$ where, again without confusion, we omit the $\pm$ sign, $$\begin{array}{rl}
\bar{H}_1=& R_u-\bar{\p}^{-1}[{\mathcal P}_1(\bar{\p}, V)V^2]_x
-\frac{P(\bar{\p}+V)_xV}{\bar{\p}(\bar{\p}+V)}-R_1, \vspace{2.5mm}\\
\bar{H}_i=&-\frac{P(\bar{\p})_{x_i}}{\bar{\p}}-\frac{P(\bar{\p}+V)_{x_i}V}{\bar{\p}(\bar{\p}+V)}-\bar{\p}^{-1}[{\mathcal
P}_1(\bar{\p}, V)V^2]_{x_i}-R_i,~~ 2\le i\le n.
\end{array}
$$
Background profile.
-------------------
For later use, we include the following known estimates on the background planar wave, cf. [@[W-Y3]]. By the definition of $W(x_1)$, we know for any integer $N$, $$\begin{array}{rl}
\displaystyle{\sup_{x_1>0}|W (x_1)-\p_+|}+&\displaystyle{\sup_{x_1<0}}|W (x_1)-\p_-|\leq
C|\p_+-\p_-|(1+x_1^2)^{-N},\vspace{2.5mm}\\
|\P^h_{x_1}W (x_1)|\leq &C|\p_+-\p_-|(1+x_1^2)^{-N},\ \ (h>0).
\end{array}$$ Recall that we have assumed in (\[l11\]) for any multi-index $\beta$, $$|\P^{\beta}_{x'}(\delta_0(x')-\delta_*)|\leq C(1+|x'|^2)^{-N}.$$ First, let us recall the results about the one-dimensional bipolar Euler-Poisson system (\[l3\]).
(see [@GHL]) Let $(\tilde{\p}^\pm, \tilde{u}_1^\pm)(x_1, 0)$ be the initial data of one-dimensional bipolar Euler-Poisson system (\[l3\]) and fix an integer $m\geq 2$. If there exists a small positive constant $E_\p$ such that the initial data $(\tilde{\p}^\pm, \tilde{u}_1^\pm)(x_1,
0)$ satisfy $$\begin{array}{rl}
&\dis{ |\p_+-\p_- |+ \|\int^{x_1}_{-\infty}(\tilde{\p}^{\pm}(z, 0)-W (z))dz \|_{L^2}}\vspace{2.5mm}\\
&+ \|\tilde{\p}^{\pm}(\cdot, 0)-W (\cdot) \|_{H^{m+1}}
+ \|\tilde{u}^{\pm}_1(\cdot, 0)-\psi(\cdot, 0) \|_{H^{m+1}}\leq E_\p,
\end{array}
$$ and $$\int^{x_1}_{-\infty}(\tilde{\p}^{\pm}(z,
0)-W(z))dz,\tilde{u}^\pm+P(W)_{x_1}\in L^1(\R),$$ then (\[l3\]) has global classical solution $(\tilde{\rho}^\pm,\tilde{u}^\pm, E)$ with $$\begin{aligned}
&&\|\partial^k(\tilde{\rho}^\pm-W)(x_1,t) \|_{L^p(\R^1_{x_1})} \leq
CE_\p(1+t)^{-\frac12(1-\frac1p)-\frac{k+1}{2}},\vspace{2.5mm}\\
&&\|\partial^k(\tilde{u}^\pm+P(W_{x_1}))(x_1,t) \|_{L^p(\R^1_{x_1})} \leq
CE_\p(1+t)^{-\frac12(1-\frac1p)-\frac{k+2}{2}}\end{aligned}$$ for any integer $k\leq m+1$ if $p=2$, and $k\leq m$ if $p=\infty$. Moreover, there exists a positive constant $\beta$ such that $$\|(\tilde{\rho}^{+} -
\tilde{\rho}^{-},\tilde{E})\|_{H^{m}(\R^1_{x_1})} \leq CE_\p
e^{-\beta t}.$$
Note that $(\tilde{\p}^\pm, \tilde{u}_1^\pm)$ is an intermediate state we constructed to approximate the one-dimensional diffusion wave $W$. The assumptions on the initial data $(\tilde{\p}^\pm,
\tilde{u}_1^\pm)(x_1, 0)$, with $\tilde{\p}^\pm(x_1, 0)$ connecting the two end states $\rho_\pm$, can be more regular than the assumptions on the initial date of the original problem .
Next, from the definition of the planar diffusion waves $(\bar\rho^{\pm}, \bar u^{\pm})$ in (\[l4\]), we can readily have
Under the assumptions in Theorem 2.1, the planar diffusion waves $(\bar\rho^{\pm}, \bar u^{\pm})$ defined in (\[l4\]) satisfy $$\begin{array}{rl}
\displaystyle{ \sup_{x^\prime} \|\P^{\alpha}(\bar{\p}^{\pm}_{x_1},
\bar{u}^{\pm}_1)(\cdot, x', t) \|_{L^2(\R^1_{x_1})}} \leq &
CE_\p(1+t)^{-{1+|{\alpha}|\over2}- {1\over4}},\vspace{2.5mm}\\
\|\P^{\alpha}(\bar{\p}^{\pm}_{x_1},
\bar{u}^{\pm}_1)(\cdot, t) \|_{L^\infty(\R^n)} \leq &
CE_\p(1+t)^{-{1+|{\alpha}|\over2}}
\end{array}
$$ for any multi-index ${\alpha}$ with $|{\alpha}|\leq m-1$, and $$\|\bar \p^{+} - \bar \p^{-}\|_{H^{m}(\R^1_{x_1})} \leq CE_\p
e^{-\beta t}.$$ In addition, for $2\le i\le n$, $$\begin{array}{rl}
\|\P^{\alpha}(\bar{\p}^{\pm}_{x_i},
(\bar{u}^{\pm}_1)_{x_i})(t) \|_{L^2(\rm \R^n)}\leq CE_\p e^{- t},\vspace{2.5mm}\\[2.5mm]
\|\P^{\alpha}(\bar{\p}^{\pm}_{x_i},
(\bar{u}^{\pm}_1)_{x_i})(t) \|_{L^\infty(\rm \R^n)}\leq CE_\p e^{-
t}.
\end{array}
$$
Note here again that we can increase the regularity of the assumptions on initial date of the one-dimensional problem to get sufficient estimates on the planar diffusion wave.
Estimates on $K=V^{+} - V^{-}$.
================================
In this section, we mainly give the estimate of $K=V^+-V^-$. Recall the linearized equation (\[l15\]) for $V^{\pm}$, we see that they are coupled by $\nabla\varphi$, which is expressed by the Riesz potential as in (\[l12\]). i.e., $ \nabla \varphi = \nabla
\Delta^{-1} K$. So we need to have some good estimates on $K$, thus $\nabla\varphi$, before the estimate of $V^{+}$ and $V^{-}$. To begin with, we give a lemma on the relation of $ \nabla \varphi $ and $K$.
If $K\in H^{l}(\R^n)$ for any integer $l>1$, then $\nabla \varphi
\in W^{l,6}(\R^n)$.
[**Proof.**]{} Note that $\nabla\varphi$ be expressed by the Riesz potential $$\nabla \varphi = \nabla \Delta^{-1} K = \mathcal R*K,$$ where $\hat {\mathcal R} = |2\pi \xi|^{-1}$ thus $\mathcal R= {1\over |x|^{n-1}}$. Here $n=3$ is the space dimension. Then by Hardy-Littlewood-Sobolev inequality [@ST] we have $$\label{l21}
\|\nabla \varphi \|_{L^{6}} = \| \mathcal R *K \|_{L^{6}} \leq C \| K\|_{L^{2}},$$ and similarly, for any multi index $|\r|\leq k$, $$\label{l22}
\|\P^{\r}\nabla \varphi \|_{L^{6}} = \| \mathcal R * \P^{\r} K \|_{L^{6}} \leq C \| \P^{\r}K\|_{L^{2}},$$ that is, $\nabla \varphi \in W^{l,6}(\R^n)$ if $K\in H^{l}(\R^n)$.
By Sobolev injection, this lemma automatically indicates $$\nabla \varphi \in L^{\infty}(\R^n).$$
Now we start to estimate $K$. The equation for $K$, from (\[l15\]), is $$K_{tt} - \Delta(a^{+}K) + K_{t} = F^{+}-F^{-}
-\mathrm{div}[(\bar\p^{+} +\bar\p^{-} +V^{+} +V^{-} )\nabla\varphi]
+ \Delta[(a^{+}-a^{-})V^{-}] ,$$ where $F^{\pm}$ on the right hand side are defined in (\[l18\]). This equation can also be written as $$\label{l17}
\begin{array}{rl}
&K_{tt} - \Delta(a^{+}K) + K_{t} + (\bar\p^{+} +\bar\p^{-} +V^{+} +V^{-} )K
\vspace{2mm}\\
= & F^{+}-F^{-} - \nabla (\bar\p^{+} +\bar\p^{-} +V^{+} +V^{-} )\cdot \nabla\varphi
+ \Delta[(a^{+}-a^{-})V^{-}] ,
\end{array}$$ note here the last term on the left hand side is a good term, which ensures the closure of energy estimate for $K$. Note that the lack of such term in equation of $V^{\pm}$ is the main difficulty in energy estimate thus we will use the frequency decomposition method introduced in [@lwy09].
To proceed, we first give the a priori assumption $$\label{l19}
{\mathcal M}(t)= \quad \quad\quad\quad\quad \quad \quad\quad\quad\quad
\quad \quad\quad\quad\quad \quad \quad\quad\quad\quad$$ $$\max \Big\{ \sup\limits_{0\leq s\leq t, |{\alpha}|\leq k-2,
p\geq 2}(1+s)^{\frac{n}{2}(1-\frac{1}{p})+\frac{|{\alpha}|+1}{2}}
\|\partial^{\alpha}_xV^{\pm} (\cdot,s) \|_{L^p},
\sup\limits_{0\leq
s\leq t, |{\alpha}|=k,
k-1}(1+s)^{\frac{n}{4}+\frac{|{\alpha}|+1}{2}} \|\partial^{\alpha}_xV^{\pm} (\cdot,s)\|_{L^2},$$ $$\sup\limits_{0\leq s\leq t, |{\alpha}|\leq
k-2, p\geq 2}(1+s)^{\frac{n}{2}(1-\frac{1}{p})+\frac{|{\alpha}|+2}{2}} \|\partial^{\alpha}_xU^{\pm} (\cdot,s)\|_{L^p} ,
\sup\limits_{0\leq s\leq t, |{\alpha}|=k,
k-1}(1+s)^{\frac{n}{4}+\frac{k+1}{2}} \|\partial^{\alpha}_xU^{\pm}
(\cdot,s)\|_{L^2}\Big\}.$$ Under the assumption in Lemma 2.3 and the above a priori assumption, it is easy to check that for any multi-indies ${\alpha}$ and $\gamma$, the nonlinear terms in $F^{\pm}$ satisfy $$\label{l20}
\left\{
\begin{array}{lcr}
\|\P^{\alpha}_y F^i \|_{L^p(\R^n_y)} \leq CE_\rho e^{-s},~ ~ |{\alpha}|\leq k ,\vspace{2.5mm}\\
\|\P^{\r}_yF^{ij} \|_{L^p(\R^n_y)} \leq C{\mathcal M}^2
(1+s)^{-(n+1+\frac{|\r|}{2})+\frac{n}{2p}}, ~ ~ |\r|\leq k-2.
\end{array}
\right.$$ Here we should indicate that the above estimates hold for $p\geq1$.
Now we perform energy estimates. Multiply equation (\[l17\]) by $K$ and integrate the resultant equation over $\R^n$, we have $$\begin{aligned}
&&\frac{d}{dt}\int_{\R^n}(\frac12K^2+K K_{t})dx- \int_{\R^n}
\Delta(a^{+}K)Kdx + \int_{\R^n}(\bar\p^{+} +\bar\p^{-} +V^{+} +V^{-}
)K^2dx-\int_{\R^n}K^2_tdx\\
&=&\int_{\R^n} K( F^{+}-F^{-} )dx - \int_{\R^n}\nabla (\bar\p^{+}
+\bar\p^{-} +V^{+} +V^{-} )\cdot \nabla\varphi Kdx
+ \int_{\R^n} \Delta[(a^{+}-a^{-})V^{-}] Kdx.\end{aligned}$$ Note $$\int_{\R^n} - \Delta(a^{+}K) K dx= \int_{\R^n} \nabla (a^{+}K) \cdot \nabla
Kdx
= \int_{\R^n} a^{+} |\nabla K|^{2} dx - {1\over2} \int_{\R^n} (\Delta
a^{+})K^{2}dx.$$ Moreover, using Cauchy-Schwarz’s and Young’s inequality, we have $$\begin{aligned}
\int_{{\R}^n} \nabla (\bar\p^{+} +\bar\p^{-} +V^{+} +V^{-} )\cdot
\nabla\varphi K dx &\leq& \| \nabla\varphi \|_{L^{\infty}} \|
\nabla (\bar\p^{+} +\bar\p^{-} +V^{+} +V^{-} ) \|_{L^{2}} \| K
\|_{L^{2}}\\
&\leq& C \| \nabla (\bar\p^{+} +\bar\p^{-} +V^{+} +V^{-} )
\|_{L^{2}} \ \| K \|_{H^{1}}^{2} \\
&\leq& C(E_{\p}+ {\mathcal M}(t)) (1+t)^{-{5\over 4} } \ \| K
\|_{H^{1}}^{2} ,\\
\int_{\R^n} \Delta[(a^{+}-a^{-})V^{-}] Kdx &= &
\int_{\R^n} \nabla[(a^{+}-a^{-})V^{-}] \cdot \nabla Kdx \\
&\leq &C\| \nabla[(a^{+}-a^{-})V^{-}]\|_{L^{2}} \| \nabla K \|_{L^{2}}\\
& \leq &CE_\p e^{-\beta t} {\mathcal
M}(t)((1+t)^{-\frac n4-\frac12} +(1+t)^{-\frac n4-1}) \| \nabla K
\|_{L^{2}},\end{aligned}$$ since the Sobolev norm of $a^{+}-a^{-}$ has same exponential decay as $\bar\p^{+} - \bar\p^{-}$ in Lemma 2.3, and further $$\begin{aligned}
\int_{{\R}^n} K F^{\pm}dx \leq \varepsilon_{0} \int_{{\R}^n} K^{2} +
C(\varepsilon_{0}) \int_{{\R}^n} (F^{\pm})^{2} \leq \varepsilon_{0}
\int_{{\R}^n} K^{2} + C(\varepsilon_{0})( E_{\p}^{2}+ {\mathcal
M}(t)^{4}) (1+t)^{-{5\over 2}n-4},\end{aligned}$$ here we have used (\[l20\]) in the above estimate. Combine above estimates and good decay properties of $\bar\p^{\pm}$ in Lemma 2.3, we have $$\label{l23}
\begin{array}{rl}
&{\dis {d \over dt} \int_{{\R}^n} ({1 \over 2} K^{2} + K_{t}K)dx-\int_{{\R}^n} K_{t}^{2} + \int_{{\R}^n}( K^{2}dx + |\nabla K|^{2})dx }\vspace{2mm} \\
\leq &C( E_{\p}^{2}+ {\mathcal M}(t)^{4}) (1+t)^{-{5\over 2}n-4}.
\end{array}$$ Here and in the subsequent we use the fact that $\bar\p^{\pm}+V^{\pm}$ is strictly positive and bounded from below.
Next, multiply equation (\[l17\]) by $K_{t}$ and integrate the resultant equality over $\R^n$, we have $$\begin{aligned}
&&{1\over 2}{d\over dt}\int_{{\R}^n} K_{t}^{2}dx - \int_{{\R}^n}
\Delta(a^{+}K)K_{t}dx + \int_{{\R}^n} K_{t}^{2}dx + \int_{{\R}^n}
(\bar\p^{+} +\bar\p^{-} +V^{+} +V^{-} )K K_{t}dx\\
&= &\int_{{\R}^n} K_{t}(F^{+}-F^{-})dx - \int_{{\R}^n} \nabla
(\bar\p^{+} +\bar\p^{-} +V^{+} +V^{-} )\cdot \nabla\varphi K_{t}dx
+ \int_{{\R}^n} \Delta[(a^{+}-a^{-})V^{-}] K_{t}dx.\end{aligned}$$ It is easy to compute $$\begin{aligned}
&&\int_{{\R}^n} - \Delta(a^{+}K) K_{t}dx = \int_{{\R}^n} \nabla
(a^{+}K) \cdot \nabla K_{t}dx\\
&= &\int_{{\R}^n} a^{+} {1\over2} \P _{t}|\nabla K|^{2}dx
+ \int_{{\R}^n} \nabla a^{+} \cdot ( K \nabla K_{t})dx\\
&=&\int_{{\R}^n} a^{+} {1\over2} \P _{t}|\nabla K|^{2}dx
+ \int_{{\R}^n} \nabla a^{+} \cdot \P_{t}( K \nabla K)dx
- \int_{{\R}^n} \nabla a^{+} \cdot ( K_{t} \nabla K)dx\\
&=& {d \over dt} \int_{{\R}^n} a^{+} {1\over2} |\nabla K|^{2}dx
- \int_{{\R}^n}\P_{t}a^{+} {1\over2} |\nabla K|^{2}dx
- {d \over dt} \int_{{\R}^n}{1\over2} \Delta a^{+} K^{2}dx \\
& &
+ \int_{{\R}^n}{1\over2} \P_{t}\Delta a^{+} K^{2}dx
- \int_{{\R}^n} \nabla a^{+} \cdot ( K_{t} \nabla K)dx,\end{aligned}$$ in which the last term on the right satisfies $$\begin{aligned}
|\int_{{\R}^n} \nabla a^{+} \cdot ( K_{t} \nabla K)dx| & \leq &
C\| \nabla a^{+} \|_{L^{\infty}} \| K_{t} \|_{L^{2}} \| \nabla K \|_{L^{2}}\\
& \leq & \varepsilon_{1} \| K_{t} \|_{L^{2}} +
C(\varepsilon_{1}) E_{\p}^{2} (1+t) ^{-{3\over 2}} \| \nabla K \|_{L^{2}}^{2}.\end{aligned}$$ Next, note also that $\bar\p^{\pm}+V^{\pm}$ is strictly positive and bounded $$\begin{aligned}
&&\int_{{\R}^n} (\bar\p^{+} +\bar\p^{-} +V^{+} +V^{-} )K K_{t}dx \\
&= &{d \over dt} \int_{{\R}^n} {1\over2} (\bar\p^{+} +\bar\p^{-} +V^{+} +V^{-} ) K^{2}dx
- \int_{{\R}^n} {1\over2} \P_{t}(\bar\p^{+} +\bar\p^{-} +V^{+} +V^{-} ) K^{2} dx.\end{aligned}$$ Also note $$\begin{aligned}
\int_{{\R}^n} \nabla (\bar\p^{+} +\bar\p^{-} +V^{+} +V^{-} )\cdot
\nabla\varphi \ K_{t} &\leq& C\| \nabla\varphi \|_{L^{\infty}}
\| \nabla (\bar\p^{+} +\bar\p^{-} +V^{+} +V^{-} ) \|_{L^{2}} \|
K_{t}
\|_{L^{2}}\\
&\leq& \varepsilon_{2} \| K_{t} \|_{L^{2}}^{2} +C(\varepsilon_{2})
( E_{\p}^{2}+ \mathcal M^{2}) (1+t) ^{-{3\over 2}} \| K
\|_{L^{2}},\\
\int_{{\R}^n} \Delta[(a^{+}-a^{-})V^{-}] K_{t}dx &\leq&
\varepsilon_{3} \| K_{t} \|_{L^{2}}^{2} +C(\varepsilon_{3})
( E_{\p}^{2}+ \mathcal M^{2}) t^{-\beta t},\end{aligned}$$ and $$\begin{aligned}
\int_{{\R}^n} K_{t} F^{\pm} dx&\leq &\varepsilon_{0} \int K_{t}^{2}dx
+ C(\varepsilon_{0}) \int_{{\R}^n} (F^{\pm})^{2}dx \\
&\leq&
\varepsilon_{0} \int_{{\R}^n} K_{t}^{2} dx+ C(\varepsilon_{0})(
E_{\p}^{2}+ {\mathcal M}^{4}) (1+t)^{-{5\over 2}n-4},\end{aligned}$$ then combine above estimates to get $$\label{l24}
\begin{array}{rl}
&{\dis {d \over dt} \int_{{\R}^n} ( K_{t}^{2} + K^{2} +|\nabla K|^{2})dx
+\int_{{\R}^n} K_{t}^{2}dx} \vspace{2mm}\\
\leq & {\dis CE_{\p} \int_{{\R}^n} (K^{2} + |\nabla K|^{2} )dx
+ C( E_{\p}^{2}+ {\mathcal M}^{4}) (1+t)^{-{5\over 2}n-4}.}
\end{array}$$ Then multiply (\[l23\]) by ${1\over 4}$ then add to (\[l24\]), and if we assume that $ E_{\p} + \mathcal M$ is small enough, then $${d \over dt} \int_{{\R}^n}(K^{2} + |\nabla K|^{2} + K_{t}^{2})dx
+ \int_{{\R}^n}(K^{2} + |\nabla K|^{2} + K_{t}^{2})
\leq C ( E_{\p}^{2}+ {\mathcal M}^{4}) (1+t)^{-{5\over 2}n-4},$$ by Gronwall’s inequality, we have $$\int_{{\R}^n}K^{2} + |\nabla K|^{2} + K_{t}^{2}
\leq C ( E_{\p}^{2}+ {\mathcal M}^{4}) (1+t)^{-{5\over 2}n-4}.$$ then $$\|K\|_{L^{2}} \leq C ( E_{\p} + {\mathcal M}^{2}) (1+t)^{-{5\over 4}n-2}.$$ For higher order derivatives (see also the estimate on high frequency part of $V^{\pm}$ below), take $\P^{\gamma}$ on both side of equation , multiply by $\P^{\gamma}K$ and $\P^{\gamma}K_{t}$ and integrate, respectively, then combine as above estimates for lower oder derivative. Note that the last term on the right hand side of $ \Delta[(a^{+}-a^{-})V^{-}] $ has already second order derivative and the a priori assumption has control of derivatives up to $k$-th order, so we can only carry out the computation for derivatives with order $|\gamma|\leq k-2$. Similar arguments as above, we have, for $|\gamma|\leq k-2$, $$\label{l25}
\| \P^{\gamma}K \|_{L^{2}} \leq C ( E_{\p} + {\mathcal M}^{2}) (1+t)^{-{5\over 4}n-2- {|\gamma|\over 2}} ,$$ and for $|\gamma|= k-1$, $$\label{sp1}
\|\P^\gamma_xK \|_{L^2} \leq C \epsilon_0 (1+t)^{-{5\over 4}n-1- {k\over 2}} .$$ Combine the estimates (\[l21\])-(\[l22\]), (\[l25\])-(\[sp1\]) and using Lemma 3.1, we have the results of (iii) in Theorem 1.1.
Approximate Green Function and $L^p$ Estimates on the Low Frequency Component.
==============================================================================
In this section, we will give approximation Green function of the equation to $V^{\pm}$ as in [@lwy09], which is used to get $L^p$ estimates on the low frequency component of $V^{\pm}$. Recall the linearized equation (\[l15\]), $$\label{1027-1}
V^{\pm}_{tt}-\triangle(a^{\pm}(x,t)V^{\pm})+ V^{\pm}_{t}= F^{\pm} \mp \mathrm{div}[( \bar{\p}^{\pm}+V^{\pm} )\nabla\varphi].$$ We slightly abuse notations by dropping ‘$\pm$’ sign without confusion in the following estimates. Note the main difference of to the linearized equation in [@lwy09] is that we need to consider the coupling term ‘$\mp \mathrm{div}[(
\bar{\p}^{\pm}+V^{\pm} )\nabla\varphi]$’ in the present setting.
Approximate Green Function.
----------------------------
In this subsection, we study the approximate Green function for (\[1027-1\]). For convenience of readers, we briefly repeat the construction of the approximate Green function in below.
Let $G(x,t; y, s)$ be the approximate Green function for the homogeneous part of (\[1027-1\]) which meets the basic requirement $$G(x,t; y, t)=0,\ \ G_t(x,t; y, t)=\delta (x-y),$$ where $\delta $ is the Dirac function. Multiplying (\[1027-1\]) whose variables are now changed to $(y, s)$ by $G$ and integrating with respect to $y$ and $s$ over the region ${\R}^n\times [0, t]$ to get (note here and below we dropped the notation ‘$\pm$’) $$\label{l27}
\begin{array}{rl}
V(x,t) =&\dis{\int_{{\R}^n}G_s(x,t; y, 0)V(y, 0)dy
}-\dis{\int_{{\R}^n}G(x,t; y, 0)(
V+V_s)(y, 0)dy}\vspace{2.5mm}\\
&+\dis{\int^t_0\int_{{\R}^n} (G_{ss}-a^{+}\triangle_y G-
G_s )(x,t; y, s)V(y, s)dy ds} \vspace{2.5mm}\\
&-\dis{\int^t_0\int_{{\R}^n}G(x,t; y, s)
F(y, s) dy ds}\vspace{2.5mm}\\
&+\dis{\int^t_0\int_{{\R}^n}G(x,t; y, s) \mathrm{div} [ (\bar\p
+ V ) \nabla \varphi](y, s) dy ds}.
\end{array}$$ If $a (y, s)$ is a constant and $G$ is the Green function of the homogeneous part of (\[1027-1\]), then the third integral in above is zero. However, when $a (y, s)$ is not a constant, it is difficult to give an explicit expression of the Green function. Therefore, we will use the approximate Green function constructed in [@[W-Y3]; @lwy09]. The idea is to first consider the linear partial differential equation $$\label{v5-1}
\P_{tt}V-\mu\triangle V+ V_t=0,$$ where $\mu$ is a bounded parameter with $C_0<\mu<C_1$, and denote its Green function by $G^\sharp(\mu; x,t)$, whose Fourier transform $$\hat{G}^\sharp(\mu;\xi,t)=\frac{e^{\l_+t}-e^{\l_-t}}{\l_+-\l_-},$$ where $$\l_\pm(\xi)\equiv \frac{1}{2}(-1\pm \sqrt{1-4\mu |\xi|^2}).$$ Denote $\hat{G}^\sharp= \hat{E}^+ + \hat{E}^-$ with $$\hat{E}^+=\eta_0e^{\l_+t}, ~~ \hat{E}^-=\eta_0e^{\l_-t},
~~\eta_0=(\l_+ - \l_-)^{-1}.$$
The approximate Green function is defined by $$\label{p6}
G(x,t; y, s)=G^\sharp(a (y, \sigma(t,s)); x-y, t-s),$$ with $a (y, \sigma(t,s))=P^\prime(\bar{\rho} (y, \sigma(t,s)))$, and the function $\sigma(t,s)$ is chosen such that $\sigma(t, s)\in C^3([2,
\infty]\times [0, \infty])$, $$\sigma(t, s)=\left\{
\begin{array}{rl}
& s, \quad s>t/2,\vspace{2.5mm}\\
&t/2, \quad s\leq t/2-1,
\end{array}\right.$$ and $$\sum_{1\leq l_1+l_2\leq 3} |\P^{l_1}_t\P^{l_2}_s\sigma(t,
s) |\leq C, ~~s\in (t/2-1, t/2).$$ Notice that $\sigma^{-1}(t, s)\leq C(1+t)^{-1}$ for $t>2$ so that we have by Lemma 2.1 $$\label{p7}
(1+t) |\P_s a (y, \sigma(t,s)) |+(1+t)^2 |\P^2_s a (y,
\sigma(t,s)) |\leq CE_\rho,$$ where $E_\rho$ is defined in Lemma 2.1.
Notice that the decay of the derivatives of the function $a (y,\sigma(t,s))$ with respect to time will be used in the following analysis. Recall that the approximate Green function defined in (\[p6\]) is not symmetric with respect to the variables $(x,t)$ and $(y,s)$. However, straightforward calculation gives their relations as $$\label{p9}
\P_{x_i}G=-\P_{y_i}G + \P_a(G^\sharp)~a_{x_i}, ~~\P_t G=-\P_s G +
\P_a(G^\sharp)~(a_s+a_t).$$
Denote the low frequency component in the approximate Green function $G(x,t;y,s)$ by $$ G_L(x,t; y, s)=\chi(D_x)G(x,t; y,s),$$where $\chi(D_x)$, $D_x=\frac{1}{\sqrt{-1}}\P_x
=\frac{1}{\sqrt{-1}}(\P_{x_1},\cdots, \P_{x_n}) $, is the pseudo-differential operator with symbol $\chi(\xi)$ as a smooth cut-off function satisfying $$\chi(\xi)=\left\{\begin{array}{ll}1,&|\xi|<\varepsilon,\vspace{2.5mm}\\0,&|\xi|>2\varepsilon,
\end{array}\right.$$ for some chosen constant $\varepsilon$ in $(0, \varepsilon_0)$ with $\varepsilon_0=\frac{1}{2}\min\Big\{1, \sqrt{\frac{1}{4C_1}}\Big\}$, $C_{1}$ is the upper bound of $\mu$ in . Moreover, we have $$\begin{array}{rl}
G_L(x,t;y,s)=&\frac{1}{(2\pi)^{ n}}\dis{\int_{\R^n}} \chi(\xi)
e^{\sqrt{-1}(x-y)\xi} \hat{G}^\sharp(a(y, \sigma(t,s)),\xi,t-s)
d\xi\vspace{2.5mm}\\
=&G^\sharp_L(a(y, \sigma(t,s)); x-y, t-s).
\end{array}$$
Therefore, it is direct to have the following proposition [@lwy09].
For $q\in [1, \infty]$ and any indices $h, l, \alpha$ and $\beta$, we have $$\begin{array}{rl}
\displaystyle{\sup_{y}} \|\P^{\alpha}_x \P^\b_y \P^l_s \P^h_t
G_L(\cdot,t; y,s) \|_{L^q(\R^n_x)} \leq
C(1+t-s)^{-\frac{n}{2}(1-\frac{1}{q})-\frac{2\min(l+h,1)+|{\alpha}|+|\b|}{2}},\vspace{2.5mm}\\
\displaystyle{\sup_{x}} \|\P^{\alpha}_x \P^\b_y \P^l_s \P^h_t G_L(x,t;
\cdot,s) \|_{L^q(\R^n_y)} \leq
C(1+t-s)^{-\frac{n}{2}(1-\frac{1}{q})-\frac{2\min(l+h,1)+|{\alpha}|+|\b|}{2}}.
\end{array}$$
$L^p$ Estimates on the Low Frequency Component.
-----------------------------------------------
In this subsection, we will establish the $L^p$ estimates on the low frequency component by using the approximate Green function. Assume that $|{\alpha}|\leq k$ in this section. To derive the $L^p$ estimates for the low frequency part, recall (\[l27\]), and set $$\begin{aligned}
&&I^{{\alpha}}_1= \chi(D_x)\int_{{\R}^n}\P^{\alpha}_{x}G_s(x, t; y, 0)V(y,0)dy
= \dis{\int_{{\R}^n}\P^{\alpha}_{x}(G_L)_s(x, t; y,0)V(y,0)dy},\\
&&I^{\alpha}_2=-\dis{\int_{{\R}^n}\P^{\alpha}_{x}G_L(x, t; y, 0)(V+V_s)(y,
0)dy},\\
&&I^{{\alpha}}_3= \int^t_0\int_{{\R}^n}\P^{\alpha}_{x}R_{G_L}(x, t; y, s)~ V(y,
s)dyds,\\
&&I^{{\alpha}}_4= - \dis{\int^t_0\int_{{\R}^n}\P^{\alpha}_{x} G_L(x, t; y,
s) F (y, s)dy ds},\\
&&I^{{\alpha}}_5 = \dis{\int^t_0\int_{{\R}^n}\P^{\alpha}_{x}G_{L}(x,t; y, s)
\mathrm{div} [ (\bar\p + V ) \nabla
\varphi](y, s) dy ds} ,\end{aligned}$$ where $$R_G\equiv G_{ss}(x, t; y, s)-a^{+}(y, s)\triangle_y G(x, t; y, s)-
G_{s}(x, t; y, s),$$ and $$\chi(D_x)R_G=R_{G_L}.$$ Since $$(G^\sharp_{tt}-a \triangle G^\sharp+G^\sharp_t)(a(y, s); x-y,
t-s)=0,$$ we have $$\begin{array}{rl}
&R_{G_L}(x, t; y, s)\\
=& \Big[G^\sharp_{L;0, 0}(a(y, s); x-y, t-s)a_s(y,
\sigma)^2-2G^\sharp_{L;0, n+1}(a(y, s); x-y, t-s)a_s(y, \sigma)\\
&+G^\sharp_{L;0}(a(y, s); x-y, t-s) a_{ss}(y, \sigma) -
G^\sharp_{L;0}(a(y, s); x-y,
t-s) a_s(y, \sigma)\\
&+a(y,s) \Big(\sum\limits_{i=1}^n [G^\sharp_{L;0, i}(a(y, s); x-y,
t-s)a_{y_i}(y,
\sigma)\\
&-G^\sharp_{L;0, 0}(a(y, s); x-y, t-s)((a)^2_{y_i})(y, \sigma)] +
G^\sharp_{L;0}(a(y, s); x-y,
t-s)\triangle_{y}a(y,\sigma) \Big) \Big]\vspace{2.5mm}\\
& + [(a(y,
\sigma)-a(y, s))
\triangle G^\sharp_{L}(a(y, s); x-y, t-s) ]\vspace{2.5mm}\\
=: &R^1_{G_L}+R^2_{G_L}.
\end{array}$$ Here $R^i_{G_L}$, $i=1,2$, is the corresponding term in the above summation in the above equation. To denote the derivatives, we use the notations $G^\sharp_{L;0}(a;x,t)=\partial_aG^\sharp_{L}(a;x,t)$, $G^\sharp_{L;i}(a;x,t)=\partial_{x_i}G^\sharp_{L}(a;x,t)$, $G^\sharp_{L;n+1}(a;x,t)=\partial_tG^\sharp_{L}(a;x,t)$, $G^\sharp_{L;0,i}(a;x,t)=\partial_a\partial_{x_i}G^\sharp_{L}(a;x,t)$, and $G^\sharp_{L;0,n+1}(a;x,t)=\partial_a\partial_tG^\sharp_{L}(a;x,t)$ etc..
Then, set $X_L(x, t)=\chi(D_x)X(x, t)$, from above notations we have $$\P^{\alpha}_{x}X_L(x,t)=I^{{\alpha}}_1+I^{{\alpha}}_2+I^{{\alpha}}_3+I^{\alpha}_4 +I^{{\alpha}}_5.$$ We will estimate the right hand side of above term by term. The terms from $I^{{\alpha}}_1$ to $I^{{\alpha}}_4$ are similar to the estimates in [@lwy09]: by Proposition 4.1, it is straightforward to obtain $$\begin{array}{rl}
\|I^{\alpha}_1 \|_{L^p(\R^n_x)} \leq
&C(1+t)^{-\frac{n}{2}(1-\frac{1}{p})-\frac{|{\alpha}|+2}{2}} \|V_0 \|_{L_1}.
\end{array}$$ For $I^{\alpha}_2$, set $$\tilde{\nu}_0 (y)=\nu_t (y, 0)+\nu (y, 0),$$ where $\nu$ (with ${\pm}$ omitted) is defined in (\[l6\]). Then $$\begin{array}{rl}
|I^{\alpha}_2 |= |\dis{\int_{{\R}^n}}\P_{y_1}\P^{\alpha}_{x}G_L(x,t;
y, 0) \tilde{\nu}_0 (y)dy |.
\end{array}$$ Also by using Propositions 4.1, we have $$\begin{array}{rl}
\|I^{\alpha}_2 \|_{L^p(\R^n_x)} \leq
C(1+t)^{-\frac{n}{2}(1-\frac{1}{p})-\frac{|{\alpha}|+1}{2}} \|\tilde{\nu}_0 \|_{L_1} .
\end{array}$$
We now turn to estimate the term $I^{\alpha}_3$ which is the error coming from the approximate Green function. For illustration, we only consider $$J^{\alpha}_1=\int^t_{0}\int_{{\R}^n}\P^{\alpha}_{x}G^\sharp_{L;0}(a^{+}(y,\sigma), x-y, t-s)a^{+}_s(y,\sigma)V(y, s)dyds,$$ and $$J^{\alpha}_2=\int^t_{0}\int_{{\R}^n}\P^{\alpha}_{x}R^2_{G_L}(x,t; y, s)V(y, s)dy ds,$$ because the other terms in $I^{\alpha}_3$ can be estimated similarly. Note that (\[p7\]) gives $$|a^{+}_s(y, \sigma)|\leq CE_\rho (1+t)^{-1},$$ then we have, for $|\r|\leq k-2$, $$\begin{array}{rl}
\|J^\r_1 \|_{L^p(\R^n_x)}\leq &
\dis{\int^t_{0} \|\int_{{\R}^n}}\P^\r_{x}G^\sharp_{L;0}(a(y,\sigma),
x-y, t-s) ~a_s(y,\sigma)~V(y, s)dy \|_{L^p{(\R^n_x)}}ds\vspace{2.5mm}\\
\leq &CE_\rho{\mathcal
M}\Big[\dis{\int^{t/2}_0}(1+t-s)^{-\frac{n}{2}(1-\frac{1}{p})-\frac{|\r|}{2}}(1+t)^{-1} (1+s)^{-\frac{n}{2}(1-\frac{1}{1})-1/2}ds\vspace{2.5mm}\\
&+\dis{\int^t_{t/2}}(1+t-s)^{-\frac{n}{2}(1-\frac{1}{1})} (1+t)^{-1}
(1+s)^{-\frac{n}{2}(1-\frac{1}{p})-\frac{|\r|+1}{2}}ds\Big]\vspace{2.5mm}\\
\leq & CE_\rho{\mathcal
M}(1+t)^{-\frac{n}{2}(1-\frac{1}{p})-\frac{|\r|+1}{2}},
\end{array}$$ and for $|\r|= k-1$ and $k$, $$\begin{array}{rl}
\|J^\r_1 \|_{L^p(\R^n_x)} \leq &CE_\rho{\mathcal
M}\dis{\int^{t/2}_0}(1+t-s)^{-\frac{n}{2}(1-\frac{1}{p})-\frac{|\r|}{2}}
(1+t)^{-1} (1+s)^{-\frac{n}{2}(1-\frac{1}{1})-1/2}ds\vspace{2.5mm}\\
&+CE_\rho{\mathcal M}\dis{\int^t_{t/2}}(1+t-s)^{-\frac{|\r|+2-k}{2}}
(1+t)^{-1}
(1+s)^{-\frac{n}{2}(1-\frac{1}{p})-\frac{k-2+1}{2}}ds\vspace{2.5mm}\\
\leq & CE_\rho{\mathcal
M}(1+t)^{-\frac{n}{2}(1-\frac{1}{p})-\frac{|\r|+1}{2}}.
\end{array}$$ For $J^{\alpha}_2$, since $$\begin{array}{rl}
|a (y, s)-a (y,\sigma)|\leq\int^\sigma_s |a_\tau (y, \tau)|d\tau
\leq\left\{
\begin{array}{rl}
&CE_\rho\Theta(t, s), s<t/2,\vspace{2.5mm}\\
&0, \quad s\geq t/2,
\end{array}\right.
\end{array}$$ where $$\Theta(t, s)=(1+t-s)(1+t)^{-1+1/h}(1+s)^{-1/h},$$ and $h$ can be any positive integer. By using Proposition 4.1, we have $$\|J^{\alpha}_2 \|_{L^p(\R^n_x)} \leq CE_\rho{\mathcal M}
\dis{\int^{t/2}_0}
(1+t-s)^{-\frac{n}{2}(1-\frac{1}{p})-\frac{|{\alpha}|+2}{2}}\Theta(t,
s)(1+s)^{-1/2}ds.$$ By noticing that $$\begin{array}{rl}
\dis{\int^{t/2}_0}(1+t)^{-1+1/h}(1+s)^{-1/h}(1+s)^{-1/2}ds
=&(1+t)^{-1+1/h}(1+s)^{\frac{1}{2}-\frac{1}{h}} \Big|^{t/2}_0\vspace{2.5mm}\\
\leq & C(1+t)^{-1+1/h}(1+t)^{\frac{1}{2}-\frac{1}{h}}\ =\ C(1+t)^{-\frac{1}{2}},
\end{array}$$ we obtain $$\|J^{\alpha}_2 \|_{L^p(\R^n_x)}\leq CE_\rho{\mathcal M}
(1+t)^{-\frac{n}{2}(1-\frac{1}{p})-\frac{|{\alpha}|+1}{2}}.$$ Thus, combine the above estimate to have $$\|I^{\alpha}_3 \|_{L^p(\R^n_x)}\leq CE_\rho{\mathcal M}
(1+t)^{-\frac{n}{2}(1-\frac{1}{p})-\frac{|{\alpha}|+1}{2}}.$$
Next for $I^{\alpha}_4$, recall that $F$ (with $\pm$ omitted) satisfies (\[l20\]) under the a priori assumption (\[l19\]), then for $|\r|\leq k-2$, $$\begin{array}{rl}
\|I^\r_4 \|_{L^p(\R^n_x)}\leq&\dis{\int^t_0 \|
\int_{{\R}^n}\P^\r_{x}G_L
~F dy \|_{L^p(\R^n_x)}ds}\vspace{2.5mm}\\
=&\dis{\int^{\frac{t}{2}}_0 \| \int_{{\R}^n}(-\sum
\P^\r_{x}\P_{y_i}G_L
~F^i +\sum\P^\r_{x}\P_{y_iy_j}G_L ~ F^{ij})dy \|_{L^p(\R^n_x)}ds}\vspace{2.5mm}\\
&+\dis{\int^t_{\frac{t}{2}} \|\int_{{\R}^n} \P^\r_{x}G_L ~F dy \|_{L^p(\R^n_x)}ds}\vspace{2.5mm}\\
\leq &CE_\rho\dis{\int^{\frac{t}{2}}_0}
(1+t-s)^{-\frac{n}{2}(1-\frac{1}{p})-\frac{|\r|+1}{2}} e^{- s} ds
+CE_\rho\dis{\int^t_{\frac{t}{2}}} (1+t-s)^{-\frac{n}{2}(1-\frac{1}{p})-\frac{|\r|+1}{2}} e^{- s} ds\vspace{2.5mm}\\
&+C{\mathcal M}^2\dis{\int^\frac{t}{2}_0 }(1+t-s)^{-\frac{n}{2}(1-\frac{1}{p})-\frac{|\r|+2}{2}} (1+s)^{-(n+1)+\frac{n}{2}} ds\vspace{2.5mm}\\
&+C{\mathcal M}^2\dis{\int^t_{\frac{t}{2}}} (1+t-s)^{-1} (1+s)^{-(n+1+\frac{|\r|}{2})+\frac{n}{2p}} ds\vspace{2.5mm}\\
\leq &C(E_\rho+{\mathcal M}^2)
(1+t)^{-\frac{n}{2}(1-\frac{1}{p})-\frac{|\r|+1}{2}}.
\end{array}$$ The cases when $|\r|= k-1$ and $k$ can be estimated similarly, and the only difference is the estimation on the terms like $$\int^t_{\frac{t}{2}} \|\int_{\R^n}\P^\r_{x}\P_{y_iy_j}G_L ~
F^{ij}dy \|_{L^p(\R^n_x)}ds,\qquad |\gamma|\le k-2.$$ On the other hand, these terms can be estimated by replacing the derivatives of $G_L$ w.r.t. $x$ to the derivatives of $G_L$ w.r.t. $y$ using (\[p9\]). Then by using integration by parts $k-2$ times to transfer the derivatives on $G_L$ to $F^{ij}$, we have by (4.8) and Proposition 4.1 that $$\begin{array}{rl}
\dis{\int^t_{\frac{t}{2}} \|\int_{\R^n}}\P^\r_{x}G_L ~\P_{y_iy_j}
F^{ij}dy \|_{L^p(\R^n_x)}ds
\leq& C{\mathcal M}^2\dis{\int^t_{\frac{t}{2}}}(1+t-s)^{-\frac{|\r|+2-(k-2)}{2}}(1+s)^{-(n+1+\frac{k-2}{2})+\frac{n}{2p}}ds\vspace{2.5mm}\\
\leq&C{\mathcal M}^2 (1+t)^{-(n+1+\frac{k-2}{2})+\frac{n}{2p}}\vspace{2.5mm}\\
\leq&C{\mathcal M}^2
(1+t)^{-\frac{n}{2}(1-\frac{1}{p})-\frac{|\r|+1}{2}}.
\end{array}$$ Therefore, we have the $L^p$ estimate on $I^{\alpha}_4$ as $$\|I^{\alpha}_4 \|_{L^p(\R^n_x)}\leq C{\mathcal M}^2
(1+t)^{-\frac{n}{2}(1-\frac{1}{p})-\frac{|{\alpha}|+1}{2}}.$$
For $I^{{\alpha}}_5$, we write $$\begin{aligned}
I^{{\alpha}}_5 &=& \int^{t\over 2}_0 \int_{{\R}^n}\P^{\alpha}_{x}
\nabla_{y}G_{L}(x,t; y, s)
\cdot [ (\bar\p + V ) \nabla \varphi](y, s) dy
ds\\
&&+ \int^t_{t\over 2}\int_{{\R}^n} \P^{\alpha}_{x}G_{L}(x,t; y, s)
\mathrm{div}[ (\bar\p + V ) \nabla \varphi](y, s) dy ds\\
&=&: I^{{\alpha}}_{5,1} + I^{{\alpha}}_{5,2},\end{aligned}$$ in which the first term satisfies $$I^{{\alpha}}_{5,1} \leq C\int^{t\over 2}_0
\| \P^{\alpha}_{x} \nabla_{y} G_{L} \|_{L^{P}}
\| (\bar\p + V ) \nabla \varphi \|_{L^{1}}ds .$$ To estimate $$\| (\bar\p + V ) \nabla \varphi \|_{L^{1}} \leq
C\| \bar\p + V \|_{L^{6\over 5}} \| \nabla \varphi \|_{L^{6}},$$ we note that the $L^{6\over 5}$ norm of $\bar\p + V( = \p, positive)$ can be controlled by its $ L^{1} $ and $L^{2}$ norms by interpolation, and also note the $ L^{1} $ norm of $\p$ (with $\pm$ omitted) is conserved because the conservation of mass, $ L^{2} $ norm can be bounded by the a priori assumption, then we have $$\| I^{{\alpha}}_{5,1} \|_{L^{P}} \leq C{\mathcal M}^2
(1+t)^{-\frac{n}{2}(1-\frac{1}{p})-\frac{|{\alpha}|+1}{2}}.$$ For the term $ I^{{\alpha}}_{6,2}$, we estimate $$I^{{\alpha}}_{5,2} \leq C \int^t_{t\over 2}
\| \P^{\alpha}_{x} G_{L} \|_{L^{1}}
\| \mathrm{div} [ (\bar\p + V ) \nabla \varphi] \|_{L^{p}}ds .$$ Note that both $\mathrm{div} \nabla \varphi =K$ and $ \nabla
\varphi$ are in $ L^{\infty}$ thus have good decay properties, then the above term decays faster than that of $ I^{{\alpha}}_{5,1} $, then we have $$\| I^{{\alpha}}_{5} \|_{L^{P}} \leq
C{\mathcal M}^2
(1+t)^{-\frac{n}{2}(1-\frac{1}{p})-\frac{|{\alpha}|+1}{2}}.$$
In summary, by combining all above estimates, we have the estimates on the low frequency component of $ V^\pm$ in the following theorem.
[T]{}[HEOREM]{} 4.2. [*For $|{\alpha}|\leq k$, we have, $$\|\P^{\alpha}V^{\pm}_L (t) \|_{L^p} \leq C(E_0+{\mathcal M}^2)
(1+t)^{-\frac{n}{2}(1-\frac{1}{p})-\frac{|{\alpha}|+1}{2}},$$ where $E_0=\max \{ \|V^{\pm}_0 \|_{L^1},
\|\tilde{\nu}^{\pm}_0 \|_{L^1}, \|(V^{\pm}_0, U^{\pm}_0) \|_{H^k},
\|V^{\pm}_t(0) \|_{H^{k-1}}, E_\p \}$.*]{}
As an immediate consequence, we have the $L^2$ estimate on the derivatives of order higher than the $k$-th for the low frequency component because $$\|\P_{x_i}\P^{\alpha}V^{\pm}_L(t) \|_{L^2}= \|\xi_i\xi^{\alpha}\chi(\xi)\hat{V^{\pm}} \|_{L^2} \leq \varepsilon \|\xi^{\alpha}\chi(\xi)\hat{V^{\pm}} \|_{L^2}=\varepsilon \|\P^{\alpha}V^{\pm}_L(t) \|_{L^2}.$$ Thus, we have the following corollary.
For any $|\r|> k$, we have $$\|\P^\r V^{\pm}_L(t) \|_{L^2}\leq C(E_0+{\mathcal M}^2)\varepsilon
(1+t)^{-\frac{n}{2}(1-\frac{1}{2})-\frac{k+1}{2}}.$$
Estimates on the High Frequency Component.
==========================================
In this section, we will carry out the energy estimates on the high frequency component. Recall the linearized equation (\[l15\]), set $\tilde{\chi}(\xi)=1-\chi(\xi)$ and $V^{\pm}_H(x, t)=\tilde{\chi}(D_x)V^{\pm}(x, t)$. By taking $\tilde{\chi}(D_x)$ on both sides of (\[l15\]) and integrating its product with $V^{\pm}_H$ and $(V^{\pm}_H)_t$ over $\R^n$ respectively, we have $$\label{p1}
\frac{d}{dt}\int_{{\R}^n}V^{\pm}_H~(V^{\pm}_H)_t dx -
\int_{{\R}^n}((V^{\pm}_H)_t)^2 dx -\int_{{\R}^n}V^{\pm}_H ~
\triangle\tilde{\chi}(a V^{\pm})dx
+\frac{d}{dt}\int_{{\R}^n}\frac{1}{2}(V^{\pm}_H)^2 dx$$ $$=\int_{{\R}^n}V^{\pm}_H\tilde{\chi} F^{\pm} dx
\mp \int_{{\R}^n}(V^{\pm}_H)_{t}\tilde{\chi}
\mathrm{div}[( \bar{\p}^{\pm}+V^{\pm} )\nabla\varphi] ,$$ and $$\label{p2}
\frac{d}{dt}\int_{{\R}^n}\frac{1}{2}((V^{\pm}_H)_t)^2 dx -\int_{{\R}^n}(V^{\pm}_H)_t ~\triangle\tilde{\chi}(a V^{\pm})dx
+\int_{{\R}^n}((V^{\pm}_H)_t)^2dx$$ $$=\dis{\int_{{\R}^n}(V^{\pm}_H)_t\tilde{\chi}F^{\pm} dx}
\mp \int_{{\R}^n}V^{\pm}_H\tilde{\chi} \mathrm{div}[( \bar{\p}^{\pm}+V^{\pm} )\nabla\varphi].$$ Again, we slightly abuse notations by dropping the $\pm$ sign without confusion in the following estimates. First for the third term on the left hand side of (\[p1\]) as follows. That is, $$\begin{array}{rl}
\dis{-\int_{{\R}^n}V_H ~\triangle\tilde{\chi}(a V)dx }
=\dis{\int_{{\R}^n}a|\nabla V_H|^2dx-\int_{{\R}^n}V_H~\nabla [\nabla
\tilde{\chi}, a]Vdx,}
\end{array}$$ where $[A, B]=A\circ B-B\circ A$ denotes the commutator. Since $$(1+t)^{1/2} \|\nabla a \|_{L^\infty}+(1+t) \|\triangle
a \|_{L^\infty}\leq CE_\rho,$$ where $E_\rho$ is defined in Theorem 2.1. It is straightforward to show that $$\int_{{\R}^n} |\nabla [\nabla \tilde{\chi}, a]V |^2dx\leq
CE^2_\rho{\mathcal M}^2 (1+t)^{-\frac{n}{2}-3},$$ where ${\mathcal M}$ is defined in (3.4). Thus $$|\int_{{\R}^n}V_H~\nabla [\nabla \tilde{\chi}, a]Vdx |
\leq \eta\int_{{\R}^n}| V_H|^2dx+ CE^2_\rho {\mathcal
M}^2(1+t)^{-\frac{n}{2}-3}.
$$ We now turn to estimate the second term on the left hand side in (\[p2\]). That is, $$\begin{array}{rl}
-\int_{{\R}^n}(V_H)_t ~\triangle\tilde{\chi}(a V)dx
=&\dis{\int_{{\R}^n}(\nabla V_H)_t ~a\nabla\tilde{\chi}(
V_H)dx+\int_{{\R}^n}(\nabla V_H)_t
[\nabla\tilde{\chi}, a] V dx}\vspace{2.5mm}\\
=&\dis{\frac{1}{2}\frac{d}{dt}\int_{{\R}^n}a|V_H|^2dx-\frac{1}{2}\int_{{\R}^n}a_t|V_H|^2dx-\int_{{\R}^n}(
V_H)_t \nabla[\nabla\tilde{\chi}, a] V dx,}
\end{array}$$ in which, we have $$|\int_{{\R}^n}a_t |\nabla(V_H) |^2 dx | \leq
CE^2_\rho(1+t)^{-2}\int_{{\R}^n} |\nabla(V_H) |^2 dx,$$ and $$|\int_{{\R}^n} (V_H)_t\nabla[\nabla \tilde{\chi}, a]Vdx | \leq
\eta\int_{{\R}^n} |(V_H)_t |^2dx+ CE^2_\rho {\mathcal
M}^2(1+t)^{-\frac{n}{2}-3}.$$
For $\int_{{\R}^n}V_H\tilde{\chi}F dx$ and $\int_{{\R}^n}(V_H)_t\tilde{\chi}F dx$ on the right hand side of (\[p1\]) and (\[p2\]), by using Lemma 2.3 and the definition of ${\mathcal M}$ in (\[l19\]), we have $$\begin{array}{rl}
\|F^i \|_{L^2} \leq &CE_\rho e^{- s},\vspace{2.5mm}\\
\|\P^{\r}F^{ij} \|_{L^2} \leq &C{\mathcal M}^2
(1+t)^{-(n+1+\frac{|\r|}{2})+\frac{n}{4}},~~ |\r|\leq k-2,
\end{array}$$ where $F$ and $F^i, F^{ij}$ defined in (\[l18\]). Further, it is straightforward to check that $$|\int_{{\R}^n}V_H\tilde{\chi}F dx |\leq
\eta\int_{{\R}^n} |V_H |^2dx + C(\eta)(E^2_\rho +{\mathcal M}^4)
(e^{- t}+ (1+t)^{-2(n+1)+\frac{n}{2}} ),$$ and $$|\int_{{\R}^n}(V_H)_t\tilde{\chi}F dx |\leq
\eta\int_{{\R}^n} |(V_H)_t |^2dx + C(\eta)(E^2_\rho +{\mathcal
M}^4) (e^{- t}+ (1+t)^{-2(n+1)+\frac{n}{2}} ).$$
For the term $\int_{{\R}^n}(V_H)_{t}\tilde{\chi} \mathrm{div}[(
\bar{\p}+V)\nabla\varphi] $, we have $$\begin{aligned}
& & \int_{{\R}^n}(V_H)_{t}\tilde{\chi} \mathrm{div}[(
\bar{\p}+V)\nabla\varphi]dx \\
&\leq&
\varepsilon \int_{{\R}^n} (V_H)_{t}^{2}dx + C(\varepsilon) \int_{{\R}^n} (\mathrm{div}[( \bar{\p}+V)\nabla\varphi]
)^{2}dx\\
&=&\varepsilon \int_{{\R}^n} (V_H)_{t}^{2}dx +
C(\varepsilon) \int_{{\R}^n} [\nabla( \bar{\p}+V)\cdot \nabla\varphi
+(\bar{\p}+V)K
]^{2}dx\\
&\leq& \varepsilon \int_{{\R}^n} (V_H)_{t}^{2}dx +
2 C(\varepsilon) \Big( \| \nabla( \bar{\p}+V)\|^{2}_{L^{2}}
\| \nabla\varphi \|^{2}_{L^{\infty}}
+ \|\bar{\p}+V\|_{L^{\infty}}^{2} \|K\|_{L^{2}}^{2}
\Big)\\
&\leq& \varepsilon \int_{{\R}^n} (V_H)_{t}^{2} dx+
C(\varepsilon) (E_{\p}^{2} + \mathcal M^{2}) (1+t) ^{-{3\over 2}n - |{\alpha}|-2}.\end{aligned}$$ The term $ \int_{{\R}^n}V_H\tilde{\chi} \mathrm{div}[( \bar{\p}+V
)\nabla\varphi]dx $ can be estimated similarly.
To close the energy estimate, one needs the following important fact about the high frequency part: $$\int_{{\R}^n} |\nabla V_H |^2dx\geq
\varepsilon\int_{{\R}^n} |V_H |^2dx.$$ This is a Poincaré type inequality which holds only for the high frequency part in the whole space. By integrating (\[p1\]) and (\[p2\]) over $[0, t]$ and multiplying (\[p1\]) by some suitably chosen constant $0<\lambda<1$, when $\eta$ is small, the combination of above estimates give $$\begin{array}{rl}
&\dis{\int_{{\R}^n} (|V_H|^2 + |(V_H)_t|^2 + |\nabla V_H|^2
)(t)dx
+ \mu\int^t_0\int_{{\R}^n} (|V_H|^2 + |(V_H)_s|^2 + |\nabla V_H|^2 ) dxds }\vspace{2.5mm}\\
\leq&\dis{C \Big[\int_{{\R}^n} (|V_H|^2 + |(V_H)_t|^2 + |\nabla
V_H|^2 )(0)dx+(E^2_\rho+{\mathcal
M}^4)\int^t_0(1+s)^{-\frac{n}{2}-3}ds\Big],}
\end{array}$$ for some positive $\mu$. Denote $${\mathcal F}(t) = \int_{{\R}^n} (|V_H|^2 + |(V_H)_t|^2 + |\nabla
V_H|^2 ) dx.$$ Then the above inequality gives $${\mathcal F}(t) +\mu\int^t_0{\mathcal F}(s)ds\leq C ({\mathcal
F}(0)+(E^2_\rho+{\mathcal
M}^4)\int^t_0(1+s)^{-\frac{n}{2}-3}ds ).$$ By using the Gronwall inequality, we have $${\mathcal F}(t) \leq Ce^{-\mu t} ({\mathcal F}(0)+(E^2_\rho+{\mathcal
M}^4)\int^t_0e^{\mu s}(1+s)^{-\frac{n}{2}-3}ds ).$$ Hence, we have $$\begin{array}{rl}
& \| V_H(t) \|^2_{H^1} + \|(V_H)_t(t) \|^2_{L^2} \vspace{2.5mm}\\
\leq &e^{-\mu t}( \| V_H(0) \|_{H^1} +
\|(V_H)_t(0) \|_{L^2})\vspace{2.5mm}
+ C(E^2_\rho+{\mathcal M}^4)(1+t)^{-\frac{n}{2}-3}\vspace{2.5mm}\\
\leq &C(E^2_0+{\mathcal M}^4)(1+t)^{-\frac{n}{2}-3},
\end{array} $$ where $E_0$ is defined in Theorem 4.2.
Next, we will derive the energy estimates on the higher order derivatives of the high frequency component, that is, $ \int_{{\R}^n}|\P^{\alpha}V_H|^2 + |\P^{\alpha}(V_H)_t|^2 + |\nabla\P^{\alpha}V_H|^2 dx $ for $0<
|{\alpha}|\leq k-1$. In the rest of this section, we assume $0<
|{\alpha}|\leq k-1$. The estimation can be obtained by induction on $|\alpha|$. Assume that $$\int_{{\R}^n} (|\P^\r V_H|^2 + |\P^\r
(V_H)_t|^2 + |\nabla\P^\r V_H|^2 ) dx \leq C(E^2_0+{\mathcal
M}^4)(1+t)^{-\frac{n}{2}-(|\r|+3)} $$ holds for any multi-index $\r$ with $|\r| < |{\alpha}|$, we want to prove $$\label{p13}
\int_{{\R}^n} (|\P^{\alpha}V_H|^2
+ |\P^{\alpha}(V_H)_t|^2 + |\nabla\P^{\alpha}V_H|^2 )dx \leq C(E^2_0+{\mathcal
M}^4)(1+t)^{-\frac{n}{2}-(|{\alpha}|+3)}.$$
Taking $\P^{\alpha}\tilde{\chi}$ on (\[l15\]), neglecting the $\pm$ sign without confusing, and integrating its product with $\P^{\alpha}V_H$ and $\P^{\alpha}(V_H)_t$ over $\R^n$ respectively, we have $$\label{p3}
\frac{d}{dt}\int_{{\R}^n}\P^{\alpha}V_H\P^{\alpha}(V_H)_t dx -
\int_{{\R}^n}|\P^{\alpha}(V_H)_t|^2 dx -\int_{{\R}^n}\P^{\alpha}V_H
~\triangle\P^{\alpha}\tilde{\chi}(a V)dx
+\frac{d}{dt}\int_{{\R}^n}\frac{1}{2}|\P^{\alpha}V_H|^2 dx$$ $$=\int_{{\R}^n}\P^{\alpha}V_H~\tilde{\chi}\P^{\alpha}F dx \mp
\int_{{\R}^n}\P^{\alpha}V_H~\tilde{\chi}\P^{\alpha}\mathrm{div}[(
\bar{\p}^{\pm}+V^{\pm} )\nabla\varphi] ,$$ and $$\label{p4}
\frac{d}{dt}\int_{{\R}^n}\frac{1}{2}|\P^{\alpha}(V_H)_t|^2 dx
-\int_{{\R}^n}\P^{\alpha}(V_H)_t ~\triangle\tilde{\chi}\P^{\alpha}(a V)dx
+\int_{{\R}^n}|\P^{\alpha}(V_H)_t|^2 dx$$ $$=\int_{{\R}^n}\P^{\alpha}(V_H)_t~\tilde{\chi}\P^{\alpha}F dx\mp
\int_{{\R}^n}\P^{\alpha}(V_H)_{t}~\tilde{\chi}\P^{\alpha}\mathrm{div}[(
\bar{\p}^{\pm}+V^{\pm} )\nabla\varphi] .$$ For the third term on the left hand side of (\[p3\]), we have $$\dis{-\int_{{\R}^n}\P^{\alpha}V_H ~\triangle\tilde{\chi}\P^{\alpha}(a V)dx }=\dis{\int_{{\R}^n}a(\P^{\alpha}\nabla V_H)^2dx -\int_{{\R}^n}\P^{\alpha}V_H
~\nabla [\nabla\tilde{\chi}\P^{\alpha}, a] Vdx.}$$ Since $$\|\P^\beta_x a \|_{L^\infty}\leq CE_\rho(1+t)^{-|\beta|/2},$$ it holds that $$|\int_{{\R}^n}\P^{\alpha}V_H ~\nabla [\nabla\tilde{\chi}\P^{\alpha}, a]
Vdx |\leq \eta\int_{{\R}^n}|\P^{\alpha}V_H|^2dx+C_\eta E^2_\rho{\mathcal
M}^2(1+t)^{-\frac{n}{2}-3-|\alpha|}.$$ Similarly, for the second term on the left hand side of (\[p4\]), we have $$\dis{-\int_{{\R}^n}\P^{\alpha}(V_H)_t ~\triangle\tilde{\chi}\P^{\alpha}(a V)dx}
=\dis{\frac{d}{dt}\int_{{\R}^n}\frac{a}{2}|\nabla \P^{\alpha}V_H|^2dx
-\int_{{\R}^n}\frac{a_t}{2}|\nabla \P^{\alpha}V_H|^2dx}$$ $$~\hspace{3cm} \dis{-\int_{{\R}^n}\P^{\alpha}(V_H)_t~\nabla[\nabla\P^\alpha\tilde{\chi},
a] Vdx,}$$ where $$|\int_{{\R}^n}\P^{\alpha}(V_H)_t ~\nabla [\nabla\tilde{\chi}\P^{\alpha}, a]
Vdx |\leq \eta\int_{{\R}^n}|\P^{\alpha}(V_H)_t|^2dx+C_\eta
E^2_\rho{\mathcal M}^2(1+t)^{-\frac{n}{2}-3-|\alpha|}.$$ For the terms $\int_{{\R}^n}\P^{\alpha}V_H\tilde{\chi}\P^{\alpha}F dx$ and $\int_{{\R}^n}\P^{\alpha}(V_H)_t\tilde{\chi}\P^{\alpha}F dx$ on the right hand side of (\[p3\]) and (\[p4\]), we only estimate the second one because the estimation on the first is easier. Notice that the estimation on the terms with derivatives of order less or equal to $|{\alpha}|+1$ follows directly from the definition of ${\mathcal M}$ in (\[l19\]). Thus, we consider the terms with derivatives of order higher than $|{\alpha}|+1$. Firstly, by using the expression (\[l15\]) for $F$, we have $$\label{p15}
\begin{array}{rl}
F=&\tilde{Q}+\triangle ({\mathcal P}_1(\bar{\p}, V)V^2)\vspace{2.5mm}\\
=& [(R_\rho)_t+R_\rho ]-(1
+\partial_t)(V\bar{u}_1)_{x_1}-\mathrm{div}((\bar{\p}+V)_tU)
-\mathrm{div}((\bar{\p}+V)H)+\triangle ({\mathcal P}_1(\bar{\p},
V)V^2).
\end{array}$$ Since (\[l7\]) implies $$\label{p16}
{\rm div}U=-(\bar{\p}+V)^{-1} (V_{t}+(\bar{u}+U)\cdot\nabla V+
(U\cdot\nabla)\bar{\p}+V{\rm div}\bar{u}-R_\p ),$$ substituting (\[p16\]) in (\[p15\]), by the definition of $H$, we have $$F=(\bar{u}+U)\cdot\nabla((\bar{u}+U)\cdot\nabla V) + \triangle
({\mathcal P}_1(\bar{\p}, V)V^2) + {\mathcal R},$$ where ${\mathcal R}$ denotes the remainder which contains derivatives of $U$ and $K$ with order at most 1. Thus, $\P^{\alpha}{\mathcal R}$ has derivatives with order at most $|{\alpha}|+1~(\leq k)$. Then $$\int_{{\R}^n}\P^{\alpha}(V_H)_t ~ \tilde{\chi}\P^{\alpha}F dx=N_1 +N_2+N_3,$$ with $$\begin{array}{rl}
N_1=&\dis{\int_{{\R}^n}}\P^{\alpha}(V_H)_t
\tilde{\chi}\P^{\alpha}((\bar{u}+U)\cdot\nabla((\bar{u}+U)\cdot\nabla
V) ) dx,\vspace{2.5mm}\\
N_2=&\dis{\int_{{\R}^n}}\P^{\alpha}(V_H)_t
\tilde{\chi}\P^{\alpha}\triangle ({\mathcal P}_1(\bar{\p}, V)V^2)dx,\vspace{2.5mm}\\
N_3=&\dis{\int_{{\R}^n}}\P^{\alpha}(V_H)_t
\P^{\alpha}{\mathcal R}dx.
\end{array}$$
For $N_1$, we have $$\begin{array}{rl}
N_1=& \dis{\int_{{\R}^n}}\P^{\alpha}(V_H)_t
\tilde{\chi}(\bar{u}+U)\cdot\nabla((\bar{u}+U)\cdot\nabla\P^{\alpha}V)
dx+\{\cdots\}\vspace{2.5mm}\\
=&\dis{\int_{{\R}^n}}\P^{\alpha}(V_H)_t
~(\bar{u}+U)\cdot\nabla((\bar{u}+U)\cdot\nabla\P^{\alpha}V_H)
dx+\dis{\int_{{\R}^n}}\P^{\alpha}(V_H)_t
~[(\bar{u}+U)\cdot\nabla)^2,\tilde{\chi}]\P^{\alpha}V dx+\{\cdots\}\vspace{2.5mm}\\
=&\dis{-\frac{d}{dt}\int_{{\R}^n}}\frac{1}{2}|(\bar{u}+U)\cdot\nabla\P^{\alpha}(V_H)|^2dx
-\dis{\int_{{\R}^n}}(\bar{u}+U)_t\cdot\nabla\P^{\alpha}(V_H)(\bar{u}+U)\cdot\nabla\P^{\alpha}(V_H)dx\\
&-\dis{\int_{{\R}^n}}\nabla(\bar{u}+U)\cdot\nabla\P^{\alpha}(V_H)_t(\bar{u}+U)\cdot\nabla\P^{\alpha}(V_H)dx+\dis{\int_{{\R}^n}}\P^{\alpha}(V_H)_t
~[(\bar{u}+U)\cdot\nabla)^2,\tilde{\chi}]\P^{\alpha}V dx+\{\cdots\}\vspace{2.5mm}\\
=:
&-\dis{\frac{d}{dt}\int_{{\R}^n}}\frac{1}{2}|(\bar{u}+U)\cdot\nabla\P^{\alpha}(V_H)|^2dx+N_{1,1}+N_{1,2}+N_{1,3}+\{\cdots\}.
\end{array}$$ Here and in the subsequent of this section, we use $\{\cdots\}$ to denote the terms with derivatives of order at most $|{\alpha}|+1$. It is easy to see that $$|N_{1,1}+N_{1,2}+N_{1,3}+\{\cdots\} | \leq C(E_0+{\mathcal
M}^3)(1+t)^{-\frac{n}{2}-(|{\alpha}|+3)}.$$ Then for $N_2$, we have $$\begin{array}{rl}
N_2=&\dis{\int_{{\R}^n}}\P^{\alpha}(V_H)_t ~\tilde{\chi}\P^{\alpha}\triangle
({\mathcal
P}(\bar{\p}, V)V^2)dx\vspace{2.5mm}\\
=&\dis{\int_{{\R}^n}}\P^{\alpha}(V_H)_t ~\tilde{\chi}{\mathcal
P}^\prime_V~\P^{\alpha}\triangle V~V^2 + 2 \P^{\alpha}(V_H)_t
~\tilde{\chi}{\mathcal
P}(\bar{\p}, V)~V~\P^{\alpha}\triangle V dx+ \{\cdots\}\vspace{2.5mm}\\
=:& N_{2,1} + N_{2,2} + \{\cdots\}.
\end{array}$$ By noticing that $$\begin{array}{rl}
N_{2,1}=&\dis{\int_{{\R}^n}}\P^{\alpha}(V_H)_t ~\tilde{\chi}{\mathcal P}^\prime_V~\P^{\alpha}\triangle V~V^2 dx\vspace{2.5mm}\\
=&\dis{\int_{{\R}^n}}\P^{\alpha}(V_H)_t ~(V^2{\mathcal P}^\prime_V)~\P^{\alpha}\triangle V_H dx + \dis{\int_{{\R}^n}}\P^{\alpha}(V_H)_t
~[\tilde{\chi},{\mathcal P}^\prime_V V^2]~\P^{\alpha}\triangle V_L dx,\vspace{2.5mm}\\
=& -\frac{d}{dt}\dis{\int_{{\R}^n}}({\mathcal P}^\prime_V
V^2)~|\nabla\P^{\alpha}V_H|^2 dx+ O_1,
\end{array}$$ with $$O_1\leq \eta \int_{\R^n}\P^\alpha (V_H)|^2dx+C_\eta (E_0+{\mathcal
M}^3)(1+t)^{-\frac{n}{2}-(|{\alpha}|+3)}.$$ Similarly, we have $$N_{2,2}=-2\frac{d}{dt}\dis{\int_{{\R}^n}}({\mathcal P}V)~|\nabla\P^{\alpha}V_H|^2 dx+ O_2,$$ with $$O_2\leq \eta \int_{\R^n}|\P^\alpha (V_H)|^2dx+C_\eta (E_0+{\mathcal
M}^3)(1+t)^{-\frac{n}{2}-(|{\alpha}|+3)}.$$
We still need to consider the terms from the expansion of $\P^{\alpha}\mathrm{div}[( \bar{\p}^{\pm}+V^{\pm} )\nabla\varphi], \ 0<|{\alpha}|\leq
k-1$:
- If all the derivatives $\P^{\alpha}\mathrm{div}$ are taken on $ \bar{\p}^{\pm}+V^{\pm}$, it can be bounded by the a priori assumption and the fact that $\nabla\varphi \in L^{\infty}$.
- If all the derivatives $\P^{\alpha}\mathrm{div}$ are taken on $\nabla\varphi $ then $$\P^{\alpha}\mathrm{div} \nabla\varphi = \P^{\alpha}K,$$ and recall the good decay properties of $\P^{\alpha}K$ for $|{\alpha}|\leq k-1$ in Section 3, then we have better decay on these terms.
- Other terms can be estimated similarly.
Again to close the energy estimate, we now use the fact that $$\dis{\int_{{\R}^n}} |\nabla_x \P^{\alpha}V_H |^2dx\geq
\epsilon\dis{\int_{{\R}^n}} |\P^{\alpha}V_H |^2dx.$$ By integrating (\[p3\]) and (\[p4\]) over $[0, t]$ and multiplying (\[p3\]) by some suitably chosen constant $0<\lambda<1$, the combination of above estimates give (\[p13\]). Therefore, we have the following estimates on the high frequency component.
Under the assumption of Theorem 1.1, we have, for $|{\alpha}|\leq k-1$, $$\|\P^{\alpha}V^{\pm}_H \|_{H^1} + \|\P^{\alpha}(V^{\pm}_H)_t \|_{L^2} \leq
C(E_0+{\mathcal M}^{3/2})
(1+t)^{-\frac{n}{4}-\frac{|{\alpha}|+3}{2}},$$ where $E_0$ and ${\mathcal M}$ are defined in Theorem 4.2 and (\[l19\]), respectively.
Proof of Theorem 1.1.
=====================
In the previous two sections, we obtain the following estimates on the low frequency component by using the approximate Green function and the high frequency component by using the energy method respectively, $$\label{p6.1}
\|\P^{\alpha}V_L^{\pm} (t) \|_{L^p}\leq C(E_0+{\mathcal M}^2)
(1+t)^{-\frac{n}{2}(1-\frac{1}{p})-\frac{|{\alpha}|+1}{2}},~~|{\alpha}|\leq k,$$ and $$\label{p6.2}
\|\P^{\alpha}V_H^{\pm} \|_{H^1} + \|\P^{\alpha}(V_H^{\pm})_t \|_{L^2}\leq
C(E_0+{\mathcal
M}^{3/2})(1+t)^{-\frac{n}{4}-\frac{|{\alpha}|+3}{2}},~~|{\alpha}|\leq k-1.$$ It remains to combine (\[p6.1\]) and (\[p6.2\]) to close the a priori assumption (\[l19\]).
Firstly, by taking $p=2$ in (\[p6.1\]) and combining with (\[p6.2\]), we have $$\|\P^{\alpha}V^{\pm}(t) \|_{L^2}\leq C(E_0+{\mathcal
M}^{3/2})(1+t)^{-\frac{n}{4}-\frac{|{\alpha}|+1}{2}},~~|{\alpha}|\leq k.$$ Next, by using the Sobolev embedding theorem, from (\[p6.2\]), we have, for $|{\alpha}|\leq k-2$, $$\begin{array}{rl}
\|\P^{\alpha}V_H^{\pm}(t) \|_{L^\infty} \leq & \|\P^{\alpha}V_H^{\pm}(t) \|_{H^2}
\leq \|\P^{\alpha}V_H^{\pm}(t) \|_{L^2} + \|\nabla\P^{\alpha}V_H^{\pm}(t) \|_{H^1}\vspace{2.5mm}\\
\leq &C(E_0+{\mathcal M}^{3/2})(1+t)^{-\frac{n}{4}-\frac{|{\alpha}|+3}{2}}.
\end{array}$$ Moreover, for $n=3$, it holds that $-\frac{n}{4}-\frac{|{\alpha}|+3}{2}\leq-\frac{n}{2}-\frac{|{\alpha}|+1}{2}$. Thus, $$\label{p6.5}
\|\P^{\alpha}V_H^{\pm}(t)\|_{L^\infty} \leq C(E_0+{\mathcal
M}^{3/2})(1+t)^{-\frac{n}{2}-\frac{|{\alpha}|+1}{2}},~~|{\alpha}|\leq k-2.$$ Then, the interpolation of (\[p6.2\]) and (\[p6.5\]) leads to $$\label{p6.6}
\|\P^{\alpha}V_H^{\pm}(t) \|_{L^p}\leq C(E_0+{\mathcal
M}^{3/2})(1+t)^{-\frac{n}{2}(1-\frac{1}{p})-\frac{|{\alpha}|+1}{2}},~~|{\alpha}|\leq
k-2.$$ Combining (\[p6.1\]) with (\[p6.6\]) then gives $$\|\P^{\alpha}V^{\pm}(t) \|_{L^p}\leq C(E_0+{\mathcal
M}^{3/2})(1+t)^{-\frac{n}{2}(1-\frac{1}{p})-\frac{|{\alpha}|+1}{2}},~~|{\alpha}|\leq
k-2.$$
Now, we turn to estimate $U^{\pm}$ by using the equation (\[l16\]). Note that $$U^{\pm}(x,t)=e^{- t}U^{\pm}(x,0)+\int^t_0e^{-(t-s)}
((\bar{\p}^{\pm})^{-1}\nabla (a^{\pm} V^{\pm})+\bar{H}^{\pm} \pm
\nabla\varphi )(x,s)ds,$$ and it is easy to check that, for $|\r|\leq
k-2$, $$\left\{
\begin{array}{lcr} \|\P^\r \bar{H}^{\pm}(s) \|_{L^\infty}\leq
C(E_0+{\mathcal M}^2)
(1+s)^{-(n+1+\frac{|\r|+1}{2}) },\vspace{2.5mm}\\
\|\P^\r ((\bar{\p}^{\pm})^{-1}\nabla_x(a^{\pm}V^{\pm})(s) \|_{L^\infty}\leq
C(E_\rho+{\mathcal M}_V)
(1+s)^{-\frac{n}{2} -\frac{|\r|+2}{2}}, \vspace{2.5mm}\\
\| \P^\r \nabla\varphi \| _{L^\infty}\leq
\| \P^\r K \| _{H^{1}} \leq C(E_{\rho}+ \mathcal M^{2})
(1+t)^{-{5\over 4} n -2-{|\gamma| \over 2}},
\end{array}
\right.$$ thus, $$\label{A4}
\|\P^\r U^{\pm}(t) \|_{L^\infty} \leq C(E_\rho+{\mathcal M}_V+{\mathcal
M}^2)(1+t)^{-\frac{n}{2}-\frac{|\r|+2}{2}},~~|\r|\leq
k-2.$$
Next, for the $L^2$-norm, we use energy estimate. Multiply (\[l16\]) by $\bar \rho^{\pm} U^{\pm}$ and integrate, we have $$\label{A1}
\int_{{\R}^n} \bar \rho^{\pm} U^{\pm} U^{\pm}_{t}dx
+ \int_{{\R}^n} \nabla(a^{\pm}V^{\pm}) \cdot U^{\pm}dx
+ \int_{{\R}^n}\bar \rho^{\pm} (U^{\pm})^{2}dx
= \int_{{\R}^n} \bar H^{\pm} \bar \rho^{\pm} U^{\pm}dx \pm
\int_{{\R}^n} \bar \rho^{\pm} \nabla \phi \cdot U^{\pm}dx.$$ Note $$\int_{{\R}^n} \bar \rho^{\pm} U^{\pm} U^{\pm}_{t} dx= {d\over dt}
\int_{{\R}^n} \bar \rho^{\pm} (U^{\pm})^{2}dx
- \int_{{\R}^n} \P_{t} \bar \rho^{\pm} (U^{\pm})^{2}dx,$$ in which the second term is small, and $$\label{A2}
| \int_{{\R}^n} \nabla(a^{\pm}V^{\pm}) \cdot U^{\pm}dx|
\leq \varepsilon \int_{{\R}^n} (U^{\pm})^{2} + C(\varepsilon)
\mathcal M^{2} (1+t)^{-{n\over 2} -(1+1)} ,$$ in which the second term has the decay rate of $\|\nabla V^{\pm}\|
^{2}_{L^{2}}$, (which is the key observation that $U^{\pm}$ has better decay than $V^{\pm}$).
The first term on the right hand side of (\[A1\]), the nonlinear term, can be estimated as in $K$, which has better decay than (\[A2\]). For the last term with $\nabla \varphi$, we have $$\begin{aligned}
|\int_{{\R}^n} \bar \rho^{\pm} \nabla \phi \cdot U^{\pm}dx| \leq \|
\bar \rho^{\pm} \|_{L^{3}} \| \nabla \phi \|_{L^{6}} \| U^{\pm}
\|_{L^{2}} &\leq& \| \bar \rho^{\pm} \|_{L^{2}}^{1/2} \| \bar
\rho^{\pm} \|_{L^{6}}^{1/2}
\| E\|_{L^{2}} \| U^{\pm} \|_{L^{2}}\\
&\leq& \varepsilon \| U^{\pm} \|_{L^{2}}^{2} + C(\varepsilon)
\mathcal M^{2} (1+t)^{-{5\over 4}n -2-{1\over 4}}.\end{aligned}$$
Perform same estimates for $\P^{\alpha}(|\alpha|\leq k)$, and use the similar argument as for $V^{\pm}$, we can get $$\label{A3}
\| \P^{\alpha} U^{\pm} \|_{L^{2}} \leq \mathcal M
(1+t)^{-{n\over 4 } - {|\alpha|+2 \over 2}}, \hspace{3mm} |\alpha|\leq k.$$
Interpolate (\[A4\]) and (\[A3\]) to have $$ \| \P^{\alpha} U^{\pm} \|_{L^{p}} \leq \mathcal M
(1+t)^{-{n\over 2 }(1-{1\over p}) - {|\alpha|+2 \over 2}}, \hspace{3mm} |\alpha|\leq k-2.$$Then combine the above estimates to get
Under the assumption of Theorem 1.1, if the initial data $(V^{\pm}_0, U^{\pm}_0)$ satisfies that $$\begin{array}{rl}
|\p_+-\p_- |+ \|\nu^{\pm}(\cdot, 0) \|_{L^2\cap
L^1}+ \|\nu^{\pm}_t(\cdot, 0) \|_{L^2} + \|V^{\pm}_0 \|_{H^k\cup L^1} + \|V^{\pm}_t(0)
\|_{H^{k-1}} +
\|U^{\pm}_0 \|_{H^k}\leq\epsilon_0,
\end{array}$$ where $\epsilon_0>0$ is a small constant, then there exists a unique global classical solution $(V^{\pm}, U^{\pm})\in
C([0,\infty),H^{k})\cap C^1((0,\infty), H^{k-1})$ to . Moreover, we have $$\left\{ \begin{array}{rl}
\|\P^\r_xV^{\pm} \|_{L^p} \leq &C(1+t)^{-\frac{n}{2}(1-\frac{1}{p})-\frac{|\r|+1}{2}}, ~~ |\r|\leq k-2, \vspace{2.5mm}\\
\|\P^\r_xV^{\pm} \|_{L^2} \leq
&C(1+t)^{-\frac{n}{4}-\frac{|\r|+1}{2}}, ~~ |\r|= k-1,~k,\vspace{2.5mm}\\
\|\P^\r_xU^{\pm} \|_{L^p} \leq
&C(1+t)^{-\frac{n}{2}(1-\frac{1}{p})-\frac{|\r|+2}{2}}, ~~|r|\leq k-2, \vspace{2.5mm}\\
\|\P^\r_xU^{\pm} \|_{L^2} \leq
&C(1+t)^{-\frac{n}{4}-\frac{k+1}{2}}, ~~ |\r|= k-1,~k.
\end{array} \right.$$
This theorem implies (\[l19\]) then closed the a priori assumption, and then it yields the main results (i) and (ii) in Theorem 1.1.
Before concluding this paper, we point out that even though the above discussion is for the space dimension $n=3$, other higher dimensional cases can be considered similarly.
[**Acknowledgements:**]{} The research of Li is partially supported by the National Science Foundation of China (Grant No. 11171223), the Ph.D. Program Foundation of Ministry of Education of China (Grant No. 20133127110007) and the Innovation Program of Shanghai Municipal Education Commission (Grant No. 13ZZ109). The research of Liao is partially supported by National Natural Science Foundation of China (Nos.11171211, 11301182), Science and Technology commission of Shanghai Municipality (No. 13ZR1453400).
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[^1]: Corresponding author. [*E-mail address: [email protected]*]{}
|
---
address: |
Petersburg Nuclear Physics Institute,\
National Research Center “Kurchatov Institute”,\
Gatchina, 188300, Russia\
E-mail: [email protected]
author:
- 'Y. I. AZIMOV'
title: |
Vladimir Naumovich Gribov:\
Pieces of biography[^1]
---
This talk I would like to begin with some personal recollections. I was lucky to work with V.N. Gribov for 20 years. It was not [*under*]{} Gribov, but just [*with*]{}. We have several common papers, several of my papers were made according to Gribov’s suggestions. But most of my work were formally independent of Gribov. Nevertheless, all my work of those 20 years, in more or less detail, went through discussions with Gribov.
Of course, physics was not the only interest of Gribov, but, undoubtedly, it was his main interest producing very strong emotions. One could discuss with Gribov various physics problems, even rather far from his own studies (a couple of most bright examples will be mentioned below). But those discussions were not easy. Gribov was hearing very attentively and was ready to immediately jump into battle if something seemed incorrect to him. He had a rare feature: his arguments were very hard to reject even when his position appeared unjust after all (I think nobody can be just in all cases). The same situation was both in private discussions and at seminars. That is why talks at Gribov’s seminar were very difficult for speakers. As a result, some people feared to speak at his seminar. However, if a person were sufficiently brave, the talk appeared very useful for the speaker. After such trial the author himself began to better understand his own work and results.
Gribov’s name and his papers (at least some of them) are well known today. But his biography is less known. Here I present its main lines.
Vladimir (Russian nickname Volodya) Gribov was born on March 25, 1930 in Leningrad (St.Petersburg, before 1914 and now). His farther died in 1938, at the time of the Great Terror in the Soviet Union. “Luckily”, his death was the result of a decease, and not because of repression (in which case the whole family could be repressed as well). But the situation was not simple: the mother stayed alone with two small children, Volodya and his younger sister.
Gribov’s mother worked in one of Leningrad theaters (not as an actress). When Germany attacked the Soviet Union in 1941, the family was evacuated from Leningrad together with the theater. They were moving with the theater over Siberia, Far East, Urals. Nevertheless, Volodya was continuing his school education, started in 1937, without any delay. Only in summer 1945 the family was able to return to Leningrad (to enter the city after the blockade, every person needed to have a special permit). In 1947 Volodya finished here the school course. The natural question arose: what to do after that?
Volodya was growing up in theatrical environment, and he dreamed to become an actor, best of all, a cinema actor. However, when being in senior school classes, he had a possibility to expose himself to filming. As appeared, under camera he became “frozen” and lost his natural mobility. After that, one of professional actors advised him to choose some other speciality. At school, Volodya was quite successful in physics and mathematics. He preferred the former.
In 1947 Gribov enrolled in Physical Faculty of the Leningrad University. He was a student together with D.V. Volkov (later, the full member of the Ukrainian Academy of Sciences) and G.M. Eliashberg (now the full member of the Russian Academy of Sciences, the chief researcher of the Landau Institute for Theoretical Physics). In 1950 Volkov was included into the special group to study nuclear physics in more detail, and in 1951 was transferred to the Kharkov University. Gribov also wished to be in that group, but was rejected.
In 1952 Gribov presented his diploma thesis considering interaction of two electrons in Quantum Electrodynamics (under supervision of Yu.V. Novozhilov). It was evaluated very high. According to decision of the examination commission, Bulletin of the Leningrad University published the paper [@2e] based on this diploma work. It was the first publication signed by V.N. Gribov. Thus, in summer 1952 he was graduated from the University with [*diploma cum laude*]{}.
The year 1952 in the Soviet Union was not favorable for persons of jewish origin, one of which was Vladimir Naumovich Gribov. It was the last year for the trial of the Jewish Antifashist Committee (after its end, more than 100 persons were sentenced, more than 20 of them were shot). It was also the preparatory year for the “case of physicians” (“killers in white coats”) which was officially opened in January 1953. In such situation the only job available for Gribov, a young physicist with [*diploma cum laude*]{}, appeared the position of a physics teacher in an evening school, organized for working people who could not, by any reason, complete their school education in the childhood. The salary was low, and year later Gribov found additional part-time work.
Nevertheless, Volodya wished and was continuing to do science. He was able to contact with Professor L.E. Gurevich. To the beginning of 1954, they together prepared two papers, on properties of matter in external fields, electric and gravitation. The papers were submitted to Journal of Experimental and Theoretical Physics and accepted for publication [@el; @gr]. In addition, Gribov participated in the theoretical seminar of the Leninigrad Physico-Technical Institute (now the A.F. Ioffe Physico-Technical Institute, further PTI) headed by I.M. Shmushkevich and K.A. Ter-Martirosyan.
At last, in May 1954 (after Stalin’s death and extinction of the “case of physicians”), Gribov was able to be employed by PTI as the senior laboratory assistant in the group for nuclear theory. It was headed by I.M. Shmushkevich, who was simultaneously an informal head of the Theoretical Department as a whole. Now Gribov’s progress was very fast. A year later he received the higher position of the junior scientist. In March 1956 he presented his dissertation for the degree of Candidate of Sciences (analog of PhD). It was concerned with interactions of neutrons with nuclei [@dif; @rot]. The problem was suggested by K.A. Ter-Martirosyan, but at the presentation of the work he emphasized that methods for calculations were invented by Gribov himself.
After the defence of the dissertation, Shmushkevich and Ter-Martirosyan organized contacts of Gribov with L.D. Landau and I.Ya. Pomeranchuk. Gribov began to go regularly to Moscow for participating in Landau’s seminar. Initially, Landau was sceptical in respect to Gribov (he said: “I know one Gribov, the theatrical actor, and this is enough”). But he rapidly changed his mind. In 1958, when the Scientific Council of PTI discussed the next higher position for Gribov, the senior scientist, it received the very favorable recommendation from Landau. Later, Gribov always considered Landau as his main Teacher in theoretical physics.
In 1957, Shmushkevich invited Gribov to give lectures for students of the Leningrad Polytechnical Institute, where he was the Chair of the Theoretical Physics Department. Later, Gribov began to lecture in his [*alma mater*]{}, in the Leningrad University, and in 1968 he became the Professor of the University (the highest scientific degree, Doctor of Sciences, necessary for this, Gribov received in 1964). In parallel, Gribov presented lectures at various Physics Schools, both in the Soviet Union and (later) abroad.
Gribov’s scientific work after receiving the Candidate degree became also very active and self-reliant. More and more often he appeared a source of ideas for his colleagues. For instance, in 1958 he investigated three-pion decays of the K-meson. Pair-energy distributions in the decay were shown to depend on pion-pion scattering length [@np-tau; @jetp-tau]. For several years after that, the series of papers were published by members of Shmushkevich’s group (including Gribov himself) on various inelastic near-threshold reactions. They could allow to obtain information on hadron interactions ([*e.g.*]{}, pion-pion ones) unreachable in conventional ways. Regretfully, corresponding experimental attempts, as appeared, could not give definite results at those times because of insufficient quality of experimental technique.
One of his further directions of interest (probably, under influence of Pomeranchuk) was high-energy behavior of strong interactions. It was generally assumed for long time to be similar to classical diffraction of light on black screen. Gribov showed that such behavior would be inconsistent with analytic properties of strong-interaction amplitudes [@np-dif]. This work of 1961 initiated international interest to his activity. Trying to overcome this diffraction difficulty, Gribov, partly in collaboration with Pomeranchuk, developed reggeology, method of Regge poles and, later, cuts as well. This direction was also actively supported by his colleagues in PTI. He became one of leaders of reggeology not only in the Soviet Union, but in the whole world as well.
At the same time, Gribov tried to use the known quantum field theories as test-grounds for studying their high-energy properties. In Quantum Electrodynamics, as he showed, a special role play the so-called “double-logarithmic terms”. That is why the group of enthusiasts of this approach (G.V. Frolov, V.G. Gorshkov, and L.N. Lipatov first of all) was informally called “the double-logarithmic academy”. Their attempts were crowned by papers of Gribov and Lipatov [@gl1; @gl2] on summation of those “double-logarithmic terms”, very famous now (they became a basis for studying the evolution of partons).
Gribov could efficiently discuss even such problems in which he was not previously active. For instance, he did not work himself with weak interactions and, in particular, with neutrinos, but when encountered with Pontecorvo’s idea of neutrino oscillations Gribov immediately began to construct the corresponding formalism. This resulted in the joint paper of V.N. Gribov and B.M. Pontecorvo [@gr-pont], considered now to be classic in the neutrino physics.
There was one more, less known example. In his talk at one of Gribov’s seminars, Ya.B. Zeldovich explained that a charged rotating black hole should lose energy by radiation, so its rotation should slow down. When it stops, radiation, according Zeldovich, would stop as well. Here Gribov interrupted him by statement that this is not correct, radiation would continue. Zeldovich waved away this statement, and the seminar talk was continued without discussion of Gribov’s suggestion (it seems, however, that their discussions on this problem went on later, but Zeldovich stayed rigid). At that seminar, I restored for myself the lines of Gribov’s thought as follows. Gribov, as a physicist, had grown up mainly on quantum physics (in difference with Zeldovich), and he was aware quite well about the Schwinger effect: strong enough electric field, even homogeneous and static ([*i.e.*]{}, large gradient of electromagnetic potential), generates electron-positron pairs due to quantum tunneling. Near a black-hole horizon, there is very large gradient of the gravitational potential, which should analogously produce particle-antiparticle pairs.
Some time later, Zeldovich again talked at Gribov’s seminar, now about the famous paper of Hawking. In particular, Zeldovich said: “Volodya Gribov had tried to assure me that radiation of a black hole should continue, but I did not believe”. That is how he lost the interesting and important result.
Gribov ever tried to organize his whole knowledge into some consistent picture. As a rule, such approach is very useful. But not always. It may be not quite adequate if the picture needs strong changes. For Gribov, such was the case of quarks. The idea of quarks was publicized by M. Gell-Mann and G. Zweig in 1964 on the base of resonance spectroscopy. It was further supported by the discovery of scaling in deep-inelastic scattering (in 1968) and, especially, by the unexpected discovery of $J/\psi$ (in 1974). However, Gribov did not believe yet in quarks and, correspondingly, was sceptical in respect to Quantum Chromodynamics (QCD), which arose in 1971–1972. It was not a question of taste; for his position Gribov had definite rational arguments. At that time, I asked him once on reasons of his disbelieving in quarks. He answered: “The quarks should be strong-interacting objects; therefore, each of them should be surrounded by a pion cloud. Where is it?” (I should confess, that a clear consistent answer to this question is still absent. Note also that such negative relation to quarks did not prevent Gribov from suggesting to E.M. Levin and L.L. Frankfurt in early days of quarks to compare meson and baryon cross sections in terms of quarks [@lf].)
However, in 1976, after discovery of charmed particles, predicted by the quark picture, Gribov changed his mind. He began to study quarks and QCD very intensively. And in 1977 he was able to find a new feature of QCD, known now as the Gribov horizon or Gribov copies [@gr-YM1; @gr-YM2]. It is interesting to note that many–many people worked with QCD to that moment, but existence of the horizon stayed unnoticed. Initially, Gribov hoped that it is just the horizon which determines the origin of confinement of quarks and gluons. But soon he came to conclusion of its insufficiency. Since then he worked hard trying various ways to understand and describe a still unknown mechanism of confinement.
The administrative career of Gribov may look successful. To 1962 the Gatchina site of PTI had the working nuclear reactor and the proton accelerator under construction. There appeared necessity to have there a separate Theoretical Department. Gribov was suggested to organize it. Later, in 1969, after the death of I.M. Shmushktvich, Gribov was returned to the central part of PTI and became the head of its Theoretical Department. In 1971 the Gatchina site of PTI was transformed to be a new institute, Leningrad (now Petersburg) Nuclear Physics Institute (LNPI, now PNPI). All the activity on atomic nuclei and elementary particles was transferred from PTI to the new Institute. Its Theoretical Department was headed, of course, by Gribov. He was a rather good administrator, but, in my opinion, he disliked administrative duties which interfered with his scientific work. Meanwhile, those duties increased along with the Department becoming more populous. In 1980 Gribov moved from LNPI to L.D. Landau Institute of Theoretical Physics in Chernogolovka (near Moscow). There were several reasons for this step, and one of them, as I think, was his wish to diminish necessary administrative duties.
As a physicist, V.N. Gribov was recognized world-wide. He was a speaker at various conferences and schools in the Soviet Union and, sometimes, even abroad. Many foreign physicists, being in the Soviet Union, were eager to visit LNPI for discussions with Gribov. For his scientific work Gribov won various prizes, but he was most strongly proud to be the first recipient of the L.D. Landau Prize established in 1971 by the Academy of Sciences of the USSR, which honored his Teacher. In the same year, 1971, he became a member of the American Academy of Arts and Sciences. In the next year, 1972, he was elected (after several unsuccessful attempts) to be a corresponding member of the Academy of Sciences of the USSR. However, official position in respect to him was demonstrated by the fact that Gribov has never been elected to be a full member of the Soviet Academy.
After 1980, Gribov obtained the possibility to be part-time in Budapest, where his second wife, Julia Nyiri, worked in Physics Institute. This made somewhat easier his contacts with West physicists. And after 1990, longer abroad trips became possible, to be a visiting Professor of various Institutes and Universities, both in Europe and in the US.
Active and intensive work of Gribov was unexpectedly interrupted by the acute stroke in 1997 during one of scientific conferences. He was taken to a hospital and, after stabilization of his state, was transported to another hospital, in Budapest. Even there he tried to continue investigations of confinement. The medical treatment looked successful, and the physicians planned that Gribov would be able soon to go home. However, on August 13, 1997 he passed away. His grave in Budapest is marked by the memorial that reminds a beautiful fading flower, with the simple epitaph
[VLADIMIR GRIBOV\
FIZIKUS$~$\
1930–1997]{}
on the basement.
After Gribov’s death, his works have not been forgotten. Just opposite, many of his papers are republished. His lectures. which were mainly written up in Russian, but not always published, are now collected, translated into English and published as books. Therefore, his results become available and well-known even for younger generations of physicists. If judging by references, the most famous and operative of those results seem to be the Gribov copies (they are especially essential now in lattice calculations) and DGLAP equations for evolution of partons (here G stays just for Gribov). There established are various stipends and prizes called by Gribov’s name.
And yet there are Gribov’s last papers or even notes. They are mainly not completed or, at least, not quite understood by the world community. Meanwhile, they tried various ways to solve the problem of confinement, one of the hottest problems in strong interactions. In some sense the situation may be similar to the fate of Einstein’s last ideas. During his life, they looked to be out of mainstream, but now many of them feed new theoretical approaches. Such future is not excluded also for Gribov’s last ideas. Look forward...
Acknowledgments {#acknowledgments .unnumbered}
===============
This work is supported by the Russian Science Foundation (Grant No.14-22-00281).
[0]{} V. N. Gribov, “On the interaction of two electrons”, [*Vest. Leningrad. Gos. Univ.*]{} [**3**]{}, 10 (1953) \[in Russian\]. L. E. Gurevich, V. N. Gribov, “Dielectric losses in ionic dielectrics in strong electric fields”, [*Zh. Eksp. Teor. Fiz.*]{} [**29**]{}, 629 (1955) \[[*Sov. Phys. JETP*]{} [**2**]{}, 565 (1956)\]. V. N. Gribov, L. E. Gurevich, “On the theory of the stability of a layer located at the superadiabatic temperature gradient in the gravitational field”, [*Zh. Eksp. Teor. Fiz.*]{} [**31**]{}, 854 (1956) \[[*Sov. Phys. JETP*]{} [**4**]{}, 720 (1957)\]. V. N. Gribov, “Effect of diffuseness of the nuclear boundary on neutron scattering”, [*Zh. Eksp. Teor. Fiz.*]{} [**32**]{}, 647 (1957) \[[*Sov. Phys. JETP*]{} [**5**]{}, 537 (1957)\]. V. N. Gribov, “Excitation of rotational states in the interaction berween neutrons and nuclei”, [*Zh. Eksp. Teor. Fiz.*]{} [**32**]{}, 842 (1957) \[[*Sov. Phys. JETP*]{} [**5**]{}, 688 (1957)\]. V. N. Gribov, “Angular distribution in reactions involving the formation of three low energy particles, with application to $\tau^+$ meson decay”, [*Nucl. Phys.*]{}, [**5**]{}, 653 (1958). V. N. Gribov, “Angular distribution in the reactions $K^+ \to 2\pi^+ +\pi^-$ and $K^+ \to 2\pi^0+\pi^+$”, [*Zh. Eksp. Teor. Fiz.*]{} [**34**]{}, 749 (1958) \[[*Sov. Phys. JETP*]{} [**7**]{}, 514 (1958)\]. V. N. Gribov, “Asymptotic behaviour of the scattering amplitude at high energies”, [*Nucl. Phys.*]{}, [**22**]{}, 249 (1961). V. N. Gribov and L. N. Lipatov, “Deep inelastic $ep$ scattering in perturbation theory”, [*Yad.Fiz.*]{} [**15**]{}, 781 (1972) \[[*Sov. J. Nucl. Phys.*]{} [**15**]{}, 438 (1972)\]. V. N. Gribov and L. N. Lipatov, “$e^+e^-$ pair annihilation and deep inelastiv $ep$ scattering in perturbation theory”, [*Yad.Fiz.*]{} [**15**]{}, 1218 (1972) \[[*Sov. J. Nucl. Phys.*]{} [**15**]{}, 675 (1972)\]. V. N. Gribov and B. Pontecorvo, “Neutrino astronomy and lepton charge”, [*Phys.Lett.*]{} [**493**]{} (1969). E. M. Levin and L. L. Frankfurt, “The quark hypothesis and relations between cross sections at high energies”, [*Pisma ZhETF*]{} [**2**]{}, 105 (1965) \[[*JETP Lett.*]{} [**2**]{}, 65 (1965)\]. V. N. Gribov, “Instability of nonabelian gauge theories and impossibility of choice of Coulomb gauge”, [*Proceedings of the 12th LNPI Winter School on Nuclear and Elementary Particle Physics*]{}, v.1, pp. 147-162, Leningrad 1977 (in Russian); for English translation, see SLAC-TRANS-0176. V. N. Gribov, “Quantization of nonabelian gauge theories”, [*Nucl.Phys.*]{} [**B139**]{}, 1 (1978).
[^1]: ased on the invited talk at the 4th .. ribov emorial orkshop “heoretical hysics of entury” (ribov-85), 17-20 une, 2015, hernogolovka, ussia. o appear in the roceedings.
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abstract: 'The aperture mass has been shown in a series of recent publications to be a useful quantitative tool for weak lensing studies, ranging from cosmic shear to the detection of a mass-selected sample of dark matter haloes. Quantitative analytical predictions for the aperture mass have been based on a number of simplifying assumptions. In this paper, we test the reliability of these assumptions and the quality of the analytic approximations, using ray-tracing simulations through a cosmological density field generated by very large N-body simulations. We find that those analytic predictions which take into account the non-linear evolution of the matter distribution, such as the dispersion of the aperture mass and the halo abundance, are surprisingly accurately reproduced with our numerical results, whereas the predictions for the skewness, based on quasi-linear theory, are rather imprecise. In particular, we verify numerically that the probability distribution of the aperture mass decreases exponentially for values much larger than the rms. Given the good overall agreement, comparisons between the observed distribution of the aperture mass and the theoretical values provide a powerful tool for testing cosmological models.'
author:
- Katrin Reblinsky
- Guido Kruse
- Bhuvnesh Jain
- Peter Schneider
date: 'Received ; accepted '
title: 'Cosmic shear and halo abundances: analytical versus numerical results'
---
=3.33pt plus 5.4pt minus 1.11pt
Introduction
============
The gravitational distortion of light bundles from distant sources in the universe provides a unique means to investigate (the statistical properties of) the intervening mass distribution. Being observable through the image distortion of the distant faint blue galaxy population, this cosmic shear effect offers the opportunity to study statistical properties of the large-scale structure. In contrast to almost all other methods for investigating the large scale structure (LSS) – with CMB being the only exception – no assumption about the relation between dark and luminous mass is required.
Based on the assumption that the intrinsic orientation of the background galaxies is random, a net alignment in the observed galaxy images can be attributed to the tidal field (shear). Hence, the alignment pattern of the galaxy images directly reflects the properties of the mass distribution. For example, two-point statistical measures of the galaxy ellipticities (understood here and in the following as two-component quantities, with an amplitude and an orientation) can be expressed directly in terms of the power spectrum of the mass distribution, convolved with a filter function (Blandford et al. 1991; Miralda-Escudé 1991; Kaiser 1992; Jain & Seljak 1997; Bernardeau et al. 1997; Schneider et al. 1998, hereafter SvWJK; Kaiser 1998; van Waerbeke et al. 1999; Jain et al. 1999, hereafter JSW; and references therein). The power spectrum completely characterises a Gaussian random density field, and so the two-point statistics, like the two-point correlation function of galaxy ellipticities or the rms shear in an aperture, suffices to extract the statistical information contained in the distorted galaxy images. Whereas the earlier of the aforementioned papers concentrated mainly on predictions for the cosmic shear based on the linear evolution of the cosmic density field, it was pointed out by Jain & Seljak (1997) that even on scales as large as one degree, the non-linear evolution significantly affects the expected amplitude of the cosmic shear.
The non-linear evolution transforms an initially Gaussian field into a non-Gaussian one, and thus the cosmic shear on small angular scales is expected to display significant non-Gaussian features. As pointed out by Bernardeau et al. (1997) and SvWJK, the skewness of the resulting cosmic shear field is a sensitive measure of the density factor $\Omega_0$, since in quasi-linear perturbation theory the skewness is independent of the normalisation of the initial power spectrum. In order to define the skewness, mass reconstruction algorithms such as those developed for cluster reconstructions (Kaiser & Squires 1993; Seitz & Schneider 1996, Seitz et al. 1998a; Lombardi & Bertin 1998a,b; and references therein) can be employed to reconstruct the projected density field. This density field, appropriately spatially filtered, can then be used to calculate the skewness. In contrast to the two-point statistical measures mentioned above, which are defined directly in terms of the observable image ellipticities, this measurement of skewness is more indirect. This causes estimates of the statistical error from the data itself not to be straightforward. On the other hand, the aperture mass $M_{\rm ap}$, introduced as a measure for cosmic shear in SvWJK, is a scalar quantity directly defined in terms of the image ellipticities, and can thus be easily used for defining a skewness, as well as a second order statistics, the rms of $M_{\rm ap}(\theta)$, where $\theta$ is the angular scale of the circular aperture (definitions are given in Sect. 2 below). In particular, in contrast to the two-point ellipticity correlation function and the rms shear in an aperture, for which the filter with which the power spectrum of the projected density field is measured is broad, the corresponding filter function for the rms of $M_{\rm ap}(\theta)$ is very narrow, and can be approximated very accurately by a delta “function”, so that $M_{\rm
ap}(\theta)$ directly measures the power of the projected density at wavelength $\ell\approx 4.25/\theta$ (Bartelmann & Schneider 1999). The skewness defined in terms of $M_{\rm ap}$ has been considered in SvWJK.
As in the evolution of the three-dimensional density field, where highly non-linear structures like clusters of galaxies form, the projected mass density attains strongly non-Gaussian features, e.g., the projection of collapsed haloes. As a scalar quantity, the aperture mass is ideally suited to probe the full probability distribution of the projected mass density; in particular, values of $M_{\rm
ap}(\theta)$ far out in the non-Gaussian tail signal the presence of massive dark matter haloes. Therefore, peaks in the distribution of $M_{\rm ap}$ can be used to search for such haloes, independent of their luminous properties (Schneider 1996, hereafter S96). Indeed, a first application of $M_{\rm ap}$ to large scale structure simulations by Reblinsky & Bartelmann (1999) revealed that the detection of dark matter haloes through the aperture mass is more reliable in terms of completeness and spurious detections and suffers less from projection effects than optically selected cluster samples. Assuming that highly significant peaks are caused by such haloes, one can predict their abundance (i.e., number of peaks above a certain threshold per unit solid angle) by combining the spatial abundance as predicted by Press & Schechter (1974) theory with an assumed density profile, such as the universal dark matter profile found by Navarro et al. (1996, 1997; combined NFW). This idea has been put forward by Kruse & Schneider (1999a; hereafter KS1), who found that, depending on the cosmological model and the redshift distribution of background galaxies, of order 10 such haloes per square degree will be detectable in deep ground-based optical images, with a signal-to-noise ratio larger than 5. Given the rapid evolution of wide-field imaging, a mass-selected sample of dark matter haloes is now well within reach. Indeed, a first example of a shear-detected mass concentration has recently been found by Erben et al. (1999).
All of these predictions on cosmic shear are made using simplifying assumptions in order to make analytic progress. JSW tested some of these assumptions using ray-tracing simulation through a cosmic density field generated by very large N-body simulations; similar tests have been carried out by van Waerbeke et al. (1999). They found that the major approximations made in these analytic treatments, namely the so-called “Born-approximation” (which projects the density fields along “straight lines”), and the neglect of non-linear terms in the propagation equations, are very well satisfied. In particular, the twist of light bundles, which vanishes identically in the usual analytical treatments, is indeed very small. The predictions on the projected power spectrum using the approximation for the fully non-linear power spectrum of Peacock & Dodds (1996) agree very well with the numerical results, whereas the predictions concerning the skewness are less precise, indicating a breakdown of quasi-linear perturbation theory on small scales.
In this paper, we extend the study of JSW to the particular application of the aperture mass statistics. We use the same simulations as JSW, resulting in a table of shear and projected mass density as a function of angular position. From the shear, we can simulate observations of the aperture mass as a function of position on the “data” field, and investigate its statistical properties. In particular, we calculate the probability distribution of $M_{\rm
ap}(\theta)$, which we find to be highly non-Gaussian, and from that we study the dispersion, skewness and kurtosis of the distribution. In agreement with JSW, we find that the dispersion is accurately predicted by analytic theory, whereas the skewness predictions can differ substantially from the numerical results. Of particular interest is the kurtosis, since it enters the determination of the uncertainty of the dispersion measurement due to cosmic variance (SvWJK). We find that the kurtosis is a slowly decreasing function of angular scale $\theta$, and attains values of $\sim 3$ even on scales as large as $10'$; hence, the cosmic variance will be the major source of statistical error in the measurement of the power spectrum of the projected density field from $M_{\rm ap}$. The analytic predictions on the abundance of significant peaks of $M_{\rm ap}$, and thus presumably of dark matter haloes, turns out to be remarkably precise, given the strong assumptions made. We confirm the high number density of haloes detectable with this method. The shape of the probability distribution of $M_{\rm ap}$ in the highly non-Gaussian tail, predicted by Kruse & Schneider (1999b; hereafter KS2), can also be confirmed to be well approximated by an exponential function.
The aperture mass measure $M_{\rm ap}$ {#aperturemass}
======================================
In this section, we briefly summarise the properties of the aperture mass, i.e., its definition, its relation to the shear, and its signal-to-noise ratio. For more details, the reader is referred to S96 and SvWJK.
$M_{\rm ap}$ statistics
-----------------------
We define the spatially filtered mass inside a circular aperture of angular radius $\theta$ around the point $\mbox{\boldmath$\zeta$}$ by $$M_{\rm ap} (\mbox{\boldmath$\zeta$}):=\int \d^2 \vartheta
\ \kappa(\mbox{\boldmath$\vartheta$})
\ U(\vert \mbox{\boldmath$\vartheta-\zeta$} \vert),
\label{mapt}$$ where the continuous weight function $U(\vartheta)$ vanishes for $\vartheta>\theta$. If $U(\vartheta)$ is a compensated filter function, $$\int_0^{\theta} \d \vartheta \ \vartheta \ U(\vartheta)=0,$$ one can express $M_{\rm ap}$ in terms of the tangential shear $\gamma_{\rm t}(\mbox{\boldmath$\xi$}; \mbox{\boldmath$\zeta$})$ at position $\mbox{\boldmath$\xi + \zeta$}$ relative to $\mbox{\boldmath$\zeta$}$ as $$M_{\rm ap} (\mbox{\boldmath$\zeta$})=\int \d^{2} \xi \
\gamma_{\rm t} (\mbox{\boldmath$\xi$}; \mbox{\boldmath$ \zeta$})
\ Q(\vert \mbox{\boldmath$\xi$} \vert),
\label{mapshear}$$ (Fahlmann et al. 1994; S96), where $$\gamma_{\rm t}(\mbox{\boldmath$\xi$}; \mbox{\boldmath$\zeta$})
= -{\rm Re}(\gamma(\mbox{\boldmath$\xi+\zeta$}) \, {\rm e}^{-2{\rm i} \phi}),
\label{tanshear}$$ and $\phi$ is the polar angle of $ \mbox{\boldmath$\xi$}$. The function $Q$ is related to $U$ by $$Q(\vartheta) = \frac{2}{\vartheta^2} \ \int_0^{\vartheta}
\d \vartheta^{\prime} \ \vartheta^{\prime} \ U(\vartheta^
{\prime})
\ - U(\vartheta) .$$
We use the filter function for $l=1$ from the family given in SvWJK: writing $U(\vartheta)=u(\vartheta/\theta)/\vartheta^2$, and $Q(\vartheta)=q(\vartheta/\theta)/\vartheta^2$, we take $$\label{SvWJK98}
u(x)=\frac{9}{\pi} \left(1 - x^{2} \right)
\left( \frac{1}{3} - x^{2} \right),$$ and $$q(x)=\frac{6}{\pi}x^2(1-x^2),$$ with $u(x)=0=q(x)$ for $x>1$.
Signal-to-noise ratio {#sn_section}
---------------------
An estimate of the shear field $\gamma$, and thus of the aperture mass $M_{\rm ap}(\mbox{\boldmath$\vartheta$})$ through Eq. (\[mapshear\]), is provided by the distortions of images of faint background galaxies. The complex ellipticity of galaxy images is defined in terms of second moments of the surface-brightness tensor (e.g., Tyson et al. 1990; Kaiser & Squires 1993). Specifically, we use here the ellipticity parameter $\epsilon$ (Schneider 1995; Seitz & Schneider 1997), which is defined such that for sources with elliptical isophotes of axis ratio $r\le1$, the modulus of the source ellipticities is given as $|\epsilon^{({\rm s})}|=(1-r)/(1+r)$, and the phase of the $\epsilon^{({\rm s})}$ is twice the position angle of the major axis.
The complex image ellipticity $\epsilon$ can then be calculated in terms of the source ellipticity $\epsilon^{({\rm s})}$ and the reduced shear $g\equiv\gamma\,(1-\kappa)^{-1}$ by the transformation (Seitz & Schneider 1997) $$\epsilon=\frac{\epsilon^{({\rm s})} + g}{1+g^{*} \epsilon^{({\rm
s})}}\;.
\label{eps}$$ This relation is valid only for noncritical clusters. For critical clusters, it has to be replaced by a different transformation. However, as we are mainly interested in the weak lensing regime, the above relation is sufficient here.
It has been demonstrated (Schramm & Kayser 1995; Seitz & Schneider 1997) that the ellipticity $\epsilon$ of a galaxy image is an unbiased estimate of the local reduced shear, provided that the intrinsic orientations of the sources are random. In the case of weak lensing, $\kappa\ll1$, one then has $$\langle \epsilon \rangle =g \approx \gamma$$ by averaging (\[eps\]) with the probability distribution of the source ellipticities.
As for the tangential shear component $\gamma_{\rm t}$ occurring in (\[tanshear\]), a similar quantity for the image ellipticities can be defined. Consider a galaxy image $i$ at a position $\mbox{\boldmath$\vartheta$}_i+\mbox{\boldmath$\zeta$}$ relative to the point $\mbox{\boldmath$\zeta$}$ with a complex image ellipticity $\epsilon_i$. In analogy to (\[tanshear\]) the tangential ellipticity $\epsilon_{{\rm t}i}(\mbox{\boldmath$\vartheta_i$};
\mbox{\boldmath$\zeta$})$ of this galaxy is then given by $$\epsilon_{{\rm
t}i}(\mbox{\boldmath$\vartheta$}_i;\mbox{\boldmath$\zeta$})=-{\rm
Re} \left( \epsilon_i
(\mbox{\boldmath$\vartheta$}_i+\mbox{\boldmath$\zeta$})
\ {\rm e}^{-2 {\rm i} \phi_i} \right),
\label{epsi_t}$$ where $\phi_i$ is the polar angle of $\mbox{\boldmath$\vartheta$}_i$.
We can now estimate the integral (\[mapshear\]) by a discrete sum over galaxy images, $$M_{\rm
ap}(\mbox{\boldmath$\zeta$})=\frac{1}{n}\,\sum_{i}\epsilon_{{\rm
t}i}(\mbox{\boldmath$\vartheta$}_i;\mbox{\boldmath$\zeta$})\,
Q(|\mbox{\boldmath$\vartheta$}_i|),
\label{map_1}$$ where $n$ is the number density of galaxy images. The discrete dispersion $\sigma_{\rm d}$ of the aperture mass $M_{\rm ap}(\mbox{\boldmath$\zeta$})$ is found by squaring (\[map\_1\]) and taking the expectation value in the absence of lensing, which leads to $$\label{disp_d}
\sigma_{\rm
d}^{2} = \frac{\sigma_\epsilon^2}{2\,n^2}
\sum_{i}Q^2(|\mbox{\boldmath$\vartheta$}_i|),$$ where $\sigma_\epsilon^2=\langle \vert \epsilon^{({\rm s})} \vert^2 \rangle$. Performing an ensemble average of Eq. (\[disp\_d\]) leads to the continuous dispersion $\sigma_{\rm c}$ $$\label{disp_c}
\sigma_{\rm c}^{2} (\theta) = \frac{\pi \sigma_{\epsilon}^2}{n}
\int_{0}^{\theta} {\rm d} \vartheta \ \vartheta \ Q^{2}(\vartheta ).$$ Finally, the [*signal-to-noise*]{} ratio $S$ at position $\mbox{\boldmath$\zeta$}$ is $$S(\mbox{\boldmath$\zeta$}) \equiv \frac{M_{\rm
ap}(\mbox{\boldmath$\zeta$})}{\sigma_{\rm d}} =
\frac{\sqrt{2}}{\sigma_\epsilon} \
\frac{\sum_i\epsilon_{{\rm t}i}(\mbox{\boldmath$\vartheta$}_i
;\mbox{\boldmath$\zeta$}) \
Q(|\mbox{\boldmath$\vartheta$}_i|)}
{\left[\sum_iQ^2(|\mbox{\boldmath$\vartheta$}_i|)\right]^{1/2}}\;.
\label{eq_sn}$$
In the simulations, we draw the source ellipticities from a Gaussian probability distribution, $$\label{distrib}
p_{\rm s}(|\epsilon^{({\rm s})}|)=
\frac{1}{\pi\sigma_{\epsilon}^2
\left[1-\exp\left(-\sigma_\epsilon^{-2}\right)\right]}\,
\exp\left(-\frac{|\epsilon^{({\rm s})}|^2}{\sigma_\epsilon^2}\right)\;,$$ where the width of the distribution is chosen as $\sigma_\epsilon=0.2$. Throughout this paper, we assume that the number density of the background sources is $n=30\,{\rm arcmin}^{-2}$.
Analytical work done with $M_{\rm ap}$ {#map_anal}
--------------------------------------
The aperture mass has been considered in the framework of blank field surveys in a variety of earlier publications. Introduced as a convenient statistics for cosmic shear, SvWJK have calculated the rms of $M_{\rm ap}$ as a function of angular scale, using the Peacock & Dodds (1996) approximation for the non-linear evolution of the power spectrum of density fluctuations. Like other two-point statistics, the dispersion of $M_{\rm ap}$ is an integral over the power spectrum of the projected mass distribution, weighted by a filter function. The filter function corresponding to $\langle M_{\rm
ap}^2\rangle$ is very narrow and can be well approximated by a delta function (Bartelmann & Schneider 1999). Hence, $\langle M_{\rm
ap}^2(\theta)\rangle$ reproduces the shape of the projected power spectrum and, depending on the cosmological model and the redshift distribution of the sources, it reveals a broad peak at $\theta\sim
1'$. One convenient property of the aperture mass is that the correlation function of $M_{\rm ap}$ of two apertures spatially separated by $\Delta\theta$ quickly decreases and already achieves values of $10^{-2}$ for $\Delta\theta\sim 2\theta$. This means that measurements of $M_{\rm ap}$ from a large consecutive area can be considered independent if the apertures are densely laid out on this data field; this is in contrast to the rms shear in apertures which is strongly correlated, and thus must be obtained from widely separated regions on the sky.
Being a scalar quantity, $M_{\rm ap}$ can also be used for higher-order statistical measures of the cosmic shear. SvWJK calculated the skewness of $M_{\rm ap}$, using Eulerian perturbation theory for the evolution of the three-dimensional density contrast $\delta$. In agreement with Bernardeau et al. (1997) they found that the skewness is a sensitive function of the cosmic density factor $\Omega_0$, and is in this approximation independent of the normalisation of the power spectrum.
A measurement of the dispersion of $M_{\rm ap}$ is affected by two main sources of statistical error: the intrinsic ellipticity distribution of the source galaxies, and cosmic variance. To estimate the latter, one needs to know the kurtosis of $M_{\rm ap}$ which cannot easily be determined analytically.
Values of $M_{\rm ap}$ much larger than its rms probe the highly non-Gaussian regime of the projected density field. From its definition, one sees that large values of $M_{\rm ap}$ are expected if the aperture is centred on a density peak with a size comparable to the filter scale $\theta$. Therefore, the aperture map can be used to search for such density peaks, presumably collapsed dark matter haloes, in blank field imaging surveys. In this way it is possible to obtain a mass-selected sample of such haloes (S96). Simple analytical arguments in S96 suggest that dark matter haloes with an approximately isothermal profile are detectable with a signal-to-noise ratio larger than 5 if their velocity dispersion exceeds $\sim
600$ km/s, assuming a number density of background sources of $n\sim
30$ arcmin$^{-2}$. Indeed, this theoretical expectation was verified in the lensing investigation of the cluster MS1512+36 (Seitz et al.1998b). This cluster has a velocity dispersion of about $\sim 600$ km/s, as obtained from strong lensing modelling and from spectroscopy of cluster members, and is detected in the weak lensing analysis with very high statistical significance.
Assuming that the high signal-to-noise peaks of $M_{\rm ap}$ are due to collapsed dark matter haloes, one can attempt to estimate the abundance of such peaks using analytic theory. KS1 have calculated the number density of haloes with aperture mass larger than $M_{\rm ap}$, $N(> M_{\rm ap}, \theta)$, assuming (1) that dark matter haloes are distributed according to Press & Schechter (1974) theory which yields the number density of collapsed haloes as a function of halo mass and redshift, and (2) that the azimuthally-averaged projected density profiles of these haloes can be described by the projection of the universal halo density profile found in numerical simulations by NFW. Depending on the cosmological model and on the redshift distribution of the faint galaxies, the number density of peaks of $M_{\rm ap}$ with a signal-to-noise ratio larger than 5 was estimated to be $\ga
10$ per square degree, and the redshift distribution of these haloes is strongly dependent on the behaviour of the linear growth factor for density perturbations, and thus on $\Omega_0$. This abundance is encouraging, since it allows one to obtain samples of haloes selected by their mass properties alone (for a first example, see Erben et al.1999).
Using the same model, KS2 have calculated the probability distribution of $M_{\rm ap}$ for values of $M_{\rm ap}$ much larger than its rms, assuming that this non-Gaussian tail of the probability distribution is dominated by dark matter haloes. They found that the distribution is very well described by an exponential; i.e., the tail is much broader than for a Gaussian.
All these analytic predictions are based on a number of approximations and simplifying assumptions. In Sect. 4 below we shall compare these analytic results with those found in ray-tracing simulations through a cosmological mass distribution obtained from very large N-body calculations, as described in the next section.
Generation of shear maps with ray-tracing simulations {#ray_tracing}
=====================================================
Simulated shear maps due to weak lensing by large-scale structure are made by performing ray tracing simulations through the dark matter distribution produced by N-body simulations (JSW). The N-body simulations used are a set of adaptive particle-particle/particle-mesh (AP$^3$M) simulations. The long-range component of the gravitational force is computed by solving Poisson’s equation on a grid. The grid calculation is supplemented with a short range correction computed either by a direct sum over neighbouring particles, or, in highly clustered regions, by combining a calculation on a localised refinement mesh with a direct sum over a smaller number of much closer neighbours. The parameters used by the N-body simulations are given in Table \[cosmo\].
The simulations were run with a parallel adaptive AP$^3$M code (Couchman et al. 1995; Pearce & Couchman 1997) kindly made available by the Virgo Supercomputing Consortium (e.g. Jenkins et al. 1998). They followed $256^3$ particles using a force law with softening length $l_{\rm soft}\simeq 30\ h^{-1}$kpc at $z=0$ (the force is $\sim 1/2$ its $1/r^2$ value at one softening length and is almost exactly Newtonian beyond two softening lengths). $l_{\rm soft}$ was kept constant in physical coordinates over the redshift range of interest to us here. The simulations were carried out using 128 or 256 processors on CRAY T3D machines at the Edinburgh Parallel Computer Centre and at the Garching Computer Centre of the Max-Planck Society. These simulations have previously been used for studies of strong lensing by Bartelmann et al. (1998), for studies of dark matter clustering by Jenkins et al. (1998), and for studies of the relation between galaxy formation and galaxy clustering by Kauffmann et al. (1999a,b), and Diaferio et al. (1999).
The ray tracing simulations of weak lensing from which we use the convergence and shear maps were computed by JSW. They used a multiple lens-plane calculation that implements the discrete recursion relations for the position of a given photon and for the Jacobian matrix of the lens mapping at this position (Schneider & Weiss 1988; Schneider et al. 1992; see Seitz et al. 1994 for a thorough justification for this approach). Aside from the distance factors, the main input into the recursion relations is the shear matrix at each lens plane. The ray tracing algorithm consists of three parts: constructing the dark matter lens planes, computing the shear matrix on each plane, and using these to evolve the photon trajectory from the observer to the source. The details involved at each step are as follows:
1\. The dark matter distribution between source and observer is projected onto $20-30$ equally spaced (in comoving distance) lens planes. The particle positions on each plane are interpolated onto a grid of size $2048^2$. Since the three-dimensional mass distribution is taken from a single realisation of the evolution of the LSS, the projected mass distributions of consecutive lens planes are correlated. In order to decorrelate them, the projection is carried out along a randomly chosen one of the three coordinate axes; in addition, the origin of the coordinate system in each lens plane is translated by a random vector and the lens plane is rotated by a random angle. In this way, the projected mass distributions of consecutive lens planes are as independent as possible, given the restriction of only a single realisation of the 3-d matter distribution.
2\. On each plane, the shear matrix is computed on a grid by Fourier transforming the projected density and using its Fourier space relation to the shear. The inverse Fourier transform is then used to return to real space.
3\. The photons are started on a regular grid on the first lens plane. Perturbations along the line of sight distort this grid and are computed using the relation between deflection angle and projected density. Once we have the photon positions, we interpolate the shear matrix onto them and solve the recursion relations for the Jacobian of the mapping from the $n$-th lens plane to the first plane.
4\. Solving the recursion relations up to the source plane yields the Jacobian matrix at these positions. Note that the ray tracing is done backwards from the observer to the source, thus ensuring that all the photons reach the observer. The first lens plane is the image plane and has the unperturbed photon positions. All sources are assumed to be at a redshift of $z_{\rm s}=1$.
There are two kinds of resolution limitations in the ray-tracing simulations. The first reflects the finite size and resolution of our N-body simulations, the second the use of finite grids when computing deflection angles and shear tensors on the lens planes. At the peak redshift of the lensing contribution, both effects give a small scale resolution of order $0.2'$. However, since the lens efficiency is not very sharply peaked, effects at other redshifts also enter. Thus depending on the statistical measure being used, the small scale resolution lies in the range $\sim 0.2'-0.4'$.
On large scales the finite box-size of the N-body simulations sets the upper limit on the angular scales available. The angular size of our simulation box at $z=1$ is about 3$^\circ$. Thus on scales comparable to $1^\circ$, only a few sample regions are available, leading to large fluctuations across different realisations. We therefore restrict our considerations to apertures with radius $\theta\le 10'$ using one realisation for every cosmological model. For the $\tau$CDM model, we use ten different realisations of the ray tracing simulations (i.e., they differ in the direction of projections, the translation and rotation of the projected matter distribution in the individual lens planes) to estimate the cosmic variance.
[@rllll@]{}\
& & & &\
\[10pt\]\
$N_{\rm par}$ & $256^3$ & $256^3$ & $256^3$ & $256^3$\
$l_{\rm soft} [h^{-1}$ kpc\] & 36 & 36 & 30 & 30\
$\Gamma$ & 0.5 & 0.21 & 0.21 & 0.21\
$L_{\rm box} [h^{-1}$ Mpc\] & 85 & 85 & 141 & 141\
$\Omega_0$ & 1.0 & 1.0 & 0.3 & 0.3\
$\Lambda_0$ & 0.0 & 0.0 & 0.7 & 0.0\
$H_0$ \[km/s/Mpc\] & 50 & 50 & 70 & 70\
$\sigma_8$ & 0.6 & 0.6 & 0.9 & 0.85\
$m_{\rm p} 10^{10} h^{-1}{\rm M}_{\sun}$ & 1.0 & 1.0 & 1.4 & 1.4\
field size \[$^{\circ}$\] & 2.7&2.7&3.4&3.9\
\
Application of $M_{\rm ap}$ to simulated shear maps
===================================================
For each of the shear maps generated as described in the last section, we create a 2-dim. “$M_{\rm ap}$ map” by simulating “observations” of $M_{\rm ap}$ as a function of position on the 2-dim. shear maps. The probability distribution function of $M_{\rm ap}$ (PDF) and some of its moments are then calculated for every $M_{\rm ap}$ map and compared to the analytical model.
It is most instructive to consider two different sets of simulated maps: in the first, we neglect noise from the intrinsic ellipticity distribution of the background sources and compute $M_{\rm ap}$ directly from the shear values on the grid according to Eq. (\[mapshear\]). We do this either in the limit of weak lensing, i.e. we use (\[mapshear\]) directly, or we replace $\gamma_{\rm t}$ in (\[mapshear\]) by the reduced shear $g_{\rm t}$, which is the quantity estimated from the observable galaxy ellipticities.
In the second set of simulations we introduce ellipticities of background galaxies according to the distribution function (\[distrib\]). The ellipticities add noise to $M_{\rm ap}$.
The noise-free results are the ones best compared to the analytic results, whereas the ones accounting for intrinsic ellipticities yield a more realistic description of the observational situation. In the following the term “without noise” will refer to the first set of $M_{\rm ap}$ simulations, while the term “with noise” will be used for the second one.
As an illustrative example, the 2-dimensional distribution of $M_{\rm
ap}$ for a standard CDM (SCDM) and an open model (OCDM) is shown in Fig. \[s\_and\_ocdm\]. In both cases high peaks in these maps correspond to haloes in the intervening matter distribution. It is possible to construct a shear-limited sample of haloes from these maps and to determine their abundance.
Comparing the two model universes, we see that the $M_{\rm ap}$ maps reflect the different growth of structure in different cosmologies. The $M_{\rm ap}$ map of the OCDM model is dominated by many isolated peaks which correspond to already collapsed dark matter haloes. The level of background noise coming from matter not yet collapsed is considerably smaller than for the SCDM model in which the structure forms later. The peaks in the SCDM model are less pronounced and isolated than in the open model.
The PDF of $M_{\rm ap}$ and its moments
---------------------------------------
Once we have computed the 2-dimensional distribution of $M_{\rm ap}$, it is straightforward to determine the one-point probability distribution function (PDF) of $M_{\rm ap}$ and its moments. The PDF contains the cosmological information. The lower order moments like rms value and skewness can be derived analytically under simplifying assumptions, but the PDF itself cannot be calculated. Therefore, ray tracing simulations provide the only tool for testing the precision of the analytical calculations.
The qualitative features of the PDF for different filter scales $\theta$ and for the four different cosmologies (Table \[cosmo\]) can be studied in Fig. \[histo\]. The first point to note is that the non-Gaussian features, namely the tail of the PDF at high $M_{\rm ap}$ values, are less pronounced for larger filter scales. This is due to the fact that the smaller filter scales are more sensitive to the already collapsed, non-linear objects. The second feature to note is the exponential decrease of the tail of $M_{\rm ap}$ which was already obtained semi-analytically in KS2. We shall discuss this feature in more detail later in this section.
We now turn to the rms value $\langle M_{\rm ap}^2 \rangle ^{1/2}$ of $M_{\rm ap}$. Fig. \[dispersion\] compares the analytical rms value of $M_{\rm ap}$ calculated using the nonlinear power spectrum of Peacock & Dodds (1996) to the rms values computed from the PDFs without noise for $\gamma_{\rm t}$ (left panel) and $g_{\rm t}$ (right panel). The comparison of the latter shows that the difference between shear and reduced shear is negligible even on filter scales as small as $\theta\sim2$ arcmin corresponding to the highly nonlinear regime of the mass distribution.
In the left panel of Fig. \[dispersion\], there is an excellent agreement between the analytic predictions and the rms values computed from simulations for the SCDM model. There is also good agreement for the $\Lambda$CDM and OCDM models, especially for the larger apertures. The notable exception is the $\tau$CDM model, for which the simulations for small filters deviate by a larger factor from the theoretical predictions.
When interpreting this difference between analytical calculation and simulation in the $\tau$CDM model, one has to keep in mind that the numerical results of Fig. \[dispersion\] are based on a single realisation. As the cosmic variance is relatively large, it is possible that the large deviation is due to the special choice of the realisation. This interpretation is supported by the fact that the mean for the 10 realisations is considerable lower than for the single realisation plotted. Furthermore, the field sizes of the simulated fields used are too small to represent a characteristic region of the universe.
The next higher moment of the PDF is the skewness, which is defined as $$\label{skew_def}
S_{3}(\theta):=\frac{\langle M_{\rm ap}^3\rangle}{\langle M_{\rm ap}^2\rangle^2},$$ for which we can perform a similar analysis as for the rms value of $M_{\rm ap}$. As pointed out by Bernardeau et al. (1997), van Waerbeke et al. (1999), and JSW, the skewness defined in analogy to (\[skew\_def\]) using a top-hat filter is a very sensitive probe of the cosmic density parameter $\Omega_0$.
The dependence of the skewness on filter scale $\theta$ is displayed in Fig. \[skew\]. Again, we compare the skewness computed from the PDF obtained from the ray tracing simulations without noise, both using $\gamma_{\rm t}$ and $g_{\rm t}$, to the skewness of $M_{\rm ap}$ obtained using quasi-linear theory (SvWJK). The error bars on the skewness for the $\tau$CDM model for 2, 5, and 10 arcmin are derived from the 10 different realisations and are centred on their arithmetic mean.
Again, the differences between the skewness obtained from simulations with $\gamma_{\rm t}$ and $g_{\rm t}$ are small, though slightly larger than for the dispersion, owing to the larger contribution from high-$\kappa$ regions to the skewness. This difference, which is of order a few percent at most, has been predicted to be small in the Appendix of SvWJK.
When comparing the skewness as determined from second-order perturbation theory for the density evolution to that obtained from simulations (either computed with $\gamma_{\rm t}$ or $g_{\rm t}$) we see that the former underpredicts the skewness by factors of up to 2. This failure of quasi-linear theory for the prediction of higher-order moments has been demonstrated previously (Jain & Seljak 1997; Gaztanaga & Bernardeau 1998). As we only determine the skewness on scales below 10 arcmin, we are in a regime where the density contrast is non-linear already. The skewness as calculated by Hui (1999) using the so-called hyper-extended perturbation theory (Scoccimarro & Frieman 1999) may provide a more accurate analytical prediction of $S_3$ than that from second-order perturbation theory.
Another point to note is the increase of the skewness towards smaller filter scales. Generally speaking, such a behaviour is expected, as the non-linear structure growth becomes more and more important for small filter scales. This increase is described insufficiently by quasi-linear theory: for the two EdS universes and even for the $\Lambda$ model on large filter scales above 5 arcmin, this increase (not the absolute value!) is predicted satisfactorily, but the slope for the open model is larger than analytic values on all scales displayed. This discrepancy between fully non-linear simulations and quasi-linear theory can be attributed to the fact that the open model is much more dominated by already collapsed, non-linear objects than all other models.
The highest moment we consider explicitly is the kurtosis $S_4$ $$\label{curt_def}
S_{4}(\Theta):=
\frac{\langle M_{\rm ap}^4\rangle}{\langle M_{\rm ap}^2\rangle^2}-3.$$ The kurtosis is not only important by itself, but also for the determination of the error of the rms value of $M_{\rm ap}$, as will be discussed. As for the skewness, the kurtosis for the noise-free simulations for both $\gamma_{\rm t}$ and $g_{\rm t}$ is plotted, and the scatter for $\tau$CDM is determined from the 10 realisations. No analytic result for $S_4$ is available; however, using third-order perturbation theory, Bernardeau (1998) has calculated the kurtosis for a top-hat filter.
We clearly see that the difference between $\gamma_{\rm t}$ and $g_{\rm t}$ becomes important for the kurtosis, at least for the smaller filter scales, since it is even more dominated by the non-Gaussian tail of the PDF than the skewness. The large error bars on the kurtosis are mainly due to large cosmic variance in combination with the small fields used; thus, the current simulations are unable to provide an accurate determination of $S_4$.
We now turn to the error bars on the rms values of $M_{\rm ap}$ in Fig. \[dispersion\]. In the right panel, they were estimated as the standard deviation from 10 different realisations for the $\tau$CDM model. The error bars in the left panel were calculated as follows:
As shown in SvWJK, an unbiased estimator of $\langle M_{\rm
ap}^2\rangle$ from a single aperture is given by $$M={(\pi\theta^2)^2\over N(N-1)}\sum_{i,j\ne i}^N Q_i\,Q_j\,
\epsilon_{{\rm t}i}\,\epsilon_{{\rm t}j} \;,$$ where $N$ is the number of galaxies in the aperture, and $Q_i$ is the value of the weight function $Q$ for the $i$-th galaxy. The dispersion of this estimator is $$\label{sigma_map_field}
\sigma^2(M) \approx S_4 \langle M_{\rm ap}^2\rangle^2
+ \left(\frac{6\sigma^2_{\epsilon}}{5\sqrt{2}N}
+\sqrt{2}\langle M_{\rm ap}^2\rangle^2
\right)^2\;,$$ where the two terms in parenthesis correspond to the noise from the intrinsic ellipticity distribution, and the Gaussian cosmic variance, respectively, whereas the term involving $S_4$ is the excess cosmic variance due to non-Gaussianity. For a collection of $N_{\rm
f}$ independent apertures, all containing the same number of galaxy images, an unbiased estimator for $\langle M_{\rm
ap}^2\rangle$ is the mean ${\cal M}$ of $M$ over these apertures, and the dispersion is $$\label{sigma_map}
\sigma\left({\cal M}\right) = \frac{\sigma (M)}{\sqrt{N_{\rm f}}}.$$
Note that this result does not assume that the density field is Gaussian. If one had a collection of $N_{\rm f}$ fields widely separated on the sky, they would be statistically independent, so that $N_{\rm f}=N$. In the opposite situation where a consecutive area on the sky is available, one can lay down apertures on that field, but they will not be statistically independent. However, as was shown in SvWJK, the $M_{\rm ap}$ values of two apertures which touch each other (i.e., with separation twice their radii), are almost uncorrelated. Whereas the fact that the two aperture masses in these two apertures are uncorrelated does [*not*]{} imply that they are [*independent*]{} (which would mean that the joint probability distribution for the values of $M_{\rm ap}$ would factorize) – as would be the case for Gaussian fields – we assume the statistical independence for estimating the effective number of fields $N_{\rm f}$ entering (\[sigma\_map\]). Thus, the error bars in the left panel of Fig. \[dispersion\] are obtained from (\[sigma\_map\]), assuming that the number of independent apertures is $N_{\rm f}=[\Theta/(2
\theta)]^2$, where $\Theta$ is the side length of the simulated shear field.
In contrast, the error bars plotted in the right panel of Fig. \[dispersion\] for the $\tau$CDM model at the three different filter scales $\theta=2,5,10$ arcmin are based on 10 different realisations of the ray-tracing simulations and allow one to obtain a rough estimate for the error from cosmic variance. Notice that the error bars are centred on the arithmetic mean of the 10 realisations and [*not*]{} on the plotted results from a single realisation.
Comparing the size of the error bars in both panels, we see that both methods give errors of the same order of magnitude even though the errors estimated from the 10 realisations are smaller than the errors from the estimator of $\langle M_{\rm ap}\rangle$. There are two possible reasons for this: first, the effective number of independent apertures is probably larger than our estimate given above, so that the error bars on the left panel in Fig. \[dispersion\] most likely overestimate the true error. Second, in the calculation of the error bars in the right panel, it was assumed that the 10 realisations are independent; but as argued in Sect. \[ray\_tracing\] it is possible that the realisations are not completely independent. This would lead to an underestimation of the cosmic variance. From Fig. \[dispersion\] these two competing effects cannot be quantified. It should be noted that at least on the largest scale plotted, the contribution of the intrinsic ellipticity distribution to the error (\[sigma\_map\]) is completely negligible compared to the cosmic variance.
Halo abundances
---------------
As already indicated in Sect. \[map\_anal\], high signal-to-noise peaks of $M_{\rm ap}$ can be identified with dark matter haloes, rendering the construction of a mass-limited (more correctly: shear-limited) sample feasible. Analytically, the halo abundances can be modelled using the Press & Schechter (1974) prediction for the mass- and redshift-dependent halo number density, and the universal density profile of NFW, while in the simulated $M_{\rm ap}$ map all connected regions above the corresponding threshold are counted as haloes. We shall consider haloes with signal-to-noise ratio $S$ larger than 5, i.e., a peak in the $M_{\rm ap}$ map is counted as a halo if $M_{\rm ap}\ge M_5\equiv 5 \sigma_{\rm c}(\theta)$.
We consider two differently constructed halo abundances in the following: The first sample is simply $N(>M_5,\theta)$, the number density of haloes with an aperture mass larger than $M_5$ for a given filter size $\theta$. The second sample selects peaks with $M_{\rm
ap}\ge M_5$ within a connected, cross sectional area of $\pi \xi_{\rm t}^2$, where $\xi_{\rm t}$ is the corresponding cross section radius; the number density of such peaks is denoted $N(>M_5,>\xi_{\rm t},\theta)$. Hence, the size of the peaks in the second sample exceeds the threshold $\xi_{\rm t}$; these peaks are expected to be more robust with respect to noise coming, e.g., from the intrinsic ellipticity distribution and measurement errors. We use a fixed value of $\xi_{\rm t}=0.6$arc minutes.
In Fig. \[no\_noise\] the number density of the two halo samples as determined from the simulations without noise are compared to the results from the analytic calculation in KS1 over a range of filter scales $2'\le \theta\le 10'$. The four panels in Fig. \[no\_noise\] refer to the four cosmological models considered. The error bars for the $\tau$CDM model at 2,5, and 10 arcmin are again obtained from the 10 different realisations centred on the arithmetic mean of the realisations.
In general, the number counts determined from simulations agree astonishingly well with the analytical results, considering the simplifying assumptions entering the latter. The deviations between simulations and analytical calculation for three of the four cosmologies, namely SCDM, OCDM, and $\Lambda$CDM, and especially for the filter scales above 5 arcmin, are less than 10 %. The largest deviation found for these three models is a factor of 2, for the $\Lambda$CDM model at smallest filter scale.
The only notable exception is the $\tau$CDM model where the deviation remains above 10 % even for the largest filter scales ($\theta=10$ arcmin). This relatively bad agreement has already been noticed for the rms value of $M_{\rm ap}$ and is probably due to the fact that the realisation plotted is not characteristic for the mean properties of that model, as also indicated by the fact that the halo abundance lies above the mean of all realisations as indicated by the error bars.
The good agreement between analytic estimates and numerical results for the halo number density are surprising, given that (a) Press-Schechter theory does not exactly reproduce the spatial number density of haloes when compared to N-body simulations, and (b) the universal density profile found by NFW has been obtained by spherical averaging, and therefore cannot account for the non-axisymmetry of their projected density. Furthermore, (c) the haloes found from the simulated $M_{\rm ap}$ are expected to be affected by projection effects (Reblinsky & Bartelmann 1999) which are completely neglected in the analytic estimates. Despite these effects which one might suspect to yield significant discrepancies, we find that the analytic estimates are very accurate.
We also investigate the halo abundance in an observationally more realistic situation in Fig. \[with\_noise\], including the noise from the intrinsic ellipticity distribution of the background sources. The plot displays the same quantities as Fig. \[no\_noise\] except for the fact that the halo abundances of the two different samples have been determined using the tangential ellipticities in the case of the simulations. The analytic estimates are obtained as in KS1. For all four cosmologies, we determined error bars using 7 different realisations of the ellipticity distribution of the background sources (\[distrib\]). The error bars from the 10 realisations shown for $\tau$CDM are slightly sub-Poissonian, as in Fig. \[no\_noise\]. As expected from the large value of the kurtosis the error coming from the intrinsic ellipticity distribution is much smaller than the error coming from the cosmic variance. On the whole, the number of detected haloes is increased in all cosmologies because, due to the steepness of the Press–Schechter mass function for massive objects, there are more objects just below the threshold than above it. So on average more objects will be lifted above the threshold by noise than brought down below it.
The tail of $M_{\rm ap}$
------------------------
In Fig. \[tail\] we compare the PDF for $M_{\rm ap}\ge M_5$ as obtained from analytic calculations (KS2) with that derived from the simulations without noise. The PDF is shown for four filter scales $\theta=2,4,6,10$ arcmin in the range $M_5\le
M_{\rm ap}\le 2 M_5$, for which the analytic results predict a nearly exponential behaviour. Indeed, the numerical PDF in the non-Gaussian tail also seems to follow an exponential rather closely, with a slope very similar to the analytic result.
In order to see how much the PDF varies between different realisations, we have plotted in Fig. \[tail1\] the PDF for $M_5\le
M_{\rm ap}\le 2 M_5$ obtained from the 10 realisations in the $\tau$CDM model, for 3 filter radii, together with their mean and the corresponding analytic prediction. We find that for the smallest filter scale $\theta=2'$, all 10 realisations are clearly below the analytic result, whereas for the larger filters, the realisation mean of the PDF agrees very well with the analytic prediction.
Remembering that the analytic predictions were made by assuming that all high values of $M_{\rm ap}$ are coming from regions close to collapsed haloes, in addition to the assumptions used for estimating the number density of $M_{\rm ap}$ peaks (Press-Schechter halo abundance and NFW density profile), this good agreement is somewhat surprising.
Conclusion
==========
We used ray-tracing simulations through N-body-generated cosmic density distributions to study the statistical properties of the aperture mass $M_{\rm ap}$ as a statistics for cosmic shear measurements and for finding dark matter haloes from their shear properties. In particular, we have compared results from these simulations with the available analytic results and found in most cases a very good agreement, except for the skewness which is the least accurate of these predictions. Whereas all other predictions tested here are based on manifestly non-linear results (like the Press-Schechter halo abundance and the Peacock & Dodds power spectrum), the skewness was estimated analytically by using second-order Eulerian perturbation theory which, on the scales considered, is not very accurate.
Comparing the results from our simulations with analytic studies, we obtain the following main results: (1) The rms of $M_{\rm ap}$ is accurately described by analytic results if the fully non-linear prescription of the power spectrum of density fluctuations is used. (2) The statistical error of this rms is dominated by cosmic variance, which in turn depends on the kurtosis of $M_{\rm
ap}$. This kurtosis turns out to be relatively large even on angular scales of $\sim 10'$, implying the need for many more measurements of $M_{\rm ap}$ than expected for a Gaussian field, for a given accuracy of the estimated projected power spectrum. (3) The skewness is only approximately described by analytic considerations based on second-order perturbation theory. (4) The predicted abundance of dark matter haloes detectable at given statistical significance is very well approximated by the semi-analytic theory which combines the Press-Schechter number density of haloes with the universal density profile of Navarro, Frenk & White. (5) Similarly, the functional form of the probability distribution of $M_{\rm ap}$ for values much higher than the rms (i.e., in the non-Gaussian tail) is found to closely follow an exponential form, of similar slope and amplitude as predicted by analytic theory which needs to assume that such high values originate due to collapsed haloes. Thus, on the whole, we find that the analytical estimates for the statistical properties of $M_{\rm ap}$ are surprisingly accurate.
We also find that our simulations are not sufficiently large for an accurate estimate of the higher-order statistical measures, owing to the finite size of the simulation box in combination with the large effect of cosmic variance.
As discussed in SvWJK, KS1, KS2, van Waerbeke et al. (1999) and Bartelmann & Schneider (1999), the aperture mass is a useful cosmic shear measure which will eventually allow one to constrain cosmological parameters, completely independent of any assumption on the relation between mass and light. For this purpose, the predictions from cosmology must be known precisely, and our results here indicate that analytic estimates are relatively accurate. Unfortunately, we found a large cosmic variance; e.g., in the estimate of the variance of the rms value of $M_{\rm ap}$, the kurtosis enters and it decreases only rather slowly with increasing filter scale. It can be expected that the first successful application of the aperture mass will be the definition of a sample of haloes defined in terms of their lensing properties only, with a first example given by Erben et al. (1999). The combination of cosmic shear information and CMB measurements can be extremely useful, as shown by Hu & Tegmark (1999), increasing the precision of the determination of cosmological parameters substantially over each of the two individual methods. Their study was based solely on the dispersion of cosmic shear, i.e., on second-order statistics. It is to be expected that a similar combination of CMB results with the PDF of $M_{\rm ap}$ will yield even more precise parameter estimates. A detailed study of this combination is expected to be very valuable, but requires a larger grid of cosmological N-body simulations.
We thank M. Bartelmann for his many valuable suggestions and a careful reading of the manuscript, as well as an anonymous referee for his constructive comments. This work was supported by the “Sonderforschungsbereich 375-95 für Astro-Teilchenphysik" der Deutschen Forschungsgemeinschaft.
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---
abstract: |
We study the growth of fractal clusters in the Dielectric Breakdown Model (DBM) by means of iterated conformal mappings. In particular we investigate the fractal dimension and the maximal growth site (measured by the Hoelder exponent $\alpha_{min}$) as a function of the growth exponent $\eta$ of the DBM model. We do not find evidence for a phase transition from fractal to non-fractal growth for a finite $\eta$-value. Simultaneously, we observe that the limit of non-fractal growth ($D\to 1$) is consistent with $\alpha_{min}
\to 1/2$. Finally, using an optimization principle, we give a recipe on how to estimate the effective value of $\eta$ from temporal growth data of fractal aggregates.
author:
- 'Joachim Mathiesen$^1$, Mogens H. Jensen$^2$, and Jan [Ø]{}ystein Haavig Bakke$^3$'
title: 'Dimensions, Maximal Growth Sites and Optimization in the Dielectric Breakdown Model'
---
Introduction
============
Laplacian growth and the formation of complex patterns has been the subject of numerous theoretical and experimental works. The classical examples are the ramified pattern appearing in a Hele-Shaw cell when a less viscous fluid is injected into a more viscous fluid [@Feder] and the fractal structures emerging from the particle aggregation in Diffusion-limited Aggregation (DLA) [@81WS]. In the latter example mono-disperse particles are released one-by-one from a remote source and diffuse until they hit and irreversibly adhere to a seed cluster at the center of coordinates. The cluster slowly expands as particles are added. Statistically, the motion of a single particle is described by the harmonic potential $U$ satisfying the Laplace equation $\Delta
U=0$ and the probability of sticking to the cluster at a specific site, $z$, is given by the harmonic measure $|\nabla U (z)|$. The formulation of DLA is contained within a more general model, the Dielectric Breakdown Model (DBM) [@84NPW; @01Ha], where the growth probability $\rho_\eta$ at the cluster interface is proportional to the harmonic measure raised to a power $\eta$, $\rho_\eta\propto|\nabla U|^\eta$. Despite intensive research in Laplacian growth, fundamental questions regarding the scaling properties still have no answer. The growth laws of DLA and DBM are extremely simple and in apparent disparity to the complex patterns they produce. The complex patterns arise from a strong correlation between the position of already aggregated particles and the influx of new particles. As the outermost tips advance the probability for particles to reach the parts left behind diminishes and the harmonic measure broadens and becomes even multifractal [@02JLMP]. For increasing values of $\eta$ the growth probability will concentrate around the tips and the fractal dimension gets closer to unity and ultimately, in the limit of infinite $\eta$, the particle cluster loses fractality. Recently, it has been speculated that in two dimensions this transition from fractal to non-fractal growth may happen at a finite critical value of $\eta$ and numerically, this value has been found to be $\eta\approx 4$ [@93SGSHL; @01Ha]. In the vicinity of such a critical point it may be safe to disregard the noise giving rise to local density fluctuations along the branches [@02H]. For that reason, the dominating stochastic component in the cluster growth is the rate at which growing tips split in two or more branches. While growing, neighboring branches compete and if one branch quickly dies after a tip-splitting the growth will stay non-fractal. It has been shown [@01H] that in the idealized case of straight growing branches, tip-splitting is suppressed for $\eta>4$ supporting that $\eta_c=4$. Based on the idealized branch growth model a renormalization group approach has been used in an expansion around $\eta_c$. Although an expansion provides important information for small values of $4-\eta$ it may provide little information on DLA ($\eta=1$). In this article we test the hypothesis of a critical point at $\eta=4$ performing extensive numerical simulations. We provide detailed figures on the dependence of the fractal dimension, $\alpha_{min}$ and the exponent $\eta$. Moreover, we propose a method for extracting effective $\eta$-exponents given either experimental or numerical data series. For that purpose we make use of iterated conformal maps [@98HL] which have proven a convenient tool for generating conformal mappings of domains of arbitrary shape [@06MPST], see section \[icm\]. In Section \[opti\] a method is proposed for extracting effective $\eta$ exponents by optimization. In section \[frac\] we present results [*pro et con*]{} a phase transition in DBM, the maximal growth sites and the fractal dimensions.
Iterated conformal mappings {#icm}
===========================
The conformal invariance of the Laplace equation reduces the problem of finding the harmonic measure, $\rho_1(z)$, around any simply connected domain in the complex plane to that of finding a conformal transformation $\omega = \Phi^{-1}(z)$ of the domain to the unit disc $$\rho_1(z)=\frac 1 {\left | \Phi^{'}(w)\right|}$$ The method of iterated conformal mappings provides a general framework to construct such transforms as well as a simple procedure to grow DLA clusters. Assume that a DLA cluster of $n$ particles is mapped to the unit disc by $\Phi_n^{-1}$. An extra particle is added to the cluster by first adding a small bump of size $\sqrt \lambda$ to the unit disc using a mapping $\varphi_{n+1}$ and subsequently applying the inverse mapping $\Phi_n$. Finally, the composed mapping $\Phi_n\circ
\varphi_{n+1}$ transforms the unit disc into a cluster of $n+1$ particles. The basic mapping $\varphi_{n+1}$ is defined by two parameters the position and size of the bump, the position, $e^{i\theta}$, is random in DLA since the measure is uniform around the circle. The size $\sqrt\lambda$ of the $n$’th bump is controlled by the condition that $$\sqrt{\lambda_0}=\sqrt\lambda_n | \Phi' (e^{i\theta})|$$ Consequently, the particles (transformed bumps) will all to linear order have the same size $\sqrt{\lambda_0}$. The full recursive dynamics is written as iterations of the basic map $$\label{eq:4}
\Phi^{(n)}(w)=\varphi_{\theta_{1},\lambda_{1}}\circ\ldots\circ\varphi_{\theta_n,\lambda_n}(w) \ .$$ Note that this structure is unusual in the sense that the order of iterates is inverted compared to standard dynamical systems.
For DBM the growth measure along the cluster interface, parameterized by $s$, is given by $$\rho_\eta(s)=\frac{\rho_1^\eta(s)}{\int\rho_1^\eta(t) dt},$$ which for $\eta\neq1$ is not conformally invariant. On the unit circle, parameterized by $\theta$, the growth measure transforms into $$\label{eq:5}
\rho_\eta(\theta)d\theta \sim
\rho_\eta(s(\theta))\left|\frac{ds}{d\theta}\right|d\theta\sim
|\Phi'(e^{i\theta})|^{1-\eta}d\theta
\ .$$ In the simulations we choose $\theta$ according to the distribution $\rho_\eta$ using standard Monte Carlo samplings of the measure. The number of samples needed for an accurate estimate of the distribution increases with $\eta$ and is chosen according to $$\label{mc}
\frac k {\sqrt{ \lambda_0}\max_s \rho_1(s)}$$ By choosing $k>1$, the site of maximal measure will on the average be visited more than once during the sampling. It turns out that there is no visible change in the scaling of the clusters when choosing $k>1$, see Fig. \[conv\] for a test of convergence as function of $k$; in the results presented below, we use $2\leq k\leq 8$.
Extracting effective $\eta$ exponents by optimization {#opti}
=====================================================
Consider an interface growing at a rate determined by some unknown function of the harmonic measure. The method of iterated conformal mappings is readily turned into a framework for estimating this function. More specifically, it is here demonstrated on numerical simulation data of the DBM that the value of $\eta$ can be extracted from a careful tracking of the cluster growth. The general idea is to utilize the iteration scheme in tracking the motion of the interface by gradually expanding the mapping, see [@06MPST] for further details. The harmonic measure is recorded as the interface evolves and from a maximum likelihood principle the $\eta$ value of the growth is extracted. The probability for growth to occur at a site $z_n$ at the interface is in a given growth step $n$ approximated by the sum $$\label{etaprob}
\rho_\eta(n,z_n)=\frac{\rho_1^\eta(n,z_n)}{\sum_z \rho_1^\eta(n,z)}$$ From this expression, more ways exist to estimate the $\eta$ value used in the simulation. Assuming that the $n$’th growth event occurred at the site $z_n$, a direct estimate of $\eta$ follows from maximizing $\rho_\eta(z_n)$ with respect to $\eta$. Naturally, this will lead to dramatic fluctuations in the estimates and therefore maximizing products of $\rho_\eta$ over several growth steps provides a better estimate, $$\label{est}
\prod_k \rho_\eta(k,z_k)$$
In Fig. \[optimum\], we show how this product varies as function of $\eta$ and with the number of factors used. With an increasing number of factors the maximum becomes more pronounced and the $\eta$ value used in the simulations is easily recovered. These products confirms that the number of Monte Carlo samples used in Eq. (\[mc\]) are appropriate and more importantly that the method is directly applicable to experimental data for estimating an effective $\eta$ value or more generally the boundary condition function determining the growth rate.
Dimension and $\alpha_{min}$ {#frac}
============================
The dimension of a cluster grown by this conformal mapping technique is determined by the first term in the Laurant expansion of $\Phi^{(n)}$, $F_1^{(n)}$, which will scale like $F_1^{(n)}
\sim n^{1/D} \sqrt{\lambda_0}$ [@98HL]. The dimension is thus estimated by a direct fit of this scaling law as demonstrated in Fig. \[laurant\] for a cluster 80000 particles and $\eta=4.0$.
Using the conformal mapping technique we have grown clusters up to sizes 80000 particles with varying values of $\eta$ in the interval $\eta \in
[1,5]$. Fig. \[dim\] shows the results for the value of the dimension versus $\eta$. As is clear from the figure, the value of the dimension decreases smoothly with $\eta$, from the DLA value $D=1.71$ for $\eta=1$ down towards $D \sim 1$ for $\eta
\to \infty$. Hastings [@01H] presented arguments in favor of an upper critical dimension $\eta_c = 4$ for which the clusters become one-dimensional. We however do not observe indications of this transition. As seen in Fig. \[dim\] it is quite clear that the data smoothly bends away before reaching the point $(\eta,D)=(4,1)$ and only approaching the one-dimensional growth in the limit of large $\eta$-values. We thus conclude that there do not exist a critical point at a finite $\eta$.
Halsey [@02H] has computed a first-order correction to $D$ for $\eta<4$, obtaining $D= 1 + \frac{1}{2} (4-\eta) + O(4-\eta)^2$. This relation predicts a linear variation of slope $\frac{1}{2}$ around $\eta_c = 4$. As seen in Fig. \[dim\] we do not observe this behavior.
It is well know that the growth measure of a DBM model exhibits multifractal properties with a spectrum of growth exponents measured by local Hoelder exponents $\alpha$ [@02JLMP]. The points of highest growth measures are characterized by the minimum $\alpha$-value, $\alpha_{min}$. We have earlier determined this value using the iterated conformal mapping technique [@03JMP] and extend it here to the DBM model. In this method, it is very easy to keep track of where the maximum growth probability is located as more particles are added. Let us assume that at the ($n$-1)’th growth step the site with the largest probability is located at the angle $\theta_{max}$ on the unit circle, i.e. for all $\theta$ $$\frac 1 {|{\Phi^{(n-1)}}' (e^{i \theta_{max}})|}\geq
\frac 1 {|{\Phi^{(n-1)}}' (e^{i \theta})|}$$ When we add a new bump in the $n$’th growth step the position of maximal probability may not change (up to reparameterization of the angle $\theta_{max}$), or move to the new bump. We can easily find the reparameterized angle and determine the new position from $$\label{pmaxn}
\rho_1^{max,n}=\max\left\{\frac 1 {|{\Phi^{(n)}}'
(\phi^{-1}_{\lambda_n,\theta_n}(e^{i\theta_{max}}))|},
\frac 1 {|{\Phi^{(n)}}' (e^{i \theta_n})|}\right\}\ .$$ If $\rho_1^{max,n}$ is located at $\theta_n$ we put $\theta_{max}=\theta_n$ in the $(n+1)$’th growth step. Using conformal mappings, we have also previously estimated the critical branching angle as a function of $\eta$ in the DBM model [@02MJ].
Fig. \[amin\] shows the results of $\alpha_{min}$ vs. $\eta$ and we observe that $\alpha_{min}$ decreases from the DLA values $\alpha_{min} =0.68$ down to $\alpha_{min} =0.5$. It is obvious that $\alpha_{min} =0.5$ corresponds to the Hoelder exponent for a line. In consistency with the results in Fig. \[dim\] we observe that the curve bends smoothly and that the one-dimensional growth is only obtained in the limit $\eta \to \infty$. The last figure, Fig. \[dimamin\], shows $\alpha_{min}$ plotted vs. $D$. By extrapolation (as indicated by the line) we see that $\alpha_{min}$ assumes its minimal value 0.5 at a dimension $D
=1.0$.
Conclusions
===========
The conclusions of our paper are twofold. Firstly, we have presented a method to extract the effective value of the growth exponent $\eta$, for a time series of a growing aggregates, assuming an underlying mechanism based on the Dielectric Breakdown Model (DBM) model. The estimate is based on a maximum likelihood method and converges rather well for the numerical data presented here. We believe this method should be directly applicable to experimental data when it is possible to extract intermediate steps in the formation of the aggregates. We urge the method to be used in for example viscous fingering experiments in random media [@04L]. Secondly, we have thoroughly investigated the scaling structure of DBM clusters as a function of the growth exponent $\eta$. Based on extensive numerical simulations we do not find support for the conjecture that the growth becomes one-dimensional at the critical value $\eta_c = 4$[@01H; @02H]. On the contrary, our results indicate that there do not exist a critical point for at finite $\eta$-value and that the scaling exponent of the maximal growth site $\alpha_{min}$ assumes its minimal value 0.5 when the growth becomes non-fractal.
Acknowledgements
================
We thank Knut Joergen Maaloy and Stephane Santucci for interesting discussions at an early stage of this work. This project was funded by *Physics of Geological Processes*, a Center of Excellence at the University of Oslo, the Danish National Research Foundation and the VILLUM KANN RASMUSSEN Foundation for support.
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---
abstract: 'In many clinical trials studying neurodegenerative diseases such as Parkinson’s disease (PD), multiple longitudinal outcomes are collected to fully explore the multidimensional impairment caused by this disease. If the outcomes deteriorate rapidly, patients may reach a level of functional disability sufficient to initiate levodopa therapy for ameliorating disease symptoms. An accurate prediction of the time to functional disability is helpful for clinicians to monitor patients’ disease progression and make informative medical decisions. In this article, we first propose a joint model that consists of a semiparametric multilevel latent trait model (MLLTM) for the multiple longitudinal outcomes, and a survival model for event time. The two submodels are linked together by an underlying latent variable. We develop a Bayesian approach for parameter estimation and a dynamic prediction framework for predicting target patients’ future outcome trajectories and risk of a survival event, based on their multivariate longitudinal measurements. Our proposed model is evaluated by simulation studies and is applied to the DATATOP study, a motivating clinical trial assessing the effect of deprenyl among patients with early PD.'
author:
- 'Jue Wang, Sheng Luo[^1], Liang Li'
bibliography:
- 'literatures\_DP.bib'
title: 'Dynamic Prediction for Multiple Repeated Measures and Event Time Data: An Application to Parkinson’s Disease'
---
**Key Words:** Area under the ROC curve, clinical trial, failure time, latent trait model.
Introduction {#sec:intro}
============
Joint models of longitudinal outcomes and survival data have been an increasingly productive research area in the last two decades [e.g., @Tsiatis:2004StatSinica]. The common formulation of joint models consists of a mixed effects submodel for the longitudinal outcomes and a semiparametric Cox submodel [@Wulfsohn1997Biometrics] or accelerated failure time (AFT) submodel for the event time [@Tseng2005Biometrika]. Subject-specific shared random effects [@Vonesh2006SIM] or latent classes [@Proust2014SMMR] are adopted to link these two submodels. Many extensions have been proposed, e.g., relaxing the normality assumption of random effects [@Brown2003Biometrics], replacing random effects by a general latent stochastic Gaussian process [@Xu2001JRSSC], incorporating multivariate longitudinal variables [@Chi2006Biometrics], and extending single survival event to competing risks [@Elashoff2007SIM] or recurrent events [@Sun2005JASA; @Liu2009JRSSC].
Joint models are commonly used to provide an efficient framework to model correlated longitudinal and survival data and to understand their correlation. A novel use of joint models, which gains increasing interest in recent years, is to obtain dynamic personalized prediction of future longitudinal outcome trajectories and risks of survival events at any time, given the subject-specific outcome profiles up to the time of prediction. For example, [@Rizopoulos2011Biometrics] proposed a Monte Carlo approach to estimate risk of a target event and illustrated how it can be dynamically updated. [@Taylor2013Biometrics] developed a Bayesian approach using a Markov chain Monte Carlo (MCMC) algorithm to dynamically predict both the continuous longitudinal outcome and survival event probability. [@Blanche2015Biometrics] extended the survival submodel to account for competing events. [@Rizopoulos2013arXiv] compared dynamic prediction using joint models v.s. landmark analysis [@VanHouwelingen2007SJS], an alternative approach for dynamically updating survival probabilities. A key feature of these dynamic prediction frameworks is that the predictive measures can be dynamically updated as additional longitudinal measurements become available for the target subjects, providing instantaneous risk assessment.
Most dynamic predictions via joint models developed in the literature have been restricted to one or two longitudinal outcomes. However, impairment caused by the neurodegenerative diseases such as Parkinson’s disease (PD) affects multiple domains (e.g., motor, cognitive, and behavioral). The heterogeneous nature of the disease makes it impossible to use a single outcome to reliably reflect disease severity and progression. Consequently, many clinical trials of PD collect multiple longitudinal outcomes of mixed types (categorical and continuous). To properly analyze these longitudinal data, one has to account for three sources of correlation, i.e., inter-source (different measures at the same visit), longitudinal (same measure at different visits), and cross correlation (different measures at different visits) [@OBrien2004JRSSC]. Hence, a joint modeling framework for analyzing all longitudinal outcomes simultaneously is essential. There is a large number of joint modeling approaches for mixed type outcomes. Multivariate marginal models (e.g., likelihood-based [@Molenberghs2005book], copula-based [@Lambert2002SIM], and GEE-based\
[@OBrien2004JRSSC]), provide direct inference for marginal treatment effects, but handling unbalanced data and more than two response variables remain open problems. Multivariate random effects models [@Verbeke2014SMMR] have severe computational difficulties when the number of random effects is large. In comparison, mixed effects models focused on dimensionality reduction (using latent variables) provide an excellent and balanced approach to modeling multivariate longitudinal data. To this end, [@He2016SMMR] developed a joint model for multiple longitudinal outcomes of mixed types, subject to an outcome-dependent terminal event. [@Luo_Wang2014SIM] proposed a hierarchical joint model accounting for multiple levels of correlation among multivariate longitudinal outcomes and survival data. [@Proust2016SIM] developed a joint model for multiple longitudinal outcomes and multiple time-to-events using shared latent classes.
In this article, we propose a novel joint model that consists of: (1) a semiparametric multilevel latent trait model (MLLTM) for the multiple longitudinal outcomes with a univariate latent variable representing the underlying disease severity, and (2) a survival submodel for the event time data. We adopt penalized splines using the truncated power series spline basis expansion in modeling the effects of some covariates and the baseline hazard function. This spline basis expansion results in tractable integration in the survival function, which significantly improves computational efficiency. We develop a Bayesian approach via Markov chain Monte Carlo (MCMC) algorithm for statistical inference and a dynamic prediction framework for the predictions of target patients’ future outcome trajectories and risks of survival event. These important predictive measures offer unique insight into the dynamic nature of each patient’s disease progression and they are highly relevant for patient targeting, management, prognosis, and treatment selection. Moreover, accurate prediction can advance design of future studies, experimental trials, and clinical care through improved prognosis and earlier intervention.
The rest of the article is organized as follows. In Section \[sec:motivating\_CT\], we describe a motivating clinical trial and the data structure. In Section \[sec:methods\], we discuss the joint model, Bayesian inference, and subject-specific prediction. In Section \[sec:dataAna\], we apply the proposed method to the motivating clinical trial dataset. In Section \[sec:simulation\], we conduct simulation studies to assess the prediction accuracy. Concluding remarks and discussions are given in Section \[sec:discussion\].
A motivating clinical trial {#sec:motivating_CT}
===========================
The methodological development is motivated by the DATATOP study, a double-blind, placebo-controlled multicenter randomized clinical trial with 800 patients to determine if deprenyl and/or tocopherol administered to patients with early Parkinson’s disease (PD) will slow the progression of PD. We refer to as placebo group the patients who did not receive deprenyl and refer to as treatment group the patients who received deprenyl. The detailed description of the design of the DATATOP study can be found in [@Shoulson1998AN].
In the DATATOP study, the multiple outcomes collected include Unified PD Rating Scale (UPDRS) total score, modified Hoehn and Yahr (HY) scale, Schwab and England activities of daily living (SEADL), measured at 10 visits (baseline, month 1, and every 3 months starting from month 3 to month 24). UPDRS is the sum of 44 questions each measured on a 5-point scale (0-4), and it is approximated by a continuous variable with integer value from 0 (not affected) to 176 (most severely affected). HY is a scale describing how the symptoms of PD progresses. It is an ordinal variable with possible values at 1, 1.5, 2, 2.5, 3, 4, and 5, with higher values being clinically worse outcome. However, the DATATOP study consists of only patients with early mild PD and the worst observed HY is 3. SEADL is a measurement of activities of daily living, and it is an ordinal variable with integer values from 0 to 100 incrementing by 5, with larger values reflecting better clinical outcomes. We have recoded SEADL variable so that higher values in all outcomes correspond to worse clinical conditions and we have combined some categories with zero or small counts so that SEADL has eight categories.
Among the 800 patients in the DATATOP study, 44 did not have disease duration recorded and one had no UPDRS measurements. We exclude them (5.6%) from our analysis and the data analysis is based on the remaining 755 patients. The mean age of patients is 61.0 years (standard deviation, 9.5 years). 375 patients are in the placebo group and 380 are in the treatment group. About 65.8% of patients are male and the average disease duration is 1.1 years (standard deviation, 1.1 years). Before the end of the study, some patients (207 in placebo and 146 in treatment) reached a pre-defined level of functional disability, which is considered to be a terminal event because these patients would then initiate symptomatic treatment of levodopa, which can ameliorate the clinical outcomes. Figure \[fig:UPDRS\_fumonth\] displays the mean UPDRS measurements over time for DATATOP patients with follow-up time less than 6 months (96 patients, solid line), 6-12 months (215 patients, dotted line), and more than 12 months (444 patients, dashed line). Figure \[fig:UPDRS\_fumonth\] suggests that patients with shorter follow-up had higher UPDRS measurements, manifesting the strong correlation between the PD symptoms and terminal event. Similar patterns are observed in HY and SEADL measurements. Such a dependent terminal event time, if not properly accounted for, may lead to biased estimates [@Henderson2000Biostatistics].
![Mean UPDRS values over time for DATATOP patients with follow-up time less than 6 months (solid line), 6-12 months (dotted line), and more than 12 months (dashed line).[]{data-label="fig:UPDRS_fumonth"}](UPDRS_fumonth.eps){width="40.00000%"}
Because levodopa is associated with possible motor complications [@Brooks2008NDT], clinicians tend to provide more targeted interventions to delay their initiation of levodopa use. To this end, in the context of DATATOP study and similar PD studies, there is an important clinically relevant prediction question: for a new patient (not included in the DATATOP study) with one or multiple visits, what are his/her most likely future outcome trajectories (e.g., UPDRS, HY, and SEADL) and risk of functional disability within the next year, given the outcome histories and the covariate information? These important predictive measures are highly relevant for PD patient targeting, management, prognosis, and treatment selection. In this article, we propose to develop a Bayesian personalized prediction approach based on a joint modeling framework consisting of a semiparametric multilevel latent trait model (MLLTM) for multivariate longitudinal outcomes and a survival model for the event time data (time to functional disability).
Methods {#sec:methods}
=======
Joint modeling framework {#sec:model_MLLTM}
------------------------
In the context of clinical trials with multiple outcomes, the data structure is often of the type $\{y_{ik}(t_{ij}), t_i, \delta_i\}$, where $y_{ik}(t_{ij})$ is the $k$th ($k=1,\ldots,K$) outcome, which can be binary, ordinal, or continuous, for patient $i$ ($i=1,\ldots,I$) at visit $j$ ($j=1,\ldots,J_i$) recorded at time $t_{ij}$ from the study onset, $t_i=min(T_i^*,C_i)$ is the observed event time to functional disability, as the minimum between the true event time $T_i^*$ and the censoring time $C_i$ which are assumed to be independent of $T_i^*$, and $\delta_i$ is the censoring indicator ($1$ if the event is observed, and $0$ otherwise). We propose to use a semiparametric multilevel latent trait model (MLLTM) for the multiple longitudinal outcomes and a survival model for the event time.
To start building the semiparametric MLLTM framework, we assume that there is a latent variable representing the underlying disease severity score and denote it as $\theta_i(t)$ for patient $i$ at time $t$ with a higher value for more severe status. We introduce the first level model for continuous outcomes, $$\begin{aligned}
y_{ik}(t)=a_k+b_k\theta_{i}(t)+\varepsilon_{ik}(t), \label{eqn:MLLTM_continuous}\end{aligned}$$ where $a_k$ and $b_k$ (positive) are the outcome-specific parameters, and the random errors $\varepsilon_{ik}(t)\sim N(0, \sigma_{\varepsilon_k}^2)$. Note that $a_k=E[y_{ik}(t)|\theta_{i}(t)=0]$ is the mean of the $k$th outcome if the disease severity score is $0$ and $b_k$ is the expected increase in the $k$th outcome for one unit increase in the disease severity score. The parameter $b_k$ also plays the role of bringing up the disease severity score to the scale of the $k$th outcome. The models for outcomes that are binary (e.g., the presence of adverse events) and ordinal (e.g., HY and SEADL) are as follows [@Fox2005BJMSP]: $$\begin{aligned}
\label{eqn:MLLTM_ordi}
&& \textnormal{logit}\big\{p(y_{ik}(t)=1|\theta_{i}(t))\}=a_k+b_k\theta_{i}(t) \nonumber \\
&& \textnormal{logit}\big\{p(y_{ik}(t)\le l|\theta_{i}(t))\big\}=a_{kl}-b_k\theta_{i}(t),\end{aligned}$$ where $l=1,2,\ldots,n_k-1$ is the $l$th level of the $k$th ordinal variable with $n_k$ levels. Note that the negative sign for $b_k$ in the ordinal outcome model is to ensure that worse disease severity (higher $\theta_{i}(t)$) is associated with a more severe outcome (higher $y_{ik}(t)$). Interpretation of parameters is similar for continuous outcomes, except that modeling is on the log-odds, not the native scale, of the data. We have selected logit link function in model , while other link functions (e.g., probit and complementary log-log) can be adopted. A major feature of models and is that they all incorporate $\theta_{i}(t)$ and explicitly combine longitudinal information from all outcomes.
To model the dependence of severity score $\theta_{i}(t)$ on covariates, we propose the second level semiparametric model $$\begin{aligned}
\label{eqn:theta}
\theta_{i}(t)=\boldsymbol{X}_{i}(t)\boldsymbol{\beta} + \boldsymbol{Z}_{i}(t)\boldsymbol{u}_i + \boldsymbol{V}_R(t) \boldsymbol{\zeta},\end{aligned}$$ where vectors $\boldsymbol{X}_{i}(t)$ and $\boldsymbol{Z}_{i}(t)$ are $p$ and $q$ dimensional covariates corresponding to fixed and random effects, respectively. They can include covariates of interest such as treatment and time. To allow additional flexibility and smoothness in modeling the effects of some covariates, we adopt a smooth time function $\boldsymbol{V}_R(t) \boldsymbol{\zeta}=\sum_{r=1}^{R}\zeta_{r}(t-\kappa_r)_+$ using the truncated power series spline basis expansion $\boldsymbol{V}_R(t) = \{ (t-\kappa_1)_+, \ldots, (t-\kappa_R)_+ \}$, where $\boldsymbol{\kappa} = \{\kappa_1, \ldots, \kappa_R \}$ are the knots, and $(t-\kappa_r)_+ = t-\kappa_r$ if $t>\kappa_r$ and 0 otherwise. Following [@Ruppert2012JCGS], we consider a large number of knots (typically 5 to 20) that can ensure the desired flexibility and we select the knot location to have sufficient subjects between adjacent knots. To avoid overfitting, we explicitly introduce smoothing by assuming that $\boldsymbol{\zeta} = (\zeta_1, \ldots, \zeta_R)' \sim N(0, \sigma_\zeta^2 \boldsymbol{I})$[@Ruppert2003semipar_book; @Crainiceanu2005JSS]. The choice of knots is important to obtain a well fitted model and should be selected with caution to avoid overfitting. Several approaches of automatic knot selection based on stepwise model selection have been proposed [@Friedman1989Technometrics; @Stone1997AOS; @Denison1998JRSSB; @Dimatteo2001Biometrika]. [@Wand2000CS] gives a good review and comparison of some of these approaches. Penalizing the spline coefficients to constrain their influence also helps to avoid overfitting [@Ruppert2003semipar_book], as in our model. Moreover, in clinical studies with same scheduled follow-up visits, the frequency of study visits needs to be accounted for in the selection of knots. For the ease of illustration, we include the nonparametric smooth function for the time variable, although our model can be extended to accommodate more nonparametric smooth functions. The vector $\boldsymbol{u}_i=(u_{i1}, \ldots, u_{iq})'$ contains the random effects for patient $i$’s latent disease severity score and it is distributed as $N(\boldsymbol{0}, \boldsymbol{\Sigma})$. Equations , and consist of the semiparametric MLLTM model, which provides a nature framework for defining the overall effects of treatment and other covariates. Indeed, if $\theta_{i}(t)=\beta_{0}+\beta_{1}x_i+\beta_{2}t+\beta_{3}x_i t + \sum_{r=1}^{R} \zeta_r (t-\kappa_r)_+ +u_{i0}+u_{i1}t$, where $x_i$ is treatment indicator (1 if treatment and 0 otherwise), then $\beta_1$ is the main treatment effect and $\beta_3$ is the time-dependent treatment effect. In this context, the null hypothesis of no overall treatment effect is $H_0: \beta_1=\beta_3=0$. Because the number of outcomes ($K$) has been reduced to one latent disease severity score, models are quite parsimonious in terms of number of random effects, which improves computational feasibility and model interpretability.
Because the semiparametric MLLTM model is over-parameterized, additional constraints are required to make it identifiable. Specifically, we set $a_{k1}=0$ and $b_k=1$ for one ordinal outcome. For the ordinal outcome $k$ with $n_k$ categories, the order constraint $a_{k1} < \ldots < a_{kl} < \ldots < a_{kn_{k}-1}$ must be satisfied, and the probability of being in a particular category is $p(Y_{ik}(t)=l)= p(Y_{ik}(t)\leq l|\theta_{i}(t))- p(Y_{ik}(t)\leq l-1|\theta_{i}(t))$. With these assumptions, the conditional log-likelihood of observing the patient $i$ data $\{y_{ik}(t_{ij})\}$ given $\boldsymbol{u}_i$ and $\boldsymbol{\zeta}$ is $l_y(\boldsymbol{\Theta}_y;\boldsymbol{y}_i,\boldsymbol{u}_i, \boldsymbol{\zeta})=\sum^{J_i}_{j=1}\sum^K_{k=1} \log p(y_{ik}(t_{ij})|\boldsymbol{u}_i, \boldsymbol{\zeta})$. For notational convenience, we let $\boldsymbol{a}=(\boldsymbol{a}'_1,\ldots,\boldsymbol{a}'_k,\ldots,\boldsymbol{a}'_K)'$, with $\boldsymbol{a}_k$ being numeric for binary and continuous outcomes and $\boldsymbol{a}_k=(a_{k1},\ldots,a_{kn_{k}-1})'$ for ordinal outcomes. We let $\boldsymbol{b}=(b_1,\ldots,b_K)'$ and $\boldsymbol{y}_{i}(t)=\{y_{ik}(t), k=1, \ldots, K\}'$ be the vector of measurements for patient $i$ at time $t$ and let $\boldsymbol{y}_i=\{\boldsymbol{y}_{i}(t_{ij}), j=1, \ldots, J_i\}$ be the outcome vector across $J_i$ visit times. The parameter vector for the longitudinal process is $\boldsymbol{\Theta}_y = (\boldsymbol{a}', \boldsymbol{b}', \boldsymbol{\beta}', \boldsymbol{\Sigma}, \sigma_{\varepsilon_k}, \sigma_\zeta)'$.
To model the survival process, we use the proportional hazard model $$\label{eqn:Cox}
h_i(t)=h_0(t)\exp\{\boldsymbol{W}_i\boldsymbol{\gamma}+\nu \theta_i(t)\},$$ where $\boldsymbol{\gamma}$ is the coefficient for time-independent covariates $\boldsymbol{W}_i$ and $h_0(\cdot)$ is the baseline hazard function. Some covariates in $\boldsymbol{W}_i$ can overlap with vector $\boldsymbol{X}_i(t)$ in model . [@Ibrahim2010JCO] gave an excellent explanation of the coefficients for those overlapped covariates. In the current context, if we denote $\boldsymbol{\beta}_o$ and $\boldsymbol{\gamma}_o$ as the coefficients for the overlapped covariates in vectors $\boldsymbol{X}_i(t)$ and $\boldsymbol{W}_i$, respectively, we have: (1) $\boldsymbol{\beta}_o$ is the covariate effect on the longitudinal latent variable; (2) $\boldsymbol{\gamma}_o$ is the direct covariate effect on the time to event; (3) $\nu\boldsymbol{\beta}_o+\boldsymbol{\gamma}_o$ is the overall covariate effect on the time to event. The association parameter $\nu$ quantifies the strength of correlation between the latent variable $\theta_i(t)$ and the hazard for a terminal event at the same time point (refer to as Model 1: shared latent variable model). Specifically, a value of $\nu=0$ indicates that there is no association between the latent variable and the event time while a positive association parameter $\nu$ implies that patients with worse disease severity tend to have a terminal event earlier, e.g., a value of $\nu=0.5$ indicates that the hazard rate of having the terminal event increases by $65\%$ (i.e., $\exp(0.5)-1$) for every unit increase in the latent variable. For prediction of subject-specific survival probabilities, a specified and smooth baseline hazard function is desired. To this end, we again adopt a truncated power series spline basis expansion $h_0(t) = \exp\{\eta_{0}+\eta_{1}t + \sum_{r=1}^{R}\xi_{r}(t-\kappa_r)_+\}$ and assume $\boldsymbol{\xi} = (\xi_1, \ldots, \xi_R )' \sim N(0, \sigma_{\xi}^2\boldsymbol{I})$ to introduce smoothing. The knot locations can be the same or different from those in equation .
In equation , different formulations can be used to postulate how the risk for a terminal event depends on the unobserved disease severity score at time $t$. For example, one can add to equation a time-dependent slope $\theta_i'(t)$, so that the risk depends on both the current severity score and the slope of the severity trajectory at time $t$ (refer to as Model 2: time-dependent slope model): $$h_i(t)=h_0(t)\exp\{\boldsymbol{W}_i\boldsymbol{\gamma}+\nu_1 \theta_i(t)+\nu_2 \theta_i'(t)\}.$$ Alternatively, one can consider the standard formulations of joint models that include only the random effects in the Cox model (refer to as Model 3: shared random effects model): $$h_i(t)=h_0(t)\exp\{\boldsymbol{W}_i\boldsymbol{\gamma}+ \boldsymbol{\nu}' \boldsymbol{u}_i \}.$$ A good summary of these various formulations in the joint modeling framework can be found in [@Rizopoulos2014JASA] and [@Yang2015SIM].
The log-likelihood of observing event outcome $t_i$ and $\delta_i$ for patient $i$ is\
$l_s(\boldsymbol{\Theta}_s; t_i, \delta_i, \boldsymbol{u}_i, \boldsymbol{\zeta}, \boldsymbol{\xi})=\log \{h_i(t_i)^{\delta_i}S_i(t_i)\}$, where the survival function $S_i(t_i)=\exp\{-\int_0^{t_i} h_i(s)ds\}$ and the parameter vector for the survival process is $\boldsymbol{\Theta}_s = (\boldsymbol{\gamma}', \nu, \eta_0, \eta_1, \sigma_\xi)'$. Note that the truncated power series spline basis expansion in modeling the smooth time function in equation and in modeling the baseline hazard function is linear function of time, which results in tractable integration in the survival function $S_i(t_i)$, and consequently, significant gain in computing efficiency. Conditional on the random effect vector $\boldsymbol{u}_i$, $\boldsymbol{y}_i$ is assumed to be independent of $t_i$. The penalized log-likelihood of the joint model for patient $i$ given random effects $\boldsymbol{u}_i$ and smoothing parameters $\sigma_\zeta$, $\sigma_\xi$ is $$\label{eqn:joint_Lik}
l(\boldsymbol{\Theta}, \boldsymbol{\zeta}, \boldsymbol{\xi};\cdot)=l_y(\boldsymbol{\Theta}_y;\boldsymbol{y}_i,\boldsymbol{u}_i, \boldsymbol{\zeta}) + l_s(\boldsymbol{\Theta}_s;t_i,\delta_i, \boldsymbol{u}_i, \boldsymbol{\zeta}, \boldsymbol{\xi}) - \frac{1}{\sigma_\zeta^2} \boldsymbol{\zeta}'\boldsymbol{\zeta} - \frac{1}{\sigma_\xi^2} \boldsymbol{\xi}'\boldsymbol{\xi},$$ where the unknown parameter vector $\boldsymbol{\Theta} = (\boldsymbol{\Theta}'_y, \boldsymbol{\Theta}'_s)'$.
Bayesian inference {#sec:BayesianInf}
------------------
To infer the unknown parameter vector $\boldsymbol{\Theta}$, we use Bayesian inference based on Markov chain Monte Carlo (MCMC) posterior simulations. The fully Bayesian inference has many advantages. First, MCMC algorithms can be used to estimate exact posterior distributions of the parameters, while likelihood-based estimation only produces a point estimate of the parameters, with asymptotic standard errors [@Dunson2007SMMR]. Second, Bayesian inference provides better performance in small samples compared to likelihood-based estimation [@Lee2004MBR]. In addition, it is more straightforward to deal with more complicated models using Bayesian inference via MCMC. We use vague priors on all elements in $\boldsymbol{\Theta}$. Specifically, the prior distributions of parameters $\nu$, $\eta_0$, $\eta_1$, and all elements in vectors $\boldsymbol{\beta}$ and $\boldsymbol{\gamma}$ are $N(0,100)$. We use the prior distribution $b_k\sim\textnormal{Uniform}(0, 10)$, $k=2, \ldots, K$, to ensure positivity. The prior distribution for the difficulty parameter $a_k$ of the continuous outcomes is $a_k\sim N(0, 100)$. To obtain the prior distributions for the threshold parameters of ordinal outcome $k$, we let $a_{k1}\sim N(0, 100)$, and $a_{kl}=a_{k,l-1}+\Delta_l$ for $l=2, \ldots, n_k-1$, with $\Delta_l\sim N(0, 100)I(0,)$, i.e., normal distribution left truncated at $0$. We use the prior distribution $\textnormal{Uniform}[-1,1]$ for all the correlation coefficients $\rho$ in the covariance matrix $\boldsymbol{\Sigma}$, and $\textnormal{Inverse-Gamma}(0.01, 0.01)$ for all variance parameters. We have investigated other selections of vague prior distributions with various hyper-parameters and obtained very similar results.
The posterior samples are obtained from the full conditional of each unknown parameter using Hamiltonian Monte Carlo (HMC) [@Duane1987Physics_Letters] and No-U-Turn Sampler (NUTS, a variant of HMC) [@hoffman-gelman:2013]. Compared with the Metropolis-Hastings algorithm, HMC and NUTS reduce the correlation between successive sampled states by using a Hamiltonian evolution between states and by targeting states with a higher acceptance criteria than the observed probability distribution, leading to faster convergence to the target distribution. Both HMC and NUTS samplers are implemented in [`Stan`]{}, which is a probabilistic programming language implementing statistical inference. The model fitting is performed in [`Stan`]{} (version $2.14.0$) [@stan-manual:2016] by specifying the full likelihood function and the prior distributions of all unknown parameters. For large dataset, [`Stan`]{} may be more efficient than [`BUGS`]{} language [@Lunn2000Winbugs] in achieving faster convergence and requiring smaller number of samples [@hoffman-gelman:2013]. To monitor Markov chain convergence, we use the history plots and view the absence of apparent trends in the plot as evidence of convergence. In addition, we use the Gelman-Rubin diagnostic to ensure the scale reduction $\widehat{R}$ of all parameters are smaller than $1.1$ as well as a suite of convergence diagnosis criteria to ensure convergence [@Gelman2013BDA3]. After fitting the model to the training dataset (the dataset used to build the model) using Bayesian approaches via MCMC, we obtain $M$ (e.g., $M=2,000$ after burn-in) samples for the parameter vector $\boldsymbol{\Theta}_0 = (\boldsymbol{\Theta}', \boldsymbol{\zeta}', \boldsymbol{\xi}')'$. To facilitate easy reading and implementation of the proposed joint model, a [`Stan`]{} code has been posted in the \[sec:WebSupp\]. Note that [`Stan`]{} requires variable types to be declared prior to modeling. The declaration of matrix $\boldsymbol{\Sigma}$ as a covariance matrix ensures it to be positive-definite by rejecting the samples that cannot produce positive-definite matrix $\boldsymbol{\Sigma}$. Please refer to the [`Stan`]{} code in the \[sec:WebSupp\] for details.
Dynamic prediction framework {#sec:IndPred}
----------------------------
We illustrate how to make prediction for a new subject $N$, based on the available outcome histories $\boldsymbol{y}^{\{t\}}_N=\{\boldsymbol{y}_N(t_{Nj}); 0 \le t_{Nj} \le t \}$ and the covariate history $\boldsymbol{X}^{\{t\}}_N=\{\boldsymbol{X}_N(t_{Nj}), \boldsymbol{Z}_N(t_{Nj}),$ $\boldsymbol{W}_N; 0 \le t_{Nj} \le t \}$ up to time $t$, and $\delta_N=0$ (no event). We want to obtain two personalized predictive measures: the longitudinal trajectories $y_{Nk}(t')$, for $k=1, \ldots, K$, at a future time point $t'>t$ (e.g., $t'=t+\Delta t$), and the probability of functional disability before time $t'$, denoted by $\pi_N(t'|t)=p(T^*_N \le t'|T^*_N>t, \boldsymbol{y}^{\{t\}}_N, \boldsymbol{X}^{\{t\}}_N)$. To do this, the key step is to obtain samples for patient $N$’s random effects vector $\boldsymbol{u}_N$ from its posterior distribution $p(\boldsymbol{u}_N|T_N^*>t, \boldsymbol{y}^{\{t\}}_N, \boldsymbol{\Theta}_0)$. Specifically, conditional on the $m$th posterior sample $\boldsymbol{\Theta}_0^{(m)}$, we draw the $m$th sample of the random effects vector $\boldsymbol{u}_N$ from its posterior distribution $$\begin{aligned}
p(\boldsymbol{u}_N|T_N^*>t, \boldsymbol{y}^{\{t\}}_N, \boldsymbol{\Theta}_0^{(m)}) &=& \frac{p(\boldsymbol{y}^{\{t\}}_N,T_N^*>t, \boldsymbol{u}_N|\boldsymbol{\Theta}_0^{(m)})}{p(\boldsymbol{y}^{\{t\}}_N,T_N^*>t|\boldsymbol{\Theta}_0^{(m)})} \propto p(\boldsymbol{y}^{\{t\}}_N, T_N^*>t,\boldsymbol{u}_N|\boldsymbol{\Theta}_0^{(m)}) \nonumber \\
&=& p(\boldsymbol{y}^{\{t\}}_N | \boldsymbol{u}_N, \boldsymbol{\Theta}_0^{(m)}) p(T_N^*>t|\boldsymbol{u}_N, \boldsymbol{\Theta}_0^{(m)}) p(\boldsymbol{u}_N|\boldsymbol{\Theta}_0^{(m)}),\end{aligned}$$ where the first equality is from Bayes theorem.
For each of $\boldsymbol{\Theta}_0^{(m)}$, $m=1, \ldots, M$, we use adaptive rejection Metropolis sampling [@Gilks1995AppStat] to draw 50 samples of random effects vector $\boldsymbol{u}_N$ and retain the final sample. This process is repeated for the $M$ saved values of $\boldsymbol{\Theta}_0$. Suppose that patient $N$ does not develop functional disability by time $t'$, then the outcome histories are updated to $\boldsymbol{y}^{\{t'\}}_N$. We can dynamically update the posterior distribution to $p(\boldsymbol{u}_N|T_N^*>t', \boldsymbol{y}^{\{t'\}}_N, \boldsymbol{\Theta}_0^{(m)})$, draw new samples, and obtain the updated predictions.
With the $M$ samples for patient $N$’s random effects vector $\boldsymbol{u}_N$, predictions can be obtained by simply plugging in realizations of the parameter vector and random effects vector $\{\boldsymbol{\Theta}_0^{(m)}, \boldsymbol{u}_N^{(m)}, m=1,\ldots,M\}$. For example, the $m$th sample of continuous outcome $y_{Nk}(t')$ is obtained from equations and : $$y_{Nk}^{(m)}(t')=a_k^{(m)}+b_k^{(m)}\left\{ \boldsymbol{X}_N(t')\boldsymbol{\beta}^{(m)} + \boldsymbol{Z}_N(t')\boldsymbol{u}_N^{(m)} + \boldsymbol{V}_R(t')\boldsymbol{\zeta}^{(m)} \right\} +\varepsilon_{Nk}^{(m)}(t'),$$ where the random errors $\varepsilon_{Nk}^{(m)}(t') \sim N (0, \sigma_{\varepsilon_k}^{2(m)})$, and each parameter is replaced by the corresponding element in the $m$th sample $\{\boldsymbol{\Theta}_0^{(m)}, \boldsymbol{u}_N^{(m)}\}$.
Similarly, the $m$th sample of ordinal outcome $y_{Nk}(t')=l$ with $l=1, 2, \ldots, n_k$ is $$\textnormal{logit}\big\{p\big(y_{Nk}^{(m)}(t') \le l\big)\big\}=a_{kl}^{(m)} - b_k^{(m)}\big\{ \boldsymbol{X}_N(t')\boldsymbol{\beta}^{(m)} + \boldsymbol{Z}_N(t')\boldsymbol{u}_N^{(m)} + \boldsymbol{V}_R(t')\boldsymbol{\zeta}^{(m)} \big\}.$$ The probability of being in category $l$ is $p\big(y_{ik}^{(m)}(t')=l\big) = p\big(y_{ik}^{(m)}(t') \le l\big) - p\big(y_{ik}^{(m)}(t') \le l-1\big)$. The $m$th sample of the hazard of patient $i$ at time $t'$ is $$h_N^{(m)}(t'|\boldsymbol{u}_N^{(m)}) = h_0^{(m)}(t')\exp\big\{\boldsymbol{W}_N\boldsymbol{\gamma}^{(m)} + \nu^{(m)}\big[\boldsymbol{X}_N(t')\boldsymbol{\beta}^{(m)} + \boldsymbol{Z}_N(t')\boldsymbol{u}_N^{(m)} + \boldsymbol{V}_R(t')\boldsymbol{\zeta}^{(m)}\big] \big\}.$$ Thus, the conditional probability of functional disability before time $t'$ is $$\begin{aligned}
\widehat{\pi}_N(t'|t) &=& \int p(T^*_N \le t'|T^*_N>t, \boldsymbol{y}^{\{t\}}_N, \boldsymbol{X}^{\{t\}}_N, \boldsymbol{u}_N) p(\boldsymbol{u}_N|T^*_N>t, \boldsymbol{y}^{\{t\}}_N, \boldsymbol{X}^{\{t\}}_N )d\boldsymbol{u}_N \nonumber \\
&\approx & \frac{1}{M}\sum_{m=1}^M p\left(T_N^* \le t'|T_N^*>t, \boldsymbol{y}^{\{t\}}_N, \boldsymbol{X}^{\{t\}}_N, \boldsymbol{u}_N^{(m)}\right) \nonumber \\
&=& \frac{1}{M}\sum_{m=1}^M \left\{1-\frac{p(T_N^* > t'|\boldsymbol{y}^{\{t\}}_N, \boldsymbol{X}^{\{t\}}_N, \boldsymbol{u}_N^{(m)})} {p(T_N^* > t|\boldsymbol{y}^{\{t\}}_N, \boldsymbol{X}^{\{t\}}_N, \boldsymbol{u}_N^{(m)})} \right\} \\
&=& \frac{1}{M}\sum_{m=1}^M \left\{1 - \exp\left(-\int_t^{t'}h_N^{(m)}(s|\boldsymbol{u}_N^{(m)})ds\right) \right\},\end{aligned}$$ where the integration with respect to $\boldsymbol{u}_N$ in the first equality is approximated using Monte Carlo method. Note that the truncated power series spline basis expansion in modeling the smooth time function in equation and in modeling the baseline hazard function results in tractable integration not only in the survival function $S_N(t_N)$, but also in the integration of hazard function in the last equality. All prediction results can then be obtained by calculating simple summaries (e.g., mean, variance, quantiles) of the posterior distributions of $M$ samples $\big\{y_{Nk}^{(m)}(t'), m=1,\ldots,M \big\}$. Note that although it may take a few hours to obtain enough posterior samples for the parameter vector $\boldsymbol{\Theta}_0$, it only takes a few seconds to obtain the prediction results for a new subject. Hence, the dynamic prediction framework and the web-based calculator (detailed in Section 4) can provide instantaneous supplemental information for PD clinicians to monitor disease progression.
Assessing predictive performance
--------------------------------
It is essential to assess the performance of the proposed predictive measures. Here, we focus on the probability $\pi(t'|t)$. Specifically, we assess the discrimination (how well the models discriminate between patients who had the event from patients who did not) using the receiver operating characteristic (ROC) curve and the area under the ROC curves (AUC) and assess the validation (how well the models predict the observed data) using the expected Brier score (BS).
### Area under the ROC curves
Following the notation in Section \[sec:IndPred\], for any given cut point $c\in(0,1)$, the time-dependent sensitivity and specificity are defined as $\textnormal{sensitivity}(c,t,t'):P\left\{\pi_i(t'|t) > c|N_i(t,t')=1, T_i^*>t\right\}$ and $\textnormal{specificity}(c,t,t'):P\left\{\pi_i(t'|t) \le c|N_i(t,t')=0, T_i^*>t\right\}$, respectively, where $N_i(t,t')=I(t < T_i^* \le t')$, indicating whether there is an event (case) or no event (control) observed for subject $i$ during the time interval $(t, t']$. In the absence of censoring, sensitivity and specificity can be simply estimated from the empirical distribution of the predicted risk among either cases or controls. To handle censored event times, [@Li2016SMMR] proposed an estimator for the sensitivity and specificity based on the predictive distribution of the censored survival time: $$\begin{aligned}
\label{eqn:simpleROC}
\widehat{P}\left\{\pi_i(t'|t) > c|N_i(t,t')=1, T_i^*>t\right\} = \frac{\sum_{i=1}^{n}\widehat{W}_i(t,t') I\{\widehat{\pi}_i(t'|t) > c \}}{\sum_{i=1}^{n}\widehat{W}_i(t,t')} \\
\widehat{P}\left\{\pi_i(t'|t) \le c|N_i(t,t')=0, T_i^*>t\right\} = \frac{\sum_{i=1}^{n}[1-\widehat{W}_i(t,t')] I\{\widehat{\pi}_i(t'|t) \le c \}}{\sum_{i=1}^{n}[1-\widehat{W}_i(t,t')]}, \nonumber\end{aligned}$$ where $\widehat{W}_i(t, t')$ is the weight to account for censoring and it is defined as $$\begin{aligned}
\widehat{W}_i(t, t') &=& I(t<t_i \le t')\delta_i + I(t<t_i \le t')(1-\delta_i)P\{T_i^* < t'|T_i^* \ge t_i, \widehat{\pi}_i(t'|t)\} \nonumber \\
&=& I(t<t_i \le t')\delta_i + I(t<t_i \le t')(1-\delta_i)\left[1-\frac{P\{T_i^* \ge t'|\widehat{\pi}_i(t'|t)\}}{P\{T_i^* \ge t_i|\widehat{\pi}_i(t'|t)\}} \right].\end{aligned}$$ Note that the subjects who have the survival event before time $t$ (i.e., $t_i < t$) have their estimated weight $\widehat{W}_i(t, t')=0$ and thus they play no role in equation . The conditional survival distribution $P\{T_i^* \ge \tilde{t}|\widehat{\pi}_i(t'|t)\}$, where $\tilde{t}$ can be either $t'$ or $t_i$, can be estimated using kernel weighted Kaplan-Meier method with a bandwidth $d$, which can be easily implemented in standard survival analysis software accommodating weighted data: $$P\{T_i^* \ge \tilde{t}|\widehat{\pi}_i(t'|t)\} = \prod_{s \in \Omega, s \le \tilde{t}} \left[1 - \frac{\sum_{i' \neq i}K_d\{\widehat{\pi}_{i'}(t'|t), \widehat{\pi}_i(t'|t)\}I(T_{i'}=s)\delta_{i'}} {\sum_{i' \neq i}K_d\{\widehat{\pi}_{i'}(t'|t), \widehat{\pi}_i(t'|t)\}I(T_{i'}\ge s)} \right],$$ where $\Omega$ is the set of distinct $t_i$’s with $\delta_i=1$ and $K_d$ is the kernel function, e.g., uniform and Gaussian kernels. Specifically, we use uniform kernel in this article.
With the estimation of sensitivity and specificity, the time-dependent ROC curve can be constructed for all possible cut points $c\in(0, 1)$ and the corresponding time-dependent $\textnormal{AUC}(t, t')$ can be estimated using standard numerical integration methods such as Simpson’s rule.
### Dynamic Brier score
The Brier score (BS) developed in survival models can be extended to joint models for prediction validation [@Sene2016SMMR; @Proust2014SMMR]. The dynamic expected BS is defined as $E[(D(t'|t) -\pi(t'|t))^2]$, where the observed failure status $D(t'|t)$ equals to 1 if the subject experiences the terminal event within the time interval $(t, t']$ and 0 if the subject is event free until $t'$. An estimator of BS is $$\widehat{\textnormal{BS}}(t, t') = \frac{1}{N_t} \sum_{i=1}^{N_t} \widehat{G}_i(t, t')\left(D_i(t, t')-\pi_i(t'|t)\right)^2,$$ where $N_t$ is the number of subjects at risk at time $t$, and the weight $\widehat{G}_i(t, t')= \frac{I(t_i>t')}{\hat{S}_0(t')/\hat{S}_0(t)} + \frac{I(t<t_i \le t')\delta_i}{\hat{S}_0(t_i)/\hat{S}_0(t)}$ is to account for censoring with $\hat{S}_0$ denoting the Kaplan-Meier estimate [@Sene2016SMMR].
AUC and BS complement each other by assessing different aspects of the prediction. AUC has a simple interpretation as a concordance index, while BS accounts for the bias between the predicted and true risks. In general, $\textnormal{AUC}=1$ indicates perfect discrimination and $\textnormal{AUC}=0.5$ means no better than random guess, while $\textnormal{BS} = 0$ indicates perfect prediction and $\textnormal{BS} = 0.25$ means no better than random guess. [@Blanche2015Biometrics] provides excellent illustration of AUC and BS.
Application to the DATATOP study {#sec:dataAna}
================================
In this section, we apply the proposed joint model and prediction process to the motivating DATATOP study. For all results in this section, we run two parallel MCMC chains with overdispersed initial values and run each chain for $2,000$ iterations. The first $1,000$ iterations are discarded as burn-in and the inference is based on the remaining $1,000$ iterations from each chain. Good mixing properties of the MCMC chains for all model parameters are observed in the trace plots. The scale reduction $\widehat{R}$ of all parameters are smaller than $1.1$.
In order to validate the prediction and compare the performance of candidate models, we conduct a 5-fold cross-validation, where 4 partitions of the data are used to train the model and the left-out partition is used for validation and model selection. Then we fit the final selected model to the whole dataset, except that 2 patients are set aside for subject-specific prediction purpose. The covariates of interest included in equation are baseline disease duration, baseline age, treatment (active deprenyl only), time, and the interaction term of treatment and time. We allow a flexible and smooth disease progression along time by using penalized truncated power series splines with 7 knots at the location $\boldsymbol{\kappa} = (1.2, 3, 6, 9, 12, 15, 18)$ in months, to ensure sufficient patients within each interval. Specifically, euqation is $$\begin{aligned}
\theta_i(t_{ij}) &=& \beta_{0} + \beta_{1}\textnormal{duration}_i + \beta_{2}\textnormal{age}_i + \beta_{3}\textnormal{trt}_i + \beta_{4}t_{ij} \nonumber \\
&& + \beta_{5}(\textnormal{trt}_i \times t_{ij}) + \sum_{r=1}^{7}\zeta_{r}(t_{ij}-\kappa_{r})_+ + u_{i0} + u_{i1}t_{ij},\end{aligned}$$ where the random effects $(u_{i0}, u_{i1})' \sim N_2(0, \boldsymbol{\Sigma})$ with $\boldsymbol{\Sigma} = \{(\sigma_1^2, \rho\sigma_1\sigma_2), (\rho\sigma_1\sigma_2, \sigma_2^2)\}$ and $\boldsymbol{\zeta} \sim N(0, \sigma_{\zeta}^2\boldsymbol{I})$ to avoid overfitting.
For the survival part, three different formulations are considered as discussed in Section \[sec:model\_MLLTM\]. For instance, the shared latent variable model (Model 1) is $h_i(t) = h_0(t)\exp (\gamma_1\textnormal{duration}_i + \gamma_2\textnormal{age}_i + \gamma_3\textnormal{trt}_i + \nu \theta_i(t))$. The proposed time-dependent slope model (Model 2) and shared random effects model (Model 3) can be obtained by replacing $\nu \theta_i(t)$ with $\nu_1 \theta_i(t)+\nu_2 \theta_i'(t)$ and $\boldsymbol{\nu}' \boldsymbol{u}_i$, respectively. The baseline hazard $h_0(t)$ is similarly approximated by penalized splines $h_0(t) = \exp\{\eta_{0}+\eta_{1}t + \sum_{r=1}^{7}\xi_{r}(t-\kappa_r)_+\}$ and $\boldsymbol{\xi} \sim N(0, \sigma_{\xi}^2\boldsymbol{I})$. In addition, we compared the proposed model with two standard predictive models for time to event data, (1) a widely used univariate joint model (refer to as Model JM), where the continuous UPDRS is used as the longitudinal outcome regressing on same covariates of interest and the survival part is constructed in the same structure, and (2) a naive Cox model adjusted for time-independent covariates including all baseline characteristics as well as UPDRS, HY and SEADL scores.
We compare the performance of all candidate models in terms of discrimination and validation using 5-fold cross-validation and present AUC and BS score in Table \[tab:AUC\_BS\] and Web Table \[tab:S1\]. All of the three formulations of the proposed MLLTM joint model outperform the univariate Model JM (except $\textnormal{AUC}(t=3, t'=9)$) and naive Cox model with larger AUC and smaller BS in most of the scenarios, suggesting that the MLLTM model accounting for multivariate longitudinal outcomes are preferable in terms of prediction. The three formulations have very similar performance with close AUC and BS. Model 1 is selected as our final model, because it leads to a straightforward interpretation of the overall covariate effect described in Section \[sec:model\_MLLTM\] and it is more intuitive to use the trajectory of latent variable $\theta_i(t)$ to predict the time to event as in Model 1, instead of using time-dependent slope $\theta'_i(t)$ or random effects $\boldsymbol{u}_i$ as in Models 2 and 3. The results also suggest that AUC increases by using more follow up measurements, e.g., in Model 1, conditional on the the measurement history up to month 3 (i.e., $t=3$), when $t'=15$, $\textnormal{AUC}(t=3, t'=15)=0.744$, while AUC increase to $\textnormal{AUC}(t=12, t'=15)=0.766$, indicating that conditional on the measurement history up to month 12, our model has 0.766 probabilities to correctly assign higher probability of functional disability by month 15 to more severe patients (who had functional disability earlier) than less severe patients (who had functional disability later). Meanwhile, BS decreases from $\textnormal{BS}(3,15)=0.216$ to $\textnormal{BS}(12,15) = 0.108$, i.e., the mean square error of prediction decreases from 0.216 to 0.108, suggesting better prediction in terms of validation.
[rrcccccccccc]{} & & & & & &\
& & AUC & BS & AUC & BS & AUC & BS & AUC & BS & AUC & BS\
3 & 9 & 0.754 & 0.136 & 0.759 & 0.138 & 0.761 & 0.139 & 0.757 & 0.140 & 0.736 & 0.139\
& 12 & 0.744 & 0.204 & 0.744 & 0.200 & 0.744 & 0.200 & 0.739 & 0.203 & 0.725 & 0.203\
& 15 & 0.744 & 0.216 & 0.742 & 0.212 & 0.744 & 0.211 & 0.726 & 0.218 & 0.719 & 0.212\
& 18 & 0.775 & 0.171 & 0.766 & 0.163 & 0.772 & 0.167 & 0.728 & 0.186 & 0.720 & 0.185\
\
6 & 9 & 0.789 & 0.078 & 0.806 & 0.078 & 0.806 & 0.078 & 0.770 & 0.081 & 0.721 & 0.094\
& 12 & 0.764 & 0.159 & 0.778 & 0.154 & 0.775 & 0.154 & 0.732 & 0.164 & 0.705 & 0.173\
& 15 & 0.763 & 0.183 & 0.771 & 0.178 & 0.771 & 0.178 & 0.725 & 0.194 & 0.697 & 0.194\
& 18 & 0.786 & 0.158 & 0.773 & 0.154 & 0.769 & 0.159 & 0.726 & 0.175 & 0.701 & 0.175\
\
12 & 15 & 0.766 & 0.108 & 0.787 & 0.103 & 0.782 & 0.102 & 0.695 & 0.124 & 0.647 & 0.155\
& 18 & 0.758 & 0.149 & 0.739 & 0.147 & 0.723 & 0.153 & 0.700 & 0.161 & 0.663 & 0.163\
Parameter estimates based on Model 1 are presented in Table \[tab:inf\] and Web Table \[tab:S2\] (outcome-specific parameters only). To illustrate the subject-specific predictions, we set aside two patients from the DATATOP study and predict their longitudinal trajectories as well as the probability of functional disability at a clinically relevant future time point, conditional on their available measurements. A more severe Patient 169 with clinically worse longitudinal measures and earlier development of functional disability as well as a less severe Patient 718 are selected. Patient 169 had 8 visits with mean UPDRS 42.6 (SD 7.7), median HY 2, median SEADL 80, and developed functional disability at month 16. In contrast, Patient 718 had 9 visits with mean UPDRS 15.6 (SD 3.1), median HY 1, median SEADL 95, and was censored at month 21. Figure \[fig:UPDRS\] displays the predicted UPDRS trajectories for these two patients, based on different amounts of data. When only baseline measurements are used for prediction, the predicted UPDRS trajectory is biased with wide uncertainty band. For example, Patient 169 had a relatively low baseline UPDRS value of 33 and our model based only on baseline measurements tends to underpredict the future UPDRS trajectory ($t_i=0$, the first plot in upper panels). However, Patient 169’s higher UPDRS values of 41 and 40 at months 1 and 3, respectively, subsequently shift up the prediction and tend to overpredict the future trajectory ($t_i=3$ months, the second plot in upper panels). By using more follow-up data, predictions are closer to the true observed values and the 95% uncertainty band is narrower ($t_i=6$ or $12$ months, the last two plots in upper panels). Patient 169’s predicted UPDRS values after 12 months are above 40 and increase rapidly, indicating a higher risk of functional disability in the near future. In comparison, the predicted UPDRS values for Patient 718 are relatively stable because his/her observed UPDRS values are relatively stable.
[lrrrr]{} & Mean & SD &\
\
Int & $-$0.738 & 0.338 & $-$1.385 & $-$0.081\
Duration (months) & 0.021 & 0.004 & 0.014 & 0.028\
Age (years) & 0.024 & 0.005 & 0.014 & 0.035\
Trt (deprenyl) & $-$0.108 & 0.099 & $-$0.304 & 0.099\
Time (months) & 0.021 & 0.025 & $-$0.028 & 0.070\
Trt $\times$ Time & $-$0.089 & 0.010 & $-$0.109 & $-$0.071\
$\rho$ & 0.310 & 0.044 & 0.226 & 0.393\
$\sigma_1$ & 1.328 & 0.051 & 1.230 & 1.430\
$\sigma_2$ & 0.116 & 0.006 & 0.104 & 0.128\
$\sigma_{\varepsilon}$ & 5.081 & 0.074 & 4.933 & 5.226\
\
\
Duration (months) & $-$0.009 & 0.004 & $-$0.017 & $-$0.002\
Age (years) & $-$0.034 & 0.006 & $-$0.045 & $-$0.024\
Trt (deprenyl) & $-$0.608 & 0.118 & $-$0.846 & $-$0.375\
$\nu$ & 0.692 & 0.039 & 0.618 & 0.769\
![Predicted UPDRS for Patient 169 (upper panels) and Patient 718 (lower panels). Solid line is the mean of 2000 MCMC samples. Dashed lines are the 2.5% and 97.5% percentiles range of the 2000 MCMC samples. The dotted vertical line represents the time of prediction $t$.[]{data-label="fig:UPDRS"}](UPDRS.eps){width="100.00000%"}
The predicted probability being in each category for outcomes HY and SEADL are presented in Web Figures \[fig:S1\] and \[fig:S2\], respectively. Please refer to the \[sec:WebSupp\] for the interpretation. Besides the predictions of longitudinal trajectories, it is more of clinical interest for patients and clinicians to know the probability of functional disability before time $t'>t$: $\pi_i(t'|t)$, conditional on the patient’s longitudinal profiles up to time $t$ and the fact that he/she did not have functional disability up to time $t$. The predicted probabilities for Patients 169 and 718 based on various amount of data are presented in Figure \[fig:failure\]. A similar pattern is that the prediction becomes more accurate if more data are used. With such predictions, clinicians are able to precisely track the health condition of each patient and make better informed decisions individually. For example, based on the first 12 months’ data, for Patient 169, the predicted probabilities in the next 3, 6, 9 and 12 months are 0.21, 0.46, 0.78 and 0.97 (the last plot of upper panels), while for Patient 718, the probabilities are 0.02, 0.06, 0.13 and 0.30 (the last plot of lower panels). Patient 169 has higher risk of functional disability in the next few months and clinicians may consider more invasive treatments to control the disease symptoms before the functional disability is developed.
![Predicted conditional failure probability for Patient 169 (upper panels) and Patient 718 (lower panels). Solid line is the mean of 2000 MCMC samples. Dashed lines are the 2.5% and 97.5% percentiles range of the MCMC samples.[]{data-label="fig:failure"}](failure.eps){width="100.00000%"}
To facilitate the personalized dynamic predictions in clinical setting, we develop a web-based calculator available at <https://kingjue.shinyapps.io/dynPred_PD>. A screenshot of the user interface is presented in Web Figure \[fig:S3\]. The calculator requires as input the PD patients’ baseline characteristics and their longitudinal outcome values up to the present time. The online calculator will then produce time-dependent predictions of future health outcomes trajectories and the probability of functional disability, in addition to the $95\%$ uncertainty bands. Moreover, additional data generated from more follow-up visits can be input to obtain updated predictions. The calculator is a user friendly and easily accessible tool to provide clinicians with dynamically-updated patient-specific future health outcome trajectories, risk predictions, and the associated uncertainty. Such a translational tool would be relevant both for clinicians to make informed decisions on therapy selection and for patients to better manage risks.
Simulation studies {#sec:simulation}
==================
In this section, we conduct an extensive simulation study to investigate the prediction performance of the probability $\pi(t'|t)$ using the proposed Model 1. We generate 200 datasets with samples size $n = 800$ subjects and six visits, i.e., baseline and five follow-up visits ($J_i=6$), with the time vector $\boldsymbol{t}_i = (t_{i1}, t_{i2}, \ldots, t_{i6})' = (0, 3, 6, 12, 18, 24)$. The simulated data structure is similar to the motivating DATATOP study, and it includes one continuous outcome and two ordinal outcomes (each with 7 categories).
Data are generated from the following models: $\theta_i(t_{ij}) = \beta_0 + \beta_1x_{i1} + \beta_2t_{ij} + \beta_3x_{i1}t_{ij} + u_{i0} + u_{i1}t_{ij}$ and $h_i(t) = h_0\exp\{\gamma x_{i2} + \nu \theta_i(t)\}$, where the longitudinal and survival submodels share the latent variable as in proposed Model 1. Covariate $x_{i1}$ takes value 0 or 1 each with probability 0.5 to mimic treatment assignment and covariate $x_{i2}$ is randomly sampled integer from 30 to 80 to mimic age. We set coefficients $\boldsymbol{\beta} = (\beta_0, \beta_1, \beta_2, \beta_3)' = (-1, -0.2, 0.8, -0.2)'$, $\gamma=-0.12$ and $\nu=0.75$. For simplicity, baseline hazard is assumed to be constant with $h_0 = 0.1$. Parameters for the continuous outcome are $a_1 = 15$, $b_1=7$ and $\sigma_\varepsilon = 5$. Parameters for the ordinal outcomes are $\boldsymbol{a}_2 = (0, 1, 2, 4, 5, 6)$, $\boldsymbol{a}_3 = (-1, 1, 3, 4, 6, 8)$, $b_2=1$ and $b_3=1.2$. We assume that random effects vector $\boldsymbol{u}_i = (u_{i0}, u_{i1})'$ follows a multivariate normal distribution $N_2(0, \boldsymbol{\Sigma})$, where $\boldsymbol{\Sigma} = \{(\sigma_1^2, \rho\sigma_1\sigma_2), (\rho\sigma_1\sigma_2, \sigma_2^2) \}$ with $\sigma_1=1.5$, $\sigma_2=0.15$ and $\rho=0.4$. The independent censoring time is sampled from $\textnormal{Uniform}(10, 24)$.
From each simulated dataset, we randomly select 600 subjects as the training dataset and set aside the remaining 200 subjects as the validation dataset. Web Table \[tab:S3\] displays bias (the average of the posterior means minus the true values), standard deviation (SD, the standard deviation of the posterior means), coverage probabilities (CP) of 95% equal tail credible intervals (CI), and root mean squared error (RMSE) of model inference based on the training dataset. The results suggest that the model fitting based on the training dataset provides parameter estimates with very small biases and RMSE and the CP being close to the nominal level 0.95. Using MCMC samples from the fitted model and available measurements up to time $t$, we make prediction of $\pi_i(t'|t)$ for each subject in the validation dataset.
Web Table \[tab:S4\] compares the time-dependent AUC based on various amount of data from Model 1, Model JM and naive Cox model. When 3 or 6 months data are available, Model 1 outperforms Model JM and Cox with high discriminating capability and higher AUC values above 0.9. In general, AUC is increasing with more available data, e.g., $\textnormal{AUC}(3, 12) = 0.920$ and $\textnormal{AUC}(6, 12) = 0.930$.
From each of the 200 simulation datasets, we randomly select 20 subjects to plot the bias between the predicted event probability $\pi(t'|t)$ from Model 1 and the true event probability with $t'=9$ (upper panels) and $t'=12$ (lower panels) in Web Figure \[fig:S4\]. When more data are available, bias is decreasing as more bias is within the region of $[-0.2, 0.2]$. For example, with only baseline data, 5.8% and 21.7% of bias for the predictions of $\pi(t'=9|t=0)$ and $\pi(t'=12|t=0)$, respectively, are outside the range. With up to three months’ data, 3.4% and 13.7% of bias for the predictions of $\pi(t'=9|t=3)$ and $\pi(t'=12|t=3)$, respectively, are outside the range. With up to six months’ data, the prediction is precise with only 1.2% and 7.7% of bias for the prediction of $\pi(t'=9|t=6)$ and $\pi(t'=12|t=6)$, respectively, being outside the range.
Discussion {#sec:discussion}
==========
Multiple longitudinal outcomes are often collected in clinical trials of complex diseases such as Parkinson’s disease (PD) to better measure different aspects of disease impairment. However, both theoretical and computational complexity in modeling multiple longitudinal outcomes often restrict researchers to a univariate longitudinal outcome. Without careful analysis of the entire data, pace of treatment discovery can be dramatically slowed down.
In this article, we first propose a joint model that consists of a semiparametric multilevel latent trait model (MLLTM) for the multiple longitudinal outcomes by introducing a continuous latent variable to represent patients’ underlying disease severity, and a survival submodel for the event time data. The latent variable modeling effectively reduces the number of outcomes and has improved computational feasibility and model interpretability. Next we develop the process of making personalized dynamic predictions of future outcome trajectories and risks of target event. Extensive simulation studies suggest that the predictions are accurate with high AUC and small bias. We apply the method to the motivating DATATOP study. The proposed joint models can efficiently utilize the multivariate longitudinal outcomes of mixed types, as well as the survival process to make correct predictions for new subjects. When new measurements are available, predictions can be dynamically updated and become more accurate and efficient. A web-based calculator is developed as a supplemental tool for PD clinicians to monitor their patients’ disease progression. For subjects with high predicted risk of functional disability in the near future, clinicians may consider more targeted treatment to defer the initiation of levodopa therapy because of its association with motor complications and notable adverse events [@Brooks2008NDT]. Although the dynamic prediction framework has utilized only three longitudinal outcomes in the DATATOP study, it can be broadly applied to similar studies with more longitudinal outcomes.
There are some limitations in our proposed dynamic prediction framework that we will address in the future study. First, the semiparametric MLLTM submodel assumes a univariate latent variable (unidimensional assumption), which may be reasonable for small number of outcomes. However, for large number of longitudinal outcomes, multiple latent variables may be required to fully represent the true disease severity across different domains impaired by PD. We will develop a multidimensional latent trait model that allows multiple latent variables. Second, [@Proust-Lima2013BJMSP] and [@Proust2016SIM] proposed a flexible multivariate longitudinal model that can handle mixed outcomes, including bounded and non-Gaussian continuous outcomes. In contrast, our model only applied to normally distributed continuous outcomes. In our future research, we would like to extend the dynamic prediction framework to accommodate more general continuous outcomes including bounded and non-Gaussian variables. Third, we have chosen multivariate normal distribution for the random effects vector because it is flexible in modeling the covariance structure within and between longitudinal measures of patients and it has meaningful interpretation on correlation. In fact, misspecification of random effects and residuals has little impact on the parameters that are not associated with the random effects [@Jacqmin2007CSDA; @Rizopoulos2008Biometrika; @Mcculloch2011SS]. The impact of misspecification in the proposed modeling framework warrants further investigation. Alternatively, we will relax the normality assumption by considering Bayesian non-parametric (BNP) framework based on Dirichlet process mixture [@Escobar1994JASA].
Equation for ordinal outcome requires the proportional odds assumption. Statistical tests to evaluate this assumption in the traditional ordinal logistic regression have been criticized for having a tendency to reject the null hypothesis, when the assumption holds [@Harrell2015Book]. Tests of the proportional odds assumption in the longitudinal latent variable setting are not well established, and the consequence of violating the assumption is unclear and is worth future examination. Three different functional forms of joint models that allow various association between the longitudinal and event time responses are examined and they provide comparable predictions in the DATATOP study. Instead of selecting a final model in terms of simplicity and easy interpretation, a Bayesian model averaging (BMA) approach to combine joint models with different association structures [@Rizopoulos2014JASA] will be investigated in future study. In addition, missed visits and missing covariates exist in the DATATOP study. In this article, we assume that they are missing at random (MAR). However, the missing data issue becomes more complicated in prediction model framework because it can impact both the model inference (missing data in the training dataset) and dynamic prediction process (e.g., the new subject only has measurements of UPDRS and HY, but not SEADL). How to address this issue in the proposed prediction framework is an important direction of future research. Moreover, the online calculator is based on the DATATOP study, which may not represent PD patients at all stages and from all populations. Nonetheless, the large and carefully studied group of patients provide an important resource to study the clinical expression of PD. We will continue to improve the calculator by including more heterogeneous PD patients from different studies.
Acknowledgements {#acknowledgements .unnumbered}
================
Sheng Luo’s research was supported by the National Institute of Neurological Disorders and Stroke under Award Numbers R01NS091307 and 5U01NS043127. The authors acknowledge the Texas Advanced Computing Center (TACC) for providing high-performing computing resources.
Web Supplement {#sec:WebSupp .unnumbered}
==============
----- ------ -- -- -- -- -- -- -- -- -- --
$t$ $t'$
3 9
12
15
18
6 9
12
15
18
12 15
18
$t$ $t'$
3 9
12
15
18
6 9
12
15
18
12 15
18
----- ------ -- -- -- -- -- -- -- -- -- --
: Area under the ROC curve and Brier score (BS) for the DATATOP study.[]{data-label="tab:S1"}
[lrrrr]{} & Mean & SD &\
\
$a_1$ & 17.247 & 0.341 & 16.563 & 17.902\
$b_1$ & 7.624 & 0.207 & 7.251 & 8.035\
\
\
$a_{22}$ & 0.995 & 0.036 & 0.927 & 1.066\
$a_{23}$ & 4.087 & 0.079 & 3.935 & 4.243\
$a_{24}$ & 6.340 & 0.129 & 6.087 & 6.593\
\
\
$a_{31}$ & $-$1.462 & 0.076 & $-$1.610 & $-$1.311\
$a_{32}$ & 0.583 & 0.071 & 0.450 & 0.720\
$a_{33}$ & 3.008 & 0.088 & 2.838 & 3.181\
$a_{34}$ & 3.860 & 0.096 & 3.679 & 4.051\
$a_{35}$ & 6.020 & 0.132 & 5.770 & 6.283\
$a_{36}$ & 6.851 & 0.151 & 6.558 & 7.150\
$a_{37}$ & 8.474 & 0.203 & 8.082 & 8.874\
$b_3$ & 1.270 & 0.045 & 1.187 & 1.363\
Predicted Probability for Ordinal Outcomes {#predicted-probability-for-ordinal-outcomes .unnumbered}
==========================================
The predicted probability being in each category for outcome HY is presented in Figure \[fig:S1\]. For example, Patient 169 had HY measurements equal to $2$ at all visits. When only the baseline data are used for prediction (the first plot in upper panels), our model tends to underpredict the disease progression by assigning sizable probabilities to the less severe HY categories 1 and 1.5 even at the end of the study, possibly due to low baseline UPDRS value of 33. After month 3 visit (the second plot in upper panels), our model overpredicts disease progression by assigning abnormally high probability to the severe category 3, possibly due to higher UPDRS values at months 1 and 3. However, using the first 6 or 12 months’ data (the last two plots in upper panels), our model has good fit by correctly assigning the largest posterior probability to HY category 2 for all visits from baseline to month 12. Moreover, our model properly assigns higher probabilities to more severe categories 2.5 and 3 and negligible probabilities to less severe categories 1 and 1.5 for visits after month 12, due to the deteriorating UPDRS measure. Similar interpretation can be made to the predicted probability of being in each SEADL category displayed in Figure \[fig:S2\].
![Predicted probability of being in each HY category for Patient 169 (upper panels) and Patient 718 (lower panels). Patient 169 had HY measurements equal to $2$ at all 8 visits at months 0, 1, 3, 6, 9, 12, 15, and 16, while Patient 718 had HY measurements equal to $1$ at all 9 visits at months 0, 1, 3, 6, 9, 12, 15, and 18. []{data-label="fig:S1"}](HY.eps){width="100.00000%"}
![Predicted probability of SEADL to be observed in a given category for Subject 169 (upper panels) and Subject 718 (lower panels). Observed categories of SEADL for Subject 169 in the 8 follow-up visits are 90, 80, 80, 90, 80, 80, 80, 80 and for Subject 718 in the 9 visits are 95, 95, 95, 95, 90, 95, 95, 95, 95. []{data-label="fig:SEADL"}](SEADL.eps){width="100.00000%"}
\[fig:S2\]
[lrrrr]{} & BIAS & SD & CP & RMSE\
\
$\beta_0=-1$ & 0.007 & 0.114 & 0.945 & 0.114\
$\beta_1=-0.2$ & $-$0.010 & 0.118 & 0.970 & 0.118\
$\beta_2=0.8$ & 0.003 & 0.022 & 0.970 & 0.022\
$\beta_3=-0.2$ & $-$0.001 & 0.015 & 0.940 & 0.015\
$\sigma_1=1.5$ & 0.009 & 0.060 & 0.950 & 0.060\
$\sigma_2=0.15$ & 0.000 & 0.007 & 0.960 & 0.007\
$\rho=0.4$ & $-$0.003 & 0.048 & 0.935 & 0.048\
\
\
$\gamma=-0.12$ & $-$0.001 & 0.007 & 0.950 & 0.007\
$\nu=0.75$ & 0.005 & 0.044 & 0.930 & 0.044\
\
\
$a_1=15$ & $-$0.035 & 0.471 & 0.955 & 0.471\
$b_1=7$ & $-$0.024 & 0.183 & 0.960 & 0.184\
$\sigma_\varepsilon=5$ & $-$0.000 & 0.099 & 0.960 & 0.099\
\
$a_{22}=1$ & 0.004 & 0.066 & 0.925 & 0.066\
$a_{23}=2$ & 0.014 & 0.089 & 0.930 & 0.090\
$a_{24}=4$ & 0.028 & 0.124 & 0.940 & 0.127\
$a_{25}=5$ & 0.040 & 0.148 & 0.920 & 0.153\
$a_{26}=6$ & 0.038 & 0.169 & 0.915 & 0.173\
\
$a_{31}=-1$ & 0.004 & 0.106 & 0.950 & 0.106\
$a_{32}=1$ & 0.001 & 0.110 & 0.940 & 0.110\
$a_{33}=3$ & 0.011 & 0.131 & 0.950 & 0.132\
$a_{34}=4$ & 0.012 & 0.144 & 0.960 & 0.144\
$a_{35}=6$ & 0.023 & 0.194 & 0.930 & 0.195\
$a_{36}=8$ & 0.022 & 0.232 & 0.950 & 0.233\
$b_3=1.2$ & $-$0.000 & 0.040 & 0.965 & 0.040\
[rrcccccccc]{} $t$ & $t'$ & & Model 1 & & Model JM & & Cox & & True AUC\
3 & 9 & & 0.922 & & 0.909 & & 0.892 & & 0.934\
& 12 & & 0.920 & & 0.908 & & 0.875 & & 0.943\
& 15 & & 0.915 & & 0.903 & & 0.853 & & 0.952\
& 18 & & 0.907 & & 0.896 & & 0.830 & & 0.959\
\
6 & 9 & & 0.926 & & 0.911 & & 0.883 & & 0.930\
& 12 & & 0.930 & & 0.915 & & 0.868 & & 0.940\
& 15 & & 0.932 & & 0.916 & & 0.847 & & 0.950\
& 18 & & 0.930 & & 0.914 & & 0.825 & & 0.958\
![A screenshot of the web-based calculator for prediction.[]{data-label="fig:S3"}](web_calculator.eps){width="100.00000%"}
![Bias between the predicted failure probability $\widehat{\pi}_i(t'|\boldsymbol{y}_i^{\{t\}}, \boldsymbol{X}_i^{\{t\}})$ with true failure probability when $t'=9$ (upper panels) and $t'=12$ (lower panels) for 20 randomly selected subjects from each of the 200 simulation datasets.[]{data-label="fig:S4"}](sim_bias.eps){width="100.00000%"}
[`Stan`]{} code for the simulation study {#stan-code-for-the-simulation-study .unnumbered}
========================================
data {
int<lower=0> N_train; // Number of subjects in training data
int<lower=0> obs; // Number of observations
int subject[obs]; // Subject ID
int<lower=0> K_ordi; // number of ordinal outcomes
real Y_conti[obs];
int<lower=0> Y_ordi[obs, K_ordi];
int<lower=0> n_ordi; // Number of categories for ordinal outcomes
vector[2] zero;
real<lower=0> time[obs];
int<lower=0> treat[obs];
int<lower=0> treat_pts[N_train];
int<lower=0, upper=100> age_pts[N_train];
real tee[N_train]; // Survival time
int<lower=0> event[N_train]; // Censoring indicator
}
parameters {
vector<lower=-10, upper=10>[2] beta0;
vector<lower=-10, upper=10>[2] beta1;
vector[2] U[N_train];
real<lower=0> var1;
real<lower=0> var2;
real<lower=-1, upper=1> rho;
real<lower=0> var_conti;
real gamma;
real nu;
real h0;
real a_conti;
real<lower=0> b_conti;
real a_ordi_temp;
real<lower=0> b_ordi_temp;
vector<lower=0>[n_ordi-2] delta[K_ordi];
}
transformed parameters {
real<lower=0> sig1;
real<lower=0> sig2;
cov_matrix[2] Sigma_U;
real<lower=0> sd_conti;
vector[n_ordi-1] a_ordi[K_ordi];
vector<lower=0>[K_ordi] b_ordi;
real theta[obs];
real mu_conti[obs];
real<lower=0, upper=1> psi[obs, K_ordi, n_ordi];
vector<lower=0, upper=1>[n_ordi] prob_y[obs, K_ordi];
// construct the latent variable theta
for (i in 1:obs)
theta[i] <- beta0[1] + beta0[2]*treat[i] + U[subject[i], 1] +
(beta1[1] + beta1[2]*treat[i] + U[subject[i], 2])*time[i];
// construct the means for the continuous variables
for (i in 1:obs)
mu_conti[i] <- a_conti + b_conti*theta[i];
// construct the probability vector for the remaining ordinal variables
a_ordi[1, 1] <- 0;
for (l in 2:(n_ordi-1)) a_ordi[1, l] <- a_ordi[1, l-1] + delta[1, l-1] ;
for (k in 2:K_ordi) {
a_ordi[k, 1] <- a_ordi_temp;
for (l in 2:(n_ordi-1)) a_ordi[k, l] <- a_ordi[k, l-1] + delta[k, l-1];
}
b_ordi[1] <- 1;
for (k in 2:K_ordi) b_ordi[k] <- b_ordi_temp;
for (i in 1:obs) {
for (k in 1:K_ordi) {
for (l in 1:(n_ordi-1)) {
psi[i, k, l] <- inv_logit(a_ordi[k, l] - b_ordi[k]*theta[i]);
}
psi[i, k, n_ordi] <- 1;
prob_y[i, k, 1] <- psi[i, k, 1];
for (l in 2:n_ordi) {prob_y[i, k, l] <- psi[i, k, l] - psi[i, k, l-1];}
}
}
sd_conti <- sqrt(var_conti);
sig1 <- sqrt(var1);
sig2 <- sqrt(var2);
// construct the variance-covariance matrix
Sigma_U[1,1] <- sig1*sig1;
Sigma_U[1,2] <- rho*sig1*sig2;
Sigma_U[2,1] <- Sigma_U[1,2];
Sigma_U[2,2] <- sig2*sig2;
}
model {
real h[N_train];
real S[N_train];
real LL[N_train];
Y_conti ~ normal(mu_conti, sd_conti);
for (i in 1:obs) {
for (k in 1:K_ordi) {
Y_ordi[i, k] ~ categorical(prob_y[i, k]);
}
}
// construct random effects
U ~ multi_normal(zero, Sigma_U);
// construct survival part
for (i in 1:N_train) {
h[i] <- exp(gamma*age_pts[i] + nu*(beta0[1] + beta0[2]*treat_pts[i] + U[i, 1] +
(beta1[1] + beta1[2]*treat_pts[i] + U[i, 2])*tee[i]))*h0;
S[i] <- exp(-h0*exp(gamma*age_pts[i]+nu*(beta0[1]+beta0[2]*treat_pts[i]+U[i, 1])) *
(exp(nu*(beta1[1]+beta1[2]*treat_pts[i]+U[i, 2])*tee[i])-1) / (nu*(beta1[1]+beta1[2]*treat_pts[i]+U[i, 2])));
LL[i] <- log(pow(h[i],event[i])*S[i]); // event=1 for event; 0 for censored
}
increment_log_prob(LL);
// construct the priors
beta0 ~ normal(0, 10);
beta1 ~ normal(0, 10);
var1 ~ inv_gamma(0.01, 0.01);
var2 ~ inv_gamma(0.01, 0.01);
rho ~ uniform(-1, 1);
var_conti ~ inv_gamma(0.01, 0.01);
h0 ~ gamma(0.01, 0.01);
nu ~ normal(0, 10);
gamma ~ normal(0, 10);
for (i in 1:(n_ordi-2)) delta[1, i] ~ normal(0, 10) T[0,] ;
for (k in 2:K_ordi) {
b_ordi_temp ~ uniform(0, 10);
a_ordi_temp ~ normal(0, 10);
for (i in 1:(n_ordi-2)) delta[k, i] ~ normal(0, 10) T[0,] ;
}
}
Full Conditionals {#full-conditionals .unnumbered}
=================
For illustration purpose, we assume that there are one continuous outcome (denoted by $y_{i1}(t)$) and two ordinal outcomes (denoted by $y_{i2}(t)$ and $y_{i3}(t)$, respectively), while model (3) is formulated as $\theta_{i}(t)=\boldsymbol{X}_{i}(t)\boldsymbol{\beta} + \boldsymbol{Z}_{i}(t)\boldsymbol{u}_i $. Assuming non-informative prior distribution for the parameter vector $\boldsymbol{\Theta}$, denoted by $f(\boldsymbol{\Theta})$, the joint likelihood is $$\begin{aligned}
& & L(\boldsymbol{\Theta};\cdot) = p(\boldsymbol{y}|\boldsymbol{u}) p(\boldsymbol{u}) f(\boldsymbol{\Theta}) \\
& \propto & \prod_{i=1}^I \bigg\{\prod_{j=1}^{J_i} p\big[Y_{i1}(t_{ij}) = y_{i1}(t_{ij})\big] p\big[Y_{i2}(t_{ij}) = y_{i2}(t_{ij})\big] p\big[Y_{i3}(t_{ij}) = y_{i3}(t_{ij})\big] \bigg\} \big\{h_i(t_i)^{\delta_i} S_i(t_i) \big\} p(\boldsymbol{u}_i) \\
& = & \prod_{i=1}^I L_{y_1} L_{y_2} L_{y_3} L_S \cdot p(\boldsymbol{u}_i), \\\end{aligned}$$ where
L\_[y\_1]{} &= \_[j=1]{}\^[J\_i]{} {- }, &&\
L\_[y\_k]{} &= \_[j=1]{}\^[J\_i]{} \_[l=1]{}\^[n\_k]{} p\^[I\[Y\_[ik]{}(t\_[ij]{})=l\]]{} &&\
&= \_[j=1]{}\^[J\_i]{} \_[l=1]{}\^[n\_k]{} {p- p}\^[I\[Y\_[ik]{}(t\_[ij]{})=l\]]{}\
&= \_[j=1]{}\^[J\_i]{}, k=2,3, &&\
L\_S &= {h\_0(t\_i)}\^[\_i]{} , &&\
p(\_i) &= , &&\
[expit]{.nodecor}() &= . &&
The full conditionals of all parameters are
1. $f(a_1|\textnormal{others}) \propto N\left(\frac{\sum_{i=1}^{I}\sum_{j=1}^{J_i}\big[y_{i1}(t_{ij}) - b_1\theta_i(t_{ij})\big] }{N_T}, \frac{\sigma_\varepsilon^2}{N_T} \right);$
2. $f(b_1|\textnormal{others}) \propto N\left( \frac{\sum_{i=1}^{I}\sum_{j=1}^{J_i}\big[ y_{i1}(t_{ij}) - a_1\big ]\theta_i(t_{ij}) }{\sum_{i=1}^{I}\sum_{j=1}^{J_i} \theta_i(t_{ij})^2}, \frac{\sigma_\varepsilon^2}{\sum_{i=1}^{I}\sum_{j=1}^{J_i} \theta_i(t_{ij})^2} \right);$
3. $f(\frac{1}{\sigma_\varepsilon^2}|\textnormal{others}) \propto \textnormal{Gamma}\left( \frac{N_T}{2}+1, \frac{\sum_{i=1}^{I}\sum_{j=1}^{J_i}\big[y_{i1}(t_{ij}) - a_1 - b_1\theta_i(t_{ij})\big]^2 }{2} \right);$
4. $[\boldsymbol{a}_{2}, b_2 |\textnormal{others}] \propto \prod_{i=1}^I L_{y_2};$
5. $[\boldsymbol{a}_{3}, b_3 |\textnormal{others}]\propto \prod_{i=1}^I L_{y_3};$
6. $[\boldsymbol{\beta} |\textnormal{others}] \propto \prod_{i=1}^I L_{y_1} L_{y_2} L_{y_3} L_S;$
7. $[\boldsymbol{\gamma}, \nu |\textnormal{others}] \propto \prod_{i=1}^I L_S;$
8. $[\boldsymbol{u}_i|\textnormal{others}] \propto \bigg\{\prod_{j=1}^{J_i} p\big[Y_{i1}(t_{ij}) = y_{i1}(t_{ij})\big] p\big[Y_{i2}(t_{ij}) = y_{i2}(t_{ij})\big] p\big[Y_{i3}(t_{ij}) = y_{i3}(t_{ij})\big] \bigg\} L_S \cdot p(\boldsymbol{u}_i);$
9. $[\boldsymbol{\Sigma}|\textnormal{others}] \propto \prod_{i=1}^I p(\boldsymbol{u}_i),$
where $N_T=\sum_{i=1}^{I}J_i$.
[^1]: Corresponding author: Sheng Luo is Associate Professor, Department of Biostatistics, The University of Texas Health Science Center at Houston, 1200 Pressler St, Houston, TX 77030, USA (E-mail: [email protected]; Phone: 713-500-9554).
|
---
abstract: |
It is of some interest to understand how statistically based mechanisms for signal processing might be integrated with biologically motivated mechanisms such as neural networks. This paper explores a novel hybrid approach for classifying segments of sequential data, such as individual spoken works. The approach combines a hidden Markov model (HMM) with a spiking neural network (SNN). The HMM, consisting of states and transitions, forms a fixed backbone with nonadaptive transition probabilities. The SNN, however, implements a biologically based Bayesian computation that derives from the spike timing-dependent plasticity (STDP) learning rule. The emission (observation) probabilities of the HMM are represented in the SNN and trained with the STDP rule. A separate SNN, each with the same architecture, is associated with each of the states of the HMM. Because of the STDP training, each SNN implements an expectation maximization algorithm to learn the emission probabilities for one HMM state.
The model was studied on synthesized spike-train data and also on spoken word data. Preliminary results suggest its performance compares favorably with other biologically motivated approaches. Because of the model’s uniqueness and initial promise, it warrants further study. It provides some new ideas on how the brain might implement the equivalent of an HMM in a neural circuit.\
\
*Keywords:* Sequential data, classification, spiking neural network, STDP, HMM, word recognition
author:
- |
Amirhossein Tavanaei and Anthony S. Maida\
**\
**\
**
title: Training a Hidden Markov Model with a Bayesian Spiking Neural Network
---
Introduction {#intro}
============
In some settings, it is desirable to have a biologically motivated approach for classifying segments of sequential data, such as spoken words. This paper examines a novel hybrid approach towards such data classification. The approach uses two components. The first is the hidden Markov model (HMM) [@Bishop2006a] and the second is a biologically motivated spiking neural network (SNN) [@Maass1997a; @Samanwoy2009a] that approximates expectation maximization learning (EM) [@Nessler2009a; @Nessler2013a]. In addition to the intrinsic interest of exploring statistically based biologically motivated approaches to machine learning, the approach is also attractive because of its possible realization on special purpose hardware for brain simulation [@Modha2014a] as well as fleshing out the details of a large-scale model of the brain [@Eliasmith2012a].
HMMs are widely used for sequential data classification tasks, such as speech recognition [@Rabiner1989b]. There have been earlier efforts to build hybrid HMM/neural network models [@Bourlard1988a; @Niles1990a; @Bengio1992a]. In this work, the hybrid approach was motivated by the insight that ANNs perform well for non-temporal classification and approximation while HMMs are suitable for modeling the temporal structure of the speech signal. More recent work has used more powerful networks, such as deep belief networks and deep convolutional networks, for acoustic modeling of the speech signal [@Hinton2011a; @Abdel-Hamid2012a; @Abdel-Hamid2013a; @Sainath2013a]. While these efforts have met with considerable practical success, they are not obviously biologically motivated. In part, our work differs from the previous work in that we use a biologically motivated SNN.
Formally, an HMM consists of a set of discrete states, a state transition probability matrix, and a set of emission (observation) probabilities associated with each state. The set of trainable parameters in an HMM can be the initial state probabilities, the transition probabilities, and the emission probabilities. This paper limits itself to training the emission probabilities using an SNN. This is consistent the approaches of the above-mentioned earlier work.
Our work is directly influenced by the important prior work on how an HMM might be implemented in a cortical microcircuit was performed by [@Kappel2014a]. The cortical microcircuit is a repeated anatomical motif in the neocortex who some have argued is the next functional level of description above the single neuron [@Mountcastle1997a]. In its most simplified form, the microcolumn can be modeled as a recurrent neural network with lateral inhibition. Kappel et al. [@Kappel2014a] have recently shown that, with appropriate learning assumptions, a trainable HMM can be realized within this microcircuit. The contribution of the present work is to unwrap this microcircuit into a more discernable HMM. The motivation for our approach is to recognize the fact that there are many ways to potentially realize an HMM in the brain and we seek a model that may be developed in future work but that does not burn any bridges or make unnecessary commitments.
The motivation for this study is not so much to build the highest performing HMM-based classifier as it is to imagine how: 1) an HMM might be realized in the brain, and 2) be implemented in brain-like hardware.
Background {#sec:1}
==========
Since this research combines spike timing-dependent plasticity (STDP) learning with HMM classifiers, the next subsections provide background on each topic.
Spike timing-dependent plasticity
---------------------------------
The phenomenon of STDP learning in the brain has been known for at least two decades [@Markram2011a]. STDP modifies the connection strengths between neurons at their contact points (synapses). Spikes travel from the presynaptic neuron to the postsynaptic neuron via synapses. The strength of the synapse, represented by a scalar weight, modulates the likelihood of a presynaptic spike event causing a postsynaptic event. The weight of a synapse can be modified (plasticity) by using learning rules that incorporating information locally available at the synapse (for example, STDP).
Generically, an STDP learning rule operates as follows. If the presynaptic neuron fires briefly before the postsynaptic neuron, then the synaptic weight is strengthened. If the opposite happens, the synaptic weight is weakened. Such phenomena have been experimentally observed in many brain areas [@Dan2006a; @Corporale2008a]. A simple intuitive interpretation of this empirically observed constraint is that the synaptic strength is increased when the presynaptic neuron could have played a causal role in the firing of the postsynaptic neuron. The strength is weakened if causality is violated.
Probabilistic interpretations of STDP that could form a theoretical link to machine learning have emerged in the past decade. Most relevant to this paper are the following. Nessler et al. [@Nessler2009a; @Nessler2013a] developed a version of STDP to compute EM within a spiking neural circuit. Building on this, Kappel et al. [@Kappel2014a] built an HMM within a recurrent SNN. The recurrent SNN coded for all of the states in the HMM as well as implementing the learning. This was a significant hypothesis from a brain-simulation because of its very strong claim, that a cortical microcircuit may implement a full-blown HMM. The hypothesis is also highly, perhaps overly, committed from an engineering perspective.
The present approach seeks to use the right tool for the right job while still linking it to a biomorphic framework. Specifically, we encode states using an HMM but associate a separate copy of the modified version of a trainable Nessler-type SNN with each state. The purpose of the SNN is to learn the emission (observation) probabilities for that state. In future work, one may find other effective ways to fully encode an HMM model as an SNN, but it may not necessarily be the approach taken in [@Kappel2014a].
Hidden Markov model {#subsec:HMM}
-------------------
Successive observations of sequential data, such as occurs in speech spectrograms, are highly correlated. The correlation often drops significantly between observations that are sequentially distant. An effective way to classify sequential data is to use a markov chain of latent variables (states), otherwise known as an HMM. The HMM describes the data as a first-order Markov chain that assumes the probability of the next state is independent of all of the previous states, given the current state. Fig. \[fig:HMMDiagram\] shows a four-state, left-to-right HMM, whose initial state is . In the figure, arrows entering nodes that are labeled “” are state-transition probabilities and arrows entering nodes labeled “” represent the causal relationship between a state and an observation.
The HMM allows calculation of the probability of a given observation sequence of feature vectors $\mathbf{O} = [\mathbf{o}_1 \ldots \mathbf{o}_M]$, in a structure consisting of states $\mathbf{S} = \{s_1 \ldots s_P\}$, initial state probabilites $\mathbf{\Gamma}$, state transition probabilities $\mathbf{A}$, and emission probabilities $\mathbf{B}$. The probability of a particular observation sequence is given by $$\mathit{Pr}(\mathbf{O} | \boldsymbol\lambda) = \prod_{m=1}^{M} \mathit{Pr}(\mathbf{o}_m | s_m, \boldsymbol\lambda) ,
\label{eqn:mainHMM}$$
where $\boldsymbol\lambda$ denotes the set of model parameters and $s_m$ denotes the state of the HMM when the observation occurs. The emission probabilities are given by $\mathit{Pr}(\mathbf{o}|s, \boldsymbol\lambda)$ and these are the parameters learned by the SNN. We will have occasion to use the symbols $s_m$ versus $s_p$. The former means the state of the HMM when observation $m$ occurs. The latter simply means state $p$ of the HMM. To train and adjust the model parameters in an HMM, the expectation maximization (EM) approach (also known as the Baum-Welch algorithm in the HMM) is used [@Rabiner1993a]. In this paper we assumed fixed transition probabilities, $\mathbf{A}$, for all of the states.
Probability Computation
=======================
Bayesian computation
--------------------
In a Bayesian framework, a posterior probability distribution is obtained by multiplying the prior probability with the likelihood of the observation and renormalizing. Recent studies have shown that the prior and likelihood models of observations can be represented by appropriately designed neural networks [@Deneve2008a]. One such network is a spiking winner-take-all (WTA) network. Nessler et al. [@Nessler2009a; @Nessler2013a] showed that a version of the STDP rule embedded in an appropriate SNN can perform Bayesian computations.
Gaussian mixture model
----------------------
The most general representation of the probability distribution function in the HMM state is a finite mixture of the Gaussian distributions (GMM) with mean vector, $\boldsymbol{\mu}$, covariance matrix, $\boldsymbol{\Sigma}$, and mixture coefficients, $\boldsymbol{\pi}$. Each HMM state has its own mixture distribution. The probability of observation $\mathbf{o}_m$ occurring in HMM state $s_p$ is given by
$$\mathit{Pr}(\mathbf{o}_m | s_p) = \sum_{k=1}^K \pi_{k} \cdot \mathcal{N}(\mathbf{o}_m|\boldsymbol{\mu}_k,\mathbf{\Sigma}_k), \;\;\;\; 1\le p\le P$$
where $P$ is the number of HMM states, $\pi_{k}$ is the mixing parameter, and $K$ is the number of distributions in the mixture.
In our model, the emission distributions for the HMM states approximately implement the Gaussian mixture distributions. There is a separate mixture distribution associated with each state, corresponding to a separate SNN. The SNN learns distribution parameters via STDP.
Training Method
===============
The SNN trains the parameters for the emission distributions and each HMM state has a separate SNN as shown in Fig. \[fig:stateTransDiagram\]. Fig. \[fig:SNNarchitecture\] shows the SNN architecture in detail. Additionally, Nessler et al. [@Nessler2013a] showed their STDP learning approximates EM. Therefore, the proposed SNN architecture is able to implement the GMM in each state (described in section 4.2).
Network architecture
--------------------
The SNN has two layers of stochastic units (neurons) that generate Poisson spike trains. The $y$ units in the first layer encode input feature vectors to be classified. The second layer is composed of $z$ units that represent classification categories after the network is trained. The layers are fully feedforward connected from layer $y$ to $z$ by weights trained according to the STDP rule given in the next subsection. Besides the feedforward connections, the $z$ units obey a winner-take-all discipline implemented by a global inhibition signal initiated by any of the $z$ units.
The number of $y$ units in Fig. \[fig:SNNarchitecture\], $N$, shows the feature vector dimension. The $z$ units specify the output neurons detecting the samples in $K$ different clusters. The number of output neurons, $K$, manipulates the model flexibility in controlling the signal variety in one segment (analogous to the number of distributions in a GMM). The number of states, $P$, determines the number of segments in a sequential signal. For example, in spoken word recognition, it can be considered as the number of phoneme bigrams.
The spiking activity of a unit in the $z$ layer is governed by an inhomogeneous Poisson process. The rate parameter for this process is controlled by the postsynaptic potential (PSP) input to the $z$ unit. The PSP represents the sum of the synaptic effects coming into the $z$ unit. The instantaneous firing rate of unit $k$ is given by $r_k(t)$ and is defined below $$r_k(t) = \exp( {\mathit{psp}_k(t)} ).
\label{eq:exp_rk}$$
The $\mathit{psp}$ itself is the sum of the excitatory inputs into $k$ from the $y$ layer and a global inhibitory input. These are denoted respectively as $u_k(t)$ and $I(t)$. Thus, the $\mathit{psp}$ for unit $k$ at time $t$ is $$\mathit{psp}_k(t) = u_k(t) + I(t).
\label{eq:psp_components}$$
$u_k$ encodes the composite stimulus input signal to unit $k$. The input signal is provided by the $y$ units, which encode the input feature vector. The quantity $u_k$ is a linear weighted sum of the excitatory postsynaptic potentials (EPSPs) provided by the $y$ units as shown below $$u_k(t) = w_{k0}(t) + \sum_{i=1}^N \mathit{epsp}_i^k(t) \cdot w_{ki}(t).
\label{eq:exp_uk}$$
There are $N$ ‘excitatory’ units in the $y$ layer. $\mathit{epsp}_i^k$ denotes the component of the EPSP of $k$ that originates with unit $i$ in the $y$ layer. $w_{k0}$ is the bias weight to unit $k$. The $w_{ki}$ are the weights from units in the $y$ layer to unit $k$ in the $z$ layer. The weights and bias are time dependent because their values can change while the network is learning. $\mathit{epsp}_i^k$ is defined by $$\mathit{epsp}_i^k(t) =
\left\{
\begin{array}{ll}
1&\mathrm{if}\; i\;\mathrm{fired\;during\;interval} \;\left[t-\sigma,t\right]\\
0&\mathrm{otherwise}.
\end{array}
\right.
\label{eq:exp_epsp}$$ The quantity $\mathit{epsp}_i^k(t)$ has a value of 1 at time $t$ if and only if unit $i$ has fired in the previous $\sigma$ milliseconds. In the simulations, $\sigma=5$.
Training
--------
### GMM learning approximation by SNN
#### Posterior probability
The Gaussian distribution over the dataset, $\mathbf{y}$, is defined as
$$\mathcal{N}(\mathbf{y}|\boldsymbol{\mu},\boldsymbol{\Sigma})=\frac{1}{(2\pi )^{N/2} |\boldsymbol\Sigma|^{1/2}}e^{-0.5 (\mathbf{y}-\boldsymbol\mu)^T\boldsymbol\Sigma^{-1}(\mathbf{y}-\boldsymbol\mu)}
\label{eq:normal}$$
where $\boldsymbol\Sigma$ and $\boldsymbol\mu$ are covariance matrix and mean vector respectively. In the Gaussian mixture model with $K$ mixtures $z_1 \ldots z_K$, $z_k \in \{0, 1\}$, $\sum z_k = 1$, the probability of a sample, $\mathbf{y}_r$, is derived as follows: $$\mathit{Pr}(\mathbf{y}_r|z_k=1)=\mathcal{N}(\mathbf{y}_r|\boldsymbol\mu_k,\boldsymbol\Sigma_k),
\label{eq:p_y}$$ $$\mathit{Pr}(\mathbf{y}_r|\mathbf{z})=\prod_{k=1}^K \mathcal{N}(\mathbf{y}_r|\boldsymbol\mu_k,\boldsymbol\Sigma_k)^{z_k},
\label{eq:p_ys}$$ $$\mathit{Pr}(\mathbf{y}_r)=\sum_\mathbf{z} \mathit{Pr}(\mathbf{z})\mathit{Pr}(\mathbf{y}_r|\mathbf{z})=\sum_{k=1}^K \pi_k \mathcal{N}(\mathbf{y}_r|\boldsymbol\mu_k,\boldsymbol\Sigma_k),
\label{eq:gmm}$$ $$\mathrm{where} \ \ \sum_k \pi_k=1, \ \ \ 0\leq \pi_k \leq 1.
\label{eq:gmm_constraint}$$ From Eq. \[eq:p\_y\] and Eq. \[eq:gmm\] we have $$\mathit{Pr}(z_k=1|\mathbf{y}_r)=R(z_{kr})= \frac{\pi_k \mathcal{N}(\mathbf{y}_r|\boldsymbol\mu_k,\boldsymbol\Sigma_k)}{\sum_{j=1}^K \pi_j \mathcal{N}(\mathbf{y}_r|\boldsymbol\mu_j,\boldsymbol\Sigma_j)}.
\label{eq:R}$$ The conditional probability $\mathit{Pr}(z_k | \mathbf{y}_r)$ can also be written as $R(z_{kr})$ which represents the responsibility [@Bishop2006a]. To simplify the equations, assume that the samples are independent from each other in which $\mathbf{\Sigma}=\textbf{I}$. So, $$\mathit{Pr}(z_k=1|\mathbf{y}_r)=R(z_{kr})=\frac{\pi_k B e^{-0.5(\mathbf{y}_r-\boldsymbol\mu_k)^T(\mathbf{y}_r-\boldsymbol\mu_k)}}{\sum_{j=1}^K \pi_j B e^{-0.5(\mathbf{y}_r-\boldsymbol\mu_j)^T(\mathbf{y}_r-\boldsymbol\mu_j)}}.
\label{eq:R_simple}$$ The equation above reaches its maximum when $\mathbf{y}_r=\boldsymbol\mu_k$. Now, if we suppose the synaptic weight vector of neuron $k$, $w_{ki},$ ($i=1 \ldots N$) is an $N$ dimensional vector which approximates the $\boldsymbol\mu_k$, the similarity between the sample and mean (negation of distance measure), $-0.5(\mathbf{y}-\boldsymbol\mu)^T(\mathbf{y}-\boldsymbol\mu)$, can be replaced by similarity measure between $\textbf{y}$ and $\textbf{w}$ as $\textbf{w}^T\textbf{y}$ (projection of the sample vector on the weight vector reaches the maximum when they are in a same direction). Thus, $$R(z_{kr})=C\frac{\pi_k e^{\mathbf{w}_k^T \mathbf{y}_r}}{\sum_{j=1}^K \pi_j e^{\mathbf{w}_j^T\mathbf{y}_r}}
\label{eq:R_snn}$$ where $C$ is a constant. $\pi_k$ and $K$ denote the mixture coefficients and number of the mixture distributions, respectively.
#### STDP rule specification
In the training process of the GMM using EM, we have $$\boldsymbol\mu_k^{new}=\frac{\sum_{s=1}^M R(z_{ks})\mathbf{y}_s}{\sum_{s=1}^M R(z_{ks})},
\label{eq:em}$$ $$\pi_k^{new}=\frac{\sum_{s=1}^M R(z_{ks})}{M}
\label{eq:em_pi}$$ where $M$ is the number of training samples. In the proposed SNN, $k$ is the output neuron that has just fired. $\mathbf{w}_k$, which is supposed as $\boldsymbol\mu_k$ in the GMM and already represents the previous samples in this cluster, should be updated based on the new samples. Instead of calculating the average value of the samples (Eq. \[eq:em\]), new synaptic weight, $\mathbf{w}_k^{new}$ (or $\boldsymbol\mu_k^{new}$), is updated by $\mathbf{w}_k+f(\mathbf{y}_r)$ where $f(\mathbf{y}_r)$ has $N$ positive and negative numbers corresponding to $\mathbf{y}_r(i)=1$ and $\mathbf{y}_r(i)=0$ respectively. Thus, the new $\mathbf{w}_k$ is updated using $R(z_{kr})$ (which causes a neuron to fire) and input presynaptic spikes. For this purpose we use a modified version of the Nessler’s (2013) STDP learning rule. STDP is an unsupervised learning rule. Following [@Nessler2013a], weight adjustments occur exactly when some $z$ unit $k$ emits a spike. When a unit $k$ fires, the incoming weights to that unit are subject to learning according to the STDP rule given in Eq. \[eq:NesslerSTDP\_wts\]. For each weight, one of two weight-change events occurs, either LTP (strengthening) or LTD (weakening). The weight values are constrainted to be in the range \[-1 1\]. $$\Delta w_{ki} =
\left\{
\begin{array}{ll}
e^{-w_{ki}+1}-1 & \mathrm{if}\;\mathit{epsp}_i(t^\mathrm{f})=1\\
-1 & \mathrm{otherwise}.
\end{array}
\right.
\label{eq:NesslerSTDP_wts}$$
The first case above describes LTP (positive) and the second case describes LTD (always $-1$).
Another parameter of the GMM is the mixture coefficient $\pi_k$ which is obtained by Eq. \[eq:em\_pi\]. The bias weight of the proposed SNN, $w_{k0}$, represents average firing of the neuron $k$ over data occurrences. Therefore, if the neuron fires, its bias weight increases, otherwise it decreases. For this purpose we use a modified version of the Nessler’s (2013) STDP learning rule analogous to Eq. \[eq:NesslerSTDP\_wts\] as follows: $$\Delta w_{k0}=z_ke^{-w_{k0}+1}-1.
\label{eq:w0}$$
$\Delta w_{ki}$ denotes a weight adjustment that is modulated by another rate parameter $\eta_k$. Specifically,
$$w_{ki}^\mathrm{new} = w_{ki} + \eta_k \Delta w_{ki}.
\label{eq:wNewFormula}$$
That is, there is a rate parameter $\eta_k$ for each $z$ unit with $N_k$ as the number of times the unit has fired, starting with 1. It satisfies the constraint $$\eta_k \propto \frac{1}{N_k}.
\label{eq:etaKformula}$$
By considering the mixture coefficient in the SNN, $\pi_k^{\mathrm{snn}}$, to be defined as follows: $$\pi_k^{\mathrm{snn}}=\frac{e^{w_{k0}}}{D},
\label{eq:pi_snn}$$ $$D=\sum_j e^{w_{j0}}, \nonumber
\label{eq:pi_snn_norm}$$ the constraints on the mixture coefficients in Eq. \[eq:gmm\] are fulfilled. Finally, by combining Eq. \[eq:R\_snn\] and Eq. \[eq:pi\_snn\] we have $$R(z_{kr})=A\frac{\frac{e^{w_{k0}}}{D} e^{\mathbf{w}_k^T\mathbf{y}_r}}{\sum_{j=1}^K \frac{e^{w_{j0}}}{D} e^{\mathbf{w}_j^T\mathbf{y}_r}},
\label{eq:final1}$$ $$\mathit{Pr}_k(z \ \mathrm{fires}|\mathbf{y}_r)=R(z_{kr})=A\frac{e^{\mathbf{w}_k^T\mathbf{y}_r+w_{k0}}}{\sum_{j=1}^K e^{\mathbf{w}_j^T\mathbf{y}_r+w_{j0}}}.
\label{eq:finalgmmsnn}$$
#### Training procedure
Since there is a separate SNN for each state, the observation function $Pr(\mathbf{o}_m | s_m, \boldsymbol\lambda)$ can be trained separately for each state. The training procedure begins with a set of feature vectors and initial weights. Randomly selected feature vectors from the sample to be recognized are presented to the network for some number of training trials.
To the extent that the feature vectors are similar to previous observations, a subset of output neurons fire and the synaptic weights are updated according to Eq. \[eq:NesslerSTDP\_wts\] through Eq. \[eq:wNewFormula\]. A new feature vector, which is different from previous vectors, stimulates a new set of output neurons to fire. This strategy imposes an unsupervised learning method within the SNN to categorize the data in one state.
#### Extracting a probability value from the SNN
We will let $Pr_{\mathrm{snn}_p}(t)$ denote the probability that the SNN input at simulation step $t$ is of the category that the network recognizes. This value is the maximum of the output units after normalization as described below (simplified representation of Eq. \[eq:finalgmmsnn\]) $$Pr_{\mathrm{snn}_p}(t) = \max_{k\in K} \frac{e^{u_k(t)}}{Z},
\label{eq:SNN_output_prob}$$
where $Z$ is a normalizer.
Experiments and Results
=======================
Synthesized spatio-temporal spike patterns
------------------------------------------
This experiment modeled a pattern classification task that used four spatio-temporal spike sub-patterns to build a larger pattern. Each sub-pattern consisted of 80 neurons that simultaneously emitted Poisson spike trains. The duration of all spike trains within a sub-pattern was $T=20$ ms. This was called a spatio-temporal pattern because the 80 neurons compose the spatial dimension [@Dayan2001a]. Target patterns were obtained by concatenating the four sub-patterns.
Examples of the sub-patterns denoted A, B, C, and D are shown in Fig. \[fig:spikeTrainsExp1\]. Different instances of a specific sub-pattern, such as A, will have different spike trains because of the Poisson sampling. Each row shows a spike train for a single neuron. For each sub-pattern, twenty of the neurons fire at 340 Hz and the remaining sixty neurons fire at 50 Hz. The high frequency spike trains were deemed information-containing and the low-frequency spike trains were considered background noise.
#### Training phase
Only sub-patterns were trained. The SNN training was unsupervised according to STDP explained earlier. An SNN had 80 input units corresponding to each of the 80 spike trains forming the pattern. The network had eight output units to allow within category diversity. For each sub-pattern, one SNN was trained for ten iterations using STDP. One iteration meant that the network was allowed to run for $T=20$ ms with STDP enabled and the input neurons maintained Poisson firing rates according to their location within the sub-pattern. Synaptic weights were randomly initialized before training. Fig. \[fig:trainedSubpatternWeights\] shows average the synaptic weights after training for sub-pattern A. The weights for high-firing-rate spike trains 1–20 are clearly distinguishable from the weights for low-firing-rate spike trains 21–80. The results for training the other sub-patterns were analogous. The plot shows that information can be detected in the presence of noise. The average synaptic weights, $\textbf{w}_\mathrm{state}$, were calculated by averaging over the eight output units which is shown in Eq. \[eq:AvgW\].
$$\mathbf{w}_\mathrm{state}=\frac{1}{K}\sum_{k=1}^K \mathbf{w}_k \cdot w_{k0}.
\label{eq:AvgW}$$
Recall that the bias learns to represent the average firing rate.
Patterns to be classified were built from a sequence of the four sub-patterns. Each sub-pattern in a sequence corresponded to one HMM state. A collection of four HMMs, each with four states corresponding to the pattern length, were used to recognize four target patterns ABCD, DCBA, ABDC, and BACD.
Table \[tab:artificialDataResults\] shows the performance results on this data set. In the table, *desired* means the pattern that was presented and *recognized* means the HMM with the highest probability output. The probabilities along the diagonal (correctly classified) are much higher than the other probabilities in each column. Therefore, the proposed model shows initial promising results in categorizing a simple set of the synthesized spatio-temporal patterns.
[lrrrr]{} Recognized/desired & & & &\
& **0.442** & 0.112 & 0.227 & 0.219\
& 0.145 & **0.572** & 0.140 & 0.143\
& 0.254 & 0.124 & **0.495** & 0.126\
& 0.249 & 0.123 & 0.128 & **0.500**\
#### Details of the recognition mechanism
During the recognition phase, we used a set of four $P$-state HMMs ($P=4$) for each of the target patterns. The four trained SSNs are associated with the appropriate HMM state. The HMM with highest probability for a given input sequence was taken as the best match to the input signal. For each HMM, the probability of an observation seuquence, $\mathbf{O}$, was calculated by expanding Eq. \[eqn:mainHMM\] as follows:
\[eq:HMMrecognitionEqns\] $$\begin{aligned}
\mathit{Pr}(\mathbf{O} | \boldsymbol\lambda) &= Pr(s_m =1) \prod_{p=2}^4 a_{p-1,p} \cdot Pr( s_m = p ) \\ &
\mathit{Pr}( s_m = p) = \prod_{t=1}^{T=20}\mathit{Pr}_{\mathrm{snn}_p}(t).\end{aligned}$$
$\mathit{Pr}_{\mathrm{snn}_p}(t)$ is defined in Eq. \[eq:SNN\_output\_prob\]. The $a_{p-1,p}$’s and $a_{p,p}$’s are all set to $0.5$ (fixed transitions). All other $a_{ij}$’s are set to zero. In this experiment, total pattern duration was 80 ms corresponding to a concatenated sequence of four sub-patterns ($T=20$ ms). The state probability, $\mathit{Pr}( s_m = p)$, in Eq. \[eq:HMMrecognitionEqns\] is obtained by multiplying the particular state probabilities in $T=20$ sequential time steps (1 ms separation). $T$ is the duration for the Poisson spike trains. From Eq. \[eq:exp\_uk\] and Eq. \[eq:SNN\_output\_prob\], the state probability should have an exponential form as $$\mathit{Pr}( s_m = p)= e^{\textbf{w}_s^T \cdot \sum_{t=1}^{T=20}\textbf{y}(t)}
\label{eq:provePoisson}$$ where $\textbf{w}_s$ is the selected weight vector with maximum probability value in Eq. \[eq:SNN\_output\_prob\]. $\sum_{t=1}^{T=20}\textbf{y}(t)$ reports the *Poisson process rate* $\times\; T$ which can be interpreted as the feature values of an observation. Therefore, it reversely shows the statistical similarity between two numerical vectors $\textbf{w}_s$ and $\textbf{y}$ discussed in section (4.2.1, *posterior probability*).
The algorithm for training the hybrid HMM/SNN model and classifying the sequential patterns is shown in Fig. \[fig:modelAlgorithm\]. For this example, Lines 2 and 3 were not needed because the sub-patterns were already extracted.
xxxx1xxxxxxxxxxxxxxxxxxxx=1: HMM-SNN(N, k, P, T, signal):\
2: data = Feature-Extraction(signal, N)\
3: sub-patterns = Auto-Segmentation(data, P)\
4: For each sample in sub-patterns:\
5: spike-trains = Extract-Poisson-Spikes(pattern, T) // e.g. 80 spike trains\
6: Calculate output neuron status using Eq. \[eq:exp\_uk\]\
7: if (Training-Session):\
8: Train SNNs using Eqs. \[eq:NesslerSTDP\_wts\]-\[eq:pi\_snn\]\
9: else\
10: Select class with highest Pr using Eq. \[eq:HMMrecognitionEqns\] or Eq. \[eq:HMMrecognitionEqnsSpeech\]\
Speech Signals
--------------
This experiment extends the method to speech signal processing. A speech signal can be characterized as a number of sequential frames with stationary characteristics within the frame. A speech signal $S=f_1 f_2 \ldots f_M$ has $M$ sequential frames. In humans, the signal within the auditory nerve is the result of an ongoing Fourier analysis performed by the cochlea of the inner ear. That is, the frequencies’ energy and formants carry useful information for the speech recognition problem. In our experiments, we divided speech signals into 20 ms duration frames with 50 percent temporal overlap and converted each frame to the frequency domain (Line 2 of Fig. \[fig:modelAlgorithm\]).
xxxx1xxxxxxxxxxxxxxxxxxxx=1: Initialize the data into $P$ equal-width segments (sub-patterns).\
2: Repeat\
3: sample=1\
4: For $p$=1 to $P-1$\
5: While (distance(sample and centroid\[$p$\]) $\le$ distance(sample and centroid\[$p+1$\])\
6: sample++\
7: Segment($p$)=Sample\
8: Update centroids\
9: Until Segment change $\le$ threshold.\
#### Auto segmentation preprocessing step
The preprocessing groups the $M$ sequential frames into $P$ consecutive clusters. Let $\mathbf{O} = [\mathbf{o}_1 \ldots \mathbf{o}_M]$ denote a sequence of observations that is a member of, say, class $C_1$. The goal of the SNN is to classify $\mathbf{O}$ as a member of $C_1$ among the other possible classes. For instance, $\mathbf{O}$ can be a speech stream with $M$ 20 ms duration frames, where each $\mathbf{o}_m$ is a frame consisting of $N$ features observed at a given 20 ms time step. We shall call this a feature vector. The $M$ feature vectors (frames) should map to the $P$ categories corresponding to the HMM states.
The initial problem is to cluster the $M$ vectors into the $P$ HMM categories. For this purpose, a modified $k$-means algorithm was used. The algorithm, given in Fig. \[fig:autoSegmentationPreProc\], compares consecutive (adjacent) clusters to group the frames into $P$ sequential data segments (Line 3 of Fig. \[fig:modelAlgorithm\]). Each segment contains approximately similar feature vectors as judged by the clustering algorithm.
To illustrate, Fig. \[fig:zeroSpectrogram\] shows a spectrogram of the spoken word “zero.” This represents the power spectrum of frequencies in a signal as they vary with time. The auto segmentation result for this signal is also shown in Fig. \[fig:zeroSpectrogram\] by the vertical dashed lines. The signal has been divided into $P=10$ segments containing a varying number of speech frames. A specific segment consists of similar frames, where the signal is approximately stationary, and corresponds to one state of the HMM.
#### Converting a speech signal to a spike train
The speech signals were sampled at 8 kHz. Frame duration was taken to be 20 ms, which at an 8 kHz sampling rate contains 160 sample values. After converting to the frequency domain, this reduces to 80 sample values. The magnitudes of the 80 frequency components were converted to rate parameters for 80 Poisson spike trains (Line 5 of Fig. \[fig:modelAlgorithm\]). The simulation time for the spike trains was $T=20$ ms.
#### Classifying spoken words
Two experiments were conducted, classifying spoken words into either two or four categories. Data was selected from the Aurora dataset [@Pearce2000a]. The data set contains spoken American English digits taken from male and female speakers sampled at 8 kHz. 600 spoken digits belonging five categories “zero”, “one,” “four,” “eight”, and “nine” were selected. For each word recognized, an HMM with $P=10$ states was used. A separate SNN was associated with each HMM state. All SNNs used 80 input units and 8 output units. The network had $8\cdot81=648$ adaptive weights. These dimensions are the same as the network used in the previous experiment. Since the probability distribution functions of the states are independent of each other, the $P$ speech segments can be trained in parallel. The within-class variability for a single state is maintained by the $K=8$ output units which approximate a Gaussian mixture model. The input spike trains are obtained by the Poisson process based on the frame’s frequency amplitudes (80 feature values). The model was trained for 100 iterations. After training, the synaptic weights reflect the importance of specific frequencies and the final bias weights show the output neurons’ excitability in each state. The recognition phase in this experiment is more general than in Eq. \[eq:HMMrecognitionEqns\] such that each segment $S$ (1 through $P$=10), which is determined by a state, contains the number of samples. Thus,
\[eq:HMMrecognitionEqnsSpeech\] $$\begin{aligned}
\mathit{Pr}(\mathbf{O} | \boldsymbol\lambda) &= \mathit{Pr}(s_m =1) \prod_{p=2}^{P=10} a_{p-1,p} \cdot \mathit{Pr}( s_m = p ) \\ &
\mathit{Pr}( s_m = p) = \prod_{l\in S(p)}a_{p,p}\cdot \prod_{t=1}^{T=20} \mathit{Pr}_{\mathrm{snn}_p}^{l}(t),\end{aligned}$$
where $\mathit{Pr}_{\mathrm{snn}_p}^{l}(t)$ specifies the probability measure of sample $l$ of the sub-pattern corresponding to state $p$.
Table \[tab:binarySpeechResults\] shows the binary classification performance using different relative prior probabilities as bias parameters. Fig. \[fig:ROC\] illustrates the ROC curve of the results shown in Table \[tab:binarySpeechResults\]. The accuracy rate above 95 percent shows initial success of the model. Table \[tab:fourClassSpeechResults\] shows accuracy rates of the model in recognizing four spoken words. An average performance above 85 percent accuracy was obtained.
[lrrl]{} $\frac{P(\textrm`0\textrm')}{P(\textrm`1\textrm')}$ & FP % & TP % & Accuracy %\
0.9500 & 100.00 & 100.00 & 50.26\
0.9600 & 97.87 & 100.00 & 52.91\
0.9650 & 94.68 & 100.00 & 51.32\
0.9830 & 70.21 & 98.95 & 64.55\
0.9850 & 58.51 & 98.95 & 70.37\
0.9900 & 31.94 & 98.95 & 83.60\
0.9965 & 8.51 & 96.84 & 94.18\
0.9980 & 4.26 & 94.74 & **95.27**\
0.9990 & 3.19 & 92.63 & 94.71\
1.0000 & 3.19 & 90.53 & 93.65\
1.0030 & 1.06 & 75.79 & 87.30\
1.0101 & 0.00 & 53.68 & 76.72\
1.3333 & 0.00 & 0.00 & 49.00\
#### Summary of parameter choices
The number of input units, $N=80$, was chosen because there were 80 frequency components in the spectrogram at the sampling rate used. The number of HMM states, $P=10$, was chosen as the smallest value to qualitatively represent the variations in the acoustic structure of the spectrograms encountered. The number of output units, $K=8$, was chosen to be the same as number of distributions considered for the GMM in previous study [@Tavanaei2012a].
Discussion of results
---------------------
Previous work was conducted using a traditional support vector data description and an HMM to classify spoken digits using wavelets and frequency-based features [@Tavanaei2012a]. Accuracy rates above 90 percent were achieved which is better than the results obtained in the present experiments. However, that model does not have the biomorphic features that exist in the present model. Additionally, online learning in the current method makes the model flexible to new data occurrences and is able to be updated efficiently. Furthermore, using the SNNs which support communication via a series of the impulses instead of real numbers would be useful in VLSI implementation of the human brain functionality in sequential pattern recognition.
[lrrl]{} Class & Accuracy %\
“zero” & 81.91\
“four” & 82.98\
“eight” & 96.74\
“nine” & 80.65\
**Average** & **85.57**\
Conclusion
==========
A novel hybrid learning model for sequential data classification was studied. It consisted of an hidden Markov model combined with a spiking neural network that approximated expectation maximization learning. Although there have been other hybrid networks, to our knowledge this is the first using an Snn. The model was studied on synthesized spike-train data and also on spoken word data. Although the studies are preliminary, they demonstrate proof-of-concept in the sense that it provides a useful example of how a statistically based mechanism for signal processing may be integrated with biologically motivated mechanisms, such as neural networks.
Our approach derives from the described in [@Nessler2009a; @Nessler2013a; @Kappel2014a]. The work in [@Nessler2009a; @Nessler2013a] showed how to use STDP learning to approximate expectation maximization. A complete HMM was encoded in a recurrent spiking neural network in the work of [@Kappel2014a]. Our approach seeks a middle ground where the recurrent neural network is unwrapped into a sequence of HMM states, which each state having an associated nonrecurrent network. This leaves open the possibility of thinking about other ways to encode HMMs in brain circuitry by refining a model that does not make such extreme assumptions.
Planned future work includes continuing to study the model’s properties, both theoretically and experimentally, improving and extending the model’s range of performance, and extending the learning capabilities of the model. Conspicuously, the state transition probabilities of the present model are predetermined and fixed. Our most important goal is to learn the sequential data in an online fashion without the segmentation preprocessing phase. This would provide a basis to train the state transition probabilities.
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abstract: 'In this paper, we introduce and study a new extragradient iterative process for finding a common element of the set of fixed points of an infinite family of nonexpansive mappings and the set of solutions of a variational inequality for an inverse strongly monotone mapping in a real Hilbert space. Also, we prove that under quite mild conditions the iterative sequence defined by our new extragradient method converges strongly to a solution of the fixed point problem for an infinite family of nonexpansive mappings and the classical variational inequality problem. In addition, utilizing this result, we provide some applications of the considered problem not just giving a pure extension of existing mathematical problems.'
address:
- 'Department of Mathematics, Erzurum Technical University, Erzurum 25240, Turkey'
- 'Department of Mathematics, Ataturk University, Erzurum 25240, Turkey'
author:
- Ibrahim Karahan
- Murat Ozdemir
title: A New Iterative Projection Method for Approximating Fixed Point Problems and Variational Inequality Problems
---
Introduction
============
Throughout this paper, we assume that $H$ is a real Hilbert space whose inner product and norm are denoted by $\left\langle \cdot ,\cdot
\right\rangle $ and $\left\Vert \cdot \right\Vert $, respectively, $C$ is a nonempty closed convex subset of $H$, and $I$ is the idendity mapping on $C$. Below, we gather some basic definitions and results which are needed in the subsequent sections. Recall that a mapping $T:C\rightarrow C$ is called nonexpansive if $$\left\Vert Tx-Ty\right\Vert \leq \left\Vert x-y\right\Vert ,\text{ }\forall
x,y\in C.$$We denote by $F(T)$ the set of fixed points of $T$. For a mapping $A:C\rightarrow H,$ it is said to be
1. monotone if$$\left\langle Ax-Ay,x-y\right\rangle \geq 0,\text{ }\forall x,y\in C;$$
2. $L$-Lipschitzian if there exists a constant $L>0$ such that $$\left\Vert Ax-Ay\right\Vert \leq L\left\Vert x-y\right\Vert ,\text{ }\forall
x,y\in C;$$
3. $\alpha $-inverse strongly monotone if there exists a positive real number $\alpha >0$ such that$$\left\langle Ax-Ay,x-y\right\rangle \geq \alpha \left\Vert Ax-Ay\right\Vert
^{2},\text{ }\forall x,y\in C.$$
\[z\]It is obvious that any $\alpha $-inverse strongly monotone mapping $A$ is monotone and $\frac{1}{\alpha }$-Lipschitz continuous.
\[y\]Every $L$-Lipschitzan mapping is $\frac{2}{L}$-inverse strongly monotone mapping.
For a mapping $A:C\rightarrow H$, the classical variational inequality problem $VI\left( C,A\right) $ is to find a $x\in C$ such that$$\left\langle Ax,y-x\right\rangle \geq 0,\text{ }\forall y\in C, \label{25}$$which is the optimality condition for the minimization problem$$\min_{x\in C}\frac{1}{2}\left\langle Ax,x\right\rangle . \label{13}$$The set of solutions of $VI\left( C,A\right) $ is denoted by $\Omega ,$ i.e., $$\Omega =\left\{ x\in C:\text{ }\left\langle Ax,y-x\right\rangle \geq 0\text{, }\forall y\in C\right\} .$$
In the context of the variational inequality problem it is easy to check that$$x\in \Omega \Leftrightarrow x\in F\left( P_{C}\left( I-\lambda A\right)
\right) ,\text{ }\forall \lambda >0.$$Variational inequalities were initially studied by Stampacchia [@stam1], [@stam2]. Such a problem is connected with convex minimization problem, the complementarity problem, the problem of finding point $x\in C$ satisfying $0\in Ax$ and etc.. Fixed point problems are also closely related to the variational inequality problems. Based on this relationship, iterative methods for nonexpansive mappings have recently been applied to find the common solution of fixed point problems and variational inequality problems; see, for example [@yalis; @caku; @lita2; @wosaya; @yaliya] and the references therein. Below, we give some of them.
In 2005, Iiduka and Takahashi [@lita] proposed an iterative process as follows:$$\left\{
\begin{array}{l}
x_{1}\in C, \\
x_{n+1}=\alpha _{n}x+\left( 1-\alpha _{n}\right) TP_{C}\left( I-\lambda
_{n}A\right) x_{n},\text{ }\forall n\geq 1\text{,}\end{array}\right. \label{D}$$where $A$ is an $\alpha $-inverse strongly monotone mapping, $\left\{ \alpha
_{n}\right\} \subset \left( 0,1\right) $ and $\left\{ \lambda _{n}\right\}
\in \left( 0,2\alpha \right) $ satisfy some parameters controlling conditions. They showed that if $F\left( T\right) \cap \Omega $ is nonempty, then the sequence $\left\{ x_{n}\right\} $ generated by (\[D\]) converges strongly to some $z\in F\left( T\right) \cap \Omega $.
One year later, in 2006, by a narrow margin from the iterative process ([D]{}), Takahashi and Toyoda [@tato] introduced the following iterative process which is based on the Mann iteration [@mann]:$$\left\{
\begin{array}{l}
x_{0}\in C, \\
x_{n+1}=\alpha _{n}x_{n}+\left( 1-\alpha _{n}\right) TP_{C}\left( I-\lambda
_{n}A\right) x_{n},\text{ }\forall n\geq 0,\end{array}\right. \label{A}$$where $C$ is a nonempty closed convex subset of a real Hilbert space $H,$ $P_{C}:H\rightarrow C$ is a metric projection, $A:C\rightarrow H$ is an $\alpha $-inverse strongly monotone mapping, and $T:C\rightarrow C$ is a nonexpansive mapping. They proved that if the set of fixed points of $T$ is nonempty, then the sequence $\left\{ x_{n}\right\} $ generated by (\[A\]) converges weakly to some $z\in F\left( T\right) \cap \Omega $ where $z=\lim_{n\rightarrow \infty }P_{F\left( T\right) \cap \Omega }x_{n}.$ In the same year, Yao et. al. [@yaliya], introduced following iterative scheme for a nonexpansive mapping $S,$ and a monotone $k$-Lipschitzian continuous mapping $A$. Under the suitable conditions, they proved the strong convergence of $\left\{ x_{n}\right\} $ for a fixed $u\in H$ and a given $x_{0}\in H$ arbitrary. $$\left\{
\begin{array}{l}
x_{n+1}=\alpha _{n}u+\beta _{n}x_{n}+\gamma _{n}SP_{C}\left( x_{n}-\lambda
_{n}y_{n}\right) , \\
y_{n}=P_{C}\left( I-\lambda _{n}A\right) x_{n},\text{ }\forall n\geq 0.\end{array}\right. \label{B}$$
Lastly, Khan [@khan] and Sahu [@sahu], individually, introduced the following iterative process which Khan referred to as Picard-Mann hybrid iterative process:$$\left\{
\begin{array}{l}
x_{1}\in C, \\
x_{n+1}=Ty_{n}, \\
y_{n}=\alpha _{n}x_{n}+\left( 1-\alpha _{n}\right) Tx_{n},\text{ }\forall
n\geq 1,\end{array}\right. \label{C}$$where $\left\{ \alpha _{n}\right\} $ is a sequence in $\left( 0,1\right) .$ Picard-Mann hybrid iterative process is independent of all Picard, Mann and Ishikawa iterative processes. Khan [@khan] showed that the process ([C]{}) converges faster than all of Picard, Mann and Ishikawa iterative processes for contractions. Moreover, he proved a strong and a weak convergence theorems in Banach space for iterative process (\[C\]) with a $T$ nonexpansive mapping under the suitable conditions.
In addition to all these studies, the existence of common elements of the set of common fixed points of an infinite family of nonlinear mappings and the set of solutions of the variational inequality problem has been also considered by many authors (see [@yao1; @rabian; @wang1]). In such articles, authors usually use a mapping generated by nonexpansive mappings such as the mapping $W_{n}$ defined, as in Shimoji and Takahashi [@shi], by$$\begin{aligned}
U_{n,n+1} &=&I \notag \\
U_{n,n} &=&\mu _{n}T_{n}U_{n,n+1}+\left( 1-\mu _{n}\right) I \notag \\
U_{n,n-1} &=&\mu _{n-1}T_{n-1}U_{n,n}+\left( 1-\mu _{n-1}\right) I \notag \\
&&\vdots \notag \\
U_{n,k+1} &=&\mu _{k+1}T_{k+1}U_{n,k+2}+\left( 1-\mu _{k+1}\right) I
\label{11} \\
U_{n,k} &=&\mu _{k}T_{k}U_{n,k+1}+\left( 1-\mu _{k}\right) I \notag \\
&&\vdots \notag \\
U_{n,2} &=&\mu _{2}T_{2}U_{n,3}+\left( 1-\mu _{2}\right) I \notag \\
W_{n} &=&U_{n,1}=\mu _{1}T_{1}U_{n,2}+\left( 1-\mu _{1}\right) I \notag\end{aligned}$$where $C$ is a nonempty closed convex subset of a Hilbert space $H,$ $\mu
_{1},\mu _{2},\ldots $ are real numbers such that $0\leq \mu _{n}\leq 1,$ and $T_{1},T_{2},\ldots $ is an infinite family of self-mappings on $C$. $W_{n}$ is called $W$-mapping generated by $T_{n},T_{n-1},\ldots ,T_{1}$ and $\mu _{n},\mu _{n-1},\ldots ,\mu _{1}.$ It is clear that nonexpansivity of each $T_{i}$, $i\geq 1$, ensures the nonexpansivity of $W_{n}$.
In this paper, motivated and inspired by the above processes and independently from all of them, we introduce the following iterative process for an infinite family of nonexpansive mappings $\left\{ T_{n}\right\} $ which is based on Picard-Mann hybrid iterative process:$$\left\{
\begin{array}{l}
x_{0}\in C \\
x_{n+1}=W_{n}P_{C}\left( I-\lambda _{n}A\right) y_{n} \\
y_{n}=\left( 1-\alpha _{n}\right) x_{n}+\alpha _{n}W_{n}P_{C}\left(
I-\lambda _{n}A\right) x_{n},\text{ }\forall n\geq 0,\end{array}\right. \label{12}$$where $A:C\rightarrow H$ is an $\alpha $-inverse strongly monotone mapping, $W_{n}$ is a mapping defined by (\[11\]), $\{\lambda _{n}\}\subset \lbrack
a,b]$ for some $a,b\in (0,2\alpha )$ and $\left\{ \alpha _{n}\right\}
\subset \left[ c,d\right] $ for some $c,d\in \left( 0,1\right) $. Also, we prove that the sequence $\left\{ x_{n}\right\} $ defined by (\[12\]) converge strongly to a common element of the set of common fixed points of the infinite family $\left\{ T_{n}\right\} $ and the set of solutions of the variational inequality (\[25\]) which is the optimality condition for the minimization problem (\[13\]).
Preliminaries
=============
In this section, we collect some useful lemmas that will be used for our main result in the next section. We write $x_{n}\rightharpoonup x$ to indicate that the sequence $\left\{ x_{n}\right\} $ converges weakly to $x,$ and $x_{n}\rightarrow x$ for the strong convergence.
It is well known that for any $x\in H,$ there exists a unique point $y_{0}\in C$ such that$$\left\Vert x-y_{0}\right\Vert =\inf \left\{ \left\Vert x-y\right\Vert :y\in
C\right\} .$$We denote $y_{0}$ by $P_{C}x,$ where $P_{C}$ is called the metric projection of $H$ onto $C.$ We know that $P_{C}$ is a nonexpansive mapping. It is also known that $P_{C}$ has the following properties:
1. $\left\Vert P_{C}x-P_{C}y\right\Vert \leq \left\Vert
x-y\right\Vert ,$ for all $x,y\in H,$
2. $\left\Vert x-y\right\Vert ^{2}\geq \left\Vert
x-P_{C}x\right\Vert ^{2}+\left\Vert y-P_{C}x\right\Vert ^{2},$ for all $x\in
H,$ $y\in C,$
3. $\left\langle x-P_{C}x,y-P_{C}x\right\rangle \leq 0,$ for all $x\in H,$ $y\in C.$
It is known that a Hilbert space $H$ satisfies the Opial condition that, for any sequence $\left\{ x_{n}\right\} $ with $x_{n}\rightharpoonup x,$ the inequality$$\lim \inf_{n\rightarrow \infty }\left\Vert x_{n}-x\right\Vert <\lim
\inf_{n\rightarrow \infty }\left\Vert x_{n}-y\right\Vert$$holds for every $y\in H$ with $y\neq x.$
\[c\][@tato] Let $C$ be a nonempty closed convex subset of a real Hilbert space $H$ and $\left\{ x_{n}\right\} $ be a sequence in $H.$ Suppose that, for all $z\in C,$$$\left\Vert x_{n+1}-z\right\Vert \leq \left\Vert x_{n}-z\right\Vert$$for every $n=0,1,2,\ldots .$ Then, $\left\{ P_{C}x_{n}\right\} $ converges strongly to some $u\in C.$
[@tato] Let $C$ be a nonempty closed convex subset of a real Hilbert space $H$ and let $A$ be an $\alpha $-inverse strongly monotone mapping of $C
$ into $H.$ Then, the solution of $VI\left( C,A\right) $, $\Omega ,$ is nonempty.
For a set-valued mapping $S:H\rightarrow 2^{H}$, if the inequality$$\left\langle f-g,u-v\right\rangle \geq 0$$holds for all $u,v\in C,f\in Su,g\in Sv,$ then $S$ is called monotone mapping. A monotone mapping $S:H\rightarrow 2^{H}$ is maximal if the graph $G\left( S\right) $ of $S$ is not properly contained in the graph of any other monotone mappings. It is known that a monotone mapping $S$ is maximal if and only if, for $\left( u,f\right) \in H\times H,$ $\left\langle
u-v,f-w\right\rangle \geq 0$ for every $\left( v,w\right) \in G\left(
S\right) $ implies $f\in Su.$ Let $A$ be an inverse strongly monotone mapping of $C$ into $H,$ let $N_{C}v$ be the normal cone to $C$ at $v\in C,$ i.e.,$$N_{C}v=\left\{ w\in H:\left\langle v-u,w\right\rangle \geq 0,\forall u\in
C\right\} ,$$and define$$Sv=\left\{
\begin{array}{cc}
Av+N_{C}v & v\in C \\
\emptyset & v\notin C.\end{array}\right.$$Then, $S$ is maximal monotone and $0\in Sv$ if and only if $v\in \Omega .$
\[b\][@kirk] Let $C$ be a nonempty closed convex subset of a real Hilbert space $H,$ and $T$ be a nonexpansive self-mapping on $C.$ If $F\left( T\right) \neq \emptyset ,$ then $I-T$ is demiclosed; that is whenever $\left\{ x_{n}\right\} $ is a sequence in $C$ weakly converging to some $x\in C$ and the sequence $\left\{ \left( I-T\right) x_{n}\right\} $ strongly converges to some $y$, it follows that $\left( I-T\right) x=y.$ Here, $I$ is the identity operator of $H.$
\[a\][@schu] Let $H$ be a real Hilbert space, let $\left\{ \alpha
_{n}\right\} $ be a sequence of real numbers such that $0<a\leq \alpha
_{n}\leq b<1$ for all $n=0,1,2,\ldots ,$ and let $\left\{ x_{n}\right\} $ and $\left\{ y_{n}\right\} $ be sequences of $H$ such that$$\limsup_{n\rightarrow \infty }\left\Vert x_{n}\right\Vert \leq c,\text{ }\limsup_{n\rightarrow \infty }\left\Vert y_{n}\right\Vert \leq c\text{ and }\lim_{n\rightarrow \infty }\left\Vert \alpha _{n}x_{n}+\left( 1-\alpha
_{n}\right) y_{n}\right\Vert =c,$$ for some $c>0.$ Then,$$\lim_{n\rightarrow \infty }\left\Vert x_{n}-y_{n}\right\Vert =0.$$
\[Y\][@xu] Assume that $\left\{ x_{n}\right\} $ is a sequence of nonnegative real numbers satisfying the conditions$$x_{n+1}\leq \left( 1-\alpha _{n}\right) x_{n}+\alpha _{n}\beta _{n},\text{ }\forall n\geq 0$$where $\left\{ \alpha _{n}\right\} $ and $\left\{ \beta _{n}\right\} $ are sequences of real numbers such that$$\begin{aligned}
&\text{(i) }&\left\{ \alpha _{n}\right\} \subset \left[ 0,1\right] \text{
and }\tsum_{n=0}^{\infty }\alpha _{n}=\infty ,\text{ or equivalently }\tprod_{n=0}^{\infty }\left( 1-\alpha _{n}\right) =0\text{, \ \ \ \ } \\
&\text{(ii)}&\limsup_{n\rightarrow \infty }\beta _{n}\leq 0\text{, or }\tsum_{n}\alpha _{n}\beta _{n}<\infty \text{.}\end{aligned}$$*Then,* $\lim_{n\rightarrow \infty }x_{n}=0.$
Concerning the mapping $W_{n}$ defined by (\[11\]), we have the following lemmas in a real Hilbert space which can be obtained from Shimoji and Takahashi [@shi].
\[AA\][@shi] Let $C$ be a nonempty closed and convex subset of a real Hilbert space $H$. Let $\left\{ T_{n}\right\} $ be an infinite family of nonexpansive mappings on $C$ such that $\tbigcap_{n=1}^{\infty }F\left(
T_{n}\right) \ $is nonempty, and let $\mu _{1},\mu _{2},\ldots $ be real numbers such that $0\leq \mu _{n}\leq 1$ for all $n\in
\mathbb{N}
$. Then, for every $x\in C$ and $k\in
\mathbb{N}
$, the limit $\lim_{n\rightarrow \infty }U_{n,k}x$ exists.
By using the Lemma \[AA\], one can define the mapping $W$ on $C$ as follows:$$Wx=\lim_{n\rightarrow \infty }W_{n}x=\lim_{n\rightarrow \infty }U_{n,1}x,\text{ }\forall x\in H.$$Such a $W$ is called the $W$-mapping generated by $T_{1},T_{2},\ldots $ and $\mu _{1},\mu _{2},\ldots $. Throughout this paper, we assume that $0<\mu
_{n}\leq b<1$ for $n\geq 0$.
\[e3\][@shi] Let $C$ be a nonempty closed and convex subset of a real Hilbert space $H$. Let $\left\{ T_{n}\right\} $ be an infinite family of nonexpansive mappings on $C$ such that $\tbigcap_{n=1}^{\infty }F\left(
T_{n}\right) \ $is nonempty, and let $\mu _{1},\mu _{2},\ldots $ be real numbers such that $0\leq \mu _{n}\leq 1$ for $n\geq 0$. Then, $F\left(
W\right) =\tbigcap_{n=1}^{\infty }F\left( T_{n}\right) $.
Main result
===========
Now, we are in a position to state and prove the main result in this paper.
\[1\*\]Let $C$ be a nonempty closed convex subset of a real Hilbert space $H$, let $A:C\rightarrow H$ be an $\alpha $-inverse strongly monotone mapping and let $\left\{ T_{n}\right\} $ be an infinite family of nonexpansive self-mappings on $C$ such that $\tciFourier
:=\tbigcap_{n=0}^{\infty }F\left( T_{n}\right) \cap \Omega \neq \emptyset .$ Let $\left\{ x_{n}\right\} $ be a sequence defined by (\[12\]), where $\{\lambda _{n}\}\subset \lbrack a,b]$ for some $a,b\in (0,2\alpha )$ and $\left\{ \alpha _{n}\right\} \subset \left[ c,d\right] $ for some $c,d\in
\left( 0,1\right) $. Then, the sequence $\left\{ x_{n}\right\} $ converges strongly to a point $z\in \tciFourier $ where $z$ is the unique solution of the variational inequality (\[25\]).
We devide our proof into five steps.
**Step 1.** First, we show that $\left\{ x_{n}\right\} $ is a bounded sequence. Let $t_{n}=P_{C}\left( I-\lambda _{n}A\right) x_{n}$ and $z\in
\tciFourier $. Then, we have$$\begin{aligned}
\left\Vert t_{n}-z\right\Vert ^{2} &=&\left\Vert P_{C}\left( I-\lambda
_{n}A\right) x_{n}-z\right\Vert ^{2} \notag \\
&\leq &\left\Vert \left( I-\lambda _{n}A\right) x_{n}-\left( I-\lambda
_{n}A\right) z\right\Vert ^{2} \notag \\
&=&\left\Vert x_{n}-z-\lambda _{n}\left( Ax_{n}-Az\right) \right\Vert ^{2}
\notag \\
&\leq &\left\Vert x_{n}-z\right\Vert ^{2}-2\lambda _{n}\left\langle
x_{n}-z,Ax_{n}-Az\right\rangle +\lambda _{n}^{2}\left\Vert
Ax_{n}-Az\right\Vert ^{2} \notag \\
&\leq &\left\Vert x_{n}-z\right\Vert ^{2}+\lambda _{n}\left( \lambda
_{n}-2\alpha \right) \left\Vert Ax_{n}-Az\right\Vert ^{2} \notag \\
&\leq &\left\Vert x_{n}-z\right\Vert ^{2} \label{1}\end{aligned}$$and from (\[1\]) we get$$\begin{aligned}
\left\Vert x_{n+1}-z\right\Vert ^{2} &=&\left\Vert W_{n}P_{C}\left(
I-\lambda _{n}A\right) y_{n}-z\right\Vert ^{2} \notag \\
&=&\left\Vert W_{n}P_{C}\left( I-\lambda _{n}A\right) y_{n}-W_{n}P_{C}\left(
I-\lambda _{n}A\right) z\right\Vert ^{2} \notag \\
&\leq &\left\Vert y_{n}-z\right\Vert ^{2} \notag \\
&=&\left\Vert \left( 1-\alpha _{n}\right) \left( x_{n}-z\right) +\alpha
_{n}\left( W_{n}t_{n}-z\right) \right\Vert ^{2} \notag \\
&\leq &\left( 1-\alpha _{n}\right) \left\Vert x_{n}-z\right\Vert ^{2}+\alpha
_{n}\left\Vert W_{n}t_{n}-z\right\Vert ^{2} \notag \\
&\leq &\left( 1-\alpha _{n}\right) \left\Vert x_{n}-z\right\Vert ^{2}+\alpha
_{n}\left\Vert t_{n}-z\right\Vert ^{2} \notag \\
&\leq &\left( 1-\alpha _{n}\right) \left\Vert x_{n}-z\right\Vert ^{2} \notag
\\
&&+\alpha _{n}\left[ \left\Vert x_{n}-z\right\Vert ^{2}+\lambda _{n}\left(
\lambda _{n}-2\alpha \right) \left\Vert Ax_{n}-Az\right\Vert ^{2}\right]
\notag \\
&=&\left\Vert x_{n}-z\right\Vert ^{2}+\alpha _{n}\lambda _{n}\left( \lambda
_{n}-2\alpha \right) \left\Vert Ax_{n}-Az\right\Vert ^{2} \notag \\
&\leq &\left\Vert x_{n}-z\right\Vert ^{2}+da\left( b-2\alpha \right)
\left\Vert Ax_{n}-Az\right\Vert ^{2} \notag \\
&\leq &\left\Vert x_{n}-z\right\Vert ^{2}. \label{2*}\end{aligned}$$Therefore, the limit $\lim_{n\rightarrow \infty }\left\Vert
x_{n}-z\right\Vert $ exists and $Ax_{n}-Az\rightarrow 0.$ Hence, $\left\{
x_{n}\right\} $ is bounded and so are $\left\{ t_{n}\right\} $ and $\left\{
W_{n}t_{n}\right\} $.
**Step 2.** We will show that $\lim_{n\rightarrow \infty }\left\Vert
x_{n}-y_{n}\right\Vert =0.$ Before that, we shall show that $\lim_{n\rightarrow \infty }\left\Vert W_{n}t_{n}-x_{n}\right\Vert =0$. From Step 1, we know that $\lim_{n\rightarrow \infty }\left\Vert
x_{n}-z\right\Vert $ exists for all $z\in \tciFourier $. Let $\lim_{n\rightarrow \infty }\left\Vert x_{n}-z\right\Vert =c.$ From (\[2\*\]), since$$\left\Vert x_{n+1}-z\right\Vert \leq \left\Vert y_{n}-z\right\Vert \leq
\left\Vert x_{n}-z\right\Vert ,$$we get$$\lim_{n\rightarrow \infty }\left\Vert y_{n}-z\right\Vert =c. \label{*1}$$On the other hand, since$$\left\Vert W_{n}t_{n}-z\right\Vert \leq \left\Vert t_{n}-z\right\Vert \leq
\left\Vert x_{n}-z\right\Vert ,$$we have$$\limsup_{n\rightarrow \infty }\left\Vert W_{n}t_{n}-z\right\Vert \leq c.
\label{*2}$$Also, we know that$$\limsup_{n\rightarrow \infty }\left\Vert x_{n}-z\right\Vert \leq c
\label{*3}$$and$$\lim_{n\rightarrow \infty }\left\Vert y_{n}-z\right\Vert =\lim_{n\rightarrow
\infty }\left\Vert \left( 1-\alpha _{n}\right) \left( x_{n}-z\right) +\alpha
_{n}\left( W_{n}t_{n}-z\right) \right\Vert =c. \label{*4}$$Hence, from (\[\*2\]), (\[\*3\]), (\[\*4\]), and Lemma \[a\] , we get that$$\lim_{n\rightarrow \infty }\left\Vert x_{n}-W_{n}t_{n}\right\Vert =0.
\label{*5}$$We have also$$\left\Vert x_{n}-y_{n}\right\Vert =\alpha _{n}\left\Vert
W_{n}t_{n}-x_{n}\right\Vert .$$So, from (\[\*5\]) we obtain that$$\lim_{n\rightarrow \infty }\left\Vert x_{n}-y_{n}\right\Vert =0. \label{5.5}$$Since $A$ is Lipschitz continuous, we have $Ax_{n}-Ay_{n}\rightarrow 0.$
**Step 3.** Now, we show that $\lim_{n\rightarrow \infty }\left\Vert
Wx_{n}-x_{n}\right\Vert =0.$ Using the properties of the metric projection, since$$\begin{aligned}
\left\Vert t_{n}-z\right\Vert ^{2} &=&\left\Vert P_{C}\left( I-\lambda
_{n}A\right) x_{n}-P_{C}\left( I-\lambda _{n}A\right) z\right\Vert ^{2} \\
&\leq &\left\langle t_{n}-z,\left( I-\lambda _{n}A\right) x_{n}-\left(
I-\lambda _{n}A\right) z\right\rangle \\
&=&\frac{1}{2}\left[ \left\Vert t_{n}-z\right\Vert ^{2}+\left\Vert \left(
I-\lambda _{n}A\right) x_{n}-\left( I-\lambda _{n}A\right) z\right\Vert
^{2}\right. \\
&&\left. -\left\Vert t_{n}-z-\left[ \left( I-\lambda _{n}A\right)
x_{n}-\left( I-\lambda _{n}A\right) z\right] \right\Vert ^{2}\right] \\
&\leq &\frac{1}{2}\left[ \left\Vert t_{n}-z\right\Vert ^{2}+\left\Vert
x_{n}-z\right\Vert ^{2}-\left\Vert \left( t_{n}-x_{n}\right) -\lambda
_{n}\left( Ax_{n}-Az\right) \right\Vert ^{2}\right] \\
&=&\frac{1}{2}\left[ \left\Vert t_{n}-z\right\Vert ^{2}+\left\Vert
x_{n}-z\right\Vert ^{2}-\left\Vert t_{n}-x_{n}\right\Vert ^{2}\right. \\
&&\left. -2\lambda _{n}\left\langle t_{n}-x_{n},Ax_{n}-Az\right\rangle
-\lambda _{n}^{2}\left\Vert Ax_{n}-Az\right\Vert ^{2}\right] ,\end{aligned}$$it follows that$$\begin{aligned}
\left\Vert t_{n}-z\right\Vert ^{2} &\leq &\left\Vert x_{n}-z\right\Vert
^{2}-\left\Vert t_{n}-x_{n}\right\Vert ^{2} \notag \\
&&+2\lambda _{n}\left\langle t_{n}-x_{n},Ax_{n}-Az\right\rangle -\lambda
_{n}^{2}\left\Vert Ax_{n}-Az\right\Vert ^{2}. \label{8}\end{aligned}$$So, by using the inequality (\[8\]) and (\[2\*\]), we get$$\begin{aligned}
\left\Vert x_{n+1}-z\right\Vert ^{2} &\leq &\left( 1-\alpha _{n}\right)
\left\Vert x_{n}-z\right\Vert ^{2}+\alpha _{n}\left\Vert t_{n}-z\right\Vert
^{2} \\
&\leq &\left\Vert x_{n}-z\right\Vert ^{2}-\alpha _{n}\left\Vert
t_{n}-x_{n}\right\Vert ^{2} \\
&&+2\lambda _{n}\alpha _{n}\left\langle t_{n}-x_{n},Ax_{n}-Az\right\rangle
-\lambda _{n}^{2}\alpha _{n}\left\Vert Ax_{n}-Az\right\Vert ^{2} \\
&\leq &\left\Vert x_{n}-z\right\Vert ^{2}-d\left\Vert t_{n}-x_{n}\right\Vert
^{2} \\
&&+2\lambda _{n}\alpha _{n}\left\langle t_{n}-x_{n},Ax_{n}-Az\right\rangle
-\lambda _{n}^{2}\alpha _{n}\left\Vert Ax_{n}-Az\right\Vert ^{2}.\end{aligned}$$Since $\lim_{n\rightarrow \infty }\left\Vert x_{n+1}-z\right\Vert
=\lim_{n\rightarrow \infty }\left\Vert x_{n}-z\right\Vert $ and $Ax_{n}-Az\rightarrow 0,$ we obtain$$\lim_{n\rightarrow \infty }\left\Vert x_{n}-t_{n}\right\Vert =0. \label{*6}$$On the other hand, we have$$\begin{aligned}
\left\Vert W_{n}x_{n}-x_{n}\right\Vert &\leq &\left\Vert
W_{n}x_{n}-W_{n}t_{n}\right\Vert +\left\Vert W_{n}t_{n}-x_{n}\right\Vert \\
&\leq &\left\Vert x_{n}-t_{n}\right\Vert +\left\Vert
W_{n}t_{n}-x_{n}\right\Vert .\end{aligned}$$So, it follows from (\[\*5\]) and (\[\*6\]) that$$\lim_{n\rightarrow \infty }\left\Vert W_{n}x_{n}-x_{n}\right\Vert =0.
\label{20}$$Hence, from (\[20\]) and by the same argument as in the [ceng]{}, it follows that$$\left\Vert Wx_{n}-x_{n}\right\Vert \leq \left\Vert
Wx_{n}-W_{n}x_{n}\right\Vert +\left\Vert W_{n}x_{n}-x_{n}\right\Vert
\rightarrow 0, \label{*7}$$as $n\rightarrow \infty $.
**Step 4.** Next, we show that $$\limsup_{n\rightarrow \infty }\left[ \left\langle
W_{n}t_{n}-z,x_{n}-z\right\rangle +\left\Vert W_{n}t_{n}-z\right\Vert ^{2}\right] \leq 0,$$where $z\in \tciFourier $. But first, we need to show that the variational inequality (\[25\]) has unique solution. Indeed, suppose both $p\in C$ and $q\in C$ are solutions to (\[25\]), then$$\left\langle Ap,p-q\right\rangle \leq 0 \label{26}$$and$$\left\langle Aq,q-p\right\rangle \leq 0. \label{27}$$Combining (\[26\]) and (\[27\]), we get$$\left\langle Aq-Ap,q-p\right\rangle \leq 0. \label{28}$$Since the mapping $A$ is an inverse strongly monotone mapping, (\[28\]) implies $p=q.$ So, the uniqueness of the solution of the variational inequality (\[25\]) is proved. Next, we need to show that $\left\{
x_{n}\right\} $ converges weakly to an element of $\tciFourier $. Since $\left\{ x_{n}\right\} $ and $\left\{ W_{n}t_{n}\right\} $ are bounded sequences, there exist subsequences $\left\{ x_{n_{i}}\right\} $ of $\left\{
x_{n}\right\} $ and $\left\{ W_{n}t_{n_{i}}\right\} $ of $\left\{
W_{n}t_{n}\right\} $ such that$$\begin{aligned}
&&\limsup_{n\rightarrow \infty }\left[ \left\langle
W_{n}t_{n}-z,x_{n}-z\right\rangle +\left\Vert W_{n}t_{n}-z\right\Vert ^{2}\right] \notag \\
&=&\limsup_{i\rightarrow \infty }\left[ \left\langle
W_{n}t_{n_{i}}-z,x_{n_{i}}-z\right\rangle +\left\Vert
W_{n}t_{n_{i}}-z\right\Vert ^{2}\right] . \label{21}\end{aligned}$$Without loss of generality, we may further assume that $x_{n_{i}}\rightharpoonup p.$ From (\[\*5\]), we have $W_{n}t_{n_{i}}\rightharpoonup p
$. Hence, (\[21\]) reduces to$$\limsup_{n\rightarrow \infty }\left[ \left\langle
W_{n}t_{n}-z,x_{n}-z\right\rangle +\left\Vert W_{n}t_{n}-z\right\Vert ^{2}\right] =2\left\Vert p-z\right\Vert ^{2}$$Now, it is sufficient to show that $p$ belongs to $\tciFourier ,$ i.e., $p=z$. First, we show that $p\in \Omega .$ Let$$Sv=\left\{
\begin{array}{ll}
Av+N_{C}v & ,\text{ }v\in C, \\
\emptyset & ,\text{ }v\notin C.\end{array}\right.$$Then, $S$ is maximal monotone mapping. Let $\left( v,w\right) \in G\left(
S\right) .$ Since $w-Av\in N_{C}v$ and $t_{n}\in C,$ we get$$\left\langle v-t_{n},w-Av\right\rangle \geq 0. \label{10}$$On the other hand, from the definiton of $t_{n},$ we have that$$\left\langle x_{n}-\lambda _{n}Ax_{n}-t_{n},t_{n}-v\right\rangle \geq 0$$and hence,$$\left\langle v-t_{n},\frac{t_{n}-x_{n}}{\lambda _{n}}+Ax_{n}\right\rangle
\geq 0.$$Therefore, using (\[10\]), we get$$\begin{aligned}
\left\langle v-t_{n_{i}},w\right\rangle &\geq &\left\langle
v-t_{n_{i}},Av\right\rangle \\
&\geq &\left\langle v-t_{n_{i}},Av\right\rangle -\left\langle v-t_{n_{i}},\frac{t_{n_{i}}-x_{n_{i}}}{\lambda _{n_{i}}}+Ax_{n_{i}}\right\rangle \\
&=&\left\langle v-t_{n_{i}},Av-Ax_{n_{i}}-\frac{t_{n_{i}}-x_{n_{i}}}{\lambda
_{n_{i}}}\right\rangle \\
&=&\left\langle v-t_{n_{i}},Av-At_{n_{i}}\right\rangle +\left\langle
v-t_{n_{i}},At_{n_{i}}-Ax_{n_{i}}\right\rangle \\
&&-\left\langle v-t_{n_{i}},\frac{t_{n_{i}}-x_{n_{i}}}{\lambda _{n_{i}}}\right\rangle \\
&\geq &\left\langle v-t_{n_{i}},At_{n_{i}}-Ax_{n_{i}}\right\rangle
-\left\langle v-t_{n_{i}},\frac{t_{n_{i}}-x_{n_{i}}}{\lambda _{n_{i}}}\right\rangle .\end{aligned}$$Hence, for $i\rightarrow \infty ,$ we have$$\left\langle v-p,w\right\rangle \geq 0.$$Since $S$ is maximal monotone, we have $p\in S^{-1}0$ and hence $p\in \Omega
.$ Next, we show that $p\in F\left( W\right) .$ From (\[\*7\]), Lemma [b]{} and by using $x_{n_{i}}\rightharpoonup p$, we have that $p\in F\left(
W\right) .$ So, from Lemma \[e3\], we get $p\in \tciFourier $.
Also, Opial’s condition guarantee that the weakly subsequential limit of $\left\{ x_{n}\right\} $ is unique. Hence, this implies that $x_{n}\rightharpoonup p\in \tciFourier .$ From the uniqueness of the solution of the variational inequality, we obtain $p=z\in \tciFourier $. So, the desired conclusion$$\limsup_{n\rightarrow \infty }\left[ \left\langle
W_{n}t_{n}-z,x_{n}-z\right\rangle +\left\Vert W_{n}t_{n}-z\right\Vert ^{2}\right] \leq 0$$is obtained.
Furthermore, $p=\lim_{n\rightarrow \infty }P_{\tciFourier }x_{n}.$ Indeed, since $p\in \tciFourier ,$ we have$$\left\langle p-P_{\tciFourier }x_{n},P_{\tciFourier
}x_{n}-x_{n}\right\rangle \geq 0.$$By Lemma \[c\], $\left\{ P_{\tciFourier }x_{n}\right\} $ converges strongly to $u_{0}\in \tciFourier .$ Then, we get$$\left\langle p-u_{0},u_{0}-p\right\rangle \geq 0,$$and hence $p=u_{0}.$
**Step 5. ** Let** **$z\in \tciFourier .$ Then, we have$$\begin{aligned}
\left\Vert x_{n+1}-z\right\Vert ^{2} &=&\left\Vert W_{n}P_{C}\left(
I-\lambda _{n}A\right) y_{n}-z\right\Vert ^{2} \\
&=&\left\Vert W_{n}P_{C}\left( I-\lambda _{n}A\right) y_{n}-W_{n}P_{C}\left(
I-\lambda _{n}A\right) z\right\Vert ^{2} \\
&\leq &\left\Vert y_{n}-z\right\Vert ^{2}=\left\langle
y_{n}-z,y_{n}-z\right\rangle \\
&=&\left\langle \left( 1-\alpha _{n}\right) \left( x_{n}-z\right) +\alpha
_{n}\left( W_{n}t_{n}-z\right) ,y_{n}-z\right\rangle \\
&=&\left( 1-\alpha _{n}\right) \left\langle x_{n}-z,y_{n}-z\right\rangle
+\alpha _{n}\left\langle W_{n}t_{n}-z,y_{n}-z\right\rangle \\
&\leq &\left( 1-\alpha _{n}\right) \left\Vert x_{n}-z\right\Vert ^{2}+\alpha
_{n}\left\langle W_{n}t_{n}-z,y_{n}-z\right\rangle \\
&=&\left( 1-\alpha _{n}\right) \left\Vert x_{n}-z\right\Vert ^{2}+\alpha
_{n}^{2}\left\langle W_{n}t_{n}-z,x_{n}-z\right\rangle \\
&&+\alpha _{n}\left( 1-\alpha _{n}\right) \left\langle
W_{n}t_{n}-z,W_{n}t_{n}-z\right\rangle \\
&=&\left( 1-\alpha _{n}\right) \left\Vert x_{n}-z\right\Vert ^{2}+\alpha
_{n}\beta _{n}\end{aligned}$$ where $\beta _{n}=\alpha _{n}\left\langle W_{n}t_{n}-z,x_{n}-z\right\rangle
+\left( 1-\alpha _{n}\right) \left\Vert W_{n}t_{n}-z\right\Vert ^{2}$. Thus an application of Lemma \[Y\] combined with Step 4 yields that the sequence $\left\{ x_{n}\right\} $ defined by (\[12\]) converges strongly to the unique element $z\in \tciFourier .$
Let $C$ be a nonempty closed convex subset of a real Hilbert space $H,$ let $A:C\rightarrow H$ be an $\alpha $-inverse strongly monotone mapping and let $T$ be a nonexpansive self-mappings on $C$ such that $F\left( T\right) \cap
\Omega \neq \emptyset .$ Let $\left\{ x_{n}\right\} $ be a sequence defined by$$\left\{
\begin{array}{l}
x_{0}=x\in C \\
x_{n+1}=TP_{C}\left( I-\lambda _{n}A\right) y_{n} \\
y_{n}=\left( 1-\alpha _{n}\right) x_{n}+\alpha _{n}TP_{C}\left( I-\lambda
_{n}A\right) x_{n},\forall n\geq 0,\end{array}\right.$$where $\{\lambda _{n}\}\subset \lbrack a,b]$ for some $a,b\in (0,2\alpha )$ and $\left\{ \alpha _{n}\right\} \subset \left[ c,d\right] $ for some $c,d\in \left( 0,1\right) $. Then, the sequence $\left\{ x_{n}\right\} $ converges strongly to a point $z\in F\left( T\right) \cap \Omega $ where $z$ is the unique solution of the variational inequality (\[25\]).
Applications
============
In the first section, we state that the convex minimization problem is one of the application area of the variational inequality problems and the fixed point problems. One of the relationships between a convex minimization problem and a variational inequality problem is as follows: Let $f$ be a convex differentiable function on a nonempty closed convex subset $C$ of a real Hilbert space $H$ and $\limfunc{Argmin}_{x\in C}f\left( x\right) $ be the set of minimizers of $f$ relative to the set $C$. Then, it is known that element $x^{\ast }\in C$ is a minimizer of $f\left( x\right) $ if and only if $x^{\ast }$ satisfies the variational inequality (\[25\]). On the other hand, iterative processes are often used to minimize a convex differentiable function. Also, it is stated in Remark \[y\] that every $L$-Lipschitzian mapping is $2/L$-inverse strongly monotone mapping. Therefore, we can give the following strong convergence theorem.
Let $C$ be a nonempty closed convex subset of a real Hilbert space $H$. Let $f$ be a convex differentiable function on an open set $D$ containing the set $C$ and let $\left\{ T_{n}\right\} $ be an infinite family of nonexpansive self mappings on $C$ such that $\mathcal{G=}\bigcap_{n=0}^{\infty }F\left(
T_{n}\right) \cap \limfunc{Argmin}_{x\in C}f\left( x\right) \neq \emptyset $. Suppose that the gradient vector of $f$, $\nabla f,$ is a $L$-Lipschitz continuous operator on $D$. For an arbitrarily initial value $x_{0}\in C,$ let $\left\{ x_{n}\right\} $ be a sequence in $C$ defined by$$\left\{
\begin{array}{l}
x_{n+1}=W_{n}P_{C}\left( I-\lambda _{n}\nabla f\right) y_{n} \\
y_{n}=\left( 1-\alpha _{n}\right) x_{n}+\alpha _{n}W_{n}P_{C}\left(
I-\lambda _{n}\nabla f\right) x_{n},\forall n\geq 0,\end{array}\right.$$where $W_{n}$ is a mapping defined by (\[11\]), $\{\lambda _{n}\}\subset
\lbrack a,b]$ for some $a,b\in (0,4/L)$ and $\left\{ \alpha _{n}\right\}
\subset \left[ c,d\right] $ for some $c,d\in \left( 0,1\right) $. Then the sequence $\left\{ x_{n}\right\} $ converges strongly to an element of $\mathcal{G}$.
Considering the Remark \[y\], as in the proof of Theorem \[1\*\], if we take $A=\nabla f$, then we obtain the desired conclusion.
Next, we give another theorem for a pair of nonexpansive mapping and strictly pseudocontractive mapping. A mapping $S:C\rightarrow C$ is called $k
$-strictly pseudocontractive mapping if there exists $k$ with $0\leq k<1$ such that$$\left\Vert Sx-Sy\right\Vert ^{2}\leq \left\Vert x-y\right\Vert
^{2}+k\left\Vert \left( I-S\right) x-\left( I-S\right) y\right\Vert ^{2}$$for all $x,y\in C.$ Let $A=I-S.$ Then, it is known that the mapping $A$ is inverse strongly monotone mapping with $\left( 1-k\right) /2$, i.e.,$$\left\langle Ax-Ay,x-y\right\rangle \geq \frac{1-k}{2}\left\Vert
Ax-Ay\right\Vert ^{2}.$$
Let $C$ be a nonempty closed convex subset of a real Hilbert space $H.$ Let $\left\{ T_{n}\right\} $ be an infinite family of nonexpansive self mappings on $C$ and $S:C\rightarrow C$ be a $k$-strictly pseudocontractive mapping such that $\mathcal{H}=\bigcap_{n=0}^{\infty }F\left( T_{n}\right) \cap
F\left( S\right) \neq \emptyset .$ For an arbitrarily initial value $x_{0}\in C,$ let $\left\{ x_{n}\right\} $ be a sequence defined by$$\left\{
\begin{array}{l}
x_{n+1}=W_{n}\left( \left( I-\lambda _{n}\right) y_{n}+\lambda
_{n}Sy_{n}\right) \\
y_{n}=\left( 1-\alpha _{n}\right) x_{n}+\alpha _{n}W_{n}\left( \left(
I-\lambda _{n}\right) x_{n}+\lambda _{n}Sx_{n}\right) ,\forall n\geq 0,\end{array}\right.$$where $\{\lambda _{n}\}\subset \lbrack a,b]$ for some $a,b\in (0,1-k)$ and $\left\{ \alpha _{n}\right\} \subset \left[ c,d\right] $ for some $c,d\in
\left( 0,1\right) $. Then, the sequence $\left\{ x_{n}\right\} $ converges weakly to a point $p\in \mathcal{H}$.
Let $A=I-S.$ Then, we know that $A$ is inverse strongly monotone mapping. Also, it is clear that $F\left( S\right) =VI\left( C,A\right) .$ Since, $A$ is a mapping from $C$ into itself, we get$$\left( I-\lambda _{n}\right) x_{n}+\lambda _{n}Sx_{n}=x_{n}-\lambda
_{n}\left( I-S\right) x_{n}=P_{C}\left( I-\lambda _{n}A\right) x_{n}.$$So, from Theorem \[1\*\], we obtain the desired conclusion.
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|
---
abstract: 'In programmatic advertising, ad slots are usually sold using second-price (SP) auctions in real-time. The highest bidding advertiser wins but pays only the second highest bid (known as the [*winning price*]{}). In SP, for a single item, the dominant strategy of each bidder is to bid the true value from the bidder’s perspective. However, in a practical setting, with budget constraints, bidding the true value is a sub-optimal strategy. Hence, to devise an optimal bidding strategy, it is of utmost importance to learn the winning price distribution accurately. Moreover, a demand-side platform (DSP), which bids on behalf of advertisers, observes the winning price if it wins the auction. For losing auctions, DSPs can only treat its bidding price as the lower bound for the unknown winning price. In literature, typically censored regression is used to model such partially observed data. A common assumption in censored regression is that the winning price is drawn from a fixed variance (homoscedastic) uni-modal distribution (most often Gaussian). However, in reality, these assumptions are often violated. We relax these assumptions and propose a heteroscedastic fully parametric censored regression approach, as well as a mixture density censored network. Our approach not only generalizes censored regression but also provides flexibility to model arbitrarily distributed real-world data. Experimental evaluation on the publicly available dataset for winning price estimation demonstrates the effectiveness of our method. Furthermore, we evaluate our algorithm on one of the largest demand-side platform and significant improvement has been achieved in comparison with the baseline solutions.'
author:
- 'Aritra Ghosh[^1] ()'
- Saayan Mitra
- Somdeb Sarkhel
- Jason Xie
- Gang Wu
- Viswanathan Swaminathan
title: 'Scalable Bid Landscape Forecasting in Real-time Bidding'
---
Introduction
============
Real-time Bidding (RTB) has become the dominant mechanism to sell ad slots over the internet in recent times. In RTB, ad display opportunities are auctioned when available from the publishers (sellers) to the advertisers (buyers). When a user sees the ad that won the auction, it is counted as an [*ad impression*]{}. An RTB ecosystem consists of supply-side platforms (SSP), demand-side platforms (DSP) and an Ad Exchange. When a user visits a publisher’s page, the SSP sends a request to the Ad Exchange for an ad display opportunity which is then rerouted to DSPs in the form of a bid request. DSPs bid on behalf of the advertisers at the Ad Exchange. The winner of the auction places the Ad on the publisher’s site. Ad Exchanges usually employ second-price auction (SP) where the winning DSP only has to pay the second highest bidding price [@yuan2013real]. Since this price is the minimum bidding price DSP needs to win, it is known as the [*winning price*]{}. When a DSP wins the auction, it knows the actual winning price. However, if the DSP loses the auction, the Ad Exchange does not reveal the winning price. In that case, the bidding price provides a lower bound on the winning price. This mixture of observed and partially-observed (lower bound) data is known as [*right censored data*]{}. The data to the [*right*]{} of the bidding price is not observed since it is right censored.
For a single [*ad impression*]{} under the second price auction scheme, the dominant strategy for an advertiser is to bid the true value of the ad. In this scenario, knowing the bidding prices of other DSPs does not change a bidder’s strategy [@edelman2007internet]. However, in reality, DSPs have budget constraints with a utility goal (e.g., number of impressions, clicks, conversions). Under budget constraints, with repeated auctions, bidding the true value is no longer the dominant strategy [@balseiro2015repeated]. In this setting, knowledge of the bidding prices of other bidders can allow one to change the bid to improve its expected utility. DSP needs to estimate the cost and utility of an auction to compute the optimal bidding strategy (or bidding price) [@zhang2014optimal]. To compute the expected cost as well as the expected utility one needs to know the winning price distribution. Therefore, modeling the winning price distribution is an important problem for a DSP [@lang2012handling]. This problem is also referred to as the [*Bid landscape forecasting*]{} problem.
Learning the bid landscape from a mix of observed and partially-observed data poses a real challenge. It is not possible for DSPs to know the behavior of the winning price beyond the maximum bidding price. Parametric approaches often assume that the winning price follows some distribution. In the existing literature, Gaussian and Log-Normal distributions are often used for modeling the winning price [@wu2015predicting; @cui2011bid]. However, these simple distributions do not always capture all the complexities of real-world data. Moreover, for losing bids, the density of winning price cannot be measured directly, and hence a standard log-likelihood based estimate does not typically work on the censored data. In this scenario, a common parametric method used is [*Censored Regression*]{}, which combines the log density and the log probability for winning and losing auctions respectively [@wu2015predicting; @powell1984least]. Another common alternative is to use non-parametric survival based methods using the Kaplan-Meier (KM) estimate for censored data [@kaplan1958nonparametric]. To improve the performance of the KM estimate, clustering the input is important. Interestingly, in [[@wang2016functional]]{}, the authors proposed to grow a decision tree based on survival methods. In the absence of distributional assumptions, non-parametric methods (KM) work well. However, efficiently scaling non-parametric methods is also challenging. On the other hand, parametric methods work on strong distributional assumptions. When the assumptions are violated, inconsistency arises. For a general discussion of the censored problem in machine learning, readers are referred to [[@wang2017machine]]{}.
Learning a distribution is generally more challenging than point estimation. Thus, parametric approaches in previous research often considered point estimation [@wu2015predicting; @wu2018deep]. However, to obtain an optimal bidding strategy, one needs the distribution of the winning price. On the other hand, non-parametric approaches like the KM method computes the distribution without any assumptions. However, these methods require clustering the data to improve the accuracy of the model using some heuristics. Clustering based on feature attributes makes these methods sub-optimal impacting generalization ability for dynamic real-world ad data.
In this paper, we close the gap of violated assumptions in parametric approaches on censored data. Censored regression-based approaches assume a unimodal (often Gaussian) distribution on winning price. Additionally, it assumes that the standard deviation of the Gaussian distribution is unknown but fixed. However, in most real-world datasets these assumptions are often violated. For example, in Figure \[fig:km-estimate\], we present two winning price distributions (learned using the KM estimate) as well as fitted Gaussian distributions[^2] on two different partitions of the iPinYou dataset [@zhang2014real]. It is evident from Figure \[fig:km-estimate\] that the distributions are neither Gaussian (blue line) nor have fixed variance (red line). In this paper, we relax each of these assumptions one by one and propose a general framework to solve the problem of predicting the winning price distribution using partially observed censored data. We first propose an additional parameterization which addresses the fixed variance assumption. Further, the Mixture Density Network is known to approximate any continuous, differentiable function with enough hidden nodes [[@bishop1994mixture]]{}. We propose a Mixture Density Censored Network to learn smooth winning price distribution using the censored data. We refer to it as MCNet in the rest of the paper. Both of our proposed approaches are generalizations of the Censored Regression.
Our main contributions are as follows. The typical deployed system uses Censored regression for point estimation of the winning price. However, we argue that point estimation is not enough for an optimal bidding strategy. We improve upon the parametric Censored Regression model to a general framework under minimal assumptions. We pose Censored Regression as a solution to the winning price distribution estimation problem (instead of a point estimate). To the best of our knowledge, we are the first to apply the mixture density network on censored data for learning the arbitrary distribution of the winning price. Our extensive experiments on a real-world public dataset show that our approach vastly out-performs existing state-of-the-art approaches such as [@wu2015predicting]. Evaluation on the historical bid data from Adobe (DSP) shows the efficacy of our scalable solution. While we restricted the analysis to winning price distribution in real-time bidding, MCNet is applicable to any partially observed censored data problem.
Background & Related Work {#sec:back}
=========================
In RTB, a DSP gets bid requests from the Ad exchange. We represent the $i^{th}$ bid request by a feature vector ${\mathbf{x}}_i$, which captures all the characteristics of the bid request. Most of the elements of ${\mathbf{x}}_i$’s are categorical (publisher verticals, user’s device, etc.). If DSP wins the auction, it pays the second (winning) price. Formally, the winning price is, $${\mathbf{w}}_i= \max\{{\mathbf{b}}^{\text{Pub}}_i, {\mathbf{b}}^{\text{DSP}_1}_i, {\mathbf{b}}^{\text{DSP}_2}_i, \cdots, {\mathbf{b}}^{\text{DSP}_K}_i\}$$ where ${\mathbf{b}}^{\text{Pub}}_i$ is the floor price set by the publisher[^3] (often 0), and ${\mathbf{b}}^{\text{DSP}_1}_i, \cdots, {\mathbf{b}}^{\text{DSP}_K}_i$ are bidding prices from all other participating DSPs. We use ${\mathbf{b}}_i$ to denote the bidding price from the DSP of our interest. Here we provide an example to illustrate the winning price (in SP auction). Suppose DSPs A, B, C bid $\$1$, $\$2$, $\$3$ respectively for a bid request. DSP C then wins the auction and pays the second-highest price, i.e., $\$2$. For DSP C, the winning price is $\$2$ (observed). For losing DSPs, A, and B, the winning price is $\$3$ (which is unknown to them). In this paper, we define the winning price from the perspective of a single DSP. Learning the landscape of winning price accurately is important for an optimal bidding strategy. A DSP is usually interested in some utility ${\mathbf{u}}_i$ (e.g., clicks, impressions, conversions) for each bid request ${\mathbf{x}}_i$ and wants to maximize the overall utility using bidding strategy ${\mathcal{A}}$ and with budget ${\mathcal{B}}$. This can be represented by the following optimization problem, $\max_{{\mathcal{A}}} \sum_{i} {\mathbf{u}}_i \mbox{ s.t. } \sum_{i} cost_i\leq {\mathcal{B}}$, where $cost_i$ is the price the DSP pays, if it wins the auction. Although the variables are unknown beforehand, the expected cost and the utility can be computed using the historical bid information. Thus the problem simplifies to, $$\begin{aligned}
\max_{{\mathcal{A}}} &\sum_{i}{\mathrm{E}}[{\mathbf{u}}_i|{\mathbf{x}}_i, {\mathbf{b}}_i] \mbox{ s.t. } \sum_{i} {\mathrm{E}}[ cost_i|{\mathbf{x}}_i, {\mathbf{b}}_i]\leq {\mathcal{B}}\end{aligned}$$ Note that, the expected utility ${\mathbf{u}}_i$ is conditioned on bid request ${\mathbf{x}}_i$ and the actual bid ${\mathbf{b}}_i$. For bid request ${\mathbf{x}}_i$, we represent the winning price distribution as $P_{{\mathbf{w}}}({\mathbf{W}}_i|{\mathbf{x}}_i)$, and its cumulative distribution function (cdf) as $F_{{\mathbf{w}}}({\mathbf{W}}_i|{\mathbf{x}}_i) $. If the DSP bids ${\mathbf{b}}_i$ for ${\mathbf{x}}_i$, expected cost and expected utility (for SP auction) is, $${\mathrm{E}}[cost_i|{\mathbf{x}}_i, {\mathbf{b}}_i] = {\int_0^{{\mathbf{b}}_i} {\mathbf{w}}P_{{\mathbf{w}}}({\mathbf{W}}_i={\mathbf{w}}|{\mathbf{x}}_i)d{\mathbf{w}}} ,\quad \ {\mathrm{E}}[{\mathbf{u}}_i|{\mathbf{x}}_i,{\mathbf{b}}_i]= F_{{\mathbf{w}}}({\mathbf{b}}_i|{\mathbf{x}}_i) {\mathrm{E}}[{\mathbf{u}}_i|{\mathbf{x}}_i]$$ An example of expected utility conditioned on bid request (${\mathrm{E}}[{\mathbf{u}}_i|{\mathbf{x}}_i]$) is Click-through rate (CTR). CTR prediction is a well-studied problem in academia and the industry [@wang2017deep]. We want to point out that the expected cost (${\mathrm{E}}[cost_i|{\mathbf{x}}_i, {\mathbf{b}}_i] $) is not the same as the expected winning price (${\mathrm{E}}[{\mathbf{W}}_i|{\mathbf{x}}_i]$). The former is always lower than the latter and is equal only when ${\mathbf{b}}_i\rightarrow \infty$ (i.e., when the advertiser wins the auction with probability 1 and observe the winning price). Thus predicting the winning price distribution instead of the point estimate is important [@wang2016display]. Further, for pacing the budget, one requires an estimate of winning price distribution [@agarwal2014budget]. In [@zhang2016bid], the authors proposed an unbiased learning algorithm of click-through rate estimation using the winning price distribution. Earlier parametric methods, considered point estimation of the winning price. The censored regression-based approach assumes a standard unimodal distribution with a fixed but unknown variance to model the winning price [@wu2015predicting; @zhu2017gamma; @wang2017deep]. In another paradigm, non-parametric methods such as the KM estimator has been successful for modeling censored data [@kaplan1958nonparametric; @wang2016functional]. In the rest of the paper, we use $P$ to denote probability density function (pdf) and $\Pr$ to denote the usual probabilities. Next, we describe how Censored Regression is applied to model the winning price.
Censored Regression
-------------------
The data available to DSP is right censored by the Ad Exchange, i.e., for losing bids only a lower bound (the bidding price) of the winning price is known. However, a maximum likelihood estimator (MLE) can still work on the censored data with some assumptions. In [@wu2015predicting], the authors assume that the winning price follows a normal distribution with fixed but unknown variance $\sigma$. The authors assume a linear relationship between the mean of the normal distribution and the input feature vector. We use ${\mathbf{W}}_i$ to represent the random variable of winning price distribution of $i^{\mbox{th}}$ bid request whereas ${\mathbf{w}}_i$ is the realization of that. Thus ${\mathbf{w}}_i = \beta^T {\mathbf{x}}_i + \epsilon_i$ where $\epsilon_i$ are independent and identically distributed ([*i.i.d*]{}) from ${\mathcal{N}}(0, \sigma^2)$ and ${\mathbf{W}}_i \sim {\mathcal{N}}(\beta^T{\mathbf{x}}_i, \sigma^2)$. One can use any standard distribution in the censored regression approach. In [[@wu2018deep]]{}, the authors argue that maximal bidding price in the limit (of infinite DSPs) resembles Gumbel distribution. However, for the generality of learning from censored data, we do not constrain on any particular distribution in this paper. Moreover, the linear link function can be replaced with any non-linear function. Thus, ${\mathbf{w}}_i$ can be parameterized as ${\mathbf{w}}_i= f(\beta, {\mathbf{x}}_i)+\epsilon_i$ where $f$ can be any continuous differentiable function. With the success of deep models, in [[@wu2018deep]]{}, the authors parameterize $f(\beta, {\mathbf{x}}_i)$ with a deep network for additional flexibility. Since we know the winning price for winning auctions, likelihood is simply the probability density function (pdf) $P({\mathbf{W}}_i = {\mathbf{w}}_i) = \frac{1}{\sigma}\phi(\frac{{\mathbf{w}}_i - \beta^T{\mathbf{x}}_i}{\sigma})$ where $\phi$ is the pdf of standard normal ${\mathcal{N}}(0,1)$. Note that, ${\mathbf{W}}_i$ is the random variable associated with the winning price distribution whereas ${\mathbf{w}}_i$ is the observed winning price. For losing auctions, as we do not know the winning price, the pdf is unknown to us. However, from the lower bound on the winning price, we can compute the probability that bid ${\mathbf{b}}_i$ will lose in the auction for bid request ${\mathbf{x}}_i$, under the estimated distribution of ${\mathbf{W}}_i$ as $ \Pr({\mathbf{W}}_i>{\mathbf{b}}_i) = \Pr(\epsilon_i < \beta^T{\mathbf{x}}_i -{\mathbf{b}}_i ) = \Phi(\frac{\beta^T{\mathbf{x}}_i - {\mathbf{b}}_i}{\sigma}).$ Here $\Phi$ is the cdf for standard normal distribution. As discussed, $\phi$ and $\Phi$ can be replaced with pdf and cdf of any other distribution (with different parameterization).
Taking log of the density for winning auctions ${\mathcal{W}}$ and the log-probability for losing auctions ${\mathcal{L}}$, we get the following objective function [@wu2015predicting], $$\begin{aligned}
\beta^{\ast}, \sigma^{\ast} =& \mbox{arg}\max_{\beta, \sigma>0} \sum_{i \in {\mathcal{W}}} {\log\left(\frac{1}{\sigma} \phi(\frac{{\mathbf{w}}_i - \beta^T{\mathbf{x}}_i}{\sigma}) \right)}
+ \sum_{i \in {\mathcal{L}}} {\log\left( \Phi(\frac{\beta^T{\mathbf{x}}_i - {\mathbf{b}}_i}{\sigma}) \right)}
\label{eq:censored}
\end{aligned}$$ When the $\epsilon_i$ (noise in the winning price model) are i.i.d samples from a fixed variance normal distribution, censored regression is an unbiased and consistent estimator [@james1984consistency; @greene1981asymptotic].\
Methodology
===========
In this paper, we build on top of (Gaussian) censored regression-based approach by relaxing some of the assumptions that do not hold in practice. First, we relax the assumption of *homoscedasticity*, i.e., noise (or error) follows a normal distribution with fixed but possibly unknown variance, by modeling it as a fully parametric censored regression. Then we also relax the unimodality assumption by proposing a mixture density censored network. We describe the details of our approaches in the next two subsections.
Fully Parametric Censored Regression
------------------------------------
The censored regression approach assumes that the winning price is normally distributed with a fixed standard deviation. As we discussed, in Figure \[fig:km-estimate\], the variance of the fitted Gaussian model is not fixed. If the noise $\epsilon$ is heteroscedastic or not from a fixed variance normal distribution, the MLE is biased and inconsistent. Using a single $\sigma$ to model all bid requests, will not fully utilize the predictive power of the censored regression model. Moreover, while the point estimate (mean) of the winning price is not dependent on the estimated variance, the [*Bid landscape*]{} changes with $\sigma$. We remove the restriction of homoscedasticity in the censored regression model and pose it as a solution to the distribution learning problem.
Specifically, we assume the error term $\epsilon$ is coming from a parametric distribution conditioned on the features. This solves the problem of error term coming from fixed variance distribution. We assume the noise term $\epsilon_i$ is coming from ${\mathcal{N}}(0, \sigma_i^2)$ where $\sigma_i = \exp(\alpha^T{\mathbf{x}}_i)$. The likelihood for winning the auction is, $P({\mathbf{W}}_i ={\mathbf{w}}_i) = \frac{1}{\exp(\alpha^T{\mathbf{x}}_i)}\phi(\frac{{\mathbf{w}}_i - \beta^T{\mathbf{x}}_i}{\exp(\alpha^T{\mathbf{x}}_i)})$ where the predicted random variable ${\mathbf{W}}_i \sim {\mathcal{N}}(\beta^T{\mathbf{x}}_i , \exp(\alpha^T{\mathbf{x}}_i)^2)$ and $\phi$ is the pdf of ${\mathcal{N}}(0,1)$. In fully parametric censored regression, $\epsilon_i\sim {\mathcal{N}}(0, \exp(\alpha^T{\mathbf{x}}_i)^2)$ are not [*i.i.d*]{} samples. For losing bids, we can similarly compute the probability based on the lower bound (bidding price ${\mathbf{b}}_i$) $$\begin{aligned}
\Pr({\mathbf{W}}_i >{\mathbf{b}}_i) = P(\epsilon_i < \beta^T{\mathbf{x}}_i -{\mathbf{b}}_i ) = \Phi(\frac{\beta^T{\mathbf{x}}_i - {\mathbf{b}}_i}{\exp(\alpha^T{\mathbf{x}}_i)})
\end{aligned}$$
Under the assumption of normal but varying variance on the noise, we can still get a consistent and unbiased estimator by solving the following problem.
$$\begin{aligned}
\beta^{\ast}, \alpha^{\ast} =& \mbox{arg}\max_{\beta, \alpha} \sum_{i \in {\mathcal{W}}} {\log\left(\frac{1}{\exp(\alpha^T{\mathbf{x}}_i)} \phi(\frac{{\mathbf{w}}_i - \beta^T{\mathbf{x}}_i}{\exp(\alpha^T{\mathbf{x}}_i)}) \right)}
+\sum_{i \in {\mathcal{L}}} {\log\left( \Phi(\frac{\beta^T{\mathbf{x}}_i - {\mathbf{b}}_i}{\exp(\alpha^T{\mathbf{x}}_i)}) \right)}
\label{eq:parm_censored}
\end{aligned}$$
Mixture Density Censored Network (MCNet)
----------------------------------------
In the previous subsection, we relaxed the fixed variance problem by using a parametric $\sigma$. However, no standard distribution can model the multi-modality that we observe in real-world data. For example, in Figure \[fig:km-estimate\](b), we see mostly unimodal behavior below the max bid price. However, the probability of losing an auction is often high ($61\%$ in Figure \[fig:km-estimate\](b)). Thus even with parametric standard deviation, when we minimize the KL-divergence with a Gaussian, the mean shifts towards the middle. Inspired by the Gaussian Mixture Model (GMM) [@bishop1994mixture] we propose a Mixture Density Censored Network (MCNet). MCNet resembles a Mixture Density Network while handling partially observed censored data for learning arbitrary continuous distribution.
In a GMM, the estimated random variable ${\mathbf{W}}_i$ consists of $K$ Gaussian densities and has the following pdf, $P({\mathbf{W}}_i ={\mathbf{w}}_i) = \sum_{k=1}^{K} \frac{\pi_k({\mathbf{x}}_i)}{ \sigma_k({\mathbf{x}}_i)} \phi(\frac{{\mathbf{w}}_i-\mu_k({\mathbf{x}}_i)}{ \sigma_k^2({\mathbf{x}}_i)})$. Here $\pi_k({\mathbf{x}}), \mu_k({\mathbf{x}}), \sigma_k({\mathbf{x}})$ are the weight, mean and standard deviation for $k^{th}$ mixture density respectively where $k \in \{1, \cdots, K\}$. To model the censored regression problem as a mixture model, a straightforward way is to formulate the mean of the Gaussian distributions with a linear function. Furthermore, to impose positivity of $\sigma$, we model the logarithm of the standard deviation as a linear function. We impose a similar positivity constraint on the weight parameters. The parameters of the mixture model are (for $k\in \{1, \cdots, K\}$), $$\begin{aligned}
\mu_k({\mathbf{x}}_i) = \beta_{\mu, k}^T {\mathbf{x}}_i, \
\sigma_k({\mathbf{x}}_i) = \exp(\beta_{\sigma, k}^T{\mathbf{x}}_i),\
\pi_k({\mathbf{x}}_i) = \frac{\exp(\beta_{\pi, k}^T{\mathbf{x}}_i)}{\sum_{j=1}^{K} \exp(\beta_{\pi,j}^T{\mathbf{x}}_i)}
\end{aligned}$$ We can further generalize this mixture model and define a Mixture Density Network (MDN) by parameterizing $\pi_k({\mathbf{x}}_i), \mu_k({\mathbf{x}}_i), \sigma_k({\mathbf{x}}_i)$ with a deep network. In applications such as speech and image processing and astrophysics, MDNs have been found useful [@zen2014deep; @salimans2017pixelcnn++]. MDN can work with any reasonable choice of base distribution. MDN combines mixture models with neural networks. The output activation layer, consists of $3K$ nodes ($z_{i,k}$ for $i \in \{\mu, \sigma,\pi\}$ and $k\in \{1, \cdots, K\}$ ). We use $z_{\mu,k}, z_{\sigma,k}, z_{\pi,k}$ to retrieve the mean, standard deviation and weight parameters of $k^{th}$ density, $$\begin{aligned}
\mu_k({\mathbf{x}}_i) = z_{\mu,k}({\mathbf{x}}),\
\sigma_k({\mathbf{x}}_i) = \exp (z_{\sigma,k } ({\mathbf{x}})),\
\pi_k({\mathbf{x}}_i) = \frac{\exp(z_{\pi, k}({\mathbf{x}}_i))}{ \sum_{j=1}^{K }\exp(z_{\pi,j} ({\mathbf{x}}_i)) }\label{eq:pi}
\end{aligned}$$
MDN outputs conditional probabilities that are used for learning distribution from fully observed data [[@bishop1994mixture]]{}. For the censored problem, however, we only observe partial data. We can now extend MDN to MCNet on censored data. Instead of conditional output probabilities, MCNet outputs the probability of losing in case auction is lost. Thus, we can compute the log-likelihood function on partially observed data. Taking the likelihood for winning auctions, the corresponding negative log-likelihood for all the winning auctions is given by $\sum_{i \in {\mathcal{W}}} -\log (\sum_{k=1}^{K} \frac{\pi_k({\mathbf{x}}_i)}{\sigma_k({\mathbf{x}}_i)} \phi(\frac{{\mathbf{w}}_i-\mu_k({\mathbf{x}}_i)}{\sigma_k({\mathbf{x}}_i)}))$ where, $\phi$ is the pdf of ${\mathcal{N}}(0,1)$. For losing bids, we can similarly compute the probability of losing based on the lower bound, $\Pr({\mathbf{W}}_i >{\mathbf{b}}_i) = \sum_{k=1}^{K} \pi_k({\mathbf{x}}_i) \Phi(\frac{\mu_k({\mathbf{x}}_i)-{\mathbf{b}}_i}{\sigma_k({\mathbf{x}}_i)})$
Negative log-probability of all the losing auctions from the mixture density is, $$\begin{aligned}
\sum_{i \in {\mathcal{L}}} -\log (\sum_{k=1}^{K} \pi_k({\mathbf{x}}_i) \Phi(\frac{\mu_k({\mathbf{x}}_i)-{\mathbf{b}}_i}{\sigma_k({\mathbf{x}}_i)}))
\label{eq:losemdn}
\end{aligned}$$ where, $\Phi$ represents the cdf of ${\mathcal{N}}(0,1)$.
From Figure \[fig:km-estimate\], recall that the distribution is not unimodal and has multiple peaks. To address the multi-modality of the data we used a mixture of multiple densities. The embedded deep network in the MCNet (${\mathcal{M}}$) is trained to learn the mean and standard deviation parameters of each of the constituents of the mixture model as well as the corresponding weights. Combining all the auctions, we get the following optimization function for censored data,
$$\begin{aligned}
{\mathcal{M}}^{\ast} = \mbox{arg}\max_{{\mathcal{M}}}\sum_{i \in {\mathcal{L}}} \log (\sum_{k=1}^{K} \pi_k({\mathbf{x}}_i) \Phi(\frac{\mu_k({\mathbf{x}}_i)-{\mathbf{b}}_i}{\sigma_k({\mathbf{x}}_i)})) \notag\\
+\sum_{i \in {\mathcal{W}}} \log (\sum_{k=1}^{K} \frac{\pi_k({\mathbf{x}}_i)}{\sigma_k({\mathbf{x}}_i)} \phi(\frac{{\mathbf{w}}_i-\mu_k({\mathbf{x}}_i)}{\sigma_k({\mathbf{x}}_i)})
\label{eq:mdn}
\end{aligned}$$
where ${\mathcal{M}}$ is the neural network (parameters).
Optimization
------------
It is easy to compute gradients of Eq. \[eq:censored\], \[eq:parm\_censored\], \[eq:mdn\] with respect to all the parameters. We used Adam optimizer for stochastic gradient optimization [@kingma2014adam].
Experimental Results
====================
In this section, we discuss experimental settings, evaluation measures, and results.
Experimental Settings
---------------------
#### **Datasets:**
We ran experiments on the publicly available iPinYou dataset [[@zhang2014real]]{} as well as on a proprietary dataset collected from Adobe Adcloud (a DSP). The iPinYou dataset contains censored winning price information. Further experimentation was done on a sampled week’s data from Adobe Adcloud. iPinYou data is grouped into two subsets: session 2 (dates from 2013-06-06 to 2013-06-12), and session 3 (2013-10-19 to 2013-10-27). We experimented with the individual dates within sessions as well. For all the datasets, we allocated 60% for training, 20% for validation and rest 20% for testing. We report the average as well as the standard deviation over five iterations. Similar to previous research on the iPinYou dataset, we remove fields that are not directly related to the winning price at the onset [@wu2015predicting; @wang2016functional]. The fields used in our methods are UserAgent, Region, City, AdExchange, Domain, AdSlotId, SlotWidth, SlotHeight, SlotVisibility, SlotFormat, Usertag. Every categorical feature (e.g City), is one-hot encoded, whereas every numerical feature (e.g Ad height) is categorized into bins and subsequently represented as one-hot encoded vectors. This way, each bid request is represented as a high-dimensional sparse vector. Table \[tab:ipin\] shows the statistics of sessions in the iPinYou datasets. The number of samples and win rates for individual dates are mentioned in Table. \[tab:dates\].
Session sample feature win rate ($\%$)
--------- ------------ --------- -----------------
2 53,289,330 40,664 22.87
3 10,566,743 25,020 29.64
: Basic statistics of iPinYou Sessions
\[tab:ipin\]
#### **Evaluation Settings:**
Evaluation on partially observed data is difficult when the winning price is unknown especially for point estimation. In [[@wu2015predicting]]{}, the authors simulated new synthetic data from the original winning auctions. While the added censoring allows validating point estimate, it does not use the whole data (or the true distribution). We evaluate the performance of predicting the winning price distribution rather than the point estimate itself. Thus we use the whole data without generating simulated censoring behavior. This setting is similar to earlier work on the survival tree-based method where the authors evaluated predicting the distribution and used the original data [@wang2016functional].
#### **Parametric methods:**
We compared the Censored Regression (CR) approach with our methods: Fully parametric Censored Regression (P-CR) and Mixture Density Censored Network (MCNet). For every method, we added an L2 regularization term for each weight vector to prevent over-fitting. For MCNet, we added an additional hyper-parameter on the number of mixtures. We chose a fully connected hidden layer with 64 nodes with RelU activation function as the architecture. Our framework is general and can be extended to multiple layers. The number of mixture components was varied from 2-4 for individual dates and 2-8 for the experiments on the two sessions. We used Adam optimizer with a learning rate of $10^{-3}$. Mini-batch training was employed due to the high volume of the data and we fixed the batch size to 1024 samples. We employed early stopping on the training loss and do not observe the validation loss for early stopping. This way, all the methods are treated similarly. The L2 regularization was varied from $10^{-6} \mbox{ to } 10$ (in log scale). We implemented the parametric methods in Tensorflow [@abadi2016tensorflow]. For the initialization of weight vectors, we sampled randomly from the standard normal distribution in all our experiments.
Recently extending Censored Regression (CR), in [[@wu2018deep]]{}, the authors proposed to use deep model (DeepCR) to parameterize the mean to provide more flexibility in the point estimation. Additionally, the authors proposed to use Gumbel distribution for point estimation. Note that, MCNet generalizes the DeepCR model when using only one mixture component and Gumbel as the base distribution. We did not see much improvement when using Gumbel to parameterize mixture components with our initial experiments. With enough Gaussian mixture components, MCNet can approximate any smooth distribution. As neural architecture is not the primary motivation for this paper, we do not discuss different architectures or distributions in this paper.
#### **Non-parametric methods:**
To the best of our knowledge, parametric methods and non-parametric methods were not compared together for winning price distribution estimation in earlier research. We compared our approaches with non-parametric approaches based on Kaplan-Meier (KM) estimate and the Survival tree (ST) method. The KM and ST based methods produce winning price distributions until the maximum bid price since the winning price behavior above that is unknown. To represent a complete landscape with the probability distribution summing to one, we introduce an extra variable representing the event that the winning price is beyond the maximum bid price. For the Survival tree, we varied the tree height from 1-20.
In the ST method, the Survival tree is built by running an Expectation Maximization (EM) algorithm for each field to cluster similar attributes. If data has $F$ fields and the average number of attributes in each field is $K$, then for $n$ samples, the EM algorithm takes, ${\mathcal{O}}(FKln)$ steps to cluster features based on their density for $l$ iterations. With depth $d$, total complexity becomes ${\mathcal{O}}(FKlnd)$. Given this runtime, we could not run ST using all attributes of [*Domain*]{}, [*SlotId*]{} fields (these fields were removed in previous research [@wang2016functional]). We trimmed the [*Domain*]{} and [*SlotId*]{} features by combining the attributes that appeared less than $10^3$ times. We created [*“other domains"*]{} and [*“other slot ids"*]{} bins for these less frequent attributes. This improved the time complexity and made the method viable. But for the CR-based methods, we could easily relax this threshold and trimmed both the features where the attributes appeared less than 10 times in the dataset in either session. For a fair comparison, whenever we use the same feature trimming in the parametric methods as ST, we denote using $\mbox{CR}^{\ast}, \mbox{P-CR}^{\ast},\mbox{MCNet}^{\ast}$. Note that parametric methods can scale easily whereas the non-parametric ST method cannot.
#### **Baseline Method:**
We also propose a simple baseline method and compare it with other methods. The baseline algorithm picks a winning price randomly conditioned on the win rate. We denote this as the Random Strategy (RS). Formally, let the maximum bid price be $z$ and probability of a win be $p$. Then, the probability that the winning price is $w$ is given by $$\begin{aligned}
P({\mathbf{W}}=w) &= \frac{p}{z} \mbox{ if } w\in[0,z], \mbox{ and } 0 \mbox{ if } w <0\
\mbox{and }&\int_z^{\infty} \Pr({\mathbf{W}}=w)dw = 1-p
\end{aligned}$$ Thus with probability $1-p$, it predicts the event that winning price is greater than max bid price and with probability $p$ it draws from ${\mathcal{U}}(0,z)$ where ${\mathcal{U}}$ is the Uniform distribution.
Evaluation Measure
------------------
Our objective is to learn the distribution of the winning price, rather than the point estimate. Hence, we choose Average Negative Log Probability ([[*ANLP* ]{}]{}) as our evaluation measure similar to [@wang2016functional]. [[*ANLP* ]{}]{}is defined as, $$\begin{aligned}
\mbox{{{\it ANLP }}}= &-\frac{1}{N}\Big(\sum_{i\in {\mathcal{W}}}\log \Pr({\mathbf{W}}_i ={\mathbf{w}}_i)+
\sum_{i\in {\mathcal{L}}} \log \Pr({\mathbf{W}}_i \geq {\mathbf{b}}_i)\Big)
\end{aligned}$$ where ${\mathcal{W}}$ represents the set of winning auctions, ${\mathbf{w}}_i$ represents winning price of the $i^{\mbox{th}}$ winning auction, ${\mathcal{L}}$ is the set of losing auctions, ${\mathbf{b}}_i$ is the bidding price for the $i^{\mbox{th}}$ losing auction, and $|{\mathcal{W}}| + |{\mathcal{L}}| = N$.
Note that, we computed pdf for winning auctions and probability (or the CDF) for losing auctions while optimizing. While the CDF represents the probability of the event, density does not represent probability. Additionally, bid prices are an integer. The KM method estimates the probability on those discrete points. However, parametric approaches estimate a continuous random variable whose probability at any discrete point is $0$. To treat losing bids and winning bids similarly in evaluation, we use quantization trick on the continuous random variable [@gersho2012vector]. For the parametric approaches, the estimate ${\mathbf{W}}_i$ is a continuous random variable. We discretized the random variable ${\mathbf{W}}_i$ as follows, ${\mathbf{W}}_i^{\mbox{bin}} = l, \mbox{ if } {\mathbf{W}}_i \in (l-0.5,l+0.5]$ where $l$ is an integer. Thus, for winning auctions ${\mathcal{W}}$ with winning price ${\mathbf{w}}_i$, quantized probability is, $$\begin{aligned}
\Pr({\mathbf{W}}_i^{\mbox{bin}} ={\mathbf{w}}_i) & =\Pr({\mathbf{W}}_i\leq {\mathbf{w}}_i +0.5) - \Pr({\mathbf{W}}_i\leq {\mathbf{w}}_i-0.5)
\end{aligned}$$ For losing auctions ${\mathcal{L}}$, the quantized probability is, $\Pr({\mathbf{W}}_i^{\mbox{bin}} \geq {\mathbf{b}}_i) =\Pr({\mathbf{W}}_i\geq {\mathbf{b}}_i-0.5)$. Using quantization technique, winning bids and losing bids are treated similarly for all methods.
Experimental Results
--------------------
\[tab:dates\]
\[tab:adcloud\]
In this section, we discuss quantitative results on iPinYou sessions 2 and 3. In Table \[tab:dates\], we provide average [[*ANLP* ]{}]{}over different dates as well as the standard deviation (std) numbers. In figure \[fig:session\], we mention the result on each session as a whole. Moreover, we plot how number of mixture components as well as tree depth affect the result for MCNet and ST respectively in Figure \[fig:component\]. In sessions 2 and 3 where we include all the dates, we also added the ST method for comparison. As ST did not run with large feature space, we also added $\mbox{CR}^{\ast}, \mbox{P-CR}^{\ast},\mbox{MCNet}^{\ast}$ for parity (where number of feature was small for all methods).
From Table \[tab:dates\], it is evident, P-CR improves upon CR on most dates (except with low volume dates) asserting the violation of fixed standard deviation assumption. While for P-CR, improvement is around $5\%$-$10\%$, MCNet improves CR by more than $30\%$ on all dates. Improvement of MCNet re-verify our assumption about the multi-modal nature of the winning price distribution. CR performs better than both RS as well as KM. This is expected as the non-parametric KM estimate does not use any features. However, KM improves RS by around [$10\%$]{} on all dates. ST improves CR and P-CR significantly implying the significance of non-parametric estimators.
In Figure \[fig:session\], one can see similar trends over CR, P-CR, and MCNet. With feature trimming, $\mbox{MCNet}^{\ast}$ performs similarly to ST methods. This is expected as both ST and MCNet can predict arbitrary smooth distributions. Although, when the MCNet approach is restricted to fewer features ($\mbox{MCNet}^{\ast}$) on the average it performs similarly to ST, the benefits of parametric methods come from the fact that parametric approaches are scalable to large feature as well as input space. It may be observed that the performance of MCNet improves ST by more than $10\%$ on both sessions. While we used only one hidden layer for MCNet, any deep network can be used to parameterize the mixture density network for potentially improving the MCNet results even further.
In Figure \[fig:component\](a), we plot [[*ANLP* ]{}]{}for different depths of the decision trees. It can be observed that for ST, the performance saturates around depth 15. In Figure \[fig:component\](b), we also show how the varying number of mixture components impacts [[*ANLP* ]{}]{}. On the larger dataset of Session 2, [[*ANLP* ]{}]{}stabilizes to a low value at 4 mixture components. However, for session 3, [[*ANLP* ]{}]{}starts increasing beyond 6 mixture components, implying over-fitting.
#### **Results on Adobe AdCloud Dataset:**
We also tested our methods on Adobe Advertising Cloud (DSP) offline dataset. We collected a fraction of logs for one week. The number of samples was $31,772,122$ and the number of features was $33,492$. It had similar categorical as well as real-valued features. We use the same featurization framework and represented each bid request with a sparse vector. In Table \[tab:adcloud\], we report the [[*ANLP* ]{}]{}results, using the same experimental setup. Note that, MCNet improves CR by $25\%$ while it improves ST by more than $10\%$. In this dataset, we do see only marginal improvement over using P-CR.
Discussion & Future Work
========================
In this paper, we particularly focus on one of the central problems in RTB, the winning price distribution estimation. In practice, DSP depends on the estimated bid landscape to optimize it’s bidding strategy. From a revenue perspective, an accurate bid landscape is of utmost importance. While, non-parametric methods can estimate arbitrary distributions, in practice, it is challenging to scale on large datasets. On the other hand, widely used parametric methods, such as Censored Regression in its original form is highly restrictive. We proposed a novel method based on Mixture Density Networks to form a generic framework for estimating arbitrary distribution under censored data. MCNet generalizes a fully parametric Censored regression approach with the number of mixture components set to one. Additionally, Censored regression is a special case of fully parametric censored regression where the standard deviation is fixed. We provided extensive empirical evidence on public datasets and data from a leading DSP to prove the efficacy of our methods. While the mixture of (enough) Gaussian densities can approximate any smooth distribution, further study is needed on the choice of base distribution. A more subtle point arises when learning with censored data as we do not observe any winning price beyond the maximum bidding price. Without any assumptions on the distribution, it is not provably possible to predict the behavior in the censored region. Non-parametric methods only learn density within the limit of maximum bidding price while under strong assumptions of standard distributions, censored regression predicts the behavior of winning price in the censored region. Although MCNet can approximate any smooth distribution, beyond the maximum bidding price, it leads to non-identifiability similar to the KM estimate. It would be interesting to explore combining MCNet with distributional assumptions where the winning price is never observed.
[10]{} \[1\][`#1`]{} \[1\][https://doi.org/\#1]{}
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[^1]: This work was conducted while the first author was doing an internship at Adobe Research, USA
[^2]: We fit the unimodal Gaussian minimizing KL divergence with the estimated KM distribution. We would like to point out, although in Figure \[fig:km-estimate\](b) winning price density is unimodal (within the limit), the probability of winning price beyond the max bid price is high ($0.61$). Thus the fitted Gaussian has a mean of $350$ and std dev of $250$ further from the peak at $75$.
[^3]: For simplicity, we view the floor price by the publisher as a bid from an additional DSP.
|
---
abstract: 'In this paper, we are interested in continuous time models in which the index level induces some feedback on the dynamics of its composing stocks. More precisely, we propose a model in which the log-returns of each stock may be decomposed into a systemic part proportional to the log-returns of the index plus an idiosyncratic part. We show that, when the number of stocks in the index is large, this model may be approximated by a local volatility model for the index and a stochastic volatility model for each stock with volatility driven by the index. This result is useful in a calibration perspective : it suggests that one should first calibrate the local volatility of the index and then calibrate the dynamics of each stock. We explain how to do so in the limiting simplified model and in the original model.'
---
****\
$\,$
*Benjamin Jourdain*[^1] and *Mohamed Sbai$\,^1$*
Introduction {#introduction .unnumbered}
============
[F]{}rom the early eighties, when trading on stock index was introduced, quantitative finance faced the problem of efficiently pricing and hedging index options along with their underlying components. Many advances have been made for single stock modeling and a variety of solutions to escape from the very restrictive model has been deeply investigated (such as local volatility models, models with jumps or stochastic volatility models). However, when the number of underlyings is large, index option pricing, or more generally basket option pricing, remains a challenge unless one simply assumes constantly correlated dynamics for the stocks. The problem then is the impossibility of fitting both the stocks and the index smiles.
We try to address this issue by making the dynamics of the stocks depend on the index. The natural fact that the volatility of the index is related to the volatilities of its underlying components has already been accounted for in the works of Avellaneda *et al.* [@Avellaneda] and Lee *et al.* [@Lee]. In the first paper, the authors use a large deviation asymptotics valid for small values of the product of the maturity by the square of the volatility to reconstruct the local volatility of the index from the local volatilities of the stocks. They express this dependence in terms of the implied volatilities using the results of Berestecky *et al.*([@Berestycki1],[@Berestycki2]). In the second paper, the authors reconstruct the Gram-Charlier expansion of the probability density of the index from the stocks using a moments-matching technique. Both papers consider local volatility models for the stocks and a constant correlation matrix but the generalization to stochastic volatility models or to varying correlation coefficients is not straightforward.
Another point of view is to say that the volatility of a composing stock should be related to the index level, or say to the volatility of the index, in some way. This is not astonishing since the index represents the move of the market and reflects the view of the investors on the state of the economy. Moreover, it is consistent with equilibrium economic models like CAPM. Following this idea, we propose a new modeling framework in which the volatility of the index and the volatilities of the stocks are related. We show that, when the number of underlying stocks tends to infinity, our model reduces to a local volatility model for the index and to a stochastic volatility model for the stocks where the stochastic volatility depends on the index level. This asymptotics is reasonable since the number of stocks composing an index is usually large. As a consequence, the correlation matrix between the stocks in our model is not constant but stochastic and we show that it is consistent with empirical studies. Finally, we address calibration issues and we show that it is possible, within our framework, to fit both index and stocks smiles. The method we introduce is based on the simulation of SDEs nonlinear in the sense of McKean, and non-parametric estimation of conditional expectations.
This paper is organized as follows. In Section 1, we specify our model for the index and its composing stocks and in Section 2 we study the limiting model when the number of underlying stocks goes to infinity. Section 3 is devoted to calibration issues. Numerical results are presented in Section 4 and the conclusion is given in Section 5.
$\,$\
We thank Lorenzo Bergomi, Julien Guyon and all the equity quantitative research team of Societe Generale CIB for numerous fruitful discussions and for providing us with the market data.
$\,$\
Model Specification
===================
An index is a collection of stocks that reflects the performance of a whole stock market or a specific sector of a market. It is valued as a weighted sum of the value of its underlying components. More precisely, if $I^M_t$ stands for the value at time $t$ of an index composed of $M$ underlyings, then $$I^M_t=\sum_{j=1}^M w_j S^{j,M}_t,$$ where $S^{j,M}_t$ is the value of the stock $j$ at time $t$ and the weightings $(w_j)_{j=1\dots M}$ are given constants[^2].
Unless otherwise stated, we always work under a risk-neutral probability measure. In order to account for the influence of the index on its underlying components, we specify the following stochastic differential equations for the stocks $$\forall j \in \{1,\dots,M\}, \quad
\frac{dS^{j,M}_t}{S^{j,M}_t}=(r-\delta_j) dt + \beta_j
\,\sigma(t,I^{M}_t) dB_t + \eta_j(t,S^{j,M}_t) dW^j_t,\;S^{j,M}_0=s^j_0$$ where
- $r$ is the short interest rate,
- $s^j_0$ is the initial value of the stock $j$,
- $\delta_j \in [0,\infty[$ is the continuous dividend rate of the stock $j$,
- $\beta_j$ is the usual beta coefficient of the stock $j$ that quantifies the sensitivity of the stock returns to the index returns (see the seminal paper of Sharpe [@sharpe]). It is defined as $\frac{Cov(r_{j},r_I)}{Var(r_I)}$ where $r_{j}$ (respectively $r_{I}$) is the rate of return of the stock $j$ (respectively of the index),
- $(B_t)_{t\in[0,T]},(W^1_t)_{t\in[0,T]},\dots,
(W^M_t)_{t\in[0,T]}$ are independent Brownian motions,
- the functions $\sigma,\eta_1,\dots,\eta_M:[0,T]\times{\mathbb{R}}\to{\mathbb{R}}$ satisfy the usual Lipschitz and growth assumptions that ensure existence and strong uniqueness of the solutions (see for example Theorem 5.2.9 of [@KaratzasShreve]) :
\[hyp:LIP\] $\exists K$ such that $\forall (t,s_1,s_2) \in
[0,T]\times {\mathbb{R}}^M\times {\mathbb{R}}^M,$ $$\begin{array}{l}
{\displaystyle}\sum_{j=1}^M \left|s^j_1 \sigma\left(t,\sum_{k=1}^M w_k
s^{k}_1\right)\right| + \left|s^j_1
\eta_j(t,s^j_1)\right| \leq K\left(1+|s_1|\right)\\[3mm]
{\displaystyle}\sum_{j=1}^M \left|s^j_1 \sigma\left(t,\sum_{k=1}^M w_k
s^{k}_1\right)-s^j_2 \sigma\left(t,\sum_{k=1}^M w_k
s^{k}_2\right)\right| \leq K|s_1-s_2|\\[3mm]
{\displaystyle}\sum_{j=1}^M \left|s^j_1 \eta_j(t,s^j_1)-s^j_2
\eta_j(t,s^j_2)\right|\leq K|s_1-s_2|.
\end{array}$$
As a consequence, the index satisfies the following stochastic differential equation : $$dI^M_t=r I^M_t dt -\left(\sum_{j=1}^M \delta_j w_j S^{j,M}_t\right)
dt+ \left(\sum_{j=1}^M \beta_j w_j S^{j,M}_t\right)\sigma(t,I^{M}_t)
dB_t +\sum_{j=1}^M w_j S^{j,M}_t \eta_j(t,S^{j,M}_t) dW^j_t
\label{indexSDE}$$
Before going any further, let us make some preliminary remarks on this framework.
- We have $M$ coupled stochastic differential equations. The dynamics of a given stock depends on all the other stocks composing the index through the volatility term $\sigma(t,I^M_t)$. Since there are $M$ linearly independent assets and $M+1$ driving Brownian motions, the market is incomplete.
- Accounting for the dividends is not relevant for all types of indices. Indeed, for many performance-based indices (such as the German DAX index) dividends and other events are rolled into the final value of the index.
- The cross-correlations between stocks are not constant but stochastic : $$\rho_{ij}(t)=\frac{\beta_i \beta_j \sigma^2(t,I^{M}_t)}{{\displaystyle}\sqrt{\beta_i^2\sigma^2(t,I^{M}_t)+\eta_i^2(t,S^{i,M}_t)}\,
\sqrt{\beta_j^2\sigma^2(t,I^{M}_t)+\eta_j^2(t,S^{j,M}_t)}}$$ Note that they depend not only on the stocks but also on the index. More importantly, it is commonly observed that the more the market is volatile, the more the stocks tend to be highly correlated. This feature is reproduced here as we can easily check that an increase in the index volatility, with everything else left unchanged, produces an increase in the cross-correlations.
In a recent paper, Cizeau *et al.* [@Bouchaud1] show that it is possible to capture the essential features of stocks cross-correlations, in particular in extreme market conditions, by a simple non-Gaussian one factor model. The authors successfully compare different empirical measures of correlation with the prediction of the following model : $$r_j(t)=\beta_j r_I(t)+\epsilon_j(t)\label{bouchaud}$$
where $r_j(t)=\frac{S^j_t}{S^j_{t-1}}-1$ is the daily return of stock $j$, $r_I(t)$ is the daily return of the market and the residuals $\epsilon_j(t)$ are independent random variables following a fat-tailed distribution[^3].
Our model is in line with (\[bouchaud\]). Indeed, since the beta coefficients are usually narrowly distributed around 1, the factor $\sum_{j=1}^M \beta_j w_j S^{j,M}_t$ of $\sigma(t,I^M_t)$ in (\[indexSDE\]) is close to $I^M_t$. Moreover, since $${\mathbb{E}}\left(\left(\int_0^T\sum_{j=1}^M w_j
S^{j,M}_t \eta_j(t,S^{j,M}_t) dW^j_t\right)^2\right)\leq \sum_{j=1}^M
w_j^2\sup_{1\leq j\leq M}\int_0^T{\mathbb{E}}\left((S^{j,M}_t
\eta_j(t,S^{j,M}_t))^2\right)dt\sim \sum_{j=1}^M
w_j^2 T,$$ one can neglect the term $\sum_{j=1}^M w_j S^{j,M}_t \eta_j(t,S^{j,M}_t) dW^j_t$ in the dynamics of the index when $\sum_{j=1}^M
w_j^2$ is small. Of course, this approximation worsens when the maturity $T$ increases. The latter condition is satisfied when $M$ is large and the weighting vector $(w_1,\hdots,w_M)$ is close to the vector $(\frac{1}{M},\hdots,\frac{1}{M})$ with constant coefficients for which $\sum_{j=1}^M\frac{1}{M^2}=\frac{1}{M}$. Then, if we denote by $r_j$ the log-return of the stock $j$ and by $r_{I^M}$ the log-return of the index, both on a daily basis, we will have $$r_j=\beta_j r_{I^M}+\eta_j \Delta W^j + \text{drift},$$ where $\Delta W^j$ is an independent Gaussian noise. Consequently, in our model too, the return of a stock is decomposed into a systemic part driven by the index, which represents the market, and a residual part.
Asymptotics for a large number of underlying stocks
===================================================
The number of underlying components of an index is usually large[^4]. As discussed in the previous section, when $\sum_{j=1}^M w_j^2$ is small, one can neglect the term $\sum_{j=1}^M w_j S^{j,M}_t
\eta_j(t,S^{j,M}_t) dW^j_t$ in and derive a simplified approximate dynamics for the index. The aim of this section is to quantify the error we commit by doing so.
To be specific, consider the limit candidate $(I_t)_{t\in[0,T]}$ solution of the following SDE : $$\left\{\begin{array}{rcl} dI_t&=&{\displaystyle}(r-\delta) I_t dt + \beta I_t
\sigma(t,I_t) dB_t \\
I_0&=&{\displaystyle}i_0
\end{array}\right.
\label{Ilim}$$ where $i_0=\sum_{j=1}^M w_js^j_0$ and $\delta$ and $\beta$ are two constant parameters that will be discussed later.
In the following theorem, we give an upper bound for the $L^{2p}$-distance between $(I^{M}_t)_{t\in[0,T]}$ and $(I_t)_{t\in[0,T]}$ under mild assumption on the volatility coefficients :
\[convind\] Let $p\in{\mathbb N}^*$. Under assumption (${\mathcal{H}}$\[hyp:LIP\]) and if the following assumptions on the volatility coefficients hold,
\[hyp:bornitude\] $\exists K_b$ such that $\forall (t,s) \in [0,T]\times {\mathbb{R}}_+,\quad
|\sigma(t,s)|+|\eta_j(t,s)|\leq K_b$.
\[hyp:xsLip\] $\exists K_\sigma$ such that $\forall
(t,s_{1},s_{2}) \in [0,T]\times {\mathbb{R}}_+\times {\mathbb{R}}_+,\quad
|s_{1}\sigma(t,s_{1})-s_{2}\sigma(t,s_{2})|\leq K_\sigma|s_{1}-s_{2}|$.
then $${\mathbb{E}}\left(\sup_{0\leq t \leq T}|I^{M}_t-I_t|^{2p}\right) \leq C_T \left(\left(\sum_{j=1}^{M}
w_{j}^2\right)^{\!\!p} + \left(\sum_{j=1}^M
w_j|\beta_j-\beta|\right)^{2p}+ \left(\sum_{j=1}^M
w_j|\delta_j-\delta|\right)^{2p} \right)$$ where $${\displaystyle}C_T=8^{2p-1}
T^p (T^p+K_pK_b^{2p})C_p\exp\left(4^{2p-1}T(2^{2p-1}K_pT^{p-1}(\beta
K_\sigma)^{2p}+(2T)^{2p-1}
\delta^{2p}+r^{2p} \,T^{2p-1})\right)$$ and $$C_p=\max_{1\leq j \leq
M}|s^{j}_{0}|^{2p} \exp\left(\left(2r+(2p-1)(\max_{j \geq
1}\beta_j^2+1)K_b^2\right)pT\right).$$
According to this result proved in the appendix, the smaller $P^M_w\stackrel{\rm def}{=}\sqrt{\sum_{j=1}^Mw_j^2}$, $P^M_\beta\stackrel{\rm def}{=}\sum_{j=1}^M w_j|\beta_j-\beta|$ and $P^M_\delta\stackrel{\rm def}{=}\sum_{j=1}^M w_j|\delta_j-\delta|$, the closer $I$ and $I^M$. The first quantity $P^M_w$ is small when the weighting vector $(w_1,\hdots,w_M)$ is close to $(\frac{1}{M},\hdots,\frac{1}{M})$ and $M$ is large. Let us now discuss how to choose $\beta$ and $\delta$ minimizing $P^M_\beta$ and $P^M_\delta$. Let $Y_\beta$ and $Y_\delta$ be discrete random variables having the following probability distributions : $$\forall
j \in \{1,\dots,M\},\quad \quad
{\mathbb{P}}\left(Y_\beta=\beta_j\right)=\frac{w_j}{\sum_{i=1}^M w_i}\quad
\text{and}\quad
{\mathbb{P}}\left(Y_\delta=\delta_j\right)=\frac{w_j}{\sum_{i=1}^M w_i} .$$ Then $$P^M_\beta=\left(\sum_{i=1}^M w_i\right)\times{\mathbb{E}}\left|Y_\beta-\beta\right|
\quad \text{and} \quad P^M_\delta=\left(\sum_{i=1}^M w_i\right)\times{\mathbb{E}}\left|Y_\delta-\delta\right|.$$ Consequently, the optimal choice of the parameters is the median[^5] of $Y_\beta$ for $\beta$ and the median of $Y_\delta$ for $\delta$. Nevertheless, to preserve the interpretation of $\beta_j$ as $\frac{Cov(r_j,r_I)}{Var(r_I)}$ which is equal to $\frac{\beta\beta_j}{\beta^2}$ for the simplified index dynamics, one should take $\beta=1$. In Table \[tab:poids\], we see that on the example of the Eurostoxx index at December 21 2007, the optimal choice of $\beta$ is very close to 1 and that the quantities of interest, $(P^M_{\beta_{opt}})^2$ and $(P^M_{\beta=1})^2$ are also very close to each other.
$(P^M_w)^2$ $\beta_{opt}$ $(P^M_{\beta_{opt}})^2$ $(P^M_{\beta=1})^2$
------------- --------------- ------------------------- ---------------------
0.026 0.975 0.0173 0.0174
: Computation of $(P^M_w)^2, \beta_{opt}$ and $(P^M_{\beta_{opt}})^2$ for the Eurostoxx index at December 21, 2007. The beta coefficients are estimated on a two year history.[]{data-label="tab:poids"}
The next theorem states that, under an additional assumption on the volatility coefficients, the $L^{2p}$-distance between a stock $(S^{j,M}_{t})_{t\in [0,T]}$ and the solution of the SDE obtained by replacing $I^M$ by $I$ $$\frac{dS^{j}_t}{S^{j}_t}=(r-\delta_j) dt + \beta_j
\,\sigma(t,I_t) dB_t + \eta_j(t,S^{j}_t) dW^j_t,\;S^j_0=s^{j}_0\label{edsjlim}$$ is also controlled by $2p$-powers of $P^M_w$, $P^M_\beta$ and $P^M_\delta$. One major drawback of the limiting simplified model - is that the limit index $I_t$ is only approximately equal to the reconstructed index level $\overline{I}^M_t\stackrel{\rm def}{=}\sum_{j=1}^M w_j S^j_t$. The next result also gives an estimation of the difference between $I^M$ and $\overline{I}^M$ in terms of $P^M_w$, $P^M_\beta$ and $P^M_\delta$, which combined with the previous theorem, provides an estimation of the difference between $I$ and $\overline{I}^M$.
\[constock\] Let $p\in{\mathbb N}^*$. Under the assumptions of Theorem 1 and if
\[hyp:xetaLip\] $\exists K_\eta$ such that $\forall j,\;\forall
(t,s_{1},s_{2}) \in [0,T]\times {\mathbb{R}}_+\times {\mathbb{R}}_+,\quad
|s_{1}\eta_j(t,s_{1})-s_{2}\eta_j(t,s_{2})|\leq K_\eta|s_{1}-s_{2}|$.
$\exists K_{Lip}$ such that $\forall (t,s_{1},s_{2}) \in [0,T]\times
{\mathbb{R}}_+\times {\mathbb{R}}_+,\quad |\sigma(t,s_{1})-\sigma(t,s_{2})|\leq
K_{Lip}|s_{1}-s_{2}|$.
Then, $\forall j \in \{1,\dots, M\}$, $${\mathbb{E}}\left(\sup_{0\leq t \leq T}|S^{j,M}_t-S^j_t|^{2p}\right) \leq
\widetilde{C}^j_T \left(\left(\sum_{j=1}^{M} w_{j}^2\right)^{\!\!p}
+ \left(\sum_{j=1}^M
w_j|\beta_j-\beta|\right)^{2p}+ \left(\sum_{j=1}^M
w_j|\delta_j-\delta|\right)^{2p} \right)$$ where $$\widetilde{C}^j_{T}=6^{2p-1}K_pT^p\beta_{j}^{2p}C_{2p}^{\frac{1}{2}} K_{Lip}^{2p} \,\, e^{3^{2p-1}((r-\delta_{j})^{2p}T^{2p-1}+K_pT^{p-1}K_\eta^{2p}+
2^{2p-1}K_pT^{p-1}\beta_{j}^{2p}K_b^{2p})T}.$$ Moreover, for $\overline{I}_t^M=\sum_{j=1}^M w_j S^j_t$, one has $${\mathbb{E}}\left(\sup_{0\leq t \leq
T}|I^M_t-\overline{I}^{M}_t|^{2p}\right) \leq \widetilde{C}_T
\left(\sum_{j=1}^{M} w_{j}\right)^{2p}\left(\left(\sum_{j=1}^{M}
w_{j}^2\right)^{\!\!p}+ \left(\sum_{j=1}^M
w_j|\beta_j-\beta|\right)^{2p}+ \left(\sum_{j=1}^M
w_j|\delta_j-\delta|\right)^{2p} \right)$$ where ${\displaystyle}\widetilde{C}_T=\max_{1\leq j \leq M} \widetilde{C}^j_T$.
The proof can also be found in the appendix. In the following corollary, we consider the limit ${M \to \infty}$ supposing that the weight of the $j$-th stock, now denoted by $w_j^M$, depends on $M$.
Under the assumptions of Theorems 1 and 2 and if
\[hyp:CT\] there exists a finite constant $A$ such that ${\displaystyle}\max_{j\geq 1} \left((s^j_0)^{2} + (\beta_j)^2 +
(\delta_j)^2\right)\leq A$,
\[hyp:i0\] ${\displaystyle}I^M_0=\sum_{j=1}^Mw_j^Ms^j_0{\operatornamewithlimits}{\longrightarrow}_{M \to \infty} i_0\in(0,+\infty)$,
\[hyp:w\] ${\displaystyle}P^M_w=\sqrt{\sum_{j=1}^M (w_j^M)^2}
{\operatornamewithlimits}{\longrightarrow}_{M \to \infty} 0$,
\[hyp:beta\] ${\displaystyle}P^M_\beta=\sum_{j=1}^M w_j^M |\beta_j-\beta|
{\operatornamewithlimits}{\longrightarrow}_{M \to \infty} 0$,
\[hyp:delta\] ${\displaystyle}P^M_\delta =\sum_{j=1}^M w_j^M
|\delta_j-\delta| {\operatornamewithlimits}{\longrightarrow}_{M \to \infty} 0$,
then, for any $p\in{\mathbb{N}}^*$, one has $${\mathbb{E}}\left(\sup_{0\leq t \leq T}|I^{M}_t-I_t|^{2p}\right) {\operatornamewithlimits}{\longrightarrow}_{M \to \infty}
0\;\mbox{ and }\forall j\in{\mathbb{N}}^*,\quad {\mathbb{E}}\left(\sup_{0\leq t \leq T}|S^{j,M}_t-S^j_t|^{2p}\right) {\operatornamewithlimits}{\longrightarrow}_{M \to \infty}
0.$$ If, in addition, ${\displaystyle}\sup_M \sum_{j=1}^M w_j^M < \infty$, then ${\mathbb{E}}\left(\sup_{0\leq t \leq T}|I^{M}_t-\overline{I}^M_t|^{2p}\right) {\operatornamewithlimits}{\longrightarrow}_{M \to \infty}
0$.
Assumptions (${\mathcal{H}}$\[hyp:i0\]), (${\mathcal{H}}$\[hyp:w\]), (${\mathcal{H}}$\[hyp:beta\]) and (${\mathcal{H}}$\[hyp:delta\]) hold for instance when $w^M_j=\frac{1}{M}$ for $1\leq j\leq M$ and $s^j_0{\operatornamewithlimits}{\longrightarrow}_{j\to \infty}
i_0$, $\beta_j{\operatornamewithlimits}{\longrightarrow}_{j\to \infty}
\beta$ and $\delta_j{\operatornamewithlimits}{\longrightarrow}_{j\to \infty}
\delta$.
Simplified model {#simplified-model .unnumbered}
----------------
To sum up, we have shown that, under mild assumptions, when the number of underlying stocks is large, the original model may be approximated by the following dynamics $$\begin{array}{ll}
{\displaystyle}\forall j \in \{1,\dots,M\}, &{\displaystyle}\frac{dS^{j}_t}{S^{j}_t}=(r-\delta_j) dt + \beta_j
\,\sigma(t,I_t) dB_t + \eta_j(t,S^{j}_t) dW^j_t\\[5mm]
&{\displaystyle}\frac{dI_t}{I_t}= (r-\delta_I)dt + \sigma(t,I_t) dB_t.
\end{array}$$ Of course the distance between this limiting model and the original one increases with the maturity.
Interestingly, we end up with a local volatility model for the index and, for each stock, a stochastic volatility model decomposed into a systemic part driven by the index level and an intrinsic part. The calibration procedures presented in the next section are based on this intuition : even in the original model, we are going to calibrate $\sigma$ as if it was the local volatility function of the index.
Note that this simplified model is not valid for options written on the index together with all its composing stocks since the index is no longer an exact, but an approximate, weighted sum of the stocks. In this case, one should consider the reconstructed index $\overline{I}_t^M=\sum_{j=1}^M w_j S^j_t$ or use the original model. The simplified model can be used for options written on the stocks or on the index or even on the index together with few stocks.
Model calibration
=================
Calibration, which is how to determine the model parameters in order to fit market prices at best, is of paramount importance in practice. In the following, we try to tackle this issue for both our simplified and original models.
Simplified model
----------------
In the simplified limiting model, the only factor which influences the dynamics of a given stock is the simplified index $I_t$ which evolves according to an autonomous SDE. So it is enough to address the calibration of a given stock together with the index and we drop the index $j$ of the stock for notational simplicity. $$\begin{array}{ll}
&{\displaystyle}\frac{dS_t}{S_t}=(r-\delta) dt + \beta
\,\sigma(t,I_t) dB_t + \eta(t,S_t) dW_t,\;S_0=s_0\\[5mm]
&{\displaystyle}\frac{dI_t}{I_t}= (r-\delta_I)dt + \sigma(t,I_t) dB_t,\;I_0=i_0.
\end{array}
\label{modelsimp1stock}$$ The short interest rate and the dividend yields can be extracted from the market. The calibration of the local volatility function $\sigma$ to fit index option prices is a classical problem. According to Dupire [@dupire], if $C_I(t,K)$ denotes the market price of the call option with maturity $t$ and strike $K$ written on the index, then for $$\sigma^2(t,K)=2\frac{\frac{\partial C_I}{\partial
t}(t,K)+(r-\delta_I)K\frac{\partial C_I}{\partial K}(t,K)+\delta_I
C_I(t,K)}{K^2\frac{\partial^2 C_I}{\partial K^2}(t,K)},$$ one has $C_I(t,K)={\mathbb{E}}\left(e^{-rt}(I_t-K)^+\right)$ for all $t,K>0$. Of course, in practice the market quotes call options only for a finite number of couples $(t,K)$. What seems a common practice among banks is to look for $\sigma$ in a parametric family of functions and compute the parameters minimizing the distance between these quoted prices and the call prices associated with the parametrized local volatility function. Since each practitioner may choose his favorite procedure to address this classical problem of local volatility calibration, we will not enter in more details.
We also assume that a local volatility function is associated with the stock by the same procedure and denote by $v_{loc}(t,x)$ the local variance function of the stock computed as the square of this local volatility function. So the local volatility model $${\displaystyle}\frac{d\overline{S}_t}{\overline{S}_t}=(r-\delta) dt +
\sqrt{v_{loc}(t,\overline{S}_t)}dW_t,\;\overline{S}_0=s_0\label{lvms}$$ is calibrated to the quoted prices of vanilla options written on the stock. In , by independence between $B$ and $W$, the variance of the stock at time $t$ is equal to $\beta^2\sigma^2(t,I_t)+\eta^2(t,S_t)$. According to Gyöngy [@Gyongy], if $$\forall
t,x>0,\;{\mathbb{E}}\left(\beta^2\sigma^2(t,I_t)+\eta^2(t,S_t)|S_t=x\right)=v_{loc}(t,x)$$ then and the local volatility model induce the same marginal distributions for the stock and therefore the same prices for the vanilla call options written on it : ${\mathbb{E}}\left(e^{-rt}(S_t-K)^+\right)={\mathbb{E}}\left(e^{-rt}(\overline{S}_t-K)^+\right)$ for all $t,K>0$. Hence if $$\forall
t,x>0,\;\eta(t,x)=\sqrt{v_{loc}(t,x)-\beta^2{\mathbb{E}}\left(\sigma^2(t,I_t)|S_t=x\right)},\label{eqvar}$$ then the stock dynamics in is calibrated to the quoted prices of the vanilla options written on the stock. It remains to choose the coefficient $\beta$ and the function $\eta$ so that this equality is satisfied. The fact that the law of $(S_t,I_t)$ given by and therefore the conditional expectation in depend on $(\beta,\eta)$ makes this problem difficult. Nevertheless, intuitively, when one fixes a value of $\beta$ that is not too large, one should be able to find a function $\eta$ such that is satisfied. The calibration of the stock smile seems over-parametrized and one should rely on the interpretation of $\beta$ as a regression coefficient to choose its value. This issue is discussed in the next section. Then we explain how to approximate the conditional expectation and deduce $\eta$ for a fixed value of $\beta$.\
Let us already point out that the calibration of our simplified model gives an advantage to the fit of index option prices in comparison with options written on the stocks, which is in line with the market since index options are usually very liquid in comparison with individual stock options.
### Choice of the coefficient $\beta$ {#choixbeta}
The interpretation of $\beta$ as the regression coefficient of the log-returns of the stock with respect to the log-returns of the index makes it possible to estimate this coefficient on historical data. Nevertheless, when the historical estimator $\beta_{hist}$ is large, then the difference in the r.h.s. of may become negative for some $(t,x)$ when $\beta=\beta_{hist}$. Then the square root is no longer defined and calibration for this choice of $\beta$ is no longer possible.
In Figure \[fig:vollocproblems\], we have plotted the local volatility of the stock $x\mapsto\sqrt{v_{loc}(T,s_0x)}$, the local volatility of the index $x\mapsto\sigma(T,i_0x)$, the systemic part of the volatility of the stock $x\mapsto\beta_{hist} \sigma(T,s_0x)$ and $x\mapsto\beta_{hist}
\sqrt{{\mathbb{E}}\left(\sigma^2(T,I_T)|S_T=s_0x\right)}$ when $\eta$ is set to zero (which intuitively gives the lowest local volatility function of the stock that one can obtain in our model ) as functions of the moneyness for a maturity $T=1$ year. We considered three representative components of the Eurostoxx which is composed of $M=50$ stocks : AXA, ALCATEL and CARREFOUR at December 21, 2007. We made this choice deliberately in order to point out the extreme situations that one can face :
- AXA is an example of a stock with a high historical beta coefficient ($\beta_{hist}=1.4$),
- CARREFOUR is an example of a stock with a low historical beta coefficient ($\beta_{hist}=0.7$),
- ALCATEL is an example of a stock with a high volatility level but with a rather flat smile ($\beta_{hist}=1.1$).
Clearly, we can deduce that the market is choosing a $\beta$ coefficient for both AXA and ALCATEL that is lower than the historical one whereas, for CARREFOUR, one can plug the historical $\beta$, or even a larger one, in (\[modelsimp1stock\]) and still be able to calibrate the model.
A satisfactory way to handle the estimation of the beta coefficient would be to compute an implied beta calibrated to the prices of options involving the correlation between the stock and the index. Unfortunately, no such option is liquid in the market (the most liquid correlation swaps are sensitive to an average correlation between all the stocks composing the index).
So we suggest to choose $$\beta=\min\left(\beta_{hist},\inf_{t,x>0}\frac{\sqrt{v_{loc}(t,s_0x)}}{\sigma(t,i_0x)}\right)
\label{betmin}.$$ Even if we have no proof that this choice of beta makes the calibration possible, it is sensible and we have checked that it works on the three examples of AXA, ALCATEL and CARREFOUR.
When one is interested in options written on the index together with all its components, one should use the reconstructed index level $\overline{I}^M_t=\sum_{j=1}^M w_j S^j_t$ instead of $I_t$. Of course, the reconstructed index dynamics will reproduce the quoted prices of vanilla options written on the index all the better as $\overline{I}^M_t$ is close to the calibrated limiting index level $I_t$. According to Theorems \[convind\] and \[constock\], for this latter property to hold, one needs $P^M_{\beta=1}=\sum_{j=1}^M w_j|\beta_j-1|\geq|\sum_{j=1}^Mw_j\beta_j-\sum_{j=1}^M w_j|$ to be small. When, because of the minimum in equality , $P^M_{\beta=1}$ is larger for the actual choice of coefficients $\beta$ than for the historical choice, one may take larger values of beta for stocks like CARREFOUR to decrease $P^M_{\beta=1}$ and improve the calibration of the reconstructed index.
### Estimation of the conditional expectation
The idea behind the following techniques is to circumvent the difficulty of calibrating the volatility coefficient $\eta$. Indeed, if we plug the formula in , we obtain a stochastic differential equation that is nonlinear in the sense of McKean :
$$\begin{array}{l}
{\displaystyle}\frac{dS_t}{S_t}=(r-\delta) dt + \beta \,\sigma(t,I_t) dB_t +
\sqrt{v_{loc}(t,S_t)-\beta^2{\mathbb{E}}\left(\sigma^2(t,I_t) \,|\,
S_t\right)} dW_t,\;S_0=s_0\\[5mm]
{\displaystyle}\frac{dI_t}{I_t}= (r-\delta_I)dt + \sigma(t,I_t) dB_t,\;I_0=i_0.
\end{array}
\label{modelsimpNP}$$
For an introduction to the stochastic differential equations nonlinear in the sense of McKean and to propagation of chaos, we refer to the lecture notes of Sznitman [@Sznitman] and Méléard [@Meleard]. In our case, the nonlinearity appears in the diffusion coefficient through the conditional expectation term. This makes the natural question of existence and uniqueness of a solution very difficult to handle. The case of a drift coefficient involving a conditional expectation has only been handled recently even for a constant diffusion coefficient (see for instance Talay and Vaillant [@TalayVaillant] and Dermoune [@Dermoune]). Meanwhile, it is possible to simulate such a stochastic differential equation by means of a system of $N$ interacting paths using either a non-parametric estimation of the conditional expectation or regression techniques. The advantage of the regression approach over the non-parametric estimation is that it also yields a smooth approximation of the function $x\mapsto{\mathbb{E}}\left(\sigma^2(t,I_t)
\,|\, S_t=x\right)$ whereas, with a non-parametric method, one has to interpolate the estimated function and to carefully tune the window parameter to obtain a smooth approximation.
$\,$\
**3.1.2a $\,$ Non-parametric estimation**\
$\,$
Non-parametric estimators of the conditional expectation, and more generally non-parametric density estimators, have been widely studied in the literature. We will focus on kernel estimators of the Nadaraya-Watson type (see [@Watson] and [@nadaraya]) : given $N$ observations $(S_{i,t},I_{i,t})_{i=1\dots N}$ of $(S_t,I_t)$, we consider the kernel conditional expectation estimator of ${\mathbb{E}}\left(\sigma^2(t,I_t) \,|\, S_t=x\right)$ given by
$$\frac{{\displaystyle}\sum_{i=1}^N
\sigma^2(t,I_{i,t}) K\left(\frac{x-S_{i,t}}{h_N}\right)}{{\displaystyle}\sum_{i=1}^N K\left(\frac{x-S_{i,t}}{h_N}\right)}$$ where $K$ is a non-negative kernel such that $\int_{\mathbb R} K(x)dx=1$ and $h_N$ is a smoothing parameter which tends to zero as $N\rightarrow +\infty$. This leads to the following system with $N$ interacting particles : $\forall \, 1\leq i\leq N,$ $$\begin{cases}
\frac{dS_{i,N,t}}{S_{i,N,t}}=(r-\delta) dt + \beta
\,\sigma(t,I_{i,t}) dB_{i,t} +
\sqrt{v_{loc}(t,S_{i,N,t})-\beta^2\frac{\sum_{j=1}^N
\sigma^2(t,I_{j,t}) K\left(\frac{S_{i,N,t}-S_{j,N,t}}{h_N}\right)}{
\sum_{j=1}^N
K\left(\frac{S_{i,N,t}-S_{j,N,t}}{h_N}\right)}}dW_{i,t},\;S_{i,N,0}=s_0\\[3mm]
\frac{dI_{i,t}}{I_{i,t}}= (r-\delta_I)dt + \sigma(t,I_{i,t})
dB_{i,t},\;I_{i,0}=i_0
\end{cases}\label{systpart}$$ where $(B_i,W_i)_{i\geq 1}$ is a sequence of independent two-dimensional Brownian motions. The integer $i$ indexes the sample-paths of the fixed stock that we consider. In their dynamics, the conditional expectation term has been replaced by interaction. The price in the calibrated model of a European option with maturity $T$ and payoff function $h:C([0,T],{\mathbb{R}})\to{\mathbb{R}}$ written on the stock may be approximated by $$\frac{1}{N}\sum_{i=1}^Ne^{-rT}h(S_{i,N,.})
\label{prixcal}.$$
The $2N$-dimensional SDE may be discretized using the Euler scheme. Let $n\in{\mathbb{N}}^*$ and $0=t_0< \cdots <t_n=T$ be the subdivision with step $\frac{T}{n}$ of $[0,T]$. For each $k\in \{0,\dots,n-1\}$, $\forall \, 1\leq i\leq
N,$ $$\begin{cases}
\overline{S}_{i,N,t_{k+1}}=\overline{S}_{i,N,t_k}\bigg(1+\sqrt{v_{loc}(t_k,\overline{S}_{i,N,t_k})-\beta^2\frac{\sum_{j=1}^N
\sigma^2(t_k,\overline{I}_{j,t_k})
K\left(\frac{\overline{S}_{i,N,t_k}-\overline{S}_{j,N,t_k}}{h_N}\right)}{\sum_{j=1}^N
K\left(\frac{\overline{S}_{i,N,t_k}-\overline{S}_{j,N,t_k}}{h_N}\right)}} \sqrt{\frac{T}{n}} \tilde{G}_{i,k}
\\\phantom{\overline{S}_{i,N,t_{k+1}}=\overline{S}_{i,N,t_k}\bigg(}+(r-\delta)
\frac{T}{n} + \beta \,\sigma(t_k,\overline{I}_{i,t_k})
\sqrt{\frac{T}{n}} G_{i,k}\bigg) \\[3mm]
\overline{I}_{i,t_{k+1}}=
\overline{I}_{i,t_{k}}\left(1+(r-\delta_I)\frac{T}{n} +
\sigma(t_k,\overline{I}_{i,t_k}) \sqrt{\frac{T}{n}} G_{i,k}\right)
\end{cases}$$ where $(G_{i,k})_{1\leq i \leq N,0 \leq k \leq n-1}$ and $(\tilde{G}_{i,k})_{1\leq i \leq N,0 \leq k \leq n-1}$ are independent centered and reduced Gaussian random variables.
$\,$\
**3.1.2b $\,$ Parametric estimation**\
$\,$
Another approach to estimate conditional expectations is to use parametric estimators, or projection. This idea has also been widely used and studied previously (for example in finance, one can think of the Longstaff-Schwartz algorithm for pricing American options [@LongstaffSchwartz]). Noting that the conditional expectation is a projection operator on the space of square integrable random variables, one can approximate ${\mathbb{E}}\left(\sigma^2(t,I_t) \,|\,
S_t=x\right)$ by the parametric estimator $\sum_{l=1}^L \alpha_l f_l(x)$ where $(f_l)_{l=1\dots L}$ is a functional basis and $\alpha=(\alpha_l)_{l=1\dots L}$ is a vector of parameters estimated by least mean squares : given $N$ observations $(S_{i,t},I_{i,t})_{i=1\dots N}$ of $(S_t,I_t)$, $\alpha$ minimizes $\sum_{i=1}^N \left(\sigma^2(t,I_{i,t}) -\sum_{l=1}^L \alpha_l
f_l(S_{i,t})\right)^2$.
### Numerical results
### 3.1.3a $\,$ A toy example {#a-a-toy-example .unnumbered}
We try to calibrate a stock with a local variance function $v_{loc}$ constant and equal to $v$. We choose $\sigma$ as the local volatility function of the Eurostoxx index fitted to the market at December 21, 2007.
We simulate the system of $N$ interacting paths and price call options for different strikes using . In Figure \[fig:volimp2\], we plot the implied volatility at $T=1$ obtained for independent simulations of $N=5000$ paths and see that they are indeed close to the desired volatility level $\sqrt{v}$. This example was generated with the following arbitrary set of parameters : $$S_0=100,\;\beta=0.7,\;r=0.05,\;\delta=\delta_I=0,\;\sqrt{v}=0.6,\;N=5000,\;n=20.$$
In this example and for all the following numerical experiments, we use a Gaussian kernel : $K(u)=\frac{1}{\sqrt{2 \pi}}
e^{-\frac{u^2}{2}}$. The smoothing parameter $h_N$ is set to $N^{-\frac{1}{5}}$ which is the optimal bandwidth that one obtains when minimizing the asymptotic mean square error of the Nadaraya-Watson estimator under some regularity assumptions and assuming independence of the random variables involved (see for example Bosq [@Bosq]).
$\,$\
**3.1.3b $\,$ An example with real data**
In the following, we test our model with real data. More precisely, given the local volatilities of the Eurostoxx index and of Carrefour at December 21, 2007, we simulate (\[modelsimpNP\]) by different methods for a one year maturity.
$\,$\
**An acceleration technique**
The simulation of the particle system is very time consuming : for each discretization step and for each stock particle, one has to make $N$ computations which yield a global complexity of order $O(nN^2)$ where $n$ is the number of time steps in the Euler scheme. Acceleration techniques are thus desirable. One possible method consists in reducing the number of interactions : instead of making $N$ computations for each estimation of the conditional expectation, one can neglect interactions which involve particles which are far away from each other. When the kernel used is non increasing with the absolute value of its argument, the easiest way to implement this idea is to sort the particles at each step and, whenever a contribution of a particle is lower than some fixed threshold, to stop the estimation of the conditional expectation.
Of course, by doing this, we lose in precision for the same number of interacting particles, especially for deep in/out of the money strikes. But what we gain in terms of computation time is much more important : in Figure \[fig:volimp3\], we plot the implied volatility obtained by the naive method and the method with the above acceleration technique for the same number $N=10000$ of particles. We take as threshold $\frac{1}{N}$ and set $h_N=N^{-\frac{1}{10}}$ for the bandwidth parameter[^6] and $n=20$ for the number of time steps in the Euler scheme. The computation time, on a computer with a 2.8 Ghz Intel Penthium 4 processor, is of 52 minutes for the naive method and of 5 minutes for the accelerated one.
More importantly, we see that the implied volatility $\widehat{\sigma}_{N}$ obtained by simulation of the system with $N$ interacting particles converges to the exact implied volatility $\widehat{\sigma}_{exact}$ computed from quoted option prices as $N$ tends to $\infty$ : see Figure \[fig:volimp4\] and Table \[tab:tab1\]. With a reasonable number of simulated paths, $N=200000$, the error on the implied volatility remains clearly tolerable for practitioners (of the order of 10 bp) except for a deep in the money call ($K=0.3 S_0$) where it attains 195 bp.
Moneyness ($\frac{K}{S_0}$) 0.30 0.49 0.69 0.79 0.89 0.99 1.09 1.19 1.28 1.48 1.98
----------------------------------------------------------- ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------
Error : $|\widehat{\sigma}_{N}-\widehat{\sigma}_{exact}|$ 195 36 8 5 2 1 2 9 17 32 56
: Error (in bp) on the implied volatility with $N=200000$ particles.[]{data-label="tab:tab1"}
$\,$\
**Independent particles**
Unlike the parametric method, non-parametric estimation of the conditional expectation gives the value of the intrinsic volatility $\eta$ at the simulated points only. However, using an interpolation technique, one can first reconstruct $\eta$ with $N_1$ dependent particles and then simulate $N_2$ independent paths of the $2$-dimensional stochastic differential equation . By doing so, we speed up the simulations but one has to choose carefully the size $N_1$ of the particle system in order to have a reasonable estimation of the intrinsic volatility and to tune the bandwidth parameter in order to smooth the estimation (our numerical tests were done with $N_1=1000,
N_2=100000$ and $h_{N_1}=N_1^{-\frac{1}{10}}$). In Figures \[fig:volimp5\] and \[fig:volimp6\], we plot the local volatility function $\sqrt{v_{loc}(t,x)}$ and the intrinsic volatility function $\eta(t,x)$ of the stock. This latter is used to draw independent simulations of the index along with the stock and we see in Figure \[fig:volimp7\] that the implied volatility obtained is close to the right one, especially near the money.
![Intrinsic volatility function $\eta(t,x)$ of the stock.[]{data-label="fig:volimp6"}](VolLoc.eps)
![Intrinsic volatility function $\eta(t,x)$ of the stock.[]{data-label="fig:volimp6"}](IntrinsicVolLoc.eps)
Original model
--------------
We now turn to the calibration of our original model : $$\forall j \in \{1,\dots,M\}, \quad
\frac{dS^{j,M}_t}{S^{j,M}_t}=(r-\delta_j) dt + \beta_j
\,\sigma(t,I^{M}_t) dB_t + \eta_j(t,S^{j,M}_t) dW^j_t
\mbox{ with }I^{M}_t=\sum_{i=1}^M w_i S^{i,M}_t.\label{original_model}$$
It is rather complicated to have a perfect calibration for both index and stocks within this framework. Nevertheless, Theorem \[convind\] ensures that the error of calibration of the index smile is small (at least when the maturity is not too large) when $\sigma$ is chosen as a local volatility function fitted to this smile. We also suppose that a local volatility function $\sqrt{v_{loc}^j}$ has been fitted to the market smile of each stock $j$. For the choice of the coefficients $\beta_j$, we proceed like in Section \[choixbeta\]. The coefficients $\eta_j(t,x)=\sqrt{v_{loc}^j(t,x)-\beta_j^2{\mathbb{E}}(\sigma^2(t,I^M_t)|S^{j,M}_t=x)}$ are then calibrated all at the same time using an adaptation of the non-parametric method presented above based on the simulation of $N$ interacting $(M+1)$-dimensional paths.
In comparison with the simplified model, we introduce in the calibration of the index a small error which grows with the maturity $T$. But we guarantee the additivity constraint $I^{M}_t=\sum_{i=1}^M w_i S^{i,M}_t$. Note that a similar error spoils the calibration of the reconstructed index in the simplified model (see the discussion at the end of Section \[choixbeta\]).
In what follows, we illustrate the effect of Theorems 1 and 2 and compare our models with a constant correlation model.
Illustration of Theorems 1 and 2 and comparison with a constant correlation model
=================================================================================
The objective of this section is to compare index and individual stock smiles obtained with three different models : our original model (\[original\_model\]), the simplified one (after letting $M\to \infty$) and a model with constant correlation coefficient. More precisely, we consider the following dynamics
1. The original model $$\begin{array}{l}
{\displaystyle}\forall j \in \{1,\dots,M\}, \quad
\frac{dS^{j,M}_t}{S^{j,M}_t}=rdt
+ \,\sigma(t,I^{M}_t) dB_t + \eta(t,S^{j,M}_t) dW^j_t\text{ with } I^{M}_t=\sum_{i=1}^M w_i S^{i,M}_t.
\end{array}$$
2. The simplified model $$\begin{array}{ll}
{\displaystyle}\forall j \in \{1,\dots,M\}, &{\displaystyle}\frac{dS^{j}_t}{S^{j}_t}=r dt
+ \sigma(t,I_t) dB_t + \eta(t,S^{j}_t) dW^j_t\\[5mm]
&{\displaystyle}\frac{dI_t}{I_t}= r dt + \sigma(t,I_t) dB_t.
\end{array}$$ Here we can also compute the reconstructed index $\overline{I}^M_t=\sum_{i=1}^M w_i S^{i}_t$.
3. The “market” model $$\forall j \in \{1,\dots,M\}, \frac{dS^{j}_t}{S^{j}_t}=r dt +
\sqrt{v_{loc}(t,S^j_t)} d\widetilde{W}^j_t$$ with, $\forall i \neq j,
\,d\!<\widetilde{W}^i,\widetilde{W}^j>_t=\rho \,dt$.
We deliberately dropped the dividend yields and the beta coefficients in order to simplify the numerical experiment. For the function $\sigma$, we take as previously the calibrated local volatility of the Eurostoxx. For $\eta$, which does not depend on $j$, we choose an arbitrary function of the forward moneyness and we evaluate $v_{loc}$ such that the “market” model and the simplified model yield the same implied volatility for individual stocks. According to [@Gyongy], it is enough to take $$v_{loc}(t,x)=\eta^2(t,x)+{\mathbb{E}}(\sigma^2(t,I_t) | S^1_t=x)$$ where the conditional expectation is approximated using the non-parametric method presented above.
Finally, we fix the correlation coefficient $\rho$ such that the market model and the simplified one have the same ATM implied volatility for the index.
The implied volatilities for the index and for an individual stock obtained by the three models are plotted in Figures \[fig:IndexCompare\] and \[fig:StockCompare\]. We also give the difference in basis points between the implied volatilities obtained with the simplified model and the original one in Tables \[tab:theo1\], \[tab:theo21\] and \[tab:theo22\]. The parameters we use in our numerical experiment are the following :
- $S_0^1=\dots=S_0^M=53$,
- $M$, $I_0$ and the weights $w_1, \dots, w_M$ : the same as of the Eurostoxx index at December 21, 2007,
- $r=0.045$,
- Maturity $T=1$ year,
- Number of time steps: $n=10$,
- Number of simulated paths : $N=100000$.
Moneyness ($\frac{K}{S_0}$) 0.5 0.8 0.9 0.95 1 1.05 1.1 1.2 1.3 1.55 1.85 2
--------------------------------------------------------------- ----- ----- ----- ------ ---- ------ ----- ----- ----- ------ ------ ----
$|\widehat{\sigma}_{simplified}-\widehat{\sigma}_{original}|$ 81 22 16 14 14 17 20 24 24 11 38 17
: Difference (in bp) the implied volatilities of an individual stock obtained with the simplified model and with the original model.[]{data-label="tab:theo22"}
Moneyness ($\frac{K}{I_0}$) 0.5 0.8 0.9 0.95 1 1.05 1.1 1.2 1.3 1.55 1.85 2
--------------------------------------------------------------- ----- ----- ----- ------ ---- ------ ----- ----- ----- ------ ------ ----
$|\widehat{\sigma}_{simplified}-\widehat{\sigma}_{original}|$ 81 22 16 14 14 17 20 24 24 11 38 17
: Difference (in bp) between the implied volatilities of the index obtained with the simplified model and with the original model.[]{data-label="tab:theo1"}
Moneyness ($\frac{K}{I_0}$) 0.5 0.8 0.9 0.95 1 1.05 1.1 1.2 1.3 1.55 1.85 2
---------------------------------------------------------------- ----- ----- ----- ------ --- ------ ----- ----- ----- ------ ------ ---
$|\widehat{\sigma}_{reconstruct}-\widehat{\sigma}_{original}|$ 10 5 4 3 2 1 2 5 4 1 0 0
: Difference (in bp) between the implied volatility of the reconstructed index $\overline{I}^M$ in the simplified model and the implied volatility of the index in the original model.[]{data-label="tab:theo21"}
As suggested by Theorems 1 and 2, we see that the original model and the simplified one yield implied volatility curves that are very close to each other, both for the index and for individual stocks. The difference in basis points between the implied volatilities is reasonable, especially between the reconstructed index in the simplified model and the index in the original model.
Concerning the market model, by construction, we have the same implied volatility for an individual stock as in the simplified model but the implied volatility of the index is far from the simplified one. This phenomenon is well known in practice (see [@Bakshi],[@Bollen] or[@Branger]) : the implied volatility smile of an index is much steeper than the implied volatility smile of an individual stock. The market model of constantly correlated local volatility dynamics for the stocks is unable to retrieve the shape of the index smile. A more sophisticated dependence structure between stocks is needed. Local correlation models provide an extension of the market model in this direction : the correlation at time $t$ between the Brownian motions driving the local volatility dynamics of the stocks is a function $\rho(t,I_t)$ of the index level. But the way this function $\rho$ influences the index smile is not clear at all. Somehow, our models provide another parametrization of the correlation structure in which, the function $\sigma$, that replaces the function $\rho$, can be interpreted as the local volatility of the index. Yet, the individual stocks can still be properly calibrated. $$\;$$**Application: Pricing of a worst-of option**$$\;$$ Apart from handling both the index and its composing stocks, our models are also relevant for the widespread financial products that are sensitive to correlation in the equity world, such as rainbow options.
One example of such products is the worst-of performance option whose payout is referenced to the worst performer in a basket of shares. For a basket of $M$ shares, the payoff of a call with strike $K$ and maturity $T$ writes ${\displaystyle}\left(\min_{1\leq i\leq
M}\frac{S_T^i}{S_0^i}-K\right)_+$. Our objective is to compare the prices obtained by our model to the prices obtained by the market model of constantly correlated stocks. The parameters of the numerical experiment are the same as previously and we set the correlation coefficient $\rho$ such that all the models exhibit the same ATM implied volatility for the index.
The result, as can be seen in Figure \[fig:worst\_of\], is that our prices are always lower than the market model price, especially in the money. Hence, a model with a constant correlation coefficient, calibrated in order to fit the at the money prices of options written on the index, will always overestimate the risks of worst-of options. The reason is that the correlation level needed to fit the at the money prices is very high. Note that the prices obtained with the original model and the simplified one are barely distinguishable from each other.
Conclusion
==========
In this paper, we have introduced a new model for describing the joint evolution of an index and its composing stocks. The idea behind our view is that an index is not only a weighted sum of stocks but can also be seen as a market factor that influences their dynamics. In order to have a more tractable model, we have studied the limit when the number of underlying stocks goes to infinity and we have shown that our model reduces to a local volatility model for the index and to a stochastic volatility model with volatility driven by the index for each individual stock. We have discussed calibration issues and proposed a simulation-based technique for the calibration of the stock dynamics, which permits us to fit both index and stocks smiles. The numerical results obtained on real data for the Eurostoxx index are very encouraging, especially for accelerated techniques. We have also compared our models (before and after passing to the limit) to a standard market model consisting of local volatility models for the stocks which are constantly correlated and we have seen that they lead to a steeper index smile. Finally, when considering the pricing of worst-of performance options, which are sensitive to the dependence structure between stocks, we have found that our prices are more aggressive than the prices obtained by the standard market model.
To sum up, we list some properties of our models depending on the options one wishes to handle in the Table below
Purpose Simplified model Original model
--------- ------------------ ----------------
- -
-
- - -
-
- -
-
- - -
-. -
-
: Which model to use and when.[]{data-label="tab:summary"}
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**Appendix**
In order to prove the Theorems 1 and 2, we need the following technical estimation
\[lem:moments\] Under assumption (${\mathcal{H}}$\[hyp:bornitude\]), for all $p \geq 1$, one has $$\forall j \in \{1,\dots,M\},\quad \sup_{0\leq t\leq
T}{\mathbb{E}}\left(|S^{j,M}_{t}|^{2p}\right) \leq C_p$$ where ${\displaystyle}C_p=\max_{1\leq j \leq M}|S^{j,M}_{0}|^{2p}
\exp\left(\left(2r+(2p-1)(\max_{j \geq
1}\beta_j^2+1)K_b^2\right)pT\right)$.
By Itô’s lemma one has $$\begin{array}{rcl}
{\displaystyle}|S^{j,M}_{t}|^{2p}&=&{\displaystyle}|S^{j,M}_{0}|^{2p}+\int_0^t|S^{j,M}_{s}|^{2p}((2p)(r-\delta_j)+p(2p-1)(\beta_j^2\sigma^2(s,I^M_s)+\eta_j^2(s,S^{j,M}_{s})))ds\\[4mm]
&&{\displaystyle}\quad+\int_0^t(2p)|S^{j,M}_{s}|^{2p}(\beta_j\sigma(s,I^M_s)dB_s+\eta_j(s,S^{j,M}_{s})dW^j_s)\end{array}$$ In order to get rid of the stochastic integral, we use a localization technique : let $\nu_n$ be the stopping time defined for each $n\in {\mathbb{N}}$ by $\nu_n:=\inf\{t \geq 0; |S^{j,M}_t| \geq
n\}$. Then, using (${\mathcal{H}}$\[hyp:bornitude\]), one has $$\begin{array}{rcl}
{\displaystyle}{\mathbb{E}}\left(|S^{j,M}_{t\wedge \nu_n}|^{2p}\right)\!\!&=&{\displaystyle}|S^{j,M}_{0}|^{2p}+{\mathbb{E}}\left(\int_0^{t\wedge
\nu_n}\!\!|S^{j,M}_{s}|^{2p}
((2p)(r-\delta_j)+p(2p-1)(\beta_j^2\sigma^2(s,I^M_s)+\eta_j^2(s,S^{j,M}_{s}))ds\right)\\[5mm]
&\leq &{\displaystyle}|S^{j,M}_{0}|^{2p}+\left((2p)(r-\delta_j)\mathbb{1}_{\{r-\delta_j
\geq0\}}+p(2p-1)(\beta_j^2+1)K_b^2\right)\int_0^{t}{\mathbb{E}}\left(|S^{j,M}_{s\wedge
\nu_n}|^{2p}\right)ds
\end{array}$$ So, by Gronwall’s lemma and the fact that the dividends are nonnegative, $$\forall t\leq T, {\mathbb{E}}\left(|S^{j,M}_{t\wedge \nu_n}|^{2p}\right) \leq
|S^{j,M}_{0}|^{2p}
\exp\left(\left(2rp+p(2p-1)(\beta_j^2+1)K_b^2\right)T\right)$$
Finally, Fatou’s lemma permits us to conclude : $$\sup_{0\leq t\leq T} {\mathbb{E}}\left(|S^{j,M}_{t}|^{2p}\right) \leq
|S^{j,M}_{0}|^{2p}
\exp\left(\left(2rp+p(2p-1)(\beta_j^2+1)K_b^2\right)T\right).$$
Using the SDEs (\[indexSDE\]) and (\[Ilim\]), one has $$\begin{array}{rcl}
{\displaystyle}|I_{t}^{M}-I_{t}|^{2p}&=&{\displaystyle}\big|r \int_{0}^{t}
\left(I^{M}_{s}-I_{s}\right)ds-\int_{0}^{t} \left(\sum_{j=1}^M \delta_j w_j S^{j,M}_s-\delta I_{s}\right)ds\\[3mm]
&&{\displaystyle}+\int_{0}^{t}\left(\sum_{j=1}^M \beta_j w_j S^{j,M}_s\sigma(s,I^{M}_{s})-\beta
I_{s}\sigma(s,I_{s})\right)dB_{s}+ \sum_{j=1}^{M} w_{j} \int_{0}^{t}S^{j,M}_{s}\eta_{j}(s,S^{j,M}_{s}) dW_{s}^{j}\big|^{2p}\\[7mm]
&\leq&{\displaystyle}4^{2p-1} \left(r^{2p}t^{2p-1}\int_{0}^{t}(I^{M}_{s}-I_{s})^{2p}ds+t^{2p-1}\int_{0}^{t}\left(\sum_{j=1}^M \delta_j w_j S^{j,M}_s-\delta I_{s}\right)^{2p}ds\right.\\[3mm]
&&{\displaystyle}+\left.\Big|\!\int_{0}^{t}\left(\sum_{j=1}^M \beta_j w_j S^{j,M}_s\sigma(s,I^{M}_{s})-\beta
I_{s}\sigma(s,I_{s})\right)
dB_{s}\Big|^{2p}+ \Big|\sum_{j=1}^{M}
w_{j}\!\int_{0}^{t}S^{j,M}_{s}\eta_{j}(s,S^{j,M}_{s})
dW_{s}^{j}\Big|^{2p}\right)
\end{array}$$ Hence, using the Burkholder-Davis-Gundy inequality (see Karatzas and Shreve [@KaratzasShreve] p. 166), there exists a universal positive constant $K_p$ such that $${\mathbb{E}}\left(\sup_{0\leq t \leq T}|I^{M}_t-I_t|^{2p}\right) \leq 4^{2p-1}(a_{M}+b_{M}+c_{M}+d_{M})$$ where
- ${\displaystyle}a_{M}=r^{2p} \,T^{2p-1} \int_{0}^{T} {\mathbb{E}}\big((I^{M}_{s}-I_{s})^{2p}\big)ds$
- ${\displaystyle}b_{M}=T^{2p-1}\int_{0}^{T} {\mathbb{E}}\left(\left(\sum_{j=1}^M \delta_j w_j S^{j,M}_s-\delta I_{s}\right)^{\!\!\!2p}\,\right)ds$
- ${\displaystyle}c_{M}=K_p T^{p-1} \int_{0}^{T}{\mathbb{E}}\left(\left(\sum_{j=1}^M \beta_j w_j S^{j,M}_s\sigma(s,I^{M}_{s})-\beta
I_{s}\sigma(s,I_{s})\right)^{\!\!\!2p}\,\right) ds$
- ${\displaystyle}d_{M}=K_p T^{p-1} \int_{0}^{T}{\mathbb{E}}\left(\left(\sum_{j=1}^M \left(w_j S^{j,M}_s\eta_j(s,S^{j,M}_{s})\right)^2\right)^{\!\!\!p}\,\right) ds$
The term $a_{M}$ is the easiest one to handle : $$a_{M} \leq r^{2p} \,T^{2p-1}\int_{0}^{T}{\mathbb{E}}\left(\sup_{0 \leq u \leq
s}|I^{M}_{u}-I_{u}|^{2p}\right) ds.\label{aM}$$
Next, using assumption (${\mathcal{H}}$\[hyp:bornitude\]) for the first inequality, Hölder’s inequality for the second and lemma \[lem:moments\] for the third, one gets $$\begin{array}{rcl}
d_{M}&=&{\displaystyle}K_p T^{p-1} \int_0^T \sum_{j_1=1}^{M} \cdots
\sum_{j_p=1}^{M} {\mathbb{E}}\left(\prod_{k=1}^p w_{j_k}^2
(S^{j_k,M}_{s})^2(\eta_{j_k}(s,S^{j_k,M}_{s}))^2\right) ds\\[5mm]
&\leq &{\displaystyle}K_p K_b^{2p}T^{p-1} \int_0^T \sum_{j_1=1}^{M} \cdots
\sum_{j_p=1}^{M} (\prod_{k=1}^p w_{j_k}^2) {\mathbb{E}}\left(\prod_{k=1}^p
(S^{j_k,M}_{s})^2\right) ds\\[5mm]
&\leq &{\displaystyle}K_p K_b^{2p}T^{p-1} \int_0^T \sum_{j_1=1}^{M} \cdots
\sum_{j_p=1}^{M} \prod_{k=1}^p w_{j_k}^2 \left({\mathbb{E}}\left(
(S^{j_k,M}_{s})^{2p}\right)\right)^{\frac{1}{p}} ds\\[5mm]
&\leq &{\displaystyle}K_p K_b^{2p}T^{p} C_p \left(\sum_{j=1}^{M}
w_{j}^2\right)^{\!\!p}
\end{array} \label{dM}$$
The same arguments enable us to control the term $b_M$ :
$$\begin{array}{rcl} {\displaystyle}b_M&=&{\displaystyle}T^{2p-1}\int_{0}^{T} {\mathbb{E}}\left(\left(\sum_{j=1}^M \delta_j w_j S^{j,M}_s-\delta I_{s}\right)^{\!\!\!2p}\,\right)ds\\[5mm]
&\leq& (2T)^{2p-1}{\displaystyle}\left(\int_{0}^{T} {\mathbb{E}}\left(\left(\sum_{j=1}^M \delta_j w_j S^{j,M}_s-\delta I^M_{s}\right)^{\!\!\!2p}\,\right)
+ {\mathbb{E}}\left(\left(\delta I^{M}_{s}-\delta I_{s}\right)^{2p}\right)ds\right)\\[5mm]
&\leq&(2T)^{2p-1} {\displaystyle}\int_{0}^{T} {\mathbb{E}}\left(\left(\sum_{j=1}^M
(\delta_j-\delta)
w_j S^{j,M}_s\right)^{2p}\right)ds +(2T)^{2p-1}
\delta^{2p}\int_{0}^{T} {\mathbb{E}}\left(\sup_{0 \leq u \leq
s}|I^{M}_{u}-I_{u}|^{2p}\right) ds\\[5mm]
&\leq&{\displaystyle}2^{2p-1} T^{2p}C_p\left(\sum_{j=1}^M w_j|\delta_j-\delta|\right)^{2p}+(2T)^{2p-1}
\delta^{2p}\int_{0}^{T} {\mathbb{E}}\left(\sup_{0 \leq u \leq
s}|I^{M}_{u}-I_{u}|^{2p}\right) ds.
\end{array}\label{bM}$$
For the remaining term $c_M$, we will also need the Lipschitz assumption (${\mathcal{H}}$\[hyp:xsLip\]) $$\begin{array}{rcl}
{\displaystyle}c_{M} &=&{\displaystyle}K_p T^{p-1} \int_{0}^{T}{\mathbb{E}}\left(\left(\sum_{j=1}^M \beta_j w_j S^{j,M}_s\sigma(s,I^{M}_{s})-\beta
I_{s}\sigma(s,I_{s})\right)^{\!\!\!2p}\,\right) ds\\[5mm]
&\leq& {\displaystyle}2^{2p-1} K_p T^{p-1} \left( \int_{0}^{T}
{\mathbb{E}}\left(\left(\sum_{j=1}^M (\beta_j-\beta) w_j
S^{j,M}_s\sigma(s,I^{M}_{s})\right)^{\!\!\!2p}\,\right)
+{\mathbb{E}}\left((\beta I^{M}_{s}\sigma(s,I^{M}_{s})-\beta
I_{s}\sigma(s,I_{s}))^{2p}\right)ds\right)\\[5mm]
&\leq&{\displaystyle}2^{2p-1} K_pT^{p}K_b^{2p}C_p\left(\sum_{j=1}^M
w_j|\beta_j-\beta|\right)^{2p}+2^{2p-1}K_pT^{p-1}(\beta K_\sigma)^{2p}
\int_{0}^{T} {\mathbb{E}}\left(\sup_{0 \leq u \leq
s}|I^{M}_{u}-I_{u}|^{2p}\right) ds.
\end{array}\label{cM}$$
So, combining the inequalities (\[aM\]), (\[dM\]), (\[bM\]) and (\[cM\]), one obtains $$\begin{array}{rcl}
{\displaystyle}{\mathbb{E}}\left(\sup_{0\leq t \leq T}|I^{M}_t-I_t|^{2p}\right)
&\leq&{\displaystyle}C_0\left(\left(\sum_{j=1}^{M}
w_{j}^2\right)^{\!\!p} + \left(\sum_{j=1}^M
w_j|\beta_j-\beta|\right)^{2p}+ \left(\sum_{j=1}^M
w_j|\delta_j-\delta|\right)^{2p} \right)\\[5mm]
&&{\displaystyle}+ C_1 \int_{0}^{T}{\mathbb{E}}\left(\sup_{0 \leq u
\leq
s}|I^{M}_{u}-I_{u}|^{2}\right) ds\end{array}$$ with $C_0=8^{2p-1} T^p (T^p+K_pK_b^{2p})C_p$ and $C_1=4^{2p-1}(2^{2p-1}K_pT^{p-1}(\beta
K_\sigma)^{2p}+(2T)^{2p-1}
\delta^{2p}+r^{2p} \,T^{2p-1}).$
Finally, by means of Gronwall’s lemma, we conclude that $${\mathbb{E}}\left(\sup_{0\leq t \leq T}|I^{M}_t-I_t|^{2p}\right) \leq C_T \left(\left(\sum_{j=1}^{M}
w_{j}^2\right)^{\!\!p} + \left(\sum_{j=1}^M
w_j|\beta_j-\beta|\right)^{2p}+ \left(\sum_{j=1}^M
w_j|\delta_j-\delta|\right)^{2p} \right)$$ where $$C_T=C_0 e^{C_1T}.$$
The proof is similar to the previous one : $$\begin{array}{rcl}
{\displaystyle}|S^{j,M}_{t}-S^j_t|^{2p}& \leq &{\displaystyle}3^{2p-1}
\left((r-\delta_{j})^{2p}t^{2p-1}\int_{0}^{t}(S^{j,M}_{s}-S^j_s)^{2p}ds+\left|\int_{0}^{t}(S^{j,M}_{s}\eta_{j}(s,S^{j,M}_{s})-S^{j}_{s}\eta_{j}(s,S^{j}_{s}))dW^{j}_{s}\right|^{2p}\right.\\[2mm]
&&{\displaystyle}\left.+\beta_{j}^{2p}\left|\int_{0}^{t}(S^{j,M}_{s}\sigma(s,I^{M}_{s})-S^{j}_{s}\sigma(s,I_{s}))dB_{s}\right|^{2p}\right)
\end{array}$$ hence, using the Burkholder-Davis-Gundy inequality, there exists a constant $K_p$ such that $$\begin{array}{rcl}
{\displaystyle}{\mathbb{E}}\left(\sup_{0\leq t \leq T}|S^{j,M}_{t}-S^j_t|^{2p}\right)&
\leq&{\displaystyle}3^{2p-1}
\left((r-\delta_{j})^{2p}T^{2p-1}\int_{0}^{T}{\mathbb{E}}\left(\sup_{0 \leq u \leq
s}|S^{j,M}_{u}-S^j_u|^{2}\right)ds\right.\\[2mm]
&&{\displaystyle}+K_pT^{p-1}\int_{0}^{T}{\mathbb{E}}\left((S^{j,M}_{s}\eta_{j}(s,S^{j,M}_{s})-S^{j}_{s}\eta_{j}(s,S^{j}_{s}))^{2p}\right)ds\\[2mm]
&&{\displaystyle}\left.+ K_pT^{p-1}\beta_{j}^{2p}\int_{0}^{T}{\mathbb{E}}\left((S^{j,M}_{s}\sigma(s,I^{M}_{s})-S^{j}_{s}\sigma(s,I_{s}))^{2p}\right)ds\right)
\end{array}$$ Using assumption (${\mathcal{H}}$\[hyp:xetaLip\]), one gets $$\int_{0}^{T}{\mathbb{E}}\left((S^{j,M}_{s}\eta_{j}(s,S^{j,M}_{s})-S^{j}_{s}\eta_{j}(s,S^{j}_{s}))^{2p}\right)ds
\leq K_{\eta}^{2p} \int_{0}^{T}{\mathbb{E}}\left(\sup_{0 \leq u \leq
s}|S^{j,M}_{u}-S^j_u|^{2p}\right)ds.$$ Finally, by means of lemma \[lem:moments\] and assumptions (${\mathcal{H}}$\[hyp:bornitude\]) and (${\mathcal{H}}$\[hyp:xsLip\]), $$\begin{array}{rcl}
{\displaystyle}\int_{0}^{T}{\mathbb{E}}\left((S^{j,M}_{s}\sigma(s,I^{M}_{s})-S^{j}_{s}\sigma(s,I_{s}))^{2p}\right)ds&\leq& {\displaystyle}2^{2p-1}
\int_{0}^{T}{\mathbb{E}}\left((S^{j,M}_{s})^{2p}(\sigma(s,I^{M}_{s})-\sigma(s,I_{s}))^{2p}\right)ds.\\[3mm]
&&{\displaystyle}+2^{2p-1}
\int_{0}^{T} {\mathbb{E}}\left((\sigma(s,I_{s}))^{2p}(S^{j,M}_{s}-S^{j}_{s})^{2p}\right)ds\\[5mm]
&\leq& {\displaystyle}2^{2p-1} C_{2p}^{\frac{1}{2}} K_{Lip}^{2p}T \sqrt{{\mathbb{E}}\left(\sup_{0\leq t \leq T}|I^{M}_t-I_t|^{4p}\right)}\\[3mm]
&&{\displaystyle}+2^{2p-1} K_b^{2p} \int_0^T {\mathbb{E}}\left(\sup_{0\leq t \leq T}|S^{j,M}_{s}-S^{j}_{s}|^{2p}\right) ds\\[5mm]
\end{array}$$ We deduce using Gronwall’s lemma : $${\mathbb{E}}\left(\sup_{0\leq t \leq T}|S^{j,M}_t-S^j_t|^{2p}\right) \leq
\widetilde{C}^j_T \sqrt{{\mathbb{E}}\left(\sup_{0\leq t \leq
T}|I^{M}_t-I_t|^{4p}\right)}$$ where $$\widetilde{C}^j_{T}=6^{2p-1}K_pT^p\beta_{j}^{2p}C_{2p}^{\frac{1}{2}} K_{Lip}^{2p} \,\, e^{3^{2p-1}((r-\delta_{j})^{2p}T^{2p-1}+K_pT^{p-1}K_\eta^{2p}+
2^{2p-1}K_pT^{p-1}\beta_{j}^{2p}K_b^{2p})T}.$$ We conclude by Theorem \[convind\] and the sublinearity of the square root function on ${\mathbb R}_+$.
$\,$\
We now turn to the $L^{2p}$-distance between $I^M$ and $\overline{I}^M$ : $$\begin{array}{rcl}
{\displaystyle}|I^M_t - \overline{I}^M_t|^{2p} & = &{\displaystyle}\left|\sum_{j=1}^M w_j
S^{j,M}_t-\sum_{j=1}^M w_j S^{j}_t\right|^{2p}\\[3mm]
&\leq& {\displaystyle}\left(\sum_{j=1}^M w_j |S^{j,M}_t-S^{j}_t|\right)^{2p}\\[3mm]
&\leq& {\displaystyle}\sum_{j_1=1}^M \dots \sum_{j_{2p}=1}^M \prod_{k=1}^{2p}
w_{j_k} |S^{j_k,M}_t-S^{j_k}_t|\\
\end{array}$$ So, using Hölder inequality, one has $$\begin{array}{rcl}
{\displaystyle}{\mathbb{E}}\left(\sup_{0\leq t\leq T} |I^M_t -
\overline{I}^M_t|^{2p}|\right) & \leq&{\displaystyle}\sum_{j_1=1}^M \dots
\sum_{j_{2p}=1}^M \left(\prod_{k=1}^{2p} w_{j_k}\right)
\prod_{k=1}^{2p} \left({\mathbb{E}}(\sup_{0\leq t\leq
T}|S^{j_k,M}_t-S^{j_k}_t|^{2p})\right)^{\frac{1}{2p}}\\[5mm]
&\leq &{\displaystyle}\left(\sum_{j=1}^{M} w_{j}\right)^{2p} \max_{1\leq j \leq
M}\widetilde{C}^j_T \left(\left(\sum_{j=1}^{M}
w_{j}^2\right)^{\!\!p} + \right.\\[5mm]
&& {\displaystyle}\quad \left.\left(\sum_{j=1}^M
w_j|\beta_j-\beta|\right)^{2p}+ \left(\sum_{j=1}^M
w_j|\delta_j-\delta|\right)^{2p} \right).
\end{array}$$
[^1]: Université Paris-Est, CERMICS, Projet MathFi ENPC-INRIA-UMLV. This research benefited from the support of the “Chair Risques Financiers”, Fondation du Risque.
Postal address : 6-8 av. Blaise Pascal, Cité Descartes, Champs-sur-Marne, 77455 Marne-la-Vallée Cedex 2.
E-mails : and
[^2]: The weightings are periodically updated but, as usually assumed, we suppose that, up to maturities of the options considered, they are constant. When updated, they are often chosen proportional to the market capitalizations of the stocks.
[^3]: The authors have chosen a Student distribution in their numerical experiments.
[^4]: 500 stocks for the S&P 500 index, 100 stocks for the FTSE 100 index, 40 stocks for the CAC40 index, etc.
[^5]: The median of a real random variable $X$ is any real number $m$ satisfying : $${\mathbb{P}}(X \leq m) \geq \frac{1}{2} \,\, \text{
and }\,\,{\mathbb{P}}(X \geq m) \geq \frac{1}{2}.$$ It has the property of minimizing the $L^1$-distance to $X$ : ${\displaystyle}m=\arg\min_{x\in{\mathbb{R}}}
{\mathbb{E}}|X-x|.$
[^6]: In order to smooth the estimation, one has to choose a bandwidth parameter that is greater than the theoretical optimal parameter $N^{-\frac{1}{5}}$.
|
---
author:
- |
<span style="font-variant:small-caps;">Britton J. Olson</span>\
Lawrence Livermore National Laboratory, Livermore, CA, USA\
[email protected]\
<span style="font-variant:small-caps;">Robin Williams</span>\
Atomic Weapons Establishment, United Kingdom
title: 'Richtmyer-Meshkov mixing layer growth from localized perturbations'
---
Introduction {#sec:intro}
============
Applications in engineering and science where hydrodynamic instability at an interface leads to large scale mixing often occur in non-ideal configurations. Many rigorous studies of Richtmyer-Meshkov (RM) and Rayleigh-Taylor (RT) (see [@zhou.pr.2017-1; @zhou.pr.2017-2] and the references therein) exist which assume instability growth on an interface which is statistically homogeneous in all directions. Therefore, the mean flow is inherently 1D, where the other dimensions have been collapsed, as in the case of planar and spherical instability growth. However, there are relatively few studies on interfacial instability growth where the underlying mean flow is truly multi-dimensional. Engineering applications such as ICF fill tubes and tent perturbations [@weber.pop.2017] represent mean geometries which are 3D and 2D respectively and create a challenge for engineering codes which utilizes Reynolds Averaged Navier-Stokes (RANS) closure models to represent the instability and turbulence as a subgrid scale transport model.
In these proceedings, we propose a modification to a simple RM test case which generates 2D mean flow features. This is accomplished by localizing a patch of initial perturbations on a plane (like many other studies) but which now contain an edge or a boundary. The lack of spatial homogeneity in these regions is representative of the aforementioned applications. It highlights potential modeling deficiencies in current RANS engineering approaches.
In Section \[sec:methods\] the Large-Eddy Simulation (LES) methodology is described. In Section \[sec:problem\] a modification to a standard planar RM test case is proposed which makes the mean flow multi-dimensional. High fidelity LES results over a range of grid resolutions are generated to establish a bound on mesh dependence. Two sets of initial perturbations are explored; one which resembles a strip leads to a “curtain” of mixing and one which resembles a patch leads to a “plume”.
In Section \[sec:results\] quantitative results are given comparing the vertical and horizontal mixing layer growth as a function of time. Higher order turbulence statistics such as TKE and flow anisotropy for the different resolutions and configurations are compared, as well. Finally, in Section \[sec:discussion\] we summarize the present findings and discuss briefly modeling considerations using the RANS modeling approach and conclude.
Methods {#sec:methods}
=======
Large Eddy Simulations (LES) of the three dimensional Navier-Stokes equations are solved using the Ares code developed at Lawrence Livermore National Laboratory. The full equations of motion and the numerical methods solved in the Ares code are detailed in [@olson.pof.2014] and [@sharp.llnl.1981]. For the present calculations, the infinite Reynolds number limit is assumed as in [@thornber.pof.2017] and no physical transport properties are used (viscosity, conductivity, diffusivity, etc.). Therefore, three grid resolutions are explored to quantify the grid resolution dependence on the quantities of interest. The coarse, medium, and fine mesh resolutions are given as (128$\times$128$\times$192), (256$\times$256$\times$384), and (512$\times$512$\times$768) grid points, respectively, with a domain size is given as $2.8\pi\times 2\pi \times 2\pi$.
Localized perturbation problem description {#sec:problem}
==========================================
We present here a numerical experiment setup which can be used to generate non one-dimensional mean flow, unlike that typically assumed for RM mixing layers. To do this, we use the work of Thornber et al [@thornber.pof.2017] as the initial conditions subject to modifications to localize the perturbations into patches and create edges to the mixing layer. A Mach 1.84 shock drives the mixing growth as it traverses from heavy ($\rho=3$) to light ($\rho=1$) fluids, both with ideal gases with $\gamma=5/3$.
From Thornber et al (see equation (4) in [@thornber.pof.2017]), we introduce a mask function $w(y,z)$ to the volume fraction field, $f_1$, which contains the initial perturbations which can then be written as: $$\begin{aligned}
f_1(x,y,z) = \frac{1}{2} \text{erfc} \left( \frac{\sqrt{\pi} [ x-S(y,z)w(y,z) ] }{ \delta } \right).\end{aligned}$$ The weight function is then given as
$$\begin{aligned}
w(y,z) = \frac{1}{2}\left( 1 - \tanh\left( \frac{r - r_0 } {\delta_w} \right) \right)\end{aligned}$$
where $r_0 = 2\pi/6$, $\delta_w = 2\pi/60$, and $\delta = 2\pi/32$ for all cases. For the “curtain” cases, $r=|y-\pi|$ and for the “plume” cases, $r=\sqrt{ (y-\pi)^2 + (z-\pi)^2 }$. From [@thornber.pof.2017] planar perturbations are contained in $S(y,z)$ and have a characteristic length scale of $\lambda_0$ and a characteristic growth rate of $\dot{W}$, which is given in [@thornber.pof.2017]. The resultant initial interfaces for the unmasked, curtain, and plume geometries are shown in Figure \[fig:init\].
![Plot of the equimolar plane between the heavy and light fluids, representing the interface. [**Left:**]{} Planar initial conditions form Thornber et al. [**Center:**]{} Present “curtain” geometry. [**Right:**]{} Present “plume” geometry []{data-label="fig:init"}](image10.png "fig:"){width=".3\textwidth"} ![Plot of the equimolar plane between the heavy and light fluids, representing the interface. [**Left:**]{} Planar initial conditions form Thornber et al. [**Center:**]{} Present “curtain” geometry. [**Right:**]{} Present “plume” geometry []{data-label="fig:init"}](image12.png "fig:"){width=".3\textwidth"} ![Plot of the equimolar plane between the heavy and light fluids, representing the interface. [**Left:**]{} Planar initial conditions form Thornber et al. [**Center:**]{} Present “curtain” geometry. [**Right:**]{} Present “plume” geometry []{data-label="fig:init"}](image11.png "fig:"){width=".3\textwidth"}
Results {#sec:results}
=======
The LES calculations discussed in Section \[sec:methods\] are performed up to a time of $\tau = 6$, where $\tau=\dot{W}/\lambda_0 t$. The temporal evolution of the mixing layer growth for the curtain and plume cases can be seen in Figure \[fig:curtainHistory\] and Figure \[fig:plumeHistory\], respectively.
The qualitative behavior is very similar between the two cases; small scale perturbations grow to large coherent bubbles and spikes which then lead the transitional turbulent behavior and small scale mixing of the two fluids. Both cases show a top-bottom asymmetry in the mixing layers as would be expected given the initial Atwood number of the two fluids. The extent of the vertical (height) and horizontal (width) mixing layer in the top and bottom fluids is quantified by constructing a “best fit” bounding box in each fluid around the contour of $4\left< Y_t Y_b \right> = .1$ and taking the length scales from these rectangles, as depicted in Figures \[fig:curtain\_length\] and \[fig:plume\_length\].
\
\
\
\
Figures \[fig:curtain\_length\] and \[fig:plume\_length\] show that the mixing length scales (between the three mesh resolutions) vary by less than 3% for $\tau < 3$. Larger variations later in time (especially for the “plume” case) appeared to be due to edge features of the mixing layer, to which the contouring algorithm was sensitive.
The mixing heights in the top and bottom fluids ($H_t,H_b$) appear to be following a $t^\theta$ like growth rate [@zhou.pr.2017-1; @zhou.pr.2017-2]. The mixing widths ($W_t,W_b$), however, appear to be following a totally different growth rate behavior and very little growth occurs in the lateral direction. The lack of growth creates a mixing layer with a growing aspect ratio that is clearly visualized in Figures \[fig:curtainHistory\] and \[fig:plumeHistory\].
Given that RM is a purely decaying process, its somewhat surprising that the mixing layer maintains and grows in spatial inhomogeneity with time that is not present in the standard planar case. The flow would be expected to relax to statistical self-similarity once the scale of the plumes substantially exceeds the lateral scale of the initial patch, but this condition has not been reached in the calculations presented here. To explore the mixing scales anisotropy, the Reynolds stress is computed in the principal directions, and normalized by its sum over all directions. This anisotropy is shown in Figure \[fig:aniso\](a,b).
Local maxima of the radial Reynolds stress anisotropy, $\left< u'_r u'_r \right> / \left< u'_i u'_i \right>$, in Figure \[fig:aniso\](a) are localized near the mixing region front and the initial interface location. Local maxima of the veritical Reynolds stress anisotropy, $\left< u'_z u'_z \right> / \left< u'_i u'_i \right>$, in Figure \[fig:aniso\](b) are localized near the core of mixing region. The $\left< \right>$ operator denotes $xr$-planar averages via a binning operation.
Inspection of the mean velocity magnitude reveals the source of this anisotropy as a vortex pair which which forms early on ($\tau < 2$) during the instability transition process and persists to late time. Figure \[fig:aniso\](c) and \[fig:aniso\](b) show contours of the mean velocity magnitude with velocity vectors and the mixing layer edge drawn in black and red, respectively. The persistent vortex entrains pure fluid into the mixing along the interface, creating local variations in the Reynolds stress anisotropy and supressing the lateral spreading of the mixing layer.
(0, 0) node\[inner sep=0\] ; (0.5,-3) \[color=white\] node [ $ \frac{\left<u_r^`u_r^`\right>}{ \left<u_i^`u_i^`\right> }$]{}; (0,-4.2) \[color=black\] node [ (a)]{};
(0, 0) node\[inner sep=0\] ; (0.5,-3) \[color=white\] node [ $ \frac{\left<u_z^`u_z^`\right>}{ \left<u_i^`u_i^`\right> }$]{}; (0,-4.2) \[color=black\] node [ (b)]{};
(0, 0) node\[inner sep=0\] ; (0,-3) \[color=white\] node [$|\vec{u}|$]{}; (0,-4.2) \[color=black\] node [ (c)]{};
(0, 0) node\[inner sep=0\] ; (0,-3) \[color=white\] node [$|\vec{u}|$]{}; (0,-4.2) \[color=black\] node [ (d)]{};
Discussion {#sec:discussion}
==========
The results presented here indicate that significant multi-dimensional effects are present in both the curtain and plume geometries. The mixing layer growth, anisotropy, and mean flow are independent of the mesh resolutions to within 5 and 15% for the curtain and plume geometries, respectively. The multi-dimensional effects appear during the transition process of the instability and persist as large scale vortices located at the edges of the mixing layer.
Preliminary studies (the results of which are beyond the scope of the present work and not shown here) using a simple K-L RANS [@morgan.shockwaves.2016] model to capture the patch of instability growth fail to capture turbulent transition and therefore don’t accurately predict the two-dimensional mean flow. The vertical and horizontal mixing length scales are under-predicted and over-predicted, respectively, by a factor of two. The vortex pair captured by the LES mean flow solution is absent in the RANS calculation. A modification to the RANS modeling approach and/or directly capturing the mean flow would be required to capture these complex mixing layers accurately. Subsequent work is planned to assess modeling approaches. This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.
R. Richtmyer (1960) Taylor instability in shock acceleration of compressible fluids 8, 297-319
E. Meshkov (1969) Instability of the interface of two gases accelerated by a shock wave. 4 , 101-108
Y. Zhou (2017) Rayleigh-Taylor and Richtmyer-Meshkov instability induced flow, turbulence, and mixing. I. 720-722, 1-136
Y. Zhou (2017) Rayleigh-Taylor and Richtmyer-Meshkov instability induced flow, turbulence, and mixing. II. 723-725, 1-160
C. R. Weber, et. al (2017) Improving ICF implosion performance with alternative capsule supports 24, 056302
B. J. Olson and J. A. Greenough (2014) Large eddy simulation requirements for the Richtmyer-Meshkov instability 26, 044103
R. Sharp, R. Barton (1981) HEMP advection model 17809. Lawrence Livermore Laboratory, Livermore, CA
B. Thornber, J. Griffond, O. Poujade, N. Attal, H. Varshochi, P. Bigdelou, P. Ramaprabhu, B. Olson, J. Greenough, Y. Zhou, O. Schilling, K. A. Garside, R. J. R. Williams, C. A. Batha, P. A. Kuchugov, M. E. Ladonkina, V. F. Tishkin, N. V. Zmitrenko, V. B. Rozanov, and D. L. Youngs (2017) Late-time growth rate, mixing, and anisotropy in the multimode narrowband Richtmyer-Meshkov instability: The $\theta$-group collaboration 29, 105107
B. E. Morgan and J. A. Greenough (2016) Large-eddy and unsteady RANS simulations of a shock-accelerated heavy gas cylinder 26:355-383
|
---
abstract: 'We consider multihadron production processes in different types of collisions in the framework of the picture based on dissipating energy of participants and their types. In particular, the similarities of such bulk observables like the charged particle mean multiplicity and the pseudorapidity density at midrapidity measured in nucleus-nucleus, (anti)proton-proton and electron-positron interactions are analysed. Within the description proposed a good agreement with the measurements in a wide range of nuclear collision energies from AGS to RHIC is obtained. The predictions up to the LHC energies are made and compared to experimental extrapolations.'
author:
- 'Edward K.G. Sarkisyan'
- 'Alexander S. Sakharov'
title: |
Multihadron production features\
in different reactions
---
[ address=[EP Division, Department of Physics, CERN, CH-1211 Geneva 23, Switzerland]{} ,altaddress=[Department of Physics, the University of Manchester, Manchester M13 9PL, UK]{} ]{}
[ address=[TH Division, Department of Physics, CERN, CH-1211 Geneva 23, Switzerland]{}, altaddress=[Swiss Institute of Technology, ETH-Zürich, 8093 Zürich, Switzerland ]{} ]{}
[**1.**]{} High densities and temperatures of nuclear matter reached at RHIC provide us with an exceptional opportunity to investigate the matter at extreme conditions. Bulk observables such as multiplicity and particle densities (spectra) being sensitive to the dynamics of strong interactions, are of fundamental interest. Recent measurements at RHIC revealed striking evidences in the hadron production process including similarity in such basic observables like the mean multiplicity and the midrapidity density measured in complex ultra-relativistic nucleus-nucleus (AA) collisions those obtained in relatively “elementary” interactions at the same centre-of-mass (c.m.) energy when number of participants (“wounded” nucleons [@woundN] in AA collisions) are taken into account . The observation is shown to be independent of the c.m. energy per nucleon $\Ecmn =$ 19.6 GeV to 200 GeV. Assuming similar mechanisms of hadron production in both types of interactions which then depends only on the amount of energy transformed into particles produced, one would expect the same value of the observables to be obtained in hadron-hadron collisions at close c.m. energies. However, this is not the case: comparing measurements in hadronic data to the findings at RHIC, one obtains quite lower values in hadron-hadron collisions. In the meantime, the RHIC dAu data at $\Ecmn=$ 200 GeV unambiguously point to the values of the mean multiplicity from data . Moreover, recent CuCu RHIC data show no changes in the values of the bulk variables compared to those from AuAu collisions when properly normalised to the number of participants .
The observations made earlier and the recent ones can be understood in the franework of a description proposed recently by us and considered here. This description is based on a picture when the whole process of a collision is interpreted as the expansion and break-up into particles of an initial state, in which the whole available energy is assumed to be concentrated in a small Lorentz-contracted volume. There are no any restrictions due to the conservation of quantum numbers besides energy and momentum constraints allowing therefore to link the amount of energy deposited in the collision zone and features of bulk variables in different reactions. This description resembles the Landau hydrodynamical approach to multiparticle production [@landau] which has been found to give good description of the mean multiplicity AA, pp, , $\nu$p data as well as of pseudorapidity distributions at RHIC .
As soon as a collision of two Lorentz-contracted particles leads to the full thermalization of the system before extension, one can assume that the production of secondaries is defined by the fraction of participants energy deposited in the volume of the system at the collision moment. This implies that there is a difference between results of collisions of structureless particles like electron and composite particles like proton, the latter considered to be built of constituents. Indeed, in composite particle collisions not all the constituents deposit their energy when they form the Lorentz-contracted volume of the thermalized initial state. As a result, the leading particles , formed out of those constituents which are not trapped in the interaction volume, carry away a part of energy. Meantime, colliding structureless particles are ultimately stopped as a whole in the initial state of the thermalized collision zone depositing their total energy in the Lorentz-contracted volume and this energy is wholly available for production of secondaries.
We consider a single nucleon as a superposition of three constituent quarks due to the additive quark picture . In this picture, most often only one quark from each nucleon contributes to the interaction with other quarks being spectators. Thus, the initial thermalized state is pumped in only by the energy of the interacting single quark pair and, so, only 1/3 of the entire nucleon energy is available for production of secondaries. Therefore, one expects that the resulting bulk variables like the multiplicity and rapidity distributions should show identical features in collisions at the c.m. energy $\Ecmp$ and in interactions at the c.m. energy $\Ecme \simeq\Ecmp/3$. Note that for the mean multiplicity, a similar behaviour was found in the beginning of LEP activity .
In AA collisions, more than one quark per nucleon interacts due to the large size of nucleus and the long path of interactions inside the nucleus. In central AA collisions, a contribution of constituent quarks rather than participating nucleons seem to determine the properties of produced particle distributions [@voloshin]. In headon collisions, the density of matter is almost saturated, so that all three constituent quarks from each nucleon may participate nearly simultaneously in collision depositing their energy coherently into the thermalized zone. Therefore, in the headon AA interactions at $\Ecmn$ the bulk variables are expected to have the values similar to those from pp collisions at $\Ecmp \simeq
3\, \Ecmn$. This makes the most central collisions of nuclei akin to collisions at $\Ecme \simeq \Ecmn$ in sense of the resulting bulk variables.
![ The charged particle mean multiplicity $N_{\rm ch}$ per participant pair ($N_{\rm
part}$/2) as a function of the c.m. energy. The solid and combined symbols show the multiplicity values from: most central heavy-ion (AA) collisions c.m. energy per nucleon, $\Ecmn$, measured by PHOBOS ($ \scriptstyle
\blacksquare$), NA49 ($ \scriptstyle
\bigstar$), and E895 ($\blktrianginsqr$) (see also ); collisions, by UA5 ($\blacktriangle$ for non-single diffractive, $\blacktriangledown$ for inelastic events) at $\Ecmp=$ 546 GeV and $\Ecmp=$ 200 and 900 GeV ; pp collisions (at lower $\Ecmp$) from CERN-ISR ($\bullet$) and bubble chamber experiments ($\blktriangdowninsqr$) (the latter compiled and analysed in ). The inelastic UA5 data at $\Ecmp=$ 200 GeV is due to the extrapolation in . The open symbols show the measurements: the high-energy LEP mean multiplicities ($ \scriptscriptstyle
\bigcirc$) averaged here from the data at LEP1.5 $\Ecme =$ 130 GeV and LEP2 $\Ecme
=$ 200 GeV , and the lower-energy data by DELPHI ($ \scriptstyle
\square$), TASSO ($ \scriptstyle
\triangle$), AMY ($ \scriptstyle
\diamondsuit$), JADE (+), LENA ($\star$), and MARK1 ($\timesplus$) experiments. (See refs. in for and pp/ data). The solid line shows the calculations from Eq. (\[prap0\]) based on our approach and using the corresponding fits (see text). The dashed and dotted lines show the fit to the pp/ data and the 3NLO perturbative QCD ALEPH fit to data. The arrows show the LHC expectations. []{data-label="fig:multshe"}](multshea){height=".54\textheight"}
[**2.**]{} According to our consideration, in Fig. \[fig:multshe\], we compare the c.m. energy dependence of the mean multiplicity in AA and interactions to that in pp/collisions at $\Ecmn = \Ecme =
\Ecmp/3$ from a few GeV to 200 GeV. For $\Ecme> M_{Z^0}$, we give the multiplicities averaged from the recent LEP data at $\Ecme =$ 130 GeV and 200 GeV: $23.35\pm 0.20 \pm 0.10$ and $27.62 \pm 0.11 \pm 0.16$. Figure shows also the mean multiplicity fit to pp/ data and the 3NLO pQCD ALEPH fit to data .
From Fig. \[fig:multshe\] one sees that the pp/ data are very close to the data at $\Ecme=\Ecmp/3$. This nearness decreases the already small deficit in the data as the energy increases. The deviation can be attributed to the inelasticity factor, or leading particle effect [@leadp] in pp/ collisions, which is known to decrease with the c.m. energy. Then, at lower $\Ecmp$, some fraction of the energy of spectators contributes more into the formation of the initial state as the spectators pass by. This leads to the excess of the mean multiplicity in pp/ data compared to the data as it is seen in Fig. \[fig:multshe\]. Comparing further the average multiplicities from pp/ collisions to those from AA ones, one finds that the data points are amazingly close to each other when the AA data are confronted the hadronic data at $\Ecmp =3\, \Ecmn$. The inclusion of the tripling energy factor indeed allows to describe such a fundamental variable as the mean multiplicity [*simultaneously*]{} in , pp/ and central AA collisions for all energies. This shows that the multiparticle production process in headon AA collisions is derived by the energy deposited in the Lorentz-contracted volume by a single pair of effectively structureless nucleons similar to that in annihilation and of quark-pair interactions in pp/ collisions. Note that an examination of Fig. \[fig:multshe\] reveals that not a factor 1/2 is needed to rescale $\Ecmp$ to match the AA or data as earlier was assumed for the mean multiplicity while recognised to unreasonably shift the data on the pseudorapidity density at midrapidity when compared to the AA measurements . This discrepancy finds its explanation in our consideration, within which the data on $\it
both$ the mean multiplicity and the midrapidity density ([*vide infra*]{}) are self-consistently matched for different reactions. Let us gain recall a factor 1/3 obtained earlier in for $\Ecmp$ for the pp mean multiplicity data relative to those from data, similar to our finding.
Fig. \[fig:multshe\] shows that the mean multiplicities in different reactions are close starting from the SPS $\Ecmn$ , and become particularly close at $\Ecmn \gtsim$ 50 GeV. However, at lower energies, the AA data are slightly below the and hadronic data and the nuclear data increase faster with energy than the pp and data do. On the other hand, as the c.m. energy increases above a few tens GeV, the AA data start to overshoot the data and reach the mean multiplicity values from interactions. From this one concludes on two different energy regions of the multiparticle production in AA reactions. The observations made can be understood in terms of the overlap zone and energy deposition by participants . Due to this, one would expect the differences to be more pronounced in midrapidity densities as discussed below.
![ Pseudorapidity density $\rho(0)$ of charged particles per participant pair ($N_{\rm part}/2$) at midrapidity as a function of the c.m. energy of collision. The open and combined symbols show the pseudorapidity density values c.m. energy per nucleon, $\Ecmn$, measured in the headon AA collisions by BRAHMS ($ \scriptscriptstyle
\bigcirc$), PHENIX ($ \scriptstyle
\triangle$), PHOBOS ($ \scriptstyle
\square$), and STAR ($\star$), and the density values recalculated from the measurements taken by CERES/NA45 (+), NA49 ($\triangleinsquare$), NA50 ($ \scriptstyle
\diamondsuit$) WA98 ($\timesplus$), E802, and E917 ($\timesplussquared$). The nuclear data at $\Ecmn$ around 20 GeV and the RHIC data at $\Ecmn=130$ GeV and 200 GeV are given spread horizontally for clarity. The solid symbols show the pseudorapidity density values c.m. energy $\Ecmp/3$ as measured in non-single diffractive ${\bar {\rm p}}{\rm p}$ collisions by UA1 ($ \scriptstyle
\blacksquare$), UA5 ($\blacktriangle$), CDF ($\blacktriangledown$), and from inelastic pp data from ISR ($
\scriptstyle
\bigstar$), and bubble chamber ($\bullet$) experiments (the latter as recalculated in ). The solid line connects the predictions from Eq. (\[prap0\]). The dashed line gives the fit to the calculations using the 2nd order log-polynomial fit function analogous to that used in data. The fit function from is shown by the dashed-dotted line. The dotted line shows the linear log approximation of UA5 to inelastic events . The arrows show the LHC expectations. Note that data at $\Ecme=$ 14 GeV to 200 GeV (not shown) follows the heavy-ion data . []{data-label="fig:rap0"}](rap0hepta){height=".54\textheight"}
[**3.**]{} In Fig. \[fig:rap0\], we compare the pseudorapidity densities per participant pair at midrapidity as a function of $\Ecmn$ from headon AA collisions at RHIC, CERN SPS and AGS to those of pp/ data from CERN and Fermilab plotted $\Ecmp/3$. Again one can see that up to the existing $\Ecmn$ the data from hadronic and nuclear experiments are close to each other being consistent with our interpretation. The measurements from the two types of collisions coincide at $8<\Ecmn<20$ GeV and are of the magnitude of the spread of AA data points at 200 GeV. However, above and below the 8-20 GeV region, there are visible differences in the midrapidity $\eta$-density values from AA pp data. These indicate that, in contrast to the mean multiplicity which is a more global observable, the midrapidity density depends on some additional factor. As the densities are measured in the very central $\eta$-region, where the participants longitudinal velocities are zeroed, it is natural to assume that this factor is related to the size of the Lorentz-contracted volume of the initial thermalized system determined by participants.
To take into account the corresponding correction, let us consider our picture in the framework of the Landau model which is close to our description. Then, one finds for the ratio of the normalised charged particle rapidity density $\rho(y)=(2/N_{\rm part})dN_{\rm ch}/dy$ at the midrapidity value $y=0$ in AA reaction, $\rho_{\rm NN}$, to the density $\rho_{\rm pp}$ in pp/ interaction, $${\rho_{\rm NN}(0)}/{\rho_{\rm pp}(0)}=
{2\,N_{\rm ch}}
%\sqrt
({{L_{\rm pp}}/{L_{\rm NN}}})^{1/2}
\,
/\big( {N_{\rm part}\, N^{\rm pp}_{\rm ch}} \big)
\,.
\label{rap0}
\vspace*{-.2cm}$$ Here, $N_{\rm ch}$ ($N_{\rm ch}^{\rm pp}$) is the multiplicity in AA (pp/) collision, $L= \ln [\sqrt {s}/(2m)]$, and $m$ is the participant mass, the proton mass $m_{\rm p}$ in AA reaction. According to our interpretation, we compare in the ratio (\[rap0\]) $\rho_{\rm NN}(0)$ to $\rho_{\rm pp}(0)$ at $\Ecmn = \Ecmp/3$ and consider a constituent quark of mass $\frac{1}{3}m_{\rm p}$ as a participant in pp/ collisions and a proton as an effectively structureless participant in headon AA collisions. Then, Eq. (\[rap0\]) reads: $$\rho_{\rm NN}(0)=
{2\,N_{\rm ch}}\,
\rho_{\rm pp}(0) \,
\,
\sqrt
%\big[
{1-{4 \ln 3}/{\ln\, (4 m_{\rm p}^2/s_{\rm NN})} }\,
%\big] ^{1/2}
\big/
\big( {N_{\rm part}\, N^{\rm pp}_{\rm ch}}\big)
\,.
\label{prap0}
\vspace*{-.2cm}$$
Using the fact that the transformation factor from $y$ to $\eta$ does not influence the above ratio and substituting the multiplicity values from Fig. \[fig:multshe\] and of $\rho_{\rm pp}(0)$ from Fig. \[fig:rap0\] into Eq. (\[prap0\]), one obtains the values of $\rho_{\rm NN}(0)$, displayed in Fig. \[fig:rap0\] by solid line. One can see that the correction made provides good agreement between the calculated $\rho_{\rm NN}(0)$ values and the data. Eq. (\[prap0\]) shows the importance of the correction for the participant type to be introduced as argued above. One can see that our calculations account also for different types of rise of AA data below and above SPS region. Note that the same two regions recently have been indicated by PHENIX from the ratio of the midrapidity [*transverse energy*]{} density to the pseudorapidity density. From these findings, one can expect the midrapidity transverse energy densities in pp/ and headon AA collisions to be similar due to the description proposed here. Also, the SPS transition region properties discussed by NA49 , can be treated without any additional assumptions.
[**4.**]{} To estimate $\rho_{\rm NN}(0)$ for $\Ecmn>200$ GeV, we extrapolated the values of Eq. (\[prap0\]) utilizing the function found to fit well the data. The predictions for $\rho_{\rm NN}(0)$ and the fit for data are shown in Fig. \[fig:rap0\] by dashed and dashed-dotted lines, respectively.
The obtained $\rho_{\rm NN}(0)$ show faster rise with $\Ecmn$ than $\rho_{\rm pp}(0)$. Our calculations, sharing the behaviour at SPS–RHIC energies with that up to the LHC ones, give $\rho_{\rm NN}(0)\approx 7.7$ for LHC. From the CDF fit and assuming it covers LHC energies, one finds $\rho_{\rm pp}(0)\approx 6.1$. Our $\rho_{\rm NN}(0)$ value for LHC is consistent with that of $\approx 6.1$ given in the PHENIX extrapolation within 1-2 particle error acceptable in the calculations we made. Our result is in a good agreement with the best ATLAS Monte Carlo tune . Noticing that $\Ecmn$ is near to $\Ecmp/3$ at LHC, the close values of $\rho_{\rm NN}(0)$ and $\rho_{\rm pp}(0)$, predicted for LHC by us and estimated independently in , demonstrates experimentally grounded description and predictive ability of our interpretation.
Solving Eq. (\[prap0\]) for $N_{\rm
ch}/(0.5 N_{\rm part})$ we predict the AA mean multiplicity energy dependence at $\Ecmn>200$ GeV. In this calculations, we use the fits of $\rho_{\rm pp}(0)$ and $N^{\rm pp}_{\rm ch}$ and our approximation for $\rho_{\rm NN}(0)$, all shown in Figs. \[fig:multshe\] and \[fig:rap0\]. From the resulted curve for $N_{\rm ch}/(0.5 N_{\rm part})$ given in Fig. \[fig:multshe\], one finds that the value obtained for LHC is just about $10\%$ above the $N^{\rm pp}_{\rm ch}(\Ecmp)$ fit prediction for LHC and about 3.3 times larger the AA RHIC data at $\Ecmn=$ 200 GeV. Again, this number is comparable with the estimate made by and points out to no evidence for change to another regime as the $\Ecmn$ increases by about two magnitudes from the top SPS energy. Nevertheless, one can see that the data obtained at the highest RHIC energy give a hint to some border-like behaviour of the mean multiplicity where the pp/ data saturate the nuclear data, and another transition energy region is possible to be found (as at low energies). This makes AA experiments at $\Ecmn>200$ GeV of particular interest.
[**5.**]{} At the end, let us dwell on the following.
From our description, the mean multiplicity in [*nucleon*]{}-nucleus collisions is predicted to be of the same values as that in pp/ data, and, moreover, almost no centrality dependence is expected for such type of interactions . These predictions are well confirmed by various data from hadron-nucleus collisions at $\Ecmn
\approx$ 10–20 GeV to recent RHIC dAu data at 200 GeV . The same seems to be correct also for the pseudorapidity density at midrapidity, which is already supported to be a trend . These findings remind about similar conclusions made about two decades ago .
The recent observation made at RHIC for multihadron data from CuCu collisions to not change compared to same c.m. energy AuAu data when scaled for the same participant numbers is also understood due to our description as already mentioned. Indeed, for the same number of participants, no difference in the bulk variables is expected as one moves from one type of (identical) colliding nuclei to another one at the same c.m. energy as soon as the same energy is deposited into the thermalization zone. Note that the proper definition of participants and, thus, of the energy available for particle production, as we discuss here, allows scaling within the constituent quark picture to be applied to model the multihadron data at RHIC for different observables.
One of us (EKGS) is grateful to Organizers for invitation and partial financial support.
[99]{}
A. Bia[ł]{}as , ; A. Bia[ł]{}as, W. Czyż, 2[36]{}[05]{}[905]{}. B.B. Back (PHOBOS), 2[757]{}[05]{}[28]{}, and refs. therein. P. Steinberg, 2[30]{}[04]{}[S683]{}; W. Busza, 2[35]{}[04]{}[2873]{}. G.J. Alner (UA5), . F. Abe (CDF), . I. Arsene (BRAHMS), 2[757]{}[05]{}[1]{}, and refs. therein. K. Adcox (PHENIX), 2[757]{}[05]{}[184]{}, and refs. therein. R. Noucier, talk at Int. Symp. Multiparticle Dynamics 2005: these Proceedings. G. Roland (for the PHOBOS ), M. Konno (for the PHENIX ): Quark Matter 2005. E.K.G. Sarkisyan and A.S. Sakharov, hep-ph/0410324. L.D. Landau, . E.L. Feinberg, Proc. Int. Conf. Elementary Particle Physics (Smolenice, 1985), p. 81; Relativistic Heavy Ion Physics (World Scientific, 1991), p. 341. P.A. Carruthers, LA-UR-81-2221 ; P. Steinberg, nucl-ex/0405022. M. Basile , , . V.V. Anisovich , Quark Model and High Energy Collisions (World Scientific, 2005). P.V. Chliapnikov and V.A. Uvarov, ; for review, see W. Kittel and E.A. De Wolf, Soft Multihadron Dynamics (World Scientific, 2005). S. Eremin and S. Voloshin, 2[67]{}[03]{}[064905]{}. S.V. Afanasiev (NA49), 2[66]{}[02]{}[054902]{}. J.L. Klay (for the E895 ), PhD Thesis (U.C. Davis, 2001), see . G.J. Alner (UA5), . R.E. Ansorge (UA5), . W. Thomé , . V.V.Ammosov ,; C.Bromberg , . W.M. Morse , . E. De Wolf, J.J. Dumont, F. Verbeure, . G. Alexander (OPAL), . A. Heister (ALEPH), 2[35]{}[04]{}[457]{}; P. Abreau (DELPHI), ; P. Achard (L3), 2[399]{}[04]{}[71]{}. G. Abbiendi (OPAL), 2[37]{}[04]{}[25]{}. P. Abreau (DELPHI), . P.D. Acton (OPAL), . Particle Data Group, S. Eidelman , 2[592]{}[04]{}[1]{}, and refs. therein. Compilation by O. Biebel, see [ http://www.cern.ch/biebel/www/RPP04/]{} . I.M. Dremin, J.W. Gary, 2[349]{}[01]{}[301]{}, and refs. therein. C. Adler (STAR), 2[757]{}[05]{}[102]{}, and refs. therein. F. Ceretto (for the CERES/NA45 , G. Agakichiev ), . F. Siklér (for the NA49 , J. Bächler ), . M.C. Abreau (NA50), 2[530]{}[02]{}[43]{}. M.M. Aggarwal (WA98), 2[18]{}[01]{}[651]{}. L.Ahle (E802), Phys.Rev. [C]{}[**59**]{} (1999)2173; B.Back (E917), Phys.Rev.Lett. [**86**]{} (2001) 1970. G. Arnison (UA1), . J. Whitmore , . A.M. Moraes, talk given at ATLAS Physics Week (CERN, Nov. 2004). P.K. Netrakanti, B. Mohanty, 2[70]{}[04]{}[027901]{}; B. De, S. Bhattacharyya, 2[71]{}[05]{}[024903]{}.
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abstract: '> We consider the problem of modeling temporal textual data taking endogenous and exogenous processes into account. Such text documents arise in real world applications, including job advertisements and economic news articles, which are influenced by the fluctuations of the general economy. We propose a hierarchical Bayesian topic model which imposes a “group-correlated” hierarchical structure on the evolution of topics over time incorporating both processes, and show that this model can be estimated from Markov chain Monte Carlo sampling methods. We further demonstrate that this model captures the intrinsic relationships between the topic distribution and the time-dependent factors, and compare its performance with latent Dirichlet allocation (LDA) and two other related models. The model is applied to two collections of documents to illustrate its empirical performance: online job advertisements from DirectEmployers Association and journalists’ postings on BusinessInsider.com.'
author:
- |
Baiyang Wang, Diego Klabjan\
Department of Industrial Engineering and Management Sciences,\
Northwestern University, 2145 Sheridan Road, Evanston, Illinois, USA, 60208\
[email protected]@northwestern.edu
bibliography:
- 'b1.bib'
nocite: '[@B10; @F73; @G04; @R04; @R06; @S15; @T05]'
title: Temporal Topic Analysis with Endogenous and Exogenous Processes
---
Introduction
============
Many organizations nowadays provide portals for job posting and job search, such as glassdoor.com from Glassdoor, indeed.com from Recruit, and my.jobs from DirectEmployers Association. Our work is inspired by data collected from the portal my.jobs, a website where job seekers can apply to the posted job openings through a provided link. The data collected from the website includes user clickstreams (users create accounts on the site) and attributes of job advertisements, such as their description, location, company name, and posted date.
In this paper, we investigate the relationship between economic fluctuations and the related changes in job advertisements, which can reveal the economic conditions of different time periods. More generally, this question is about the influence of any exogenous process on textual data with temporal dimensions. We adopt the perspective that the documents are organized into a certain number of topics, and study the impact of the exogenous process on the topic distribution, i.e. the relative topic proportions. Given a corpus of text documents with time stamps and a related exogenous process, the problem is to find a relationship between the topics discussed and the exogenous process. This setting is natural in an economic context; for instance, changes in macroeconomic indicators have an impact on government reports and [*Wall Street Journal*]{} news articles. Meanwhile, we also notice that for most temporal documents, the topic proportions change over time, which indicates an endogenous process of topic evolution.
With the goal of establishing topic dependency on the endogenous and exogenous processes, LDA-type topic models are especially suitable. The latent Dirichlet allocation (LDA) [@B03] is the original model. Since then, a large number of variants have been proposed, many of which can be found in Blei (2011).
Meanwhile, there has been relatively limited discussion on modeling time-dependent documents when there are relevant simultaneous exogenous processes. Many time-dependent topic models without the exogenous component have been proposed, such as the Topics over Time (ToT) model [@W06] and the dynamic topic model (DTM) [@B06], to name a few. However, to the best of our knowledge, none of these papers incorporate the effect of exogenous processes. On the other hand, the structural topic model (STM) [@R15] considers the effect of metadata, i.e. the attributes specified for each document, on the topic distribution. While STM can be applied for mining time-dependent textual data with exogenous covariates, it does not explicitly consider the time factor or the endogenous topic evolution processes of time-stamped documents.
Our approach to this problem is to incorporate both endogenous and exogenous processes into a topic model. For the endogenous part of our paper, we impose a Markovian structure on the topic distribution over time, similar to Blei and Lafferty (2006) and Dubey et al. (2014). For the exogenous process, we incorporate it into the topic distribution in each period, adjusting the endogenous topic evolution process. In this way, our model is essentially a stick-breaking truncation of a ”group-correlated” hierarchical Dirichlet process. Our model has the following contributions: (i) it addresses the question of measuring the influence of exogenous processes on the topics in related documents, (ii) it incorporates both endogenous and exogenous aspects, and (iii) it demonstrates that text mining can also have useful implications in the realm of economics, which, from the authors’ perspective, is a relatively new finding.
Section 2 offers a brief review on the topic modeling techniques related to our model. Section 3 develops our hierarchical Bayesian model and describes how to make posterior inferences with a variant of the Markov chain Monte Carlo (MCMC) technique. Section 4 studies the online job advertisements from DirectEmployers Association and journalists’ postings in finance on BusinessInsider.com with our proposed method, providing a comparison of performance with the standard LDA and STM. Section 5 suggests possible directions for the future and concludes the paper.
Review of Time-Dependent Topic Modeling
=======================================
We first introduce the standard model of LDA [@B03]. Suppose that there is a collection of documents $d_i$, $i=1,\ldots,N$ and words $\{x_{i,j}\}_{j=1}^{J_i}$ within each document $d_i$ indexed by a common dictionary containing $V$ words, where $N$ is the number of documents, and $J_i$ is the number of words in $d_i$. The LDA model is as follows, $$\begin{cases}
{\boldsymbol{\theta}}_i{\stackrel{iid}{\sim}}Dir({\boldsymbol{\alpha}}),\ {\boldsymbol{\phi}}_k {\stackrel{iid}{\sim}}Dir({\boldsymbol{\beta}}), \\
z_{i,j}|{\boldsymbol{\theta}}_i{\stackrel{iid}{\sim}}Cat({\boldsymbol{\theta}}_i),\ x_{i,j}|z_{i,j}\sim Cat({\boldsymbol{\phi}}_{z_{i,j}}).\\
\end{cases}{\refstepcounter{equation}\tag{\theequation}}$$ Here $i=1,\ldots,N$, $j=1,\ldots,J_i$, $k=1,\ldots,K$; ${\boldsymbol{\theta}}_i$ is the length-$K$ per-document topic distribution for $d_i$, ${\boldsymbol{\phi}}_k$ is the length-$V$ per-topic word distribution for the $k$-th topic, $z_i^j$ is the topic for the $j$-th word in $d_i$, and $K$ is the number of topics. $Dir(\cdot)$ denotes the Dirichlet distribution and $Cat(\cdot)$ denotes the categorical distribution, a special case of the multinomial distribution when $n_{obs}=1$.
The Dirichlet process is a class of randomized probability measures and can be applied for non-parametric modeling of mixture models. Denoting the concentration parameter by ${\alpha}$ and the mean probability measure by $H$, a realization $G$ from the Dirichlet process can be written as $G\sim DP({\gamma}, H)$. With the stick-breaking notation [@S94], we have $$G = \sum_{k=1}^\infty b_k{\delta}_{{\varphi}_k},$$ where ${\delta}_{{\varphi}_k}$ is a “delta” probability measure with all the probability mass placed at ${\varphi}_k$, ${\varphi}_k{\stackrel{iid}{\sim}}H$, $b_k=b_k'\prod_{i=1}^{k-1}(1-b_i')$, $b_k'{\stackrel{iid}{\sim}}Beta(1,{\gamma})$, $k=1,2,\cdots$. We write ${\boldsymbol{b}} = (b_1, b_2, \ldots) \sim Stick({\gamma})$. More properties of the Dirichlet process can be found in Ferguson (1973).
A hierarchical Dirichlet process (HDP) was proposed in the context of text modeling by Teh et al. (2005). The following hierarchical structure is assumed, $$\begin{cases}
G_0|{\gamma}\sim DP({\gamma}, H), \\
G_1, \ldots, G_N|({\alpha}, G){\stackrel{iid}{\sim}}DP({\alpha}, G), \\
{\boldsymbol{\phi}}_{i,j}|G_i{\stackrel{iid}{\sim}}G_i,\ x_{i,j}\sim Cat({\boldsymbol{\phi}}_{i,j}).
\end{cases}{\refstepcounter{equation}\tag{\theequation}}$$ Here $i=1,\ldots,N$, $j=1,\ldots,J_i$. The length-$V$ random vectors $G_0, G_1, \ldots, G_N$ are “random word distributions,” each of which is a draw from a Dirichlet process in (3). Moreover, each draw from a random word distribution is a length-$V$ fixed vector ${\boldsymbol{\phi}}_{i,j}$; it is the word distribution for $x_{i,j}$. The posterior inference can be achieved by different strategies of Gibbs sampling.
There are mainly two approaches in the literature of measuring endogenous topic evolution processes. One approach is to impose a finite mixture structure on the topic distribution: a dynamic hierarchical Dirichlet process (dHDP) [@R08] was proposed by adding a temporal dimension, and its variation was further applied on topic modeling with a stick-breaking truncation of Dirichlet processes [@P10]. The other approach imposes a Markovian structure. For instance, the dynamic topic model (DTM) [@B06] is as follows, $$\begin{cases}
{\boldsymbol{\phi}}_{t,k}|{\boldsymbol{\phi}}_{t-1,k}\sim N({\boldsymbol{\phi}}_{t-1,k}, {\sigma}^2I),\\
{\boldsymbol{\alpha}}_t|{\boldsymbol{\alpha}}_{t-1}\sim N({\boldsymbol{\alpha}}_{t-1}, {\delta}^2I),\\
{\boldsymbol{\theta}}_{t,i}|{\boldsymbol{\alpha}}_t{\stackrel{iid}{\sim}}N({\boldsymbol{\alpha}}_t, a^2I),\\
z_{t,i,j}|{\boldsymbol{\theta}}_{t,i}{\stackrel{iid}{\sim}}Cat(\exp({\boldsymbol{\theta}}_{t,i})),\\
x_{t,i,j}|z_{t,i,j}\sim Cat(\exp({\boldsymbol{\phi}}_{t,z_{t,i,j}})).
\end{cases}{\refstepcounter{equation}\tag{\theequation}}$$ Here $t=1, \ldots, T$, $i=1,\ldots,N_t$, $j=1,\ldots,J_{t,i}$, $k=1,\ldots,K$ ($t\ge2$ for the first two equations); $T$ is the number of time periods, $N_t$ is the number of documents in the $t$-th period, and $J_{t,i}$ is the number of words in the $i$-th document in the $t$-th period; the rest are similarly defined as in LDA. One major difference between DTM and LDA is that the topic distributions ${\boldsymbol{\theta}}_{t,i}$ and word distributions ${\boldsymbol{\phi}}_{t,k}$ are in log-scale in DTM. A variational Kalman filtering was proposed for the posterior inference. As this Markovian approach is simpler for both interpretation and posterior inference, we apply a more generalized version of it to specify the endogenous process in our model.
The structural topic model (STM) [@R15] measures the effect of metadata of each document with the logistic normal distribution. Their model for each document $d_i$ is as follows, $$\begin{cases}
{\boldsymbol{\theta}}_i|(X_i{\gamma},{\Sigma})\sim LogisticNormal(X_i{\gamma},{\Sigma}),\\ p({\boldsymbol{\phi}}_{i,k})\propto\exp(m+{\kappa}_k+{\kappa}_{g_i}+{\kappa}_{kg_i}),\\
z_{i,j}|{\boldsymbol{\theta}}_i{\stackrel{iid}{\sim}}Cat({\boldsymbol{\theta}}_i),\ x_{i,j}|z_{i,j}\sim Cat({\boldsymbol{\phi}}_{i,z_{i,j}}),
\end{cases}{\refstepcounter{equation}\tag{\theequation}}$$ where $i=1,\ldots,N$, $j=1,\ldots,J_i$; $X_i$ is the metadata matrix, ${\gamma}$ is a coefficient vector, ${\Sigma}$ is the covariance matrix, ${\boldsymbol{\phi}}_{i,k}$ is the word distribution for $d_i$ and the $k$-th topic, $m$ is a baseline log-word distribution, ${\kappa}_k$, and ${\kappa}_{g_i}$ and ${\kappa}_{kg_i}$ are the topic, group, and interaction effects; the rest are defined similarly to LDA. This model explicitly considers exogenous factors, and can be applied to find the relationship between topic distributions and exogenous processes. Below we adopt a slightly more general approach, incorporating both endogenous and exogenous factors.
Model and Algorithm
===================
Motivation: A Group-Correlated Hierarchical Dirichlet Process
-------------------------------------------------------------
We formulate our problem as follows: we are given time periods $t = 1,\ldots,T$, documents from each period $d_{t,i}$, $i=1,\ldots,N_t$, $t=1,\ldots,T$, and the indices of words $\{x_{t,i,j}\}_{j=1}^{J_{t,i}}$ within each document $d_{t,i}$ from the first word to the last. The words are indexed by a dictionary containing $V$ words in total. We begin with a hierarchical Dirichlet process in time $1$: let $G_1|{\gamma}\sim DP({\gamma}, H)$, $G_{1i}|({\alpha}_1, G_1)\sim DP({\alpha}_1, G_1)$, where $G_1$ is a baseline random word distribution for time $1$, and $G_{1i}$ is the random word distribution for document $d_{1i}$. For $G_2,\ldots G_T$, we have the following Markovian structure, $$p(G_t)|G_{t-1}\propto\exp[-d(G_t,G_{t-1})],\ t=2,\ldots,T.$$ Here $d(\cdot,\cdot)$ is some distance between two probability measures. This completes our endogenous process. To take an exogenous process $\{{\boldsymbol{y}}_t\}_{t=1}^T$ into account, we assume the following $${\widetilde{G}}_{t} = {\mathcal{M}}(G_t, {\boldsymbol{y}}_t),\ t=1,\ldots,T,$$ where ${\mathcal{M}}$ maps the endogenous baseline random word distribution $G_t$ to the realized baseline random word distribution ${\widetilde{G}}_t$ for time $t$, considering the influence of $\{{\boldsymbol{y}}_t\}_{t=1}^T$. Therefore, we further assume that each per-document random word distribution $G_{t,i}$ is sampled with mean ${\widetilde{G}}_t$ rather than $G_t$. The final model is as follows, $$\begin{cases}
G_1|{\gamma}\sim DP({\gamma}, H),\\
p(G_t)|G_{t-1}\propto\exp[-d(G_t,G_{t-1})],\\
{\widetilde{G}}_{t} = {\mathcal{M}}(G_t, {\boldsymbol{y}}_t),\\
G_{t,i}|({\alpha}_t, {\widetilde{G}}_t) {\stackrel{iid}{\sim}}DP({\alpha}_t, {\widetilde{G}}_t),\\
{\boldsymbol{\phi}}_{t,i,j}|G_{t,i}{\stackrel{iid}{\sim}}G_{t,i},\ x_{t,i,j}\sim Cat({\boldsymbol{\phi}}_{t,i,j}).
\end{cases}{\refstepcounter{equation}\tag{\theequation}}$$ Here $t = 1,\ldots,T$, $i=1,\ldots,N_t$, $j=1, \ldots, J_{t,i}$ ($t\ge2$ for the first line). Throughout this paper, our model is fully conditional on $\{{\boldsymbol{y}}_t\}_{t=1}^T$, i.e. we assume $\{{\boldsymbol{y}}_t\}_{t=1}^T$ to be fixed; this has an intuitive explanation, as our temporal documents represent a very small portion of the underlying environment, i.e. the exogenous process, so their influence on $\{{\boldsymbol{y}}_t\}_{t=1}^T$ is almost negligible.
A Group-Correlated Temporal Topic Model: Stick-Breaking Truncation
------------------------------------------------------------------
Below we consider a stick-breaking truncation of the model above, since posterior inference of the exact model can be intricate. With the stick-breaking expression of $G_1$ in (8), we have $$\begin{cases}
{\boldsymbol{\phi}}_1, {\boldsymbol{\phi}}_2, \ldots, {\stackrel{iid}{\sim}}H,\\
{\boldsymbol{\pi}}_1 = (\pi_{11},\pi_{12},\ldots)\sim Stick({\gamma}),\\
G_1 = \sum_{k=1}^\infty \pi_{1k}{\delta}_{{\boldsymbol{\phi}}_k}.
\end{cases}{\refstepcounter{equation}\tag{\theequation}}$$ Here we set $d(\cdot,\cdot)=+\infty$ if the two probability measures have different supports; this necessitates that all periods share the same topics. Our intent is that the topics should remain the same to investigate their relationships with endogenous and exogenous processes; otherwise, changes in topics can blur the relationships and possibly result in overfitting. We apply the total variation distance $d(p,q)={\lambda}\cdot\int|p-q|d\mu$ with ${\lambda}>0$, although many others can also be applied and lead to, for instance, a log-normal model in DTM, or a normal model [@D14; @Z15]. We have the following, $$\begin{cases}
G_t = \sum_{k=1}^\infty \pi_{tk}{\delta}_{{\boldsymbol{\phi}}_k},\\
\pi_{tk} = \pi_{t-1\, k} + Lap({\lambda}),\\
{\boldsymbol{\pi}}_t = (\pi_{t1},\pi_{t2},\ldots).
\end{cases}{\refstepcounter{equation}\tag{\theequation}}$$ Here $t=2,\ldots,T$, $k=1,2,\ldots$, and $Lap({\lambda})$ denotes a Laplacian distribution with scale parameter ${\lambda}$. For the exogenous part, we consider specifying the relationship between ${\boldsymbol{\pi}}_t$ and ${\widetilde{\boldsymbol{\pi}}}_t = ({\widetilde{\pi}}_{t1},{\widetilde{\pi}}_{t2},\ldots)$ such that ${\widetilde{G}}_t = \sum_{k=1}^\infty {\widetilde{\pi}}_{tk}{\delta}_{{\boldsymbol{\phi}}_k}$, $G_{t,i}=\sum_{k=1}^\infty \theta_{t,i,k}\delta_{{\boldsymbol{\phi}}_k}$. We let $${\widetilde{\boldsymbol{\pi}}}_t = {\boldsymbol{\pi}}_t + {\boldsymbol{\eta}}\cdot{\boldsymbol{y}}_t,\ t=1,\ldots,T,\ {\boldsymbol{1}}'\cdot{\boldsymbol{\eta}}= {\boldsymbol{0}}.$$ Here ${\boldsymbol{\eta}}$ is a $K\times p$ matrix which indicates the relationship between the topic distribution ${\widetilde{\boldsymbol{\pi}}}_t$ and the length-$p$ vector ${\boldsymbol{y}}_t$. However, we notice that ${\widetilde{\boldsymbol{\pi}}}_t$ and ${\boldsymbol{\pi}}_t$ are of infinite length, which creates difficulty in our inference. Therefore we adopt a stick-breaking truncation approach, i.e. we only consider $\{{\boldsymbol{\phi}}_k\}_{k=1}^K$ in our model; the probability weights for $\{{\boldsymbol{\phi}}_k\}_{k=K+1}^\infty$ in ${\boldsymbol{\pi}}_t$ will be added into $\pi_{tK}$. We note that a number of papers in topic modeling have put this approach into practice [@P10; @W11].
It has been shown [@P10] that when the truncation level $K$ is large, we may as well replace the distribution of ${\boldsymbol{\pi}}_1$ with ${\boldsymbol{\pi}}_1\sim Dir({\gamma}{\boldsymbol{\pi}}_0)$, where ${\gamma}=1$, ${\boldsymbol{\pi}}_0=(1/K,\ldots,1/K)$. We also let $H=Dir({\beta},\ldots,{\beta})$ as in the paper by Teh et al. (2005). We summarize our model, $$\begin{cases}
{\boldsymbol{\phi}}_1,\ldots,{\boldsymbol{\phi}}_K{\stackrel{iid}{\sim}}Dir({\beta},\ldots,{\beta}),\\
{\boldsymbol{\pi}}_1 \sim Dir({\gamma}{\boldsymbol{\pi}}_0),\ \pi_{tk} = \pi_{t-1\, k}+Lap({\lambda}),\\
{\widetilde{\boldsymbol{\pi}}}_t = {\boldsymbol{\pi}}_t + {\boldsymbol{\eta}}\cdot{\boldsymbol{y}}_t,\\
{\boldsymbol{\theta}}_{t,i}|({\alpha}_t, {\widetilde{\boldsymbol{\pi}}}_t) {\stackrel{iid}{\sim}}Dir({\alpha}_t{\widetilde{\boldsymbol{\pi}}}_t), \\
z_{t,i,j}|{\theta}_{t,i}{\stackrel{iid}{\sim}}Cat({\boldsymbol{\theta}}_{t,i}),\ x_{t,i,j}\sim Cat({\boldsymbol{\phi}}_{z_{t,i,j}}).
\end{cases}{\refstepcounter{equation}\tag{\theequation}}$$ Here $t = 1,\ldots,T$, $i=1,\ldots,N_t$, $j=1, \ldots, J_{t,i}$ ($t\ge2$ for the second line). The last two lines above are derived as in Teh et al. (2005). We note that here ${\boldsymbol{\phi}}_k$ is the per-topic word distribution, ${\boldsymbol{\theta}}_{t,i}$ is the per-document topic distribution, and $z_{t,i,j}$ is the actual topic for each word; they have the same meaning as in LDA.
We name our model a “group-correlated temporal topic model” (GCLDA). Here a “group” stands for all the documents within the same time period. We use the term “correlated” because the baseline topic distributions $\{{\boldsymbol{\pi}}_t\}_{t=1}^T$ for each period, controlling for $\{{\boldsymbol{y}}_t\}_{t=1}^T$, are endogenously correlated; meanwhile, the realized baseline topic distributions $\{{\widetilde{\boldsymbol{\pi}}}_t\}_{t=1}^T$ for each period are also correlated with the given exogenous process $\{{\boldsymbol{y}}_t\}_{t=1}^T$.
Sampling the posterior: An MCMC Approach
----------------------------------------
Direct estimation of the Bayesian posterior is often intractable since the closed-form expression, if it exists, can be difficult to integrate and thus, many approaches to approximate the posterior have been proposed. Monte Carlo methods, which draw a large number of samples from the posterior as its approximation, are particularly helpful. In this paper, we adopt the Markov chain Monte Carlo (MCMC) approach which constructs samples from a Markov chain and is asymptotically exact. Below we provide the Metropolis-within-Gibbs sampling approach tailored to our situation, which is a variant of the general MCMC approach. It only requires specifying the full conditionals of the unknown variables, which is covered below.
We consider sampling the following variables ${\boldsymbol{Z}}=\{z_{t,i,j}\}_{j=1}^{J_{t,i}}\,_{i=1}^{N_t}\,_{t=1}^T$, $\{{\alpha}_t\}_{t=1}^T$, $\{{\widetilde{\boldsymbol{\pi}}}_t\}_{t=1}^T$, ${\boldsymbol{\eta}}$, ${\lambda}$. We integrate out $\{{\boldsymbol{\theta}}_{t,i}\}_{i=1}^{N_t}\,_{t=1}^T$ and $\{{\boldsymbol{\phi}}_k\}_{k=1}^K$ to speed up calculation. Following Griffiths and Steyvers (2004), conditioning on all other variables listed for sampling, $$p(z_{t,i,j}=k|rest)\propto (C_{t,i,k}^{(-1)}+{\alpha}_t{\widetilde{\pi}}_{tk})\frac{C_{x_{t,i,k},k}^{(-1)}+{\beta}}{C_{k}^{(-1)}+V{\beta}}.$$ Here $C_{t,i,k}^{(-1)}$ is the count of elements in ${\boldsymbol{Z}}{\backslash}\{z_{t,i,j}\}$ which belong to $d_{t,i}$ and has values equal to $k$; $C_{x_{t,i,k},k}^{(-1)}$ is the count of elements in ${\boldsymbol{Z}}{\backslash}\{z_{t,i,j}\}$ whose values are $k$ and corresponding words are $x_{t,i,j}$; $C_k^{(-1)}$ is the count of elements in ${\boldsymbol{Z}}{\backslash}\{z_{t,i,j}\}$ whose values are $k$. Also following Griffiths and Steyvers (2004), we have for ${\alpha}_t$ and ${\widetilde{\boldsymbol{\pi}}}_t$ $$\begin{aligned}
&\qquad p({\alpha}_t,{\widetilde{\boldsymbol{\pi}}}_t|rest)\propto p({\boldsymbol{Z}}|{\alpha}_t,{\widetilde{\boldsymbol{\pi}}}_t) p({\boldsymbol{\pi}}_{1,\ldots,T}|{\gamma},{\boldsymbol{\pi}}_0)p({\alpha}_t)\\
&\propto \left[\frac{{\Gamma}({\alpha}_t)}{\prod_k{\Gamma}({\alpha}_t{\widetilde{\pi}}_{tk})}\right]^{N_t}\prod_{i=1}^{N_t}\frac{\prod_k{\Gamma}(C_{t,i,k}+{\alpha}_t{\widetilde{\pi}}_{tk})}{{\Gamma}(J_{t,i}+{\alpha}_t)} \\
&\times \exp \left[-{\lambda}\left( 1_{t<T}\cdot\|\pi_{t+1}-\pi_{t}\|_1+ 1_{t>1}\cdot\|\pi_{t}-\pi_{t-1}\|_1 \right)\right]\\
&\times p({\boldsymbol{\pi}}_1|{\gamma}{\boldsymbol{\pi}}_0)p({\alpha}_t).{\refstepcounter{equation}\tag{\theequation}}\end{aligned}$$ Here the difference between $C_{t,i,k}$ and $C_{t,i,k}^{(-1)}$ is to replace ${\boldsymbol{Z}}{\backslash}\{z_{t,i,j}\}$ with ${\boldsymbol{Z}}$. We also view ${\boldsymbol{\pi}}_t$, $\pi_{tk}$, etc. as functions of the other parameters; specifically, $\pi_{tk}={\widetilde{\pi}}_{tk}-{\boldsymbol{\eta}}_k{\boldsymbol{y}}_t$, where ${\boldsymbol{\eta}}_k$ is the $k$-th row of ${\boldsymbol{\eta}}$. This involves a transformation of variables; however, the related Jacobian determinant $\det(J)=1$, so (14) is still valid. For parameter ${\boldsymbol{\eta}}$, we have $$\begin{aligned}
&\qquad p({\boldsymbol{\eta}}|rest)\propto \prod_{t=2}^Tp({\boldsymbol{\pi}}_{t}|{\boldsymbol{\pi}}_{t-1},{\lambda})\cdot p({\boldsymbol{\eta}}) \\
&\propto \exp\left(-{\lambda}\cdot\sum_{t=2}^T\|\pi_{t}-\pi_{t-1}\|_1\right)p({\boldsymbol{\eta}}).{\refstepcounter{equation}\tag{\theequation}}\end{aligned}$$ Finally, for parameter ${\lambda}$, we have $$\begin{aligned}
&\qquad p({\lambda}|rest)\propto \prod_{t=2}^Tp({\boldsymbol{\pi}}_{t}|{\boldsymbol{\pi}}_{t-1},{\lambda})\cdot p({\lambda}) \\
&\propto {\lambda}^{(T-1)K}\exp\left(-{\lambda}\cdot\sum_{t=2}^T\|\pi_{t}-\pi_{t-1}\|_1\right) p({\lambda}). {\refstepcounter{equation}\tag{\theequation}}\end{aligned}$$ We note that (13) and (16) are full conditionals, and we can easily derive the full conditionals of ${\alpha}_t$, $\pi_{tk}$, and $\eta_k$ from (14) and (15). Since each $z_{t,i,j}|rest$ has a categorical distribution, and ${\lambda}|rest$ has a Gamma distribution with a conjugate prior, they can be updated with Gibbs updates. For ${\alpha}_t$, $\pi_{tk}$, and $\eta_k$, we replace a Gibbs update with a Metropolis update. Specifically, suppose we know $p(par|rest)$ up to a multiplicative constant, where $par$ is any length-$1$ parameter. We also assume $par=par^{(r)}$ at the $r$-th iteration. Then at the $(r+1)$-th iteration, $$par_{new} \sim q(\cdot|par^{(r)}),$$ $$par^{(r+1)} = \left\{\begin{array}{ll}par_{new}\textrm{\ with\ prob.\ }P,\\ par^{(r)}\textrm{\ with\ prob.\ }1-P,\end{array}\right. {\refstepcounter{equation}\tag{\theequation}}$$ where $P=\min\{p(par_{new}|rest) /p(par^{(r)}|rest), 1\}$, and $q(\cdot|\cdot)$ is a known conditional probability distribution such that $q(x|y)=q(y|x)$.
This completes our sampling and posterior inference. For the theoretical convergence properties of Metropolis-within-Gibbs samplers, the reader can refer to Robert and Casella (2004) and Roberts and Rosenthal (2006).
Case Studies
============
The proposed model, “GCLDA,” is demonstrated on two data sets: (1) online job advertisements from my.jobs from February to September in 2014, and (2) journalists’ postings in 2014 in the “Finance” section in BusinessInsider.com, an American business and technology news website. Our algorithm has been implemented in Java, and we compare GCLDA with LDA, ToT, and STM.
Experiment Settings
-------------------
We initialize the hyperparameters of LDA as follows: ${\boldsymbol{\alpha}}=(50/K,\ldots,50/K)$, ${\boldsymbol{\beta}}=(0.01,\ldots,$ $0.01)$, according to a rule of thumb which has been carried out in Berry and Kogan (2010) and Sridhar (2015). For ToT, we use the same ${\boldsymbol{\alpha}}$ and ${\boldsymbol{\beta}}$ and linearly space the timestamps to make computation feasible. For GCLDA, we let ${\gamma}=1$, ${\boldsymbol{\pi}}_0=(1/K,\ldots,1/K)$, ${\alpha}_t{\stackrel{iid}{\sim}}{\Gamma}(1,1)$, $p({\boldsymbol{\eta}})\propto e^{-0.01\sum|\eta_k|}$, ${\lambda}\sim{\Gamma}(1,1)$, ${\beta}=0.01$. We carry out the Metropolis-within-Gibbs algorithm as described in Section 3.3 for GCLDA, and run $5{,\!}000$ iterations of the Markov chain with $1{,\!}000$ burn-in samples for GCLDA, LDA, and ToT; for LDA, we apply the collapsed Gibbs sampling as in Griffiths and Steyvers (2004). The number of topics is set to $K=50$ for both data sets. For STM, we apply the “Spectral” initialization [@R15b] together with other default settings in the R package `stm`. We perform data cleaning, remove the stopwords, stem the documents, and keep most frequent $V$ words in each study. For the job advertisements, $V=2{,\!}000$ and covers $96.2\%$ of all words with repetition, which means that the choice of words in job advertisements is quite narrow; for the journalists’ postings, $V=3{,\!}000$ and covers $93.9\%$ of all words with repetition.
We use perplexity to compare the difference of the prediction power between LDA, ToT, and GCLDA. The perplexity for $N_{test}$ held-out documents given the training data $D$ is defined as $$perp = \exp\left\{-\dfrac{\sum_{i=1}^{N_{test}}\log p(d_{test,i}|D)}{\sum_{i=1}^{N_{test}}n_{test,i}}\right\}$$ where $d_{test,i}$ represents the $i$-th held-out document, and $n_{test,i}$ is the number of words in $d_{test,i}$. We expect the perplexity to be small when a model performs well, since this means that under the estimated model, the probability of a word in the testing documents being written [*a priori*]{} is large. We apply the “Left-to-right” algorithm [@W09] and apply point estimates for “$\Phi$” and “${\alpha}{\boldsymbol{m}}$” using the training data, as suggested in Section 3 in the same paper.
My.jobs: Online Job Advertisements
----------------------------------
The number of online job advertisements on my.jobs from February to September in 2014 amounts to $17{,\!}147{,\!}357$ in total, and the number of advertisements each day varies greatly. Therefore, we gather a stratified sample of $44{,\!}660$ advertisements with a roughly equal number of samples for each day, so that we have sampled $0.26\%$ of all the documents in total. The training data set consists of $40{,\!}449$ advertisements, and the testing data set consists of $4{,\!}211$ advertisements ($9.4\%$ of the sample). For the exogenous variable $\{{\boldsymbol{y}}_t\}_{t=1}^T$, we use the standardized Consumer Price Index from February to September in 2014, so that $p=1$, and $T=8$.
Figure 1 implies that GCLDA better predicts the words in the new documents in terms of perplexity. This is due to the fact that the introduction of both endogenous and exogenous processes allows us to make more accurate inference on the topic distributions of the documents in a given period of time. The standard errors for the perplexity in each period are also shown; we can observe that the difference is quite significant.
The $20$ most common topics are presented in Figure 2. The only axis, the x-axis, represents the degree of correlation $\rho=\eta/\pi$ for all topics, i.e. the percent change in the topic proportion given one unit change in the exogenous covariate. Here $\pi$ and $\eta$ denote the related component of $\sum {\boldsymbol{\pi}}_t/T$ and ${\boldsymbol{\eta}}$ for each topic. Table 1 lists the highest probability words sorted by their probabilities from high to low inside the five topics with highest $\rho$ in Figure 2.
[*equal opportunity*]{}: employment status disabled veteran equal
------------------------------------------------------------------------
[*health care*]{}: health care medical service provide center hospital
[*software*]{}: development experience software design application
[*secretarial*]{}: management operations ensure training perform
[*nursing*]{}: care nursing patient required clinical practice medical
: The highest probability words inside the five topics with highest $\rho$ in Figure 2.
A number of facts can be inferred from Figure 2. The topics with positive $\rho$ are those that have a positive correlation with the growth of the CPI in 2014. We can observe that two of them are supported by the U.S. government spending, namely “equal opportunity” and “health care,” the latter of which is probably related to the Affordable Care Act programs. This suggests that there is a causal relationship between the increase in government spending and the increase in the number of jobs in these categories, and the former was also an underlying factor in the growth of the CPI in 2014. We also observe that “software” and “secretarial” were moving in the same direction of CPI, while some traditional higher-paid job categories, such as engineering and marketing, were not. This partly agrees with some news articles in 2014 in that while the labor market was recovering, there was relatively lower growth in traditional higher-paid job categories[@L14; @C14].
Therefore, we posit that GCLDA could be helpful in identifying topics in temporal documents which are closely related to an exogenous process. The changes in the topic proportions may be caused by the exogenous process, or its underlying factors, as is illustrated by this example, where more demand of goods increases the CPI and creates more jobs in certain categories. The other way around is also possible; changes in an exogenous process are caused by certain kinds of news, as is demonstrated in the next example. Such relationships require a more case-specific examination.
We also compare our method with STM. Below is the STM counterpart of Figure 2. We can observe that the topics and correlation scores from GCLDA seem to be more time-related and tend to be more informative of the labor market during the period.
BusinessInsider.com: Financial News Articles
--------------------------------------------
We consider all contributions in the “Finance” section of BusinessInsider.com on all trading days in 2014. There are $15{,\!}659$ articles in total, which are divided into a training data set containing $12{,\!}527$ articles and a testing data set containing $3{,\!}132$ articles ($20\%$ of all articles). We increase the proportion of testing documents and let $T=252$ (all trading days) in order to create a more challenging scenario for GCLDA. We apply the daily price of the Chicago Board Options Exchange Market Volatility Index (VIX) as the exogenous process, measuring the volatility of the U.S. financial market. The other settings are the same as those in Section 4.2. We provide an analysis of the perplexity of LDA, GCLDA, and ToT in Figure 4. The lines are smoothed by LOESS with a span of $0.2$, as there are large fluctuations in perplexity from day to day.
Again we observe that GCLDA generates a lower perplexity for the testing documents over time, therefore improving the fitting of the topic model. Also, the perplexities obtained from LDA and ToT are largely the same.
From Figure 5 and Table 2 below, the topics that are strongly positively correlated with the VIX are generally short-term news, such as stock market news and announcements from central banks, as in the topics “FED” (the Federal Reserve), “revenue,” and “stock market,” which are indeed closely related to changes in the stock market. From our analysis, the drop in oil price and the instability in Russia and Ukraine were also major causes of fluctuations in the stock market in 2014. On the other hand, we observe that news about longer-term economic trends is not positively correlated with the VIX, such as “companies” and “labor market.” These are generally consistent with our understanding of the stock market.
In this example, we demonstrate an application of GCLDA for finding documents that are major contributors to changes in an exogenous process during a period of time. We can easily estimate the topic distribution for each document, and therefore, we can select the news articles that are mostly related to the stock market. Such a direction could possibly evoke future research. We also note that in this example, the causal relationship between the topic distribution and the stock market is bidirectional; news can change the stock market, and [*vice versa*]{}.
[*FED*]{}: rate FED inflation policies market federal expected
--------------------------------------------------------------------
[*revenue*]{}: quarter billion year revenue million earnings share
[*Ukraine*]{}: Russia Ukraine Moscow gas country president
[*stock market*]{}: market trade stock week day close morning
[*energy*]{}: oil price energies gas production crude supplies
: The highest probability words inside the five topics with highest $\rho$ in Figure 5.
We also compare our method with STM, with Figure 6 being the STM counterpart of Figure 5. Again we observe that the topics and correlation scores from GCLDA seem to be more time-related and tend to be more informative of the stock market during the period. These findings assert our view that GCLDA improves the structure of the topic model and makes it more time-dependent.
Conclusion
==========
We have developed a temporal topic model which analyzes time-stamped text documents with known exogenous processes. Our new model, GCLDA, takes both endogenous and exogenous processes into account, and applies Markov chain Monte Carlo sampling for calibration. We have demonstrated that this model better fits temporal documents in terms of perplexity, and extracts well information from job advertisements and financial news articles. We suggest that a possible direction for the future could be analyzing the contents of temporal documents so that they could predict the trends of related exogenous processes.
Acknowledgments {#acknowledgments .unnumbered}
===============
This research was conducted in collaboration with the Workforce Science Project of the Searle Center for Law, Regulation and Economic Growth at Northwestern University. We are indebted to Deborah Weiss, Director, Workforce Science Project, for introducing us to the subject of workforce and providing guidance. We are also very grateful for the help and data from DirectEmployers Association.
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---
abstract: 'This paper contributes to modeling and supervision of multi-stage centrifugal compressors coping with real-gas processes and steady to highly transient operating conditions. A novel dynamic model is derived, and the incorporation of the generic <span style="font-variant:small-caps;">Lee-Kesler-Plöcker</span> real-gas equation of state and its derivatives is presented. The model allows for embedding arbitrarily shaped performance maps, based on state-of-the-art polytropic change-of-state compressor characteristics. As the validity of these maps is a key issue for simulation and model-based monitoring, performance maps are treated as time-variant, and their shape is to be identified and monitored during operation. The proposed real-time map estimation scheme comprises an Unscented Kalman Filter and a newly proposed algorithm, referred to as Recursive Map Estimation. The combination yields a novel parameter and state estimator, which is expected to be superior if some parameters are characterized by a distinct operating point dependency. Two additional time-variant parameters are provided for monitoring: The first indicates the level of confidence in the local estimate, and the second points to drastic performance map alterations, which may be further exploited in fault detection. A modified reference simulation of a two-stage supercritical carbon dioxide compressor with known state trajectories, performance maps, and alterations demonstrates the successful application of the entire monitoring scheme, and serves for a discussion of the results.'
address: 'Technische Universität Berlin, Department of Measurement and Control, Hardenbergstr. 36a, 10623 Berlin, Germany'
author:
- Maik Gentsch
- Rudibert King
bibliography:
- 'Gentsch\_King\_MapEstimation\_bib.bib'
title: 'DRAFT: Real-Time Estimation of a Multi-Stage Centrifugal Compressor Performance Map Considering Real-Gas Processes and Flexible Operation'
---
Compressor modeling , Model-based supervision , Flexible operation , Real-gas processes , Map identification , Unscented Kalman Filter
Introduction {#sec:intro}
============
Off-design, and in particular, flexible operation of industrial plants arises as a consequence of economic interests, and the integration of volatile, highly dynamic fossil-free resources in power generation that has to be complemented for by quickly responding conventional gas turbines. Likewise, industries start to dynamically adapt their production to the current prizes of the energy market, e.g., in air separation. This again results in the dynamic operation of the air compressors used. Consequently, the design philosophy, as well as the supervision and maintenance of these machines, has to be adapted properly. Concerning supervision for flexible operation, the common “steady-state” assumption is likely to provide erroneous results, e.g., frequent false alarms, if the plant’s transient behavior comprises any dynamic within a relevant time scale. The same issue applies if inappropriate model assumptions are used, e.g., the integration of the ideal-gas equation for real-gas processes in supervision algorithms.
In this context, this paper deals with modeling (see Section \[sec:model\]) and supervision (see Section \[sec:monitoring\]) of multi-stage (centrifugal) compressors coping with real-gas processes and flexible operation. For geared compressors, these machines comprise large pipes as the connection between (compressor stage)–(compressor stage), (compressor stage)–(intercooler), (compressor stage)–(valve), etc. The strategy pursued here is picking out (multi-stage) compression units that are not interrupted by other plant components (intercooler, valves, etc.). For such compression units (Fig. \[fig:unit\]), a model structure is introduced in Section \[sec:structure\]. The other plant components could be included readily in the concept proposed, but this is not done here. Likewise, to keep the presented equations compact, additional dependence on potentially installed inlet guide vanes is discarded.
The proposed model will be applied within the model-based monitoring scheme. But it might be used for dynamic process simulations as well, as it is designed to cope with highly flexible operating conditions. However, for the simulation task, a quasi-steady-state behavior of a compressor stage is assumed, i.e., the validity of a compressor stage specific performance map remains unaffected, even for transients. The investigation in [@blieske2011centrifugal] supports this general practice, with the exception for power calculated from such a static map. A multitude of publications deals with the concrete shape of such maps, i.e., the concrete correlations between head, compression work, volume flow, and speed, or respective representatives of this compressor characteristics. In [@casey2012method], a set of equations is presented that aims to describe the performance map far away from the design point. As in the present contribution, the methodology in [@casey2012method] is based on a static dependence of the same dimensionless compressor characteristics as they are used here. Similarly, a map prediction and modification scheme is proposed in [@bayomi2013centrifugal]. The prediction is based on several models that are originally introduced by <span style="font-variant:small-caps;">Moore</span> and <span style="font-variant:small-caps;">Greitzer</span> [@moore1986theory], <span style="font-variant:small-caps;">Dixon</span> [@dixon2013fluid], and <span style="font-variant:small-caps;">Gravdahl</span> [@gravdahl1998modeling]. An exhaustive description of traditional models for axial and centrifugal compressors can be found in [@gravdahl2012compressor]. However, starting point for a simulation with the dynamic model proposed herein, is a given set of discrete operating points, which may be derived from the mentioned approaches. The current operating point within the performance map is then calculated via an interpolation scheme. This enables the integration of arbitrarily shaped maps, facilitating a high degree of freedom for the task of learning a map in the framework of the model-based monitoring proposed.
As mentioned in [@ludtke2004process], almost all process fluids that are used in centrifugal compressors have distinct real-gas behavior. This applies in particular to carbon dioxide (), and to an even greater extent, to supercritical , which has to be handled in a *Carbon Capture and Storage* application, for example. With the purpose of providing a versatile compressor model, the generic and easy-to-parametrize <span style="font-variant:small-caps;">Lee-Kesler-Plöcker</span> (LKP) real-gas model (see [@lee1975LKP]) is embedded in this contribution. The LKP model is capable of describing the thermal relations for a multitude of relevant process fluids and conceivable mixtures properly. The specific integration of the respective equation of state and its derivatives is introduced in Section \[sec:realgas\].
Subject to the existence of the instrumentation, depicted in Fig. \[fig:unit\], the monitoring scheme developed in this contribution is capable of tracking common compressor characteristics (polytropic head, efficiency, etc.) separately for each compressor stage. Moreover, several fluid temperature estimates are provided, which is desired for considerably delayed temperature measurements, as is the case for many high-pressure applications.
Since the calculation of compressor characteristics is based on stage-specific performance maps (see Section \[sec:stagemap\]), which can change, e.g., due to fouling, or which are not known exactly initially, the validity of these maps is a key issue for simulation and model-based monitoring. Therefore, performance maps are treated as time-variant, and the proposed approach aims to identify and monitor their shapes during operation. Online adaption of performance maps for centrifugal compressors has been presented in [@cortinovis2014online]. There, an automated decision unit, based on the deviation between measurements and the current map, triggers a (sequential) quadratic program from time to time to calculate the new map. Therefore, a proper set of past measurements is stored in a buffer. In contrast to this batch approach, which is not stated to be designed for real-gas processes and transient operation, the proposed algorithm in this contribution is characterized by a constant computational load and storage requirement for every time interval between two measurement samples. Likewise, this applies to the work of <span style="font-variant:small-caps;">Höckerdal</span> et al. [@hoeckerdal2011ekf]. The core concept of their contribution, as is the case for the monitoring scheme here, is the preservation of the operating point dependence of parameters via estimated grid points of a parameter map. Their approach is a joint estimation of the model states and the grid points, which are treated as extended model states, within a proper observer or filter scheme, e.g., the *Extended Kalman Filter* scheme. Although this is a very elegant approach, it is rather inappropriate for the application considered here, due to the following reasons: i) As a consequence of the real-gas model integration, every execution of the overall multi-stage compressor model is relative costly compared to more simple models; and ii) much more grid points are necessary to shape the multidimensional performance maps properly compared to the application in [@hoeckerdal2011ekf]. Considering that every grid point, treated as extended model state within the joint estimation scheme, increases the number of model executions in every estimation step, an unacceptable increase of the computational load arises for the present application.
Therefore, we will present an alternative approach to map adaption that comprises a filter for state and parameter estimation and a coupled map estimation, which will not trigger additional model executions. For the first issue, a constrained *Unscented Kalman Filter* (UKF, e.g., see [@julier2000new; @kolas2009constrained]) is applied (see Section \[sec:CUKF\]). The UKF is a real-time state estimator for nonlinear systems, applicable even if no Jacobian matrix could be achieved. In this paper, the term “real-time” refers to a situation where the measurement sampling rate is assumed to be low enough to complete all calculation steps between two measurement samples in real time. For this, all presented algorithms are formulated in an effective, recursive manner. For real-time performance map estimation, an algorithm referred to as *Recursive Map Estimation* (RME) is presented in Section \[sec:map\_estimation\], and combined with the UKF in Section \[sec:CSME\]. The overall monitoring performance is assessed in Section \[sec:results\] for a simulative experiment with known state trajectories, performance maps, and alterations. A two-stage supercritical compressor acts as the reference process.
Model Building {#sec:model}
==============
Model Structure and Nomenclature {#sec:structure}
--------------------------------
The starting point for the model-based monitoring approach is a nonlinear, dynamic system description
= , & $\smallfcn{\vec{x}}{t_0}=\vec{x}_0$ , \[eq:ZDGL\]\
= , & \[eq:Ausgangsgleichung\]
where $\vec{y} \in \mathbb{R}^{n_y}$, $\vec{x} \in \mathbb{R}^{n_x}$, $\vec{u} \in \mathbb{R}^{n_u}$, and $\vec{\theta} \in \mathbb{R}^{n_\theta}$ are the measurable outputs, the states, the inputs, and the parameters of the model, respectively. In general, all of these values are time-variant, but the model parameters are assumed to vary much slower than the other variables. To increase readability of the equations, the time argument $t$ is suppressed in what follows.
Consider the $N$-stage compression unit depicted in Fig. \[fig:unit\], with a suction pipe P0 and a discharge pipe PN. The model states and outputs might be structured as follows: $$\begin{aligned}
\renewcommand{\arraystretch}{1.25}
\vec{x} = \begin{bmatrix} \Pa{\vec{x}} \\ \Sa{\vec{x}} \\ \Pb{\vec{x}} \\ \Sb{\vec{x}} \\ \Pc{\vec{x}} \\ \vdots \\ \SN{\vec{x}} \end{bmatrix} \, , \quad
% \vec{y} &= \begin{bmatrix} (\Pa{\vec{y}})^T & (\Sa{\vec{y}})^T & (\Pb{\vec{y}})^T & (\Sb{\vec{y}})^T & (\Pc{\vec{y}})^T & \dotsm & \SN{T_s} \end{bmatrix}^T \, ,
\vec{y} = \begin{bmatrix} \Pa{\dot{m}} \\ \Pa{\vec{y}} \\ \Sa{T_s} \\ \Pb{\vec{y}} \\ \Sb{T_s} \\ \Pc{\vec{y}} \\ \vdots \\ \SN{T_s} \end{bmatrix} \, , \label{eq:states}
\renewcommand{\arraystretch}{1}\end{aligned}$$ where the individual state vectors of a single component $$\begin{aligned}
\renewcommand{\arraystretch}{1.25}
\Pj{\vec{x}} = \begin{bmatrix} \Pj{\Tr_f} \\ \Pj{\Tr_s} \\ \Pj{\vr} \end{bmatrix} \, , \quad \Pj{\vec{y}} = \begin{bmatrix} \Pj{T_s} \\ \Pj{p} \end{bmatrix} \, , \quad \Si{\vec{x}} = \begin{bmatrix} \Si{\Tr_f} \\ \Si{\Tr_s} \\ \Si{\Dmu} \\ \Si{\Dphi} \end{bmatrix} \, \label{eq:xP0}
\renewcommand{\arraystretch}{1} \end{aligned}$$ are detailed below. The known or measured model inputs are the speeds of the compressor shafts and the discharge pressure: $$\begin{aligned}
\vec{u} = \begin{bmatrix} \Sa{n} & \Sb{n} & \dotsm & \SN{n} & \PN{p} \end{bmatrix}^T\, . \label{eq:u}\end{aligned}$$ Here, $\left[~\right]^T$ denotes the transpose of a vector. Model parameters $\vec{\theta}$ result from first-principle modeling, and are specified below when they appear. The superscripts Pj and Si denote whether the respective physical quantity belongs to the suction pipe (P0), the $i$th compressor stage, the $j$th intermediate pipe, or the discharge pipe (PN). The physical quantities are the temperatures $T$, pressures $p$, and specific volumes $v$, or their dimensionless counterparts: $$\begin{aligned}
\Tr &= \frac{T}{T_c}\, , \qquad \pr = \frac{p}{p_c}\, , \qquad \vr = v\,\frac{p_c}{R\, T_c}\, ,\end{aligned}$$ respectively. For scaling, $R$ is the specific gas constant, and $T_c$ and $p_c$ are the critical temperature and pressure of the process fluid, respectively. Thus, for $(\Tr>1,\ \pr>1)$, the process fluid is at a supercritical state. The following sections contain further dimensionless thermodynamic quantities:$$\begin{aligned}
\hr = \frac{h}{R\, T_c}\, , \qquad \ur = \frac{u}{R\, T_c}\, , \qquad \cvr = \frac{c_v}{R}\, .\end{aligned}$$ $h$ is the specific enthalpy, $u$ is the specific internal energy, and $c_v$ is the specific isochoric heat capacity.
Additional variables are mass flows $\md$ and deviations ($\Delta$) from nominal compressor specific quantities $\mu$ and $\varphi$, which are introduced in Section \[sec:stagemap\]. For the temperature values modeled, a distinction is drawn between fluid temperatures and temperatures at the sensor location (for the same pipe cross-section), denoted by subscripts $f$ and $s$, respectively. This distinction is necessary, because temperature sensors are quite often shielded by thick-walled casings, especially for high-pressure applications, leading to considerably delayed measurements. Temperature values assigned to a compressor stage are discharge temperatures always; for example, $\Sa{T_f}$ and $\Sa{T_s}$ denote the discharge temperatures (the fluid and sensor position) of the first stage. Elsewhere, if a local assignment in the context of a single plant component (compressor stage or pipe) is needed, entry values are indicated with a subscript $1$, and exit values with a subscript $2$.
Calculation of Real-Gas Values {#sec:realgas}
------------------------------
For the sake of adaptability of the approach to fluids other than that considered in the example below, a generic real-gas state equation with a low parametrization effort is chosen. For a multitude of relevant process fluids (air, hydrocarbons, carbon dioxid, hydrogen, and ammonia), the LKP state equation shows good agreement with published gas property tables [@ludtke2004process]. The equation is based on the *three-parameter corresponding states* principle [@lee1975LKP] featuring reduced temperature $\Tr$, reduced pressure $\pr$, and acentric factor $\omega$. For a given real gas (mixture), its (pseudo-)critical temperature $T_c$ and pressure $p_c$, as well as its acentric factor $\omega$ and its specific gas constant $R$, determine all thermal relations for the LKP model. The thermal relations are formulated as follows: $$\begin{aligned}
&\pr = \fcn{\pr_S}{\Tr,\vr_S} \, , \quad \pr = \fcn{\pr_R}{\Tr,\vr_R}\, , \label{eq:p-T-v} \\
&\text{and} \quad \vr = \fcn{\delta}{\vr_S, \vr_R, \omomR} \, , \label{eq:v_vS_vR} \end{aligned}$$ where $\pr_S$ and $\pr_R$ are separated <span style="font-variant:small-caps;">Benedict-Webb-Rubin-Starling</span> (BWRS)-type equations for a *simple* fluid and a *reference* fluid, respectively, $\omega_R$ is the acentric factor of the *reference* fluid, and $\vr_S$ and $\vr_R$ are the reduced specific volumes of these fluids, which are used to interpolate the reduced real gas (mixture) specific volume $\vr$ according to $$\begin{aligned}
\fcn{\delta}{\vr_S,\, \vr_R,\, \omomR} := \vr_S + \omomR \cdot \left[ \vr_R - \vr_S \right]\ . \label{eq:fi}\end{aligned}$$ The common, originally stated way to resolve the thermal relations (\[eq:p-T-v\])–(\[eq:v\_vS\_vR\]) is as follows (see [@lee1975LKP]): Given a thermodynamic state $(\Tr,\pr)$, a typically multiple-step numerical procedure is applied to determine the pair $(\vr_S,\vr_R)$ that fulfills the equality constraint (\[eq:p-T-v\]). We refer to this as the *pressure explicit approach*. At this point, we recommend the method described in [@mills1980BWRS], which is excellent in terms of numerical convergence and reliability in the whole range of valid thermodynamic states[^1]. Once $(\vr_S,\vr_R)$ is determined, all thermodynamic properties (e.g., reduced specific enthalpy $\hr$) and their derivatives (e.g., isochoric pressure variation $(\partial \pr / \partial \Tr)_\vr$) can be calculated directly. Calculations in this *pressure explicit approach* are abbreviated in a respective manner, e.g., $\hr(\Tr,\pr)$ or $(\partial \pr / \partial \Tr)_\vr(\Tr,\pr)$. For the sake of completeness, a thermal state equation, such as the LKP state equation, is able to determine the deviation of the ideal-gas caloric properties only. Thus, caloric ideal-gas data, e.g., the thermal dependence of the isochoric heat capacity $c_v^{id}=c_v^{id}(T)$, is necessary to calculate absolute caloric values.
If the range of possible thermodynamic states is restricted to the gaseous and supercritical regions, and if the thermodynamic state can be determined by $(\Tr,\vr)$, we found an alternative approach, which is less computationally intensive and sufficiently accurate. Within this approach, given $(\Tr,\vr)$, but without knowing $\pr$ *a priori*, it is clear from (\[eq:p-T-v\]) that , or in an alternative mathematical description, has to be fulfilled. Treating formula (\[eq:v\_vS\_vR\]) as equality constraint, the problem boils down to a scalar root determination of $\epsilon$ with merely one independent variable, e.g., $\vr_S$. Applying a second-order root determination scheme, we found that a single step (iteration) results in a sufficiently small $\left|\epsilon\right|$ if the fluid is in a gaseous or supercritical phase and if the starting point of the root determination algorithm is set to $\vr_{S,0}=\vr$. Finally, this approach yields the (one-step solvable) solution: $$\begin{aligned}
\left(\vr_S,\ \vr_R\right) = \left( \vr_{S,1},\ \delta\left(\vr_{S,1},\vr,\frac{\omega_R}{\omega}\right) \right)\, ,\end{aligned}$$ where $$\begin{aligned}
\vr_{S,1} &= \vr - \frac{ \epsilon' + \sqrt{ \fcn{}{\epsilon'}^2 - 2\, \epsilon\, \epsilon'' } }{ \epsilon'' }\, , \label{eq:vrS1}\\[1em]
\epsilon &= \fcn{\pr_S}{\Tr,\vr} - \fcn{\pr_R}{\Tr,\vr}\, , \\
\epsilon' &= \frac{\partial\, \pr_S (\Tr,\vr_S)}{\partial\, \vr_S}\Big|_{\vr_S=\vr} - \left[1-\frac{\omega_R}{\omega}\right]\frac{\partial\, \pr_R (\Tr,\vr_R)}{\partial\, \vr_R}\Big|_{\vr_R=\vr}\, , \\
\epsilon'' &= \frac{\partial^2\, \pr_S (\Tr,\vr_S)}{\partial\, \vr_S^2 }\Big|_{\vr_S=\vr} - \left[1-\frac{\omega_R}{\omega}\right]^2\frac{\partial^2\, \pr_R (\Tr,\vr_R)}{\partial\, \vr_R^2}\Big|_{\vr_R=\vr}\, .\end{aligned}$$ Calculations based on the latter approach are abbreviated as $\hr(\Tr,\vr)$ or $(\partial \pr / \partial \Tr)_\vr(\Tr,\vr)$, for example. We refer to this as the *volume explicit approach*.
Compressor Stage Model {#sec:stagemap}
----------------------
Following the basic design philosophy for (real-gas) centrifugal compressors in [@ludtke2004process], a static dependency between dimensionless characteristic compressor numbers of a specific compressor stage is postulated. The dimensionless numbers are
& &\_p &= 2, \[eq:Psi\]\
& & &= , \[eq:phi\]\
& & &= , &\[eq:mu\]\
& & &= . &\[eq:Mu2\]
The variables used are listed in Table \[tab:characteristics\].
------------------------------ ------------- ---
Polytropic work $y_p$
Actual compression work $\Delta h$
Impeller diameter $d_2$ m
Blade speed at impeller exit $\tipspeed$
Suction volume flow $\dot{V}_s$
Sonic inlet velocity $a_1$
------------------------------ ------------- ---
: Compressor characteristics and their physical units[]{data-label="tab:characteristics"}
With these terms, the polytropic efficiency can be introduced that serves as a meaningful assessment measure, concerning not only the efficiency of the current operating point but also the health status of the respective compressor stage: $$\begin{aligned}
\eta_p = \frac{y_p}{\Delta h} = \frac{\Psi_p}{2\, \mu}\, . \label{eq:etap}\end{aligned}$$
For real compressors, a generically structured mapping function, e.g., a fixed-order polynomial, is improper for describing the multitude of possible shapes of specific compressor performance maps accurately. That is why we present an interpolation-based technique in Section \[sec:map\_estimation\] to approximate arbitrary shapes. In this section, the actual shape is irrelevant, and we focus on the solution strategy to determine the current operating point of a compressor stage within its performance map. To this end, one has to ask, “Which of the properties of $\Psi_p$, $\varphi$, $\mu$, and $\Mu$ are *a priori* accessible and determine the remaining properties uniquely?” Given the predefined model structure (see Section \[sec:structure\]), the dynamic simulation will supply the suction and discharge states, as well as the rotational speeds for every time instant. Thus, $\Mu$ and $\Psi_p$ can be calculated readily; see below. As a result, we propose an interpolation-based mapping , to uniquely determine $\varphi$ and $\mu$. With this information, the compressor map can be displayed in the conventional way as $\Psi_{p}$ as a function of $\varphi$; see Section \[sec:mapMonitor\] as well. The shape of $\mathfrak{M}$ is addressed in Section \[sec:map\_estimation\].
To determine an operating point within such a *polytropic* performance map, an iterative multiple-step procedure is unavoidable, because it is infeasible to determine the discharge temperature without prior knowledge of the operating point, and vice versa. Within the framework of the entire monitoring scheme, it turned out to be advantageous to resolve this dependence with the formulation of a dynamic problem. To this end, an energy balance for an appropriately defined hypothetical volume element featuring the artificial dynamic state variables $\Sj{\Tr_f}$ is formulated as a physically motivated approach; see Eq. (\[eq:xP0\]) as well. The model should ensure that such temperatures tend rapidly toward values that are consistent with the converged operating point. As an example for the $j$th stage, the approach results in $$\begin{aligned}
\difffrac{ \Sj{\Tr_f} }{ t} = \frac{ \left|\Sj{\dot{m}}\right| \left[\Sj{\hr_1}+\Sj{\Delta\hr} - \Sj{\hr_2}\right] \Sj{\vr_2} }{ \Sj{V_2} \Sj{\cvr_2} }\ \cdot \frac{R\, T_c}{p_c}\ . \label{eq:dTfS_dt}\end{aligned}$$ A steady discharge temperature $\Sj{\Tr_f}$ is obtained if and only if the actual reduced compression work , computed from the performance map, equals the direct reduced enthalpy increase over the compressor stage , i.e., the calculation is converged. Within this concept, shown in Fig. \[fig:stage model\] for the first stage, $\Sj{\Tr_f}$ is the temperature, corresponding to the reduced enthalpy $\Sj{\hr_2}$, of the small, fixed-size fluid volume $\Sj{V_2}$, which can be interpreted as a short pipe section, connected to the discharge side of the compressor stage.
{width="\textwidth"}
For the sake of consistency, this artificial fluid volume is under the same pressure as the actual volume within the downstream pipe. $\Sj{V_2}/\Pj{V}$ must be small enough to guarantee the desired “rapid” solution, but large enough to avoid dynamic stiffness. This trade-off is addressed within Section \[sec:CSME\], where we present an approach, which is proven to handle $\Sj{V_2}/\Pj{V}\approx 0.01$.
Here, for the $j$th compressor stage, the concrete, proposed solution strategy to determine an operating point within the performance map and the right side of Eq. (\[eq:dTfS\_dt\]), given the thermodynamic states $(\Pjm{\Tr_f},\Pjm{\vr})$ at the suction side and $(\Pj{\Tr_f},\Pj{\vr})$ at the discharge side, which is an intermediate pipe in the global scheme for $j\ne N$ (cf. Fig. \[fig:unit\]), as well as the current discharge temperature $\Sj{\Tr_f}$, is as follows:
1. Determine the reduced specific enthalpy, pressure, and speed of sound at the inlet, utilizing the *volume explicit approach* (see Section \[sec:realgas\]) \[item:calc\_h1\]
& =, z={ , , a } ,&
which, inter alia, leads to the machine Mach number $\Sj{\Mu}$ if the given compressor shaft speed $\Sj{n}$ and impeller diameter $\Sj{d_2}$ are taken into account.
2. Determine the reduced discharge pressure, utilizing the *volume explicit approach* \[item:calc\_p2\]
& = .&
3. Determine the reduced specific enthalpy, volume, and isochoric heat capacity at the discharge pipe-section, utilizing the *pressure explicit approach* (see Section \[sec:realgas\]) \[item:calc\_h2\] as $\Sj{\pr_2}$ is given from the last step
& = , z={ , , } .&
4. Calculate the polytropic volume exponent \[item:calc\_nv\]
& = - .&
5. Calculate the polytropic work \[item:calc\_yp\]
& = R T\_c ,&
which leads to the polytropic head coefficient $\Sj{\Psi_p}$.
6. Apply the mapping $\Sj{\mathfrak{M}}: (\Sj{\Mu}, \Sj{\Psi_p}) \rightarrow (\Sj{\varphi}, \Sj{\mu})$ to determine the remaining properties $\Sj{\varphi}$ and $\Sj{\mu}$. \[item:calc\_phi\]
Note that steps \[item:calc\_nv\] and \[item:calc\_yp\] are consistent with the definition of a polytropic change $\{p v^{n_v}=\text{const.}\,,\ n_v=\text{const.}\}$ according to <span style="font-variant:small-caps;">Zeuner</span> (cf. [@zeuner1866grundzuge]), which is occasionally considered an approximation for real gases only, although an approximation of a polytropic change $\{\eta_p=v \, dp/dh\,,\ \eta_p=\text{const.}\}$ according to <span style="font-variant:small-caps;">Stodola</span> (cf. [@stodola2013dampf]) is actually meant. However, according to the authors, both definitions are approximations of the real change-of-state path, and choosing <span style="font-variant:small-caps;">Zeuner</span>’s approach is inevitable in the given context, to enable real-time capability. References [@baehr1996warme pp. 385–386] and [@wettstein2014polytropic] are given as indications for further discussion on this topic.
To determine the right side of Eq. (\[eq:dTfS\_dt\]), the remaining terms, namely, the mass flow and the reduced compression work, follow from: $$\begin{aligned}
\Sj{\md} &= \frac{\Sj{\dot{V}_s}}{\Pjm{v}} = \frac{\Sj{\varphi} + \Sj{\Delta \varphi}}{\Pjm{\vr}} \, \frac{\pi}{4}\, {\Sj{d_2}}^2 \left[\pi\, \Sj{n}\, \Sj{d_2} \right] \cdot \frac{p_c}{R\, T_c}\ , \label{eq:mS1}\\
\Sj{\Delta\hr} &= \left[\Sj{\mu} + \Sj{\Delta \mu}\right] \left[\pi \, \Sj{n}\, \Sj{d_2}\right]^2 \cdot \frac{1}{R\, T_c}\ .\end{aligned}$$ Note the incorporation of the model states $\Sj{\Delta \varphi}$ and $\Sj{\Delta \mu}$, which represent deviations from the (nominal) performance map, and which will be estimated below.
For the $N$th compressor stage, the procedure is analogous, with the exception of step \[item:calc\_p2\], which is obsolete due to the given model input $\PN{p}$ (cf. Eq. (\[eq:u\])). Note that the multi-stage model can be programmed efficiently, because many interim results, e.g., the pressure within the intermediate pipe $\Sjm{\pr_2}=\Sj{\pr_1}$, and terms from the thermal relations that are not shown here appear multiple times.
Dynamic State Equations {#sec:state_eq}
-----------------------
In addition to the artificial state equations $\diff{\!\Sj{\Tr_f}}/\diff{\!t}$ (see Eq. (\[eq:dTfS\_dt\])), dynamic equations for the remaining states have to be derived. For quantities, representing the thermodynamic state within intermediate pipes, $\Pj{\Tr}$ and $\Pj{\vr}$ for $j=\{1,\dots,N-1\}$, appropriate balance equations are utilized. To ensure real-time capability, each fluid volume within such pipes is modeled as an open, well-mixed reservoir. The change of state for such a reservoir, considering real-gas behavior, and expressed in a differential manner, results in $$\begin{aligned}
c_v\, \diff{T} &= \frac{v}{V}\left[ \diff{U} - u\cdot \diff{m} + \pd{u}{v}{T}\cdot v\cdot \diff{m} \right]\, , \label{eq:dT}\\
\diff{ v} &= -\frac{v^2}{V}\cdot \diff{m}\, . \label{eq:dv}\end{aligned}$$ $V$ is the fixed-size reservoir volume, $U$ is the extensive internal energy, $m$ is the volume’s total mass, and further, $$\pd{u}{v}{T} = T \pd{p}{T}{v} - p$$ is a universal caloric relation (cf. [@baehr1996warme p. 140]). The energy and mass balance, given a single upstream input cross-section (subscript $1$) and a single downstream outlet cross-section (subscript $2$), yield $$\begin{aligned}
\diff{ U} &= h_1\cdot \diff{m_1} - h\cdot \diff{m_2} + \diff{Q} \, , \label{eq:dU}\\
\diff{ m} &= \diff{m_1} - \diff{m_2}\, , \label{eq:dm}\end{aligned}$$ where $dQ$ might be used to model additional energy transfer, e.g., heat transfer in the case of a diabatic pipe casing. The combination of Eqs. (\[eq:dT\])–(\[eq:dm\]), expressed with dimensionless quantities, leads to the dynamic evolution equations of the thermodynamic state of an intermediate pipe volume: $$\begin{aligned}
\begin{split}
\difffrac{ \Pj{\Tr_f}}{ t} = &\frac{\Pj{\vr}}{\Pj{V}\, \cvr\PjTv{f}{} } \Bigg[ \difffrac{\Pj{Q}}{t}\cdot\frac{1}{R\, T_c} + \Sj{\md}\\
&\hspace*{1.0cm} \cdot \left[\Sj{\hr_2} - \Sjp{\hr_1} \right] + \Pj{\Tr_f} \pd{\pr}{\Tr}{\vr}\!\!\!\PjTv{f}{} \\
&\hspace*{1.0cm} \cdot \left[\Sj{\md}-\Sjp{\md}\right]\Pj{\vr} \Bigg]\cdot \frac{R\, T_c}{p_c}\, ,
\end{split} \label{eq:dTfP1_dt}\\[0.5em]
\difffrac{\Pj{\vr}}{ t} = & -\frac{(\Pj{\vr})^2}{\Pj{V}} \left[\Sj{\md} - \Sjp{\md}\right] \cdot \frac{R\, T_c}{p_c}\, ,\label{eq:dvP1_dt}\end{aligned}$$ In Eqs. (\[eq:dTfP1\_dt\]) and (\[eq:dvP1\_dt\]), leakage mass flows might be considered, which is not done here. Note that many terms, including hidden ones such as $\Pj{(\vr_S,\vr_R)}$ (cf. Section \[sec:realgas\]), have already been calculated within the compressor stage model (see Section \[sec:stagemap\]).
For considering delayed temperature measurements, arbitrary sensor models can be applied. However, as sensor modeling is not the focus of this article, we choose a simple linear, first-order approach: $$\begin{aligned}
\difffrac{\Pj{\Tr_s} }{t} = \frac{1}{\Pj{\tau} } \left[ \Pj{\Tr_f} - \Pj{\Tr_s} \right]\, , \qquad \difffrac{\Sj{\Tr_s} }{t} = \frac{1}{\Sj{\tau} } \left[ \Sj{\Tr_f} - \Sj{\Tr_s} \right]\, .\label{eq:dTs_dt} $$ $\tau$ denotes time constants, which are the parameters to be adjusted later.
To fully describe the system (\[eq:ZDGL\])–(\[eq:Ausgangsgleichung\]), more variables have to be known for which balance equations cannot be formulated. Most prominently, this applies to the deviation variables $\Delta \Sj{\varphi}$ and $\Delta \Sj{\mu}$ of the performance map. Within the modeling scheme, they are treated as constants. The monitoring scheme described in Section \[sec:CUKF\] will be able to estimate such quantities. For this purpose, and for applying an *Unscented Kalman Filter*, dynamic equations must be formulated for these “constants”, namely $$\begin{aligned}
\begin{split}
\difffrac{ \Pa{\Tr_f}}{t} = \difffrac{ \Pa{\vr}}{t} = \difffrac{ \Sj{\Dphi}}{t} = \difffrac{\Sj{\Dmu}}{t} =0\, .
\end{split} \label{eq:x_const}\end{aligned}$$ The entire dynamic equation set $\vec{f}$ of the multi-stage model (cf. Eq. (\[eq:ZDGL\])) consists of Eqs. (\[eq:dTfS\_dt\]) and (\[eq:dTfP1\_dt\])–(\[eq:x\_const\]).
Output Equations
----------------
Considering the terms already calculated within the compressor stage model (Section \[sec:stagemap\]), the entire set of output equations $\vec{g}$ of the multi-stage model (cf. Eq. (\[eq:Ausgangsgleichung\])) simply comprises: $$\begin{aligned}
\begin{split}
&\Pa{\md} = \Sa{\md}\, , \quad \Pj{T_s} = \Pj{\Tr_s}\, T_c\, , \quad \Pj{p} = \Sjp{\pr_1}\, p_c\, ,\\
&\Sj{T_s} = \Sj{\Tr_s}\, T_c\, .
\end{split}\end{aligned}$$
Monitoring {#sec:monitoring}
==========
Constrained Unscented Kalman Filter {#sec:CUKF}
-----------------------------------
Because the aim is to focus on the novelty of our contribution, i.e., the modeling approach and the performance map estimation below, the reader is referred to [@julier1997new; @julier2000new] for an introduction to the well-known UKF. However, as the UKF is a fundamental part of the overall monitoring scheme, some remarks are in order. The objective of the UKF approach is to provide an estimate $\vec{\hat{x}}$ of the true, partly unmeasurable states $\vec{x}$ of a nonlinear system that is given by Eqs. (\[eq:ZDGL\]) and (\[eq:Ausgangsgleichung\]). Readers who are more interested in the basics of implementing this method might prefer the brief presentation in [@merwe2001squareroot].
There are several concepts of the UKF that differ in detail; cf. [@wu2005unscented; @kolas2009constrained; @julier2002scaled; @sarkka2007unscented]. For the multi-stage monitoring scheme of a compressor considered here, the *Constrained Unscented Kalman Filter* (CUKF) with the reformulated correction step proposed in [@kolas2009constrained] is combined with an additive noise assumption. Therefore, the following stochastic, nonlinear, discrete-time system description is derived from Eqs. (\[eq:ZDGL\]) and (\[eq:Ausgangsgleichung\]):
= + \^x , & $\vec{x}_0$ – given,\
= + \^y . &
A variable with an index $k$ denotes a discrete-time quantity; e.g., $\zk{z}$ would be an abbreviation for a time-sampled value (usually, , ), $\vec{r}^x$ is an additive system noise, and $\vec{r}^y$ represents measurement noise. $\vec{r}_k^x$ and $\vec{r}_k^y$ are stochastic, zero-mean, uncorrelated, discrete signals with time-variant covariance matrices $\mat{R}_k^x$ and $\mat{R}_k^y$, respectively. Applying the expectation operator $E\{\}$, $$\begin{aligned}
\begin{split}
&\fcnII{E}{\vec{r}_i^x (\vec{r}_j^x)^T} = \mat{R}_k^x\, \delta_{ij} \, , \quad \fcnII{E}{\vec{r}_i^y (\vec{r}_j^y)^T} = \mat{R}_k^y\, \delta_{ij}\, , \\
&\fcnII{E}{\vec{r}_i^x (\vec{r}_j^y)^T} = \mat{0} \, , \quad \forall\ i,\,j\, ,
\end{split}\end{aligned}$$ follows, where $\delta_{ii}=1$ and $\delta_{ij}=0$ for $i\ne j$.
The core of the UKF is the *Unscented Transformation* (UT). The UT gives an estimate of statistical moments, inter alia, the mean and the covariance, of a density function that is the outcome of a nonlinear transformation (via $\vec{F}$ or $\vec{G}$) of a prior density function. The estimate is based on specific representatives of the density function, called sigma points. The sigma points of a distribution of $\vec{x}$ or $\vec{y}$ are typically denoted with $\chi$ or $\gamma$, respectively. For the multi-stage compressor introduced, physical constraints must be respected. The valid domains are given in Table \[tab:constraints\]. Different types of constraint handling within the *Kalman Filter* approach are discussed in [@simon2010constraints]. In this contribution, a simple approach is chosen: Check that every sigma point calculated within the UT and the reformulated correction step is in a valid physical domain, and if not, place the entries involved on the nearest element inside the valid domain. This ad-hoc approach, called *clipping*, is an essential element of the entire monitoring algorithm, because it has a superior stabilizing effect, compared to any internal constraint handling within the general model $\vec{F}$ and $\vec{G}$.
Entry in $\chi$ represents Valid domain, such that
----------------------------------- -----------------------------------------------------
temperature $\Tr$ $\max (0.3, \Tr_l) \le \Tr\le \min (4, \Tr_u)$
specific volume $\vr$ $\frac{1}{11.7} \le \vr$
work input deviation $\Delta \mu$ $0\le \mu$ and $\eta_p=\frac{\Psi_p}{2\, \mu}\le 1$
flow deviation $\Delta \varphi$ $0\le \varphi$
: Sigma point constraints; $\Tr_l$ and $\Tr_u$ represent any known lower and upper temperature bounds for the underlying process[]{data-label="tab:constraints"}
Recursive Map Estimation {#sec:map_estimation}
------------------------
In Section \[sec:stagemap\], a generic mapping function was introduced to determine an operating point within the dimensionless performance map of a single compressor stage. The aim of this work is an estimation of this performance map, even when it changes over time, e.g., due to fouling, or when it is completely unknown from the beginning. Before going into details, the general idea of the RME is sketched in Fig. \[fig:RMEsketch\]:
{width="\textwidth"}
Assume that at time $t$ an estimate of the map exists, as is displayed in Fig. \[fig:RMEsketch\]a, for an arbitrary map with two independent variables and one dependent variable. For $t=0$, the initial guess might be a nominal map or just a horizontal plane $z(x,y)=\text{constant}$. Data of the actual map is stored for individual pairs of the independent variables $x$ and $y$ on a rectangular grid, as shown as well. By interpolation, $z$ can be calculated for every pair $(x, y)$. Now, at time $t$, with the help of the CUKF, an estimate of the process state is obtained that can be exploited to calculate an estimate of the local dependent variable $z(t)$ marked by in Fig. \[fig:RMEsketch\]. This estimate will be used in the RME to adapt the dependent variable $z$ of the map in an optimal manner, in which neighboring $z$-grid values will be more affected than distant ones, and $x$-$y$-grid values will remain in their initial position, as depicted in Fig. \[fig:RMEsketch\]b. By this, the shape of a time-invariant map can be learned, or a time-variant map can be estimated.
In this sense, for allowing almost arbitrary shapes, we define a performance map via a set of $N_\mathfrak{M}$ discrete operating points that are initialized for $t=0$, and updated for all future time instants. The corresponding coordinates $\text{M}_{{d_2},i}^\mathfrak{M}$, $\Psi_{p,i}^\mathfrak{M}$, $\varphi_i^\mathfrak{M}$, and $\mu_i^\mathfrak{M}$ for these operating points $i$ are captured in respective column vectors $\vec{\text{M}}_{d_2}^{\mathfrak{M}}$, $\vec{\Psi}_{p}^{\mathfrak{M}}$, $\vec{\varphi}^{\mathfrak{M}}$, and . Using an interpolation scheme to merge the set of grid points into a coherent map, the mapping function boils down to:
= \^ , & \[eq:map\_phi\]\
= \^ , & \[eq:map\_mu\]
where $\vec{m}^T \in \mathbb{R}^{N_\mathfrak{M}}$ is a row vector containing interpolation coefficients that depend on the point to be interpolated $(\Mu,\Psi_p)$ and (usually a subset of) grid points $\vec{\text{M}}_{d_2}^{\mathfrak{M}}$ and $\vec{\Psi}_{p}^{\mathfrak{M}}$. Further dependencies, e.g., describing the effect of potential inlet guide vanes, might be included as well. Note that the structure of $\vec{m}^T$ depends on the selected interpolation method. The proposed RME is restricted to interpolation methods, where $\vec{m}_k^T$ is not a function of the dependent variables, $\vec{\varphi}^\mathfrak{M}$ or $\vec{\mu}^\mathfrak{M}$. We utilize a bilinear interpolation method based on a rectangular, normalized interpolation grid to enable efficient programming.
To exemplify the RME, we return to the $x$-$y$-$z$ notation from the beginning of this section. Consider a mapping function , which represents one of the expressions (\[eq:map\_phi\]) or (\[eq:map\_mu\]), with a rectangular grid, as shown in Fig. \[fig:grid\].
![Rectangular grid; the significance of the colors is given in the text[]{data-label="fig:grid"}](fig/grid4.pdf)
The objective is to estimate the performance map by an optimal adjustment of $\vec{z}^\mathfrak{M}$ considering any (new) information collected. The grid vectors are arranged as follows: $$\begin{aligned}
\begin{split}
&\kbordermatrix{
& y_1^\mathfrak{M} & y_2^\mathfrak{M} & & y_{N_y}^\mathfrak{M} \\
x_1^\mathfrak{M} & z_{11} & z_{12} & \dotsm& z_{1N_y} \\
x_2^\mathfrak{M} & z_{21} & z_{22} & \dotsm& z_{2N_y} \\
& \vdots & & \ddots& \vdots \\
x_{N_x}^\mathfrak{M} & z_{{N_x}1}& z_{{N_x}2}& \dotsm& z_{{N_x}N_y} \\
}
=
\begin{bmatrix}
{\vec{z}_1^y}^T \\ {\vec{z}_2^y}^T \\ \vdots \\ {\vec{z}_{N_x}^y}^T
\end{bmatrix} \\
&\hspace*{7mm}=
\begin{bmatrix}
~{\vec{z}_1^x}~~ & ~{\vec{z}_2^x}~ & \dotsm & ~~{\vec{z}_{N_y}^x}~
\end{bmatrix} \, .
\end{split}\end{aligned}$$ With the introduced notation, it is easy to see that $$\begin{aligned}
\underbrace{
\begin{bmatrix}
z_{1j} \\ z_{2j} \\ \vdots \\ z_{{N_x}j}
\end{bmatrix}
}_{\displaystyle \vec{z}_j^x} &=
\underbrace{
\begin{bmatrix}
\smallfcn{ \vec{m}^T }{ \vec{x}^\mathfrak{M}, \vec{y}^\mathfrak{M}, x_1^\mathfrak{M}, y_j^\mathfrak{M} } \\
\smallfcn{ \vec{m}^T }{ \vec{x}^\mathfrak{M}, \vec{y}^\mathfrak{M}, x_2^\mathfrak{M}, y_j^\mathfrak{M} } \\
\vdots \\
\smallfcn{ \vec{m}^T }{ \vec{x}^\mathfrak{M}, \vec{y}^\mathfrak{M}, x_{N_x}^\mathfrak{M}, y_j^\mathfrak{M} }
\end{bmatrix}
}_{\displaystyle \mat{M}_{y_j^\mathfrak{M}}}
\cdot
\underbrace{
\begin{bmatrix}
{\vec{z}_1^x} \\ {\vec{z}_2^x} \\ \vdots \\ {\vec{z}_{N_y}^x}
\end{bmatrix}
}_{\displaystyle \vec{z}^\mathfrak{M}} \, ,\\
\underbrace{
\begin{bmatrix}
z_{i1} \\ z_{i2} \\ \vdots \\ z_{i{N_y}}
\end{bmatrix}
}_{\displaystyle \vec{z}_i^y} &=
\underbrace{
\begin{bmatrix}
\smallfcn{ \vec{m}^T }{ \vec{x}^\mathfrak{M}, \vec{y}^\mathfrak{M}, x_i^\mathfrak{M}, y_1^\mathfrak{M} } \\
\smallfcn{ \vec{m}^T }{ \vec{x}^\mathfrak{M}, \vec{y}^\mathfrak{M}, x_i^\mathfrak{M}, y_2^\mathfrak{M} } \\
\vdots \\
\smallfcn{ \vec{m}^T }{ \vec{x}^\mathfrak{M}, \vec{y}^\mathfrak{M}, x_i^\mathfrak{M}, y_{N_y}^\mathfrak{M} }
\end{bmatrix}
}_{\displaystyle \mat{M}_{x_i^\mathfrak{M}}}
\cdot
\underbrace{
\begin{bmatrix}
{\vec{z}_1^x} \\ {\vec{z}_2^x} \\ \vdots \\ {\vec{z}_{N_y}^x}
\end{bmatrix}
}_{\displaystyle \vec{z}^\mathfrak{M}} \, \label{eq:My}\end{aligned}$$ is true. After $k$ time instants, a respective amount of information $\{z_1,z_2,\dots,z_k\}$ (usually, from measurements; below, from CUKF estimates) has been collected, corresponding to $k$ independent operating points $\{(x_1,y_1),(x_2,y_2), \dots,(x_k,y_k)\}$. The information collected, stored in vector $\vec{z}_k$, will now be used to adapt the $z$-grid values, which are stored in the time-variant vector $\vec{z}_k^\mathfrak{M}$. It follows: $$\begin{aligned}
\underbrace{
\begin{bmatrix}
z_{1} \\ z_{2} \\ \vdots \\ z_{k}
\end{bmatrix}
}_{\displaystyle \vec{z}_k} &=
\underbrace{
\begin{bmatrix}
\smallfcn{ \vec{m}^T }{ \vec{x}^\mathfrak{M}, \vec{y}^\mathfrak{M}, x_1, y_1 } \\
\smallfcn{ \vec{m}^T }{ \vec{x}^\mathfrak{M}, \vec{y}^\mathfrak{M}, x_2, y_2 } \\
\vdots \\
\smallfcn{ \vec{m}^T }{ \vec{x}^\mathfrak{M}, \vec{y}^\mathfrak{M}, x_k, y_k }
\end{bmatrix}
}_{\displaystyle \mat{M}_k}
\cdot
\underbrace{
\begin{bmatrix}
{\vec{z}_{1,k}^x} \\ {\vec{z}_{2,k}^x} \\ \vdots \\ {\vec{z}_{{N_y},k}^x}
\end{bmatrix}
}_{\displaystyle \vec{z}_k^\mathfrak{M}}
+
\underbrace{
\begin{bmatrix}
e_1^z \\ e_2^z \\ \vdots \\ e_k^z
\end{bmatrix}
}_{\displaystyle \vec{e}_k^z} \, . \label{eq:zk}\end{aligned}$$ The objective is to minimize the weighted sum of squared errors ${\vec{e}_k^z}^T \mat{W}_{1,k}^z \vec{e}_k^z$, where $$\begin{aligned}
\vec{e}^z_k = \vec{z}_k - \mat{M}_k \cdot \vec{z}^\mathfrak{M}_k = \vec{z}_k - \mat{M}_k \cdot \left[ \vec{z}_*^\mathfrak{M} + \Delta\vec{z}^\mathfrak{M}_k \right]\, . \label{eq:e_map}\end{aligned}$$ If prior knowledge concerning the map is available, this is stored in $\vec{z}_*^\mathfrak{M}$. Consequently, $\vec{z}_*^\mathfrak{M}$ is fixed, and $\Delta\vec{z}^\mathfrak{M}_k$ is the actual design variable of the optimization problem, the cost function of which is: $$\begin{aligned}
\begin{split}
\fcn{J_{1,k}^z}{\Delta\vec{z}_k^\mathfrak{M}} &= {\Delta\vec{z}_k^\mathfrak{M}}^T \mat{M}_k^T \mat{W}_{1,k}^z \mat{M}_k \Delta\vec{z}_k^\mathfrak{M} \\
&\hspace*{0.3cm}+ 2 \left[ {\vec{z}_*^\mathfrak{M}}^T \mat{M}_k^T - \vec{z}_k^T \right] \mat{W}_{1,k} \mat{M}_k \Delta\vec{z}_k^\mathfrak{M}\, .
\end{split} \end{aligned}$$ A unique minimum of $J_{1,k}^z$ exists under very strict conditions only. To stress this issue, consider a situation, where all information collected relates to the green area within the grid space, shown in Fig. \[fig:grid\]. A variation of $z$-values corresponding to grid points within the red area does not affect the interpolated $z$-surface within the green area, and, thus, it has no impact on $\vec{e}^z_k$ and $J_{1,k}^z$, consequently.[^2] Within the joint estimation scheme proposed by <span style="font-variant:small-caps;">Höckerdal</span> et al., see [@hoeckerdal2011ekf], the explained issue transforms into the loss of observability of the respective grid point “states”. To overcome this issue, they suggest a specific restriction to the estimated covariance matrix of their approach, with the intention to prevent the divergence of the filter. The countermeasure here is, to extend the cost function by several regularization terms that add
- the cost of deviations to the *a priori* map ($\vec{z}_*^\mathfrak{M}$)
& = [\_k\^]{}\^T \_[2]{}\^z \_k\^ , &
- the cost of a mean gradient of the entire performance map ($\vec{z}_*^\mathfrak{M} + \Delta\vec{z}_k^\mathfrak{M}$)
& = [\_k\^]{}\^T \_g\^z \_k\^ + 2 [\_\*\^]{}\^T \_g\^z \_k\^ ,&
- and the cost of a mean curvature of the entire performance map ($\vec{z}_*^\mathfrak{M} + \Delta\vec{z}_k^\mathfrak{M}$)
& = [\_k\^]{}\^T \_c\^z \_k\^ + 2 [\_\*\^]{}\^T \_c\^z \_k\^ . &
Costs (ii) and (iii) provide the smooth shape of a characteristic performance map. Matrices $\mat{L}_g^z$ and $\mat{L}_c^z$ are computed as follows: $$\begin{aligned}
\begin{split}
\mat{L}_g^z &= \frac{1}{N_y-1} \sum\limits_{j=1}^{N_y-1} \frac{\Delta \mat{M}_{y_{j}^\mathfrak{M}}^T}{\Delta y_{j}^\mathfrak{M}} \mat{W}_{g_y}^z \frac{\Delta \mat{M}_{y_{j}^\mathfrak{M}}}{\Delta y_{j}^\mathfrak{M}} \\
&\hspace*{0.3cm} + \frac{1}{N_x-1} \sum\limits_{i=1}^{N_x-1} \frac{\Delta \mat{M}_{x_{i}^\mathfrak{M}}^T}{\Delta x_{i}^\mathfrak{M}} \mat{W}_{g_x}^z \frac{\Delta \mat{M}_{x_{i}^\mathfrak{M}}}{\Delta x_{i}^\mathfrak{M}} \, ,
\end{split}
\\
\begin{split}
\mat{L}_c^z &= \frac{1}{N_y-2} \cdot \sum\limits_{j=1}^{N_y-2} \frac{\left[ \frac{\Delta \mat{M}_{y_{j+1}^\mathfrak{M}}}{\Delta y_{j+1}^\mathfrak{M}} - \frac{\Delta \mat{M}_{y_{j}^\mathfrak{M}}}{\Delta y_j^\mathfrak{M}} \right]^T}{\frac{y_{j+2}^\mathfrak{M} - y_{j}^\mathfrak{M}}{2}} \mat{W}_{c_y}^z \\
&\hspace*{0.3cm}\cdot \frac{ \frac{\Delta \mat{M}_{y_{j+1}^\mathfrak{M}}}{\Delta y_{j+1}^\mathfrak{M}} - \frac{\Delta \mat{M}_{y_{j}^\mathfrak{M}}}{\Delta y_j^\mathfrak{M}} }{\frac{y_{j+2}^\mathfrak{M} - y_{j}^\mathfrak{M}}{2}} + \frac{1}{N_x-2} \\
&\hspace*{0.3cm} \cdot \sum\limits_{i=1}^{N_x-2} \frac{\left[ \frac{\Delta \mat{M}_{x_{i+1}^\mathfrak{M}}}{\Delta x_{i+1}^\mathfrak{M}} - \frac{\Delta \mat{M}_{x_{i}^\mathfrak{M}}}{\Delta x_i^\mathfrak{M}} \right]^T}{\frac{x_{i+2}^\mathfrak{M} - x_{i}^\mathfrak{M}}{2}} \mat{W}_{c_x}^z \frac{ \frac{\Delta \mat{M}_{x_{i+1}^\mathfrak{M}}}{\Delta x_{i+1}^\mathfrak{M}} - \frac{\Delta \mat{M}_{x_{i}^\mathfrak{M}}}{\Delta x_i^\mathfrak{M}} }{\frac{x_{i+2}^\mathfrak{M} - x_{i}^\mathfrak{M}}{2}} \, ,
\end{split}\end{aligned}$$ where (cf. Eq. \[eq:My\]) and , and $\Delta \mat{M}_{x_i^\mathfrak{M}}$ and $\Delta x_i^\mathfrak{M}$ are defined analogously. All introduced weighting matrices $\mat{W}$ are symmetric and positive definite. Note that $\mat{L}_g$ and $\mat{L}_c$ are time-invariant, and thus, they can be computed offline.
For the purpose of real-time estimation, a recursive algorithm can be derived to solve the final optimization problem $$\begin{aligned}
\begin{split}
\min\limits_{ \vec{z}^\mathfrak{M}_k } \ \Big[ &\smallfcn{J_{1,k}^z}{\vec{z}_k^\mathfrak{M}-\vec{z}_*^\mathfrak{M}} + \smallfcn{J_{2}^z}{\vec{z}_k^\mathfrak{M}-\vec{z}_*^\mathfrak{M}} \\
&+ \smallfcn{J_{3}^z}{\vec{z}_k^\mathfrak{M}-\vec{z}_*^\mathfrak{M}} + \smallfcn{J_{4}^z}{\vec{z}_k^\mathfrak{M}-\vec{z}_*^\mathfrak{M}} \Big]
\end{split}\end{aligned}$$ based on the recent optimal solution $\vec{z}^\mathfrak{M}_{k-1}$:
1. Initialize with:
& \^\_[0]{} = \_0\^z \_2\^z \_\*\^ , \_0\^z = \^[-1]{} . &
2. For $k \in \mathbb{N} \setminus \{ 0 \} $:
&\^\_[k]{} = \^\_[k-1]{} + \_k\^z \_k w\_[1,k]{}\^z ,\
&\_[k]{}\^z = \_[k-1]{}\^z - , &
where $z_k$ is the new information (the last element of $\vec{z}_k$, cf. Eq. (\[eq:zk\])), $\vec{m}_k$ is an abbreviation for , and $w_{1,k}$ is the last element of $\mat{W}_{1,k}^z$:
& \_[1,k]{}\^z = . &
Clearly, step 2 does not differ from the well-known *Recursive Least Squares* (RLS) algorithm; cf. [@ljung1999system pp. 363 ff.]. Consequently, any known issue and modification of the RLS algorithm that can be found in the literature may apply. The distinguishing feature is the initialization, step 1, where the time-invariant regularization terms are incorporated. Within the RLS approach, $\mat{P}_k^z$ is the covariance matrix of the estimation error if and the information $z_k$ is a normally distributed, uncorrelated signal. Although this does not apply here, $\mat{P}_k^z$ provides information about the uncertainty of the current estimate $\vec{z}_k^\mathfrak{M}$. A low diagonal element indicates a reliable estimate of the corresponding element in $\vec{z}_k^\mathfrak{M}$; i.e., considerable information has already been collected within the vicinity of the corresponding grid point. We refer to these diagonal elements as *uncertainty levels*.
Coupled State and Map Estimation {#sec:CSME}
--------------------------------
Thus far, the model $(\vec{f}, \vec{g})$, a state estimator (CUKF), and the RME have been presented. In this section, a combination is presented leading to a novel real-time parameter and state estimation scheme, referred to as *Coupled State and Map Estimator* (CSME), which is expected to be superior if the map parameters have a distinct operating point dependency. As the overall performance will be sensitive to some implementation details, we propose a specific algorithm, and provide design suggestions.
To avoid an extensive use of indexes, we denote the entry of vector $\vec{x}$ that corresponds to the physical quantity $z$ with $\vec{x}\{z\}$. Analogously, $\mat{P}\{z\}$ denotes the diagonal element of matrix $\mat{P}$ that corresponds to $z$. Furthermore, $\vec{D}\{\mat{P}\}$ represents a column vector containing all diagonal elements—in corresponding order—of matrix $\mat{P}$. The proposed scheme of the CSME is as follows:
1. Declare the required variables, e.g.,
& && \_0 , \_0\^x ,\_0\^y , &\
& ( & ): &&\_[d\_2]{}\^ , \_[p]{}\^ , &\
& (): &&\_\*\^ , \_[2]{}\^= w\_2\^\_[N\_]{} , &\
& && \_[g\_x]{}\^= w\_g\^\_[N\_y]{} , \_[g\_y]{}\^= w\_g\^\_[N\_x]{} ,\
& && \_[c\_x]{}\^= w\_[c]{}\^\_[N\_y]{} , \_[c\_y]{}\^= w\_[c]{}\^\_[N\_x]{} ,&\
& (): &&\_\*\^ , \_[2]{}\^= w\_2\^\_[N\_]{} , &\
& && \_[g\_x]{}\^= w\_g\^\_[N\_y]{} , \_[g\_y]{}\^= w\_g\^\_[N\_x]{} ,\
& && \_[c\_x]{}\^= w\_[c]{}\^\_[N\_y]{} , \_[c\_y]{}\^= w\_[c]{}\^\_[N\_x]{} .
The initial state $\vec{\hat{x}}_0$ of the dynamic system refers to all state variables, i.e., normalized temperatures and specific volumes, and variables describing deviations in the work input factor and flow coefficient. $\mat{I}_N$ denotes an identity matrix. Remember that $\vec{\text{M}}_{d_2}^{\mathfrak{M}}$ and $\vec{\Psi}_{p}^{\mathfrak{M}}$ are fixed time-invariant grid vectors. They have to be preset properly; i.e., they should span the entire range of possible operating points. Naturally, the number and distribution of the declared grid points determine the flexibility of the performance map and the storage requirement of the routine. Concerning this trade-off, the aim is to put the maximum compatible number of grid points within the actual domain of possible operating points. Therefore, we use a rectangular grid with normalized grid points between $\Psi_p=0$ and the expected surge line $\Psi_p=\overline{\Psi}_p(\Mu)$, near which the density of the points increases. However, if an actual operating point is found to lie outside the preset domain, one could think of applying an extrapolation scheme (the same structure as Eqs (\[eq:map\_phi\]) or Eq. (\[eq:map\_mu\])) instead of redesigning the interpolation grid.
Note that every individual compressor stage has its own performance map, and if map variations should be monitored, its own RME calculation steps. To avoid repetitions, the stage-number superscript Sj is suppressed in this section. With the scalar weights $w_2^\varphi$ and $w_2^\mu$, the user declares whether to trust the *a priori* performance map (high weights) or not (low weights). Even in cases where there is no *a priori* knowledge, they have to be declared positive. For these cases, we set $w_2^\varphi=w_2^\mu=10^{-4}$ and $\vec{\varphi}_*^\mathfrak{M}=\vec{\mu}_*^\mathfrak{M} = \mat{0}_{N_\mathfrak{M} \times 1} $. Otherwise, with $w_2=0$, the matrix inverse within the initialization step of the RME may not exist. Concerning the adjustment of the remaining weights for the presented test case below (see Section \[sec:results\]), we found a proper balance for and .
1. Initialize the CUKF and the RME:
& &&\_0{}=0 , \_0{}=0 ,\
& && \_[x\_0]{}=w\_P \_0\^x , &\
& (): && \_0\^ = \_0\^\_2\^ \_\*\^ ,\
& && \^\_0 = \^[-1]{} ,\
& (): && \_0\^ = \_0\^\_2\^ \_\*\^ ,\
& && \^\_0 = \^[-1]{} .
Most UT algorithms (cf. Section \[sec:CUKF\]) require $w_P>0$. Without prior knowledge of $\mat{P}_{x_0}$, $w_P\gg 1$ is a common choice that allows the state estimator to apply large adjustment steps during the initial phase. Since large adjustments may have a destabilizing effect on the CSME, especially if the prior performance map knowledge is very uncertain ($w_2$ small), we recommend waiting for an initial period before enabling map estimation. Following this advice, the initial transient behavior of the CUKF, configured via $w_P$, is rather irrelevant in the CSME scheme.
1. Initialize the *revised map vectors* (the explanation follows, see step (\[item:last\])):
& \_0\^ = \_0\^ , \_0\^ = \_0\^ .&
2. \[item:CSME\_k\] For $k \in \mathbb{N} \setminus \{ 0 \} $:
1. \[item:Rxk\_b\] Adjust time-variant system noise for the CUKF:
&\_[k]{}\^x{ } = w\_[R,[k]{}]{} \_0\^x{ } , &\
&\_[k]{}\^x{ } = w\_[R,[k]{}]{} \_0\^x{ } , &\
&\_[k]{}\^x{ \_f\^ } = w\_(\_[k-1]{}{ \_f\^ })\^2 , &\
& \_[k-1]{} = ,&\
& { \_i\^ } = \_[i,k-1]{}\^ , { \_i\^ } = \_[i,k-1]{}\^ .&
- The latter line should clarify that the *revised map vectors* are applied within the compressor stage model. As a reminder, $\vec{f}_{k-1}\{ \Tr_f^\text{S} \}$ was introduced in Section \[sec:stagemap\], Eq. (\[eq:dTfS\_dt\]), as an artificial model equation with the purpose of determining the converged compressor stage’s discharge temperature $\Tr_f^\text{S}$ within a small time interval. If $\vec{f}_{k-1}\{ \Tr_f^\text{S} \}=0$, then $\Tr_f^\text{S}$ is in a converged state. In contrast, if $(\vec{f}_{k-1}\{ \Tr_f^\text{S} \})^2$ is large, then $\Tr_f^\text{S}$ is far from the converged state. Enlarging the corresponding model equation uncertainty $\mat{R}_{k}^x\{ \Tr_f^\text{S} \}$ in the latter situation, enables the state estimator to apply large adjustment steps of $\vec{\hat{x}}_k\{ \Tr_f^\text{S} \}$, thus, increasing the speed of convergence. We apply $w_\Tr=10^{-2}$.
Further, it is advisable to inform the state estimator whether the current operating point lies within a certain known region (low *uncertainty level*) of the performance map. If not, the estimator should be allowed to apply larger deviations from the nominal performance map ($\vec{\hat{x}}_k\{ \Delta \varphi \}$, $\vec{\hat{x}}_k\{ \Delta \mu \}$). For instance,
& w\_[R,k]{} = \_[k-1]{}\^T &
serves this purpose, where $\vec{m}_{k-1}$ is an abbreviation for , which contains the interpolation coefficients depending on the location of the recent operating point within the performance map, which is assumed to lie within the vicinity of the current operating point. The hat symbol $\ \widehat{~}\ $ denotes the consistent calculation according to the state estimate $\vec{\hat{x}}$; i.e., respective entries from $\vec{\hat{x}}$ are used to calculate the hat marked values according to the presented formulae. Considering that $\vec{D}\{ \mat{P}_{k-1}^\mu \}$ is consistently ordered to the interpolation grid, gives the interpolated *uncertainty level* of the recent operating point. If the applied interpolation scheme for calculating the interpolation coefficients in $\vec{m}_{k-1}^T$ is comonotone (monotone between neighbored grid points), is fulfilled within the entire grid domain (no extrapolation). In this case, one could define a *Local Information Level* $$\begin{aligned}
\text{LIL}_{k} := \frac{1}{\sqrt{w_{R,k}}}\, , \quad \SI{0}{\percent} < \text{LIL}_k \le \SI{100}{\percent}
\end{aligned}$$ serving as a meaningful monitoring indicator that correlates with the amount of information collected within the vicinity of the current operating point (“amount of confidence” in the local estimation).
2. Update the state estimate and the covariance matrix considering the current measurements $\vec{y}_k$ by applying the proposed CUKF scheme:
& ( \_k, \_[x\_k]{} ) = .&
- The incorporation of the model $(\vec{f}, \vec{g})$ into the CUKF scheme is quite clear (cf. Section \[sec:CUKF\]):
& =\
& + \_[t\_[k-1]{}]{}\^[t\_k]{} , &\
& = .&
From the user’s point of view, note that we end up using a simple forward Euler method for numerical integration that provides acceptable performance in terms of stability, accuracy, and computational speed, at least with step sizes $\approx \SI{5e-2}{\second}$ for volumes $\ge \SI{e-2}{\cubic\meter}$, as chosen here. Further, since entries in the input vector $\vec{u}$ that comprises the individual shaft speeds and the last stage’s discharge pressure may arise from time-sampled measurements (cf. Eq. (\[eq:u\])), it is not advisable to ignore their changes throughout the prediction horizon, especially in cases of highly transient operation or large $t_{k}-t_{k-1}$. Therefore,
& = \_[k-1]{} + \_k&
is embedded in the numerical integration scheme.
3. If the initial transient phase of the state estimator is concluded, continue with step (\[item:next\]); otherwise, skip steps (\[item:next\])–(\[item:last\]).
- \[item:startRLS\] For a proper indication, condition
& {\_[x\_[k-1]{}]{}} \_\
& < &
may be checked, where the operator $\oslash$ denotes the Hadamard division (element-wise division).
4. \[item:next\] Concerning the weighted map estimation error (cf. Eq. (\[eq:e\_map\])), adjust the time-variant weights for the RME:
& w\_[1,k]{}\^= , w\_[1,k]{}\^= . &
- In a standard *Least Squares* approach without regularization terms, $\mat{W}_{1,k}^z = ({\mat{C}^z_k})^{-1} $ gives the optimal (minimum covariance) estimate of $\vec{z}^\mathfrak{M}_k$ if $\mat{C}^z_k$ is the true covariance matrix of the collected information $\vec{z}_k$ that arises from a stochastic, uncorrelated process. We already stated that such a premise does not apply here. The current information to be considered will arise from the CUKF estimate $\vec{\hat{x}}_k\{z\}$, which is treated as the expected mean of an unspecified distribution with an expected variance $\mat{P}_{x_k}\{ z \} $. However, the proposed weighting $w_{1,k}^\varphi$ and $w_{1,k}^\mu$ clearly indicates the underlying intention.
5. \[item:RMEupdate\] Incorporate the updated state estimates $\vec{\hat{x}}_k \{ \Delta \varphi \}$ and $\vec{\hat{x}}_k \{ \Delta \mu \}$ into the performance map applying the recursive step of the RME scheme:
& (): & \^\_[k]{} &= \^\_[k-1]{} + \_k\^\_k w\_[1,k]{}\^, &\
& & \_[k]{}\^&= \_[k-1]{}\^- , &\
& (): & \^\_[k]{} &= \^\_[k-1]{} + \_k\^\_k w\_[1,k]{}\^,\
& & \_[k]{}\^&= \_[k-1]{}\^- , &
- where the “new information” to be considered arises from
& (): & \_k = \_k\^T \_[k-1]{}\^ + \_k { } , &\
& (): & \_k = \_k\^T \_[k-1]{}\^ + \_k { } , &
and $\vec{m}_{k}$ is an abbreviation for . Note that, for instance, is embedded in the model in place of $\Sj{\varphi}$ from Eq. (\[eq:mS1\]), and $\vec{\hat{x}}_k \{ \Delta \varphi \}$ represents $\Sj{\Delta \varphi}$ in this context; i.e., the CUKF calculates the displacement in relation to the former *revised map*.
6. \[item:last\] Update the *revised map vectors* with the approach described below:
& \_k\^ = \^[-1]{} \^T \_k\^ + ,&\
& \_k\^ = \^[-1]{} \^T \_k\^ + ;&
and reset the *a priori* performance map deviations of the next iteration afterward:
& \_[k]{}{}=0 , \_[k]{}{}=0 . &
- Before the newly introduced variables are declared, the conceptual idea behind the *revised map vectors* and step (\[item:last\]) needs clarification. A sketch of this concept is shown in Fig. \[fig:updateMap\] for the simplified situation $ \widehat{\text{M}}_{d_2,{k-1}} = \widehat{\text{M}}_{d_2,{k}} = \text{M}_{d_2,i}^\mathfrak{M} $; i.e., the dimension along machine Mach number variation becomes neglectable, yielding a scalar interpolation approach along $\Psi_{p}$ only.
![Update of revised map vector $\widetilde{\vec{\varphi}}^\mathfrak{M}$ applying a scalar C^0^ continuous linear interpolation approach (dependence on $\Mu$ neglected); symbols denote *uncertainty levels* at grid points []{data-label="fig:updateMap"}](fig/revised_map2.pdf){width="1\columnwidth"}
For the current time stamp $k$, the CUKF estimates a deviation $\vec{\hat{x}}_k\{\Delta \varphi\}$ to the former *revised map*, the $\varphi$-coordinates of which are stored in $\vec{\widetilde{\varphi}}_{k-1}^\mathfrak{M}$. In step (\[item:RMEupdate\]) this deviation is incorporated into the *actual map*, the $\varphi$-coordinates of which are stored in $\vec{\varphi}_k^\mathfrak{M}$, by recursively solving the optimization problem, as described in Section \[sec:map\_estimation\]. As is the situation in Fig. \[fig:updateMap\], the estimated flow coefficient $\widehat{\varphi}_k$ is unlikely to have no bias to the actual map (normally, an interpolated surface; here, an interpolated line); i.e., . In other words, the actual map is not consistent with the estimated state. This is expected, since the CUKF presumes the map to be time-invariant. Instead of advancing the model states by $\vec{\varphi}_k$, etc., and applying a joint estimation within the CUKF scheme, which would be in accordance with the approach in [@hoeckerdal2011ekf] and raise a massive increase in the computational burden in the present context, we derived the *Coupled State and Map Estimator* and provide the revised map vectors for this purpose. These vectors, $\vec{\widetilde{\varphi}}_{k}^\mathfrak{M}$ and $\vec{\widetilde{\mu}}_{k}^\mathfrak{M}$, are consistent to the CUKF estimate, i.e., $$\begin{aligned}
\vec{m}_k^T \cdot \vec{\widetilde{\varphi}}_{k}^\mathfrak{M} = \widehat{\varphi}_k\, , \quad \vec{m}_k^T \cdot \vec{\widetilde{\mu}}_{k}^\mathfrak{M} = \widehat{\mu}_k \, ,\label{eq:NB}
\end{aligned}$$ which is assured via step (\[item:last\]). Metaphorically speaking, the estimated deviation is preserved within the revised map. Therefore, the respective CUKF states must be reinitialized for the next iteration, to be consistent with the revised situation itself. Consequently, the revised map and the CUKF estimates are capable of tracking spontaneous and wide map deviations quickly, even if the actual map, calculated within the RME scheme, is in a nearly converged state, where adjustments to changed circumstances are typically sluggish. The diverging response times of actual and revised map vectors may be exploited for fault detection. As an example for $\mu$, if condition $$\begin{aligned}
\left\lVert \vec{\mu}_j^\mathfrak{M} - \vec{\widetilde{\mu}}_j^\mathfrak{M} \right\rVert_\infty > w_{f}\, \sqrt{ \mat{P}_{x_j}\{ \Delta \mu \} } \label{eq:FD}
\end{aligned}$$ is fulfilled for $(N_f+1)$ consecutive time stamps , a drastic change in behavior, i.e., a fault at time stamp $k-N_f$, of the corresponding compressor stage is plausible.
The proposed calculation of $\vec{\widetilde{\varphi}}_{k}^\mathfrak{M}$ or $\vec{\widetilde{\mu}}_{k}^\mathfrak{M}$ minimizes the weighted sum of squared errors between $\vec{\varphi}_{k}^\mathfrak{M}$ and $\vec{\widetilde{\varphi}}_{k}^\mathfrak{M}$ or between $\vec{\mu}_{k}^\mathfrak{M}$ and $\vec{\widetilde{\mu}}_{k}^\mathfrak{M}$, respectively, subjected to (\[eq:NB\]). We found it reasonable to keep revised points with a low *uncertainty level* close to the actual map (cf. Fig. \[fig:updateMap\]). Therefore, weighting matrices $\mat{W}_k^{\widetilde{z}}$ are constructed as diagonal matrices that fulfill $\vec{D}\{ {\mat{W}_k^{\widetilde{z}}}^{-1} \} = \vec{D}\{ {\mat{P}_k^{z}} \}$. $\Az$, $\bv$, and $\bm$ are as follows:
&=
, &\
&=
, =
.&
$m_{k,{j_k}}$ is the $j_k$th element of $\vec{m}_k$, and $\vec{m}_{{j_k},k} \in \mathbb{R}^{N_\mathfrak{M}-1}$ is a subvector of $\vec{m}_{k}$, constructed by removing $m_{k,{j_k}}$, and $j_k$ is an arbitrary index that fulfills $m_{k,{j_k}}\ne0$. Be aware that—applying a C^0^ continuous interpolation method—the matrix inversion in step (\[item:last\]) can be reformulated; thus, the actual matrix to be inverted is of dimension $3\times 3$.
Results {#sec:results}
=======
Test Case {#sec:setup}
---------
For validation purposes, a reference process with known performance maps and state trajectories is mandatory, and distinct real-gas behavior is desired to emphasize the scope of this research. Therefore, a numerical, i.e., simulative experiment (SE) of a two-stage supercritical (, ) compressor acts as reference and “measurement” generator. The “measurements” are sampled at a rate of . At a reasonable effort, several modifications have been implemented compared to the model that is applied within the monitoring scheme (cf. Section \[sec:model\]), to draw a somewhat more realistic situation for the CSME:
- In contrast to the approximative but versatile real-gas model that is applied within the CSME (cf. Section \[sec:realgas\]), the -specific model presented in [@span1996new], which matches the real-gas behavior of closely, is embedded within the SE.
- The assumption of well-mixed volumes within the connecting pipes is discarded, and the delay of the temperature information due to (1D) transportation of the mass inside the pipes is considered in the SE.
- For the mapping function $\mathfrak{M}$, a C^1^ continuous (piecewise cubic) interpolation method is applied based on 350 grid points vs. C^0^ continuous (bilinear) interpolation with grid points in the CSME.
- For every time stamp, the converged discharge temperature is calculated vs. the artificial state approach according to Eq. (\[eq:dTfS\_dt\]) of Section \[sec:stagemap\], in the CSME.
The preset simulation inputs, the compressor-shaft speed and the discharge pressure, can be found in Fig. \[fig:SE\_Input\]. We do not claim to have designed a realistic operating scenario. The intention was to run the machine across varying operating points, as in a highly flexible operation, connected via transients with a supposedly realistic, non-stepwise shape. The suction conditions of the SE are fixed at , . All calculated values that are treated as “measurements” for the CSME are affected by artificial, normally distributed noise with a standard deviation of $\SI{0.1}{\bar}$, $\SI{0.1}{\kelvin}$, or $\SI{0.1}{\kg\per\second}$, respectively. The time stamp index $k$ is suppressed in this section, since the physical time, e.g., measured in *minutes*, seems more natural. Such a dependence is obvious from the following figures anyway.
{width="\textwidth"}
Process Tracking {#sec:tracking}
----------------
Concerning the tracking performance, i.e., the capability of the CSME to track reference process states, we focus on the unmeasured, in reality unknown states. The CSME is able to track the measurement values as well, while significantly reducing the (artificial) measurement noise. Some exemplary results are shown in Fig. \[fig:SE\_Track\].
{width="\textwidth"}
The delay between the fluid’s temperature in the intermediate pipe $\Pb{T_f}$ and the corresponding “measured” value $\Pb{T_s}$, both calculated in the SE, can be seen in Fig. \[fig:SE\_Track\]a for the intermediate temperature. In the SE and the CSME, a time constant of $\Pb{\tau}=\SI{10}{\second}$ is assumed for a first-order system; see Section \[sec:state\_eq\], Eq. (\[eq:dTs\_dt\]). The CSME is capable of tracking the actual temperature of interest $\Pb{T_f}$ (slightly noisy), revealing any temperature peak, which is hidden from the measurements. For the remaining temperature positions, the tracking performance is very similar; thus, they are omitted here.
Fig. \[fig:SE\_Track\]b–d shows the estimation of several compressor characteristics. Obviously, the generic real-gas model, presented in Section \[sec:realgas\], is sufficiently accurate; otherwise, the estimates would have to be biased from the reference. None of the values shown in Fig. \[fig:SE\_Track\]b–d are declared model states for the CUKF scheme; thus, the values arise from the presented formulae (see Section \[sec:stagemap\]), embedding the “direct estimates” preserved in $\vec{\hat{x}}$. Since $\vec{\hat{x}}\{ \Delta \mu \}$ enters the denominator of $$\hat{\eta}_p = \frac{\widehat{\Psi}_{p}}{2\, \widehat{\mu}} = \frac{\widehat{\Psi}_{p,k}}{2\, \left( \vec{m}_k^T \cdot\vec{\widetilde{\mu}}_{k-1}^\mathfrak{M} + \vec{\hat{x}}_k \{ \Delta \mu \} \right)}$$ (cf. Eq. (\[eq:etap\]) and step (\[item:RMEupdate\]) of Section \[sec:CSME\]), this value is particularly prone to noise transmission, as can be seen in Fig. \[fig:SE\_Track\]d. Note that the tuning parameters, e.g., $\mat{R}_0^x$, $\mat{R}^y_k$, weights, etc., are manually tuned, and thus, the existence of an alternative parameter set that provides a “better” performance is very likely.
Fault Indication and Isolation {#sec:FDI}
------------------------------
Although the term *alteration* would be much better suited in the context of this work, the common term *fault* is used consistently. To investigate the simple fault detection scheme stated in Section \[sec:CSME\] (cf. Eq. (\[eq:FD\])), the reference performance map of the *first compressor stage* is modified within the SE from $t=\SI{40}{\minute}$ on, referred to as a fault event in the present section. More specifically, the first stage’s work input factor, calculated via a reference mapping function, is increased by $0.1$ for $t\ge\SI{40}{\minute}$ in the SE. A respective, spontaneous efficiency decline can already be found in Fig. \[fig:SE\_Track\]d, while it is hard to detect an increase in the temperature measurement in Fig. \[fig:SE\_Track\]a during flexible operation. Altogether, the tracking performance seems unaffected by the fault, which indicates a correct fault isolation. Generally, fault isolation is the capability to assign a fault to the correct cause. Here, due to the lack of defining particular failure sets, the term fault isolation is used for the local assignment to a specific plant component, i.e., the correct compressor stage. Because the proposed fault detection is based on component-specific parameters, fault isolation is a straightforward task.
Situations where the fault condition (\[eq:FD\]) is fulfilled for the current time stamp () setting are referred to as *fault indication*. A fault indication is marked by a colored background in Fig. \[fig:SE\_FDI\]b.
{width="\textwidth"}
The proposed scheme clearly indicates the fault assigned to the first compressor stage. For $N_f=5$ (six consecutive fault indications), an automated fault detection would have raised the failure flag for “compressor stage 1” for the first time after the actual fault event, while failure flag “compressor stage 2” would remain deactivated for the entire experiment. Naturally, the frequency of the fault indications diminishes over time, since the actual map estimate $\vec{\mu}^\mathfrak{M}$ is never in a converged state if a failure has occurred. Consequently, the faulty behavior is gradually incorporated into the actual map estimate.
In Fig. \[fig:SE\_FDI\], the advantage of defining a time-variant threshold ($2\, \sqrt{\mat{P}_{x_k}\{ \Delta \mu \}}$) for fault indication is quite obvious. Every time the compressor stage runs into an uncertain operating range, i.e., the *Local Information Level* LIL is low (Fig. \[fig:SE\_FDI\]a), the amplitude of adaption, recognizable via $\mu_{j_\infty}^\mathfrak{M} - \widetilde{\mu}_{j_\infty}^\mathfrak{M}$ (Fig. \[fig:SE\_FDI\]b), increases, which is enhanced by step of the CSME algorithm (cf. Section \[sec:CSME\]). Large adaption steps are facilitated by large entries in $\mat{P}_{x_k}$; thus, the proposed threshold is logical. For a nominal situation, the mismatch between actual map $\vec{\mu}^\mathfrak{M}$ and revised map $\vec{\widetilde{\mu}}^\mathfrak{M}$ vanishes as additional information is collected, yielding a noisy, nearly zero-mean signal $\mu_{j_\infty}^\mathfrak{M} - \widetilde{\mu}_{j_\infty}^\mathfrak{M}$. For the faulty situation, the signal characteristic completely changes, which might serve as an indication whether for manual monitoring or some augmented and automated fault detection schemes, the investigation of which is beyond the scope of this paper.
Performance Map Monitoring {#sec:mapMonitor}
--------------------------
As denoted in Fig. \[fig:SE\_FDI\], we utilize the delayed activation of the RME, which was suggested in step of Section \[sec:CSME\]. The preset “threshold” of enables the map estimation after . For this time stamp, the initialized map estimate (dashed speed curves), free of any reasonable *a priori* shape, can be found in Fig. \[fig:RME1\]a. In this figure, polytropic work $y_p$ and suction volume flow $\dot{V}_s$ are displayed instead of their dimensionless counterparts. Thus far, the information of merely one operating point (OP) marked by has been considered. The introduced regularization terms provide the “straight” shape, since gradients ($\mat{L}_g$) and curvatures ($\mat{L}_c$) have a relative high cost to this moment.
{height="0.93\textheight"}
{height="0.93\textheight"}
The presented shape is mapped into the typical diagram. In contrast, the true map and the (actual and revised) map estimate[^3] are preserved in the dimensionless counterpart , which is not really suitable for the intended presentation here.[^4] Due to the basic model assumption that presupposes a static dependency between (cf. Section \[sec:stagemap\]), it is necessary to use an artificial, time-invariant suction condition (index $r$) for conversion ( $\rightarrow$ )[^5] according to relations (\[eq:Psi\])–(\[eq:Mu2\]) if the true performance map (solid speed curves) is to be fixed for the supervision monitor. As a result, the projected OP (related to $T_r$ and $p_r$) of the first and second stages do not lie on the same (projected) speed curve, even if the stages are mechanically coupled ($\Sa{n}=\Sb{n}$, cf. Fig. \[fig:SE\_Input\]a). Note that both shapes, true and estimated, are depicted for the same $\Psi_p$ and $\Mu$ domain.
As the operating point varies, the shape adapts quickly, as can be seen in Fig. \[fig:RME1\]b. Here, after map initialization, 55 respective measurements have been incorporated. Every , the corresponding OP estimated is drawn into the present diagram. As a result of the flexible interpolation-based mapping scheme, it seems unreasonable to expect a correct shape adjustment for map regions in which no information has been collected thus far. Consequently, the speed curve shape of the first stage does not converge correctly in the region where the surge line would be anticipated, as this region was not accessed thus far. However, near the operating points, good estimation results can be seen. Whenever a “new” region is entered, the supervisor may recognize this by the rapid decline of the LIL.
As time passes, and the amount of information collected increases (Fig. \[fig:RME1\]c, Fig. \[fig:RME2\]a), not only the estimated shape of speed curves but also the estimated shape of the efficiency map improves, which becomes obvious by the drawn levels of the efficiency estimation error $\Delta \eta_p = \left| \eta_p - \hat{\eta}_p \right|$, depicted as filled contours within the estimated speed curve shape. Concerning these levels, a systematic estimation error remains, due to the different mapping functions used in the SE and the CSME, as mentioned in Section \[sec:setup\]. The fault event at $t=\SI{40}{\minute}$, a spontaneous decline in the first stage’s $\mu$ map vectors of the reference (SE), affects neither the speed curve shapes nor the second stage’s performance map at all (Fig. \[fig:RME2\]b,c). This complies with the correct behavior, facilitating the further improvement of the second stage’s performance map beyond the fault event (cf. Fig. \[fig:RME2\]c). The low level of $\Delta \eta_p$ within the vicinity of the current (red) operating point, after the fault event (Fig. \[fig:RME2\]b), proves the fast adaptability of the *revised map*, which is presented for all time stamps in Fig. \[fig:RME1\] and Fig. \[fig:RME2\]. A further issue, already stated, can be seen in Fig. \[fig:RME2\]c for the first stage. Within regions where much informations had already been collected (high concentration of past, black OP), the estimated map is highly inflexible. However, if the fault had been detected, a reinitialization (after potential interventions) would be advisable anyway.
Conclusions {#sec:conclusion}
===========
From the methodological aspect, two main issues have been presented.
#### Model building {#model-building .unnumbered}
A novel, low-order dynamic model for centrifugal multi-stage compressors has been derived. Real-gas behavior is taken into account explicitly. To this end, the generic LKP real-gas equation of state [@lee1975LKP] is applied. Several refinements are provided to embed this equation and its derivatives properly into the overall model scheme, in terms of accuracy and computational speed. A compressor stage’s behavior arises from its possibly time-variant performance map. The proposed approach utilizes an interpolation scheme based on four grid vectors, i.e., polytropic head coefficient, flow coefficient, work input factor, and machine Mach number, which allows for the description of nearly arbitrary performance map shapes. The proposed scheme may easily be extended by further dependencies, e.g., for variable inlet guide vanes.
#### Monitoring {#monitoring .unnumbered}
The *Unscented Kalman Filter* approach and a new *Recursive Map Estimation* are combined, yielding a novel real-time estimation scheme, which is expected to be superior if the parameters to be estimated have a distinct operating point dependency, as is the case for the grid vectors of a compressor stage’s performance map. Real-time capability is addressed via
- a first-principle, but—in detail—approximate and consequently less computationally intensive model;
- a recursive formulation of all estimation steps, yielding a constant calculation workload;
- an optimal preservation of past estimates concerning the operating point dependency within fixed-size grid vectors, yielding a constant memory requirement.
As a by-product, three time-variant supplementary observations can be provided for the monitoring task in the context of monitoring of a multi-stage compressor:
1. a performance map for every compressor stage, i.e., the estimated shape of speed and efficiency curves;
2. a *Local Information Level*, indicating the reliability of estimates at the local operating point;
3. a fault indicator for every compressor stage, which might be extended for fault detection and isolation if conceivable faults have been defined.
The estimator is able to handle *a priori* knowledge optionally, whether the task is to monitor deviations from the *a priori* presumed performance map or to identify the performance map during operation.\
\
The model-based monitoring scheme was validated via numerical simulations of a two-stage carbon dioxide compressor operating in the supercritical phase of the fluid. The reference simulation, which replaces the real “measurements”, was modified considerably; e.g., the real-gas model was interchanged, and the mass transportation delay was considered. In spite of this adverse situation, the proposed estimator performed well. The estimator was capable of tracking every state or variable, whether it was measured or not, without noticeable bias. For operating ranges that have already been reached, the estimated performance maps converged correctly. Within the remaining regions, the map shape maintains its flexibility. A preset fault event was isolated (to the respective compressor stage) correctly, and the overall behavior of the estimates and fault indicators was as desired.
Subjects of future research may arise from the following:
- The request to continuously incorporate an alteration, which might be detectable with the proposed scheme already, into a nearly converged map estimate. Since the integration of common (global) forgetting factors is considered unreasonable within the given context, the approach of local forgetting might be further investigated.
- The demand to recover from an erroneous map status, which may be triggered from faulty measurements or extreme deviations between plant and model behavior, e.g., due to a short period of compressor surge. Strategies for recovering as well as surge modeling, may contribute to this issue.
- Augmented fault detection and isolation schemes, i.e., the real-time classification of conceivable failure sets.
Acknowledgment {#acknowledgment .unnumbered}
==============
This work was supported by MAN Energy Solutions SE and the Federal Ministry for Economic Affairs and Energy based on a decision by the German Bundestag as part of the ECOFLEX-Turbo project \[grant number 03ET7091T\].
[^1]: The LKP state equation is valid for $(\Tr, \pr) \in \left[0.3;\ 4\right] \times \left]0;\ 10\right]$. Due to its continuous pressure explicit formulation, it is improper to describe the multiphase region correctly. The recommended method guarantees an (always existing) solution outside this region.
[^2]: To be precise, the illustration applies only to a C^0^ continuous interpolation method, e.g., a bilinear interpolation, but the issue still exists—to a lesser extent—for C^n^ continuous interpolation functions.
[^3]: For the presentation of estimates in Fig. \[fig:RME1\] and Fig. \[fig:RME2\], the *revised* map is used (cf. Section \[sec:CSME\]).
[^4]: The $\mu$-dimension becomes visible via $\eta_p$ or, to be precise, via $\Delta \eta_p$ (filled contours, an explanation follows).
[^5]: The real-gas models differ, whether the conversion refers to the true or the estimated map; see Section \[sec:setup\].
|
---
abstract: 'We present a joint theoretical and experimental study to investigate polymorphism in $\alpha$-sexithiophene (6T) crystals. By means of density-functional theory calculations, we clarify that the low-temperature phase is favorable over the high-temperature one, with higher relative stability by about 50 meV/molecule. This result is in agreement with our thermal desorption measurements. We also propose a transition path between the high- and low-temperature 6T polymorphs, estimating an upper bound for the energy barrier of about 1 eV/molecule. The analysis of the electronic properties of the investigated 6T crystal structures complements our study.'
author:
- Bernhard
- Caterina
- Linus
- Stefan
- Claudia
title: 'Polymorphism in $\alpha$-sexithiophene crystals: Relative stability and transition path'
---
Introduction {#Sec: Introduction}
============
Organic crystalline semiconductors are promising materials for a variety of devices, ranging from light emitting diodes [@Forr2003OE; @Forr2004NAT] to photovoltaics [@Peum-Yaki-Forr2003JAP], and field-effect transistors [@Mucc2006NATM]. The possibility of synthesizing and processing these systems at low temperature and in solution is a particularly attractive feature [@Frax2006CUP]. For these reasons, organic crystals have attracted considerable interest in the last few decades. Depending on size and chemical composition of the molecular components as well as on the packing arrangement, it is possible to design systems with optimized properties.
Oligothiophenes represent a well-known family of organic crystals, which has been largely studied in view of opto-electronic applications [@dipp+93cpl; @mark+95epl]. While the shortest thiophene oligomers are not suitable for device applications, due to their large band gap, most interest is devoted to longer chains such as $\alpha$-sexithiophene (6T). This material presents, in fact, a favorable combination of relatively high charge-carrier mobility [@horo+89ssc; @akim+91apl; @katz+95cm; @Horo-Hajl-Kouk1998EPJ-AP] and visible light absorption [@oete+94jcp; @oelk+96tsf; @kane+96prb; @fich00jmc; @mark+98jpcb; @Pith+2015CGD], which makes it a very promising compound for organic electronics.
The weak interactions between 6T molecules enable the growth of two crystal structures, which are referred to as high- and low-temperature (HT and LT) phases [@Sieg+1995JMR; @Horo+1995CM]. Both exhibit herringbone packing, with either two (HT) or four (LT) molecules per unit cell. Polymorphism, i.e., the presence of two or more possible arrangements of the molecules in the solid state [@desi08cgd], is often observed in organic crystals. The coexistence of different morphologies of molecules in their crystal phases may strongly influence the properties of such materials [@bern93jpd; @cair98tcc; @lorch_growth_2015]. This has practical impact not only in condensed-matter physics and materials science [@thre95analyst; @brag+98cr], but also in biochemistry and pharmacology [@giro95thermca; @rodr+04addr], where polymorphism is known to crucially affect drug formulation and stability. Hence, a clear understanding of the fundamental mechanisms ruling this phenomenon is desired in view of tailoring molecular materials with customized features.
In this paper, we address the question of polymorphism in 6T with a joint theoretical and experimental study. Specifically, we aim to determine which of the experimentally observed phases is the more stable one. To do so, we combine a first-principle approach, based on density-functional theory (DFT) and including van der Waals interactions, with thermal desorption measurements. Moreover, we propose and analyze a possible transition path from one structure to the other, and estimate the size of the corresponding energy barrier. The information about the crystal structure is complemented by an analysis of the electronic properties of selected systems along the transition path.
The paper is organized as follows: In Sec. \[Sec: Systems and Methods\], we introduce the HT and LT polymorphs of 6T and the theoretical and experimental methods that we use. Sec. \[Sec: Relative Stability of HT and LT\] adresses the relative stability of the two known polymorphs, both theoretically and experimentally. In Sec. \[Sec: Transition Path\], we discuss the transition path and the electronic properties corresponding to several structures along that path.
Systems and Methods {#Sec: Systems and Methods}
===================
Sexithiophene crystals {#Sec: Crystal Structure}
----------------------
![(color online) Unit cells of LT (left) and HT (right) polymorphs of 6T. Lattice paramters $a$, $b$, $c$, as well as tilt ($\phi$), herringbone ($\tau$), and monoclinic angles ($\alpha / \beta$) are indicated. The dashed line marks the plane, which divides the unit cell of the LT polymorph in half.\[Fig: Crystal Structures\]](HT-LT-Structures.eps "fig:"){width=".45\textwidth"} \[Structure-Notation\]
$a$ \[Å\] $b$ \[Å\] $c$ \[Å\] $\alpha$/$\beta$ \[deg\] $\phi$ \[deg\] $\tau$ \[deg\]
-------- ----------- ----------- ----------- -------------------------- ---------------- ----------------
**HT** **5.68** **9.14** **20.67** **97.8** **48.5** **55.0**
5.75 8.88 21.01 52.1 57.2
5.79 8.75 21.17 53.9 58.3
5.82 8.62 21.34 55.7 59.4
5.89 8.37 21.68 59.3 61.6
5.93 8.24 21.85 61.1 62.7
5.96 8.11 22.01 62.9 63.8
**LT** **6.03** **7.85** **22.35** **90.8** **66.5** **66.0**
: Structural parameters of the HT and LT polymorphs as well as of intermeditate structures (see Sec. \[Sec: Construction of the Path\]). The lattice parameters of the LT polymorph are referred to the reduced unit cell (see Sec. \[Sec: Relative Stability of HT and LT\]). Lattice constants $a$, $b$ and $c$, as well as the monoclinic angles $\alpha / \beta$, the tilt angle $\phi$, and the herringbone angle $\tau$ are displayed. \[Tab:Structural Parameters\]
The building blocks of 6T crystals are planar molecules, consisting of carbon, hydrogen, and sulfur atoms, which are arranged in a chain of six rings. Sexithiophene exhibits a herringbone arrangement of the molecules in its crystal phases [@Sieg+1995JMR; @Horo+1995CM]. The monoclinic unit cells of the LT and HT polymorphs are shown in Fig. \[Fig: Crystal Structures\]. The corresponding structural parameters, based on x-ray diffraction experiments [@Sieg+1995JMR; @Horo+1995CM], are listed in Tab. \[Tab:Structural Parameters\]. In the LT phase (left), the unit cell belongs to the space group $P2_1/n$, with a monoclinic angle $\beta_{LT}=90.8^\circ$ and contains four molecules, arranged in such a way, that the long molecular axes are almost parallel to each other. In this configuration, the tilt angle $\phi_{LT}=66.5^\circ$ is defined between the long molecular axis and the $ab$ plane, with lattice parameters $a=6.03\mbox{ \AA}$ and $b=7.85\mbox{ \AA}$. The herringbone angle $\tau_{LT}$ between the short molecular axes adopts a value of $66^\circ$ (bottom-left of Fig. \[Fig: Crystal Structures\]).
The HT structure is shown on the right-hand side of Fig. \[Fig: Crystal Structures\]. Also in this case $a$ and $b$ have comparable values ($a=5.68$ Å, $b=9.14$ Å), while $c$ is much larger, being $20.67$ Å. Note that we have permuted the crystal axes to highlight the similarity with the LT phase and to facilitate the construction of the transition path between them (see Sec. \[Sec: Transition Path\]) . The space group for this structure is $P2_1/b$, with $\alpha_{HT}=97.8^\circ$. The volume of the HT unit cell is half as large as the LT one and accommodates two inequivalent molecules. Again, the long molecular axes are approximately parallel to each other. Both, the herringbone and tilt angle, that is defined with respect to the $ab$ plane, have smaller values than in the LT phase, being 55$^\circ$ and 48.5$^\circ$, respectively (see also Tab. \[Tab:Structural Parameters\]).
Computational details {#Sec: Theory}
---------------------
Total-energy and force calculations are performed within the framework of DFT. All calculations are carried out with the full-potential all-electron code `exciting` [@Gula+2014JPCM], implementing the (linearized) augmented planewave plus local orbitals method.
A planewave cutoff $G_{max}\approx4.7 \mbox{ bohr}^{-1}$ is adopted for minimizing the atomic forces. This value corresponds to $R_{min}^{MT} G_{max}=3.5$, where $R_{min}^{MT}=0.75$ bohr is the smallest muffin-tin radius, corresponding to hydrogen. Radii of 1.15 bohr and 1.80 bohr are used for carbon and sulfur, respectively. To evaluate the small energy difference between the two polymorphs, we further increase the planewave cutoff to $G_{max}\approx6.7 \mbox{ bohr}^{-1}$, corresponding to $R_{min}^{MT} G_{max}=5.0$. We use $1 \times 5 \times 3$ and $4 \times 3 \times 1$ **k**-point grids for HT and LT, respectively. These parameters ensure the convergence of total energies within 0.4 meV/molecule.
For most calculations, we adopt the local density approximation (LDA), using the Perdew-Wang exchange-correlation functional [@LDA_PW]. The internal geometry is optimized by minimizing the atomic forces until they are smaller than 25 meV/Å. The lattice parameters are thereby fixed at their experimental values (see Tab. \[Tab:Structural Parameters\]). Doing so, the internal geometry depends largely on the electrostatic interactions. These are well described by LDA, as confirmed by the agreement of the interatomic distances with those reported in Ref. [@herm+05jpca]. Similar results have been found in an earlier study of anthracene [@Kerstin_Hummer].
To check the reliability of the LDA results and obtain more accurate energy differences between the two polymorphs, we also employ the DFT-D2 method [@Grim2006JCC] on top of the Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional [@PBE] to calculate the total energies of the HT and LT polymorphs.
Experimental methods {#Sec: Exp. Methods}
--------------------
We grow 6T films by thermal evaporation in an organic molecular beam deposition vacuum chamber equipped with a beryllium window for *in situ* x-ray measurements at a base pressure of $7\cdot10^{-7}$ mbar. Cleaved KCl substrates are heated up to $420\,^\circ$C in vacuum to reduce surface contamination prior to the deposition. The films are grown with molecular deposition rates between 1 and 1.5Å/min at $50\,^\circ$C substrate temperature to a thickness of $150 \pm 20$Å. The film thickness is monitored with a quartz crystal microbalance during growth. The grown thin films are analyzed by means of x-ray diffraction in a $\theta-2\theta$ geometry, in which the reflectivity in dependence of the out-of-plane scattering vector $q_{z} = \dfrac{4\pi}{\lambda}\sin \theta$ is studied at the corresponding values of the different crystal phases. The measurements are performed at a $Cu\,K_{\alpha}$ rotating anode system with a wavelength of $\lambda$=1.5406 Å in nitrogen atmosphere. Using a temperature controlled stage we heat the substrate to evaluate molecular desorption from the decreasing intensity of the Bragg reflections of the LT and HT crystal phases. After reaching the desired temperature, we first monitor the x-ray reflection intensity of the HT phase at $q_z=0.907$Å$^{-1}$ and subsequently the reflection of the LT phase at $q_z=0.838$Å$^{-1}$.
Relative Stability of HT and LT Polymorphs {#Sec: Relative Stability of HT and LT}
==========================================
The large LT unit cell is almost symmetrical with respect to a plane parallel to the $ab$ plane, as indicated in Fig. \[Fig: Crystal Structures\]. Thus the unit cell can be approximately divided into two halves, with lattice constants $a$, $b$, and $c=22.35 \mbox{ \AA}$, containing two molecules each. The total energies of these smaller structures, obtained after minimizing the atomic forces, differ by less than 0.2 meV/molecule. Since this value is within our computational accuracy (see Sec. \[Sec: Theory\]), we can consider the LT polymorph in such reduced unit cell . This allows us to significantly decrease the computational costs, ensuring the same numercial accuracy for both polymorphs. In the remainder of this article the label LT will refer to the reduced unit cell.
The relative stability of the HT and LT polymorphs is determined by their total energies. By employing the LDA functional, we find the LT phase to be more stable by 36 meV/molecule, compared to the HT phase. This energy difference increases to 51 meV/molecule, when we explicitly account for van der Waals interaction using the DFT-D2 method. This shows that both functionals lead to the same qualitative picture.
Experimentally, we determine the more stable phase, as well as the difference in the desorption energy barrier $E_d$ of the two crystal structures, by measuring the desorption rates of both LT and HT phase crystallites at fixed temperatures. A 15 nm thick, polycrystalline 6T thin film on NaCl exhibits phase coexistence as seen from the characteristic $(006)_{LT}$ and $(003)_{HT}$ Bragg reflections that both occur in a $\Theta -2\Theta$ scan (see Figure \[Fig: ExperimentalData\], inset). Heating this 6T film to a temperature of $428 \pm 5\, K$, corresponding to the onset of molecular desorption, we observe that the intensities of the Bragg reflections drop linearly as shown in Figure \[Fig: ExperimentalData\] . Interestingly, the HT phase desorbs at a faster rate $[R^{HT} = (1.531 \pm 0.050) \cdot 10^{-}4$ s$^{- 1}]$ than the LT phase $[R^{LT} = (1.000 \pm 0.011) \cdot 10^{-}4$ s$^{- 1}]$, indicating a higher stability of the LT phase.
![(color online) Decay of the HT and LT Bragg reflection intensities over time at a substrate temperature *T*=$428 \pm 5\, K$. Inset: $\Theta-2\Theta$ scan of the monitored LT and HT reflections. \[Fig: ExperimentalData\]](experiment_v4.eps){width=".45\textwidth"}
For a quantitative analysis, we explain the differences between the temporal decay of the two Bragg intensities by a difference in desorption energy barriers $E_d$. Assuming that the Bragg intensity is directly proportional to the respective amount of HT or LT phase, the constant slope of the decay curves can be explained by molecular desorption from step edges at a constant rate without any significant morphological changes of HT and LT islands. In atomic force microscopy measurements, resolving molecular terraces of standing upright molecules, we find no distinctly different HT and LT islands, so that a similar geometry is assumed for both phases. We use an Arrhenius-type relation of the desorption rate $R = - A\, e^{- E_d/k_{B}T}$ with the molecular desorption energy $E_d$, the (constant) temperature *T*, and an attempt frequency *A*. Assuming $A = A^{LT} = A^{HT}$ for both phases, one can write $$\begin{aligned}
\ln\left(\frac{R^{HT}}{R^{LT}}\right)&=&-\frac{E_d^{HT}}{k_B T}+\frac{E_d^{LT}}{k_B T}.\end{aligned}$$ Therefore the desorption energy difference $\Delta E_d = E_d^{LT} - E_d^{HT}$ is given by $$\begin{aligned}
\Delta E=\ln\left(\frac{R^{HT}}{R^{LT}}\right)\cdot k_B T .\end{aligned}$$ From the decay rates we estimate $\Delta E_d = 15.7 \pm 3.1$ meV between the two phases. This finding is in qualitative agreement with theory, however, the energy difference is about 3 times smaller than the computed difference in relative stability. We can identify two possible sources of such discrepancy. Most important, the process of thermal desorption occurs at the surface and in particular at step edges and corners. In this case each molecule interacts only with a reduced number of nearest neighbors compared to the bulk, and therefore the resulting binding energy is intrinsically lower. Moreover, DFT calculations do not take into account thermodynamical effects. Although most of these contributions cancel out when considering energy differences, they still may lead to a systematic overestimation of the experimental values, as previously pointed out for other organic crystals [@nabo+08prb]. Overall, we claim good agreement between our theoretical and experimental results, which identify the LT phase as the more stable one. It is finally worth mentioning that our result is in contrast with a previous work based on classical force-fields calculations [@Della_2008_JPCA]. In that case, the authors found the HT polymorph to be energetically favored with respect to LT by about 15 meV/molecule. Although the absolute value of this difference is rather small, we can attribute the better accuracy and, importantly, the correct sign of our result to the inclusion of quantum effects.
Transition path between HT and LT phases {#Sec: Transition Path}
========================================
Construction of the path {#Sec: Construction of the Path}
------------------------
{width="90.00000%"}
In Sec. \[Sec: Relative Stability of HT and LT\] we have clarified the higher relative stability of the LT polymorph with respect to the HT one. However, the energy difference between the two polymorphs is not the only factor that determines the material to grow in one phase rather than the other. One aspect is that film growth not only involves thermodynamical stability, but also kinetic processes. A quantity of interest in this context is the energy required to transform one structure into the other. To this extent we propose a possible transition path. Thereby, we face the challenge of determining the intermediate structures. While a variety of methods have been proposed and employed for such purpose [@McKe-Page1993JWL], only a few of them can be applied to molecular crystals. In fact, in these systems molecules should keep their shape and be able to reorient themselves with respect to each other, while the unit cell adjusts accordingly. To fulfill these requirements, we adopt the so-called *drag* method [@Henk-Joha-Jons2002Springer]. In our case, we treat molecules as rigid, while interpolating lattice parameters, as well as herringbone and tilt angles, the latter defining the orientation of the molecules with respect to each other and to the unit cell, respectively.
Six intermediate structures are constructed and shown in Fig. \[Fig: Intermediate-Structures\_Barrier\]a, labeled from to . In Tab. \[Tab:Structural Parameters\], lattice constants, as well as herringbone and tilt angles are reported. For comparison, also the structural parameters of the HT and LT phase are shown, highlighted in bold. It is worth noting that interpolation of $a$, $b$, $c$, $\phi$ and $\tau$, indirectly determines the angles between the lattice parameters. Hence, while the initial HT and LT structures are monoclinic, the intermediate ones become triclinic. By inspecting Fig. \[Fig: Intermediate-Structures\_Barrier\]a the variation of herringbone and tilt angles is visible, as well as the change of the lattice parameters. The six intermediate structures are optimized to minimize the atomic forces, while the unit cell parameters are held fixed to the values reported in Tab. \[Tab:Structural Parameters\].
Relative stabilities and electronic structure {#Sec: Relative stabilities and electronic structure}
---------------------------------------------
**HT** **LT**
----- ---------- ------ ------- ------- ------- ------- ------ ---------
LDA **36.2** 42.2 582.3 687.4 601.1 517.4 16.8 **0.0**
vdW **51.0** **0.0**
: Energy difference per molecule of each structure with respect to the LT phase, as obtained from LDA and DFT-D2 (labeled vdW). All energies are given in meV/molecule.\[Tab: Relative Energies\]
As a next step, we evaluate the total energies of the optimized intermediate structures. In this way, we are able to estimate the energy barrier between the HT and LT phases. The results of these calculations are reported in Tab. \[Tab: Relative Energies\] and in Fig. \[Fig: Intermediate-Structures\_Barrier\]b. The relative energies with respect to the most stable polymorph, the LT phase, are displayed as a function of the reaction coordinate $\Omega$, which is defined by the lattice parameters and the angles $\tau$ and $\phi$. Overall the barrier presents a *top hat* shape. While structure resembles the HT phase and is energetically very close to it, structure is similar in energy and shape to the LT phase (see Tab. \[Tab: Relative Energies\]). On the other hand, structures - are significantly less favored, exhibiting energies more than 0.5 eV/molecule higher compared to LT. The most unfavored polymorph is structure . The total energy exceeds that of the LT phase by about 0.7 eV per molecule, which represents the apex of the dome. Considering an increase in energy by explicitely accounting for van der Waals interactions, we estimate the upper bound of the barrier to be about 1 eV. Similar results have been obtained for other organic crystals, such as *para*-sexiphenyl on a mica step-edge [@Hlaw+2008SCI].
{width="99.00000%"}
We finally present, in Fig. \[Fig: All-Bands\], the electronic properties of selected intermediate structures, compared to the HT and LT ones. These results allow us to further characterize the predicted metastable structures. In Fig. \[Fig: All-Bands\] the band structure and density of states (DOS) of the three selected intermediates , and are shown in comparison with the stable polymorphs HT and LT. We notice the typical features of organic crystals (see e.g. Ref. \[\]). In both the valence and conduction regions, subbands are arranged in pairs, according to the double multiplicity of the 6T molecules in the unit cell. These features are reflected also in the DOS, which presents in both cases relatively sharp peaks in the valence region, corresponding to the different subbands. The subbands in the conduction region are energetically closer to each other, and present overall increased dispersion compared to the valence region. The highest valence-bands (VB) and lowest conduction-bands (CB) are highlighted in green. Both polymorphs have indirect Kohn-Sham (KS) band gaps of 1.2 and 1.1 eV for LT and HT, respectively.
In the LT phase the VB bandwidth is 0.2 eV, while it is larger (0.5 eV) in the CB. For symmetry reasons, both bands are degenerate along the path from $Y$ to $C$, and exhibit a small splitting between $C$ and $Z$. The splitting is largest at the $\Gamma$-point and halfway inbetween the points $A_0$ and $Y$. In the HT phase, the bandwidth is twice as large in the VB (0.4 eV), and slightly larger in the CB (0.6 eV), compared to LT. The largest splitting is found at $\Gamma$ and $Z$, as well as halfway between $A_0$ and $Y$. Bands are degenerate between $Y$ and $C$, as well as between $B$ and $A_0$.
For comparison, we have selected those structures (, and ), which mostly differ from each other in the arrangement of the molecules and in the energetics (see Figs. \[Fig: Intermediate-Structures\_Barrier\]a and b). These intermediate structures belong to the space group $P1$, exhibiting trivial symmetry. These systems present indirect KS band gaps of 1.0 eV (), 0.9 eV (), and 1.2 eV (). These values are comparable to those of the HT (1.1 eV) and LT (1.2 eV) phases. The lower stabilities of these systems with respect to the stable HT and LT polymorphs is evident from the band structures. In particular, in structure , which is clearly unfavored in the chosen transition path, a large band splitting is observed, in both valence and conduction regions. Especially the occupied states feature sharp peaks in the DOS. This is a signature of the lower symmetry of this system, compared to the stable HT and LT polymorphs. The other intermediate structures, and , which are structurally and energetically close to HT and LT, respectively, still present subbands, and the fingerprints of reduced symmetry, such as subband degeneracy, are less pronounced. Overall, the bands exhibit low dispersion, especially for the and structures. This implies high effective carrier masses and therefore low charge-carrier mobilities in these regions. The bandwidths in the VBs decrease from 0.4 eV for and to 0.2 eV in structure . Thus, structure () shares the same bandwidth in the VB as the HT (LT) polymorph. The CB bandwidths tend to be larger with values of 0.6 eV (structure ), 0.7 eV (structure ) and 0.7 eV (structure ), thus they are 0.1 eV larger than those of the HT and LT phase, respectively.
Summary and Conclusions {#Sec: Summary}
=======================
We have presented a combined theoretical and experimental study on polymorphism in 6T crystals. We have clarified that the LT phase is favored over the HT one by about 50 meV/molecule, as obtained from DFT calculations, explicitly taking into account van der Waals interactions. This result is in agreement with our thermal desorption measurements. Our results confirm the importance of explicitly accounting for quantum effects and dispersive intermolecular interactions, to quantify the relative stability of different polymorphs in organic crystal structures.
In addition, we have proposed a transition path between the two stable 6T polymorphs, estimating the energy barrier between the HT and LT phase of about 1 eV/molecule. The results are supplemented by a thorough analysis of the electronic properties of the stable and selected intermediate structures.
Acknowledgement {#acknowledgement .unnumbered}
===============
We acknowledge fruitful discussions with Hong Li, Dmitrii Nabok, and Peter Schäfer. This work was partly funded by the German Research Foundation (DFG) through the Collaborative Research Centers SFB 658 and SFB 951. C. C. acknowledges support from the Berliner Chancengleichheitsprogramm (BCP). L. P. acknowledges support from the Studienstiftung des deutschen Volkes.
[10]{} \[1\][`#1`]{} \[1\][ ]{} \[2\]
> <span style="font-variant:small-caps;">Annotation:</span> \#2
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---
abstract: 'A remarkable feature of fluid dynamics is its relationship with classical dynamics and statistical mechanics. This has motivated in the past mathematical investigations concerning, in a special way, the “derivation” based on kinetic theory, and in particular the Boltzmann equation, of the incompressible Navier-Stokes equations (INSE). However, the connection determined in this way is usually merely asymptotic (i.e., it can be reached only for suitable limit functions) and therefore presents difficulties of its own. This feature has suggested the search of an alternative approach, based on the construction of a suitable inverse kinetic theory (IKT; Tessarotto et al., 2004-2008), which can avoid them. IKT, in fact, permits to achieve an exact representation of the fluid equations by identifying them with appropriate moment equations of a suitable (inverse) kinetic equation. The latter can be identified with a Liouville equation advancing in time a phase-space probability density function (PDF), in terms of which the complete set of fluid fields (prescribing the state of the fluid) are determined. In this paper we intend to investigate the mathematical properties of the underlying *finite-dimensional* phase-space classical dynamical system, denoted *Navier-Stokes dynamical system*, which can be established in this way. The result we intend to establish has fundamental implications both for the mathematical investigation of Navier-Stokes equations as well as for diverse consequences and applications in fluid dynamics and applied sciences.'
author:
- 'Massimo Tessarotto$^{a,b}$, Claudio Asci$^{a}$, Claudio Cremaschini$^{c,d}$, Alessandro Soranzo$^{a}$ and Gino Tironi$^{a,b}$'
title: '**Mathematical properties of the Navier-Stokes dynamical system for incompressible Newtonian fluids$^{\S }$** '
---
1 - Introduction
================
A fundamental theoretical issue in mathematical physics is the search of a possible *finite-dimensional* classical dynamical system - to be denoted as *Navier-Stokes* (NS) *dynamical system* - which uniquely advances in time the complete set of fluid fields which characterize a fluid system. In the case of an incompressible isothermal Newtonian fluid (also known as *NS fluid*) these are identified with the (mass) fluid velocity and the (non-negative) scalar fluid pressure, which in the domain of the fluid itself satisfy the *incompressible NS equations* (INSE). The reason why the determination of such a dynamical system is so important is that its existence is actually instrumental for the establishment of theorems of existence and uniqueness for the related initial-boundary value problem (*INSE problem*). In this paper we prove that, based on the inverse kinetic theory approach (IKT) developed by Tessarotto *et al.* (2004-2007 [Tessarotto2004,Ellero2005,Tessarotto2006]{}), the problem can actually be given a well-defined formulation. Main goal of the paper is the establishment of an *equivalence theorem* between the INSE problem and the NS dynamical system. Basic consequences and applications of this result are pointed out. In particular, contrary to the widespread view according to which the phase-space dynamical system characterizing the fluid fields should be infinite dimensional, here we intend to prove that a *finite-dimensional classical dynamical system* exists which advances in time the complete set of fluid fields. This is realized by the NS dynamical system.
2 - The strong stochastic INSE problem
======================================
For definiteness, let us[ assume that the complete set of fluid fields fluid fields ]{}$\left\{ Z\right\} \equiv \left\{ \rho ,\mathbf{V},p\right\} ,$[ respectively denoting the mass density, the fluid velocity and the fluid pressure, describing an NS fluid, are local strong solutions of the equations ]{} $$\begin{aligned}
\rho &=&\rho _{o}, \label{1b} \\
\nabla \cdot \mathbf{V} &=&0, \label{1ba} \\
N\mathbf{V} &=&0, \label{1bb} \\
Z(\mathbf{r,}t_{o}) &\mathbf{=}&Z_{o}(\mathbf{r}), \label{1ca} \\
\left. Z(\mathbf{r,}t)\right\vert _{\partial \Omega }
&\mathbf{=}&\left. Z_{w}(\mathbf{r,}t)\right\vert _{\partial
\Omega }, \label{1c}\end{aligned}$$[In particular]{}, Eqs. (\[1b\])-(\[1c\]) are respectively the *incompressibility, isochoricity and Navier-Stokes equations* and the initial and Dirichlet boundary conditions for $\left\{
Z\right\} ,$ with $\left\{
Z_{o}(\mathbf{r})\right\} $ and $\left\{ \left. Z_{w}(\mathbf{\cdot ,}%
t)\right\vert _{\partial \Omega }\right\} $ suitably prescribed initial and boundary-value fluid fields, defined respectively at the initial time $%
t=t_{o}$ and on the boundary $\partial \Omega .$ [In the remainder, for definiteness, we shall require that:]{}
1. [$\Omega $[ is coincides with the Euclidean space ]{}$E^{3}$ on ]{}$%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$[$^{3}$[ (*external domain*) and ]{}]{}$\partial \Omega $ with the improper plane of $%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$[$^{3};$]{}
2. $I$ is identified with the real axis $%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ (*global domain*) .[ ]{}
We shall assume that the fluid fields are continuous in $\overline{\Omega }%
\times {I}$ and fulfill the inequalities$$\begin{aligned}
&&\left. p>0,\right. \label{5aa} \\
&&\left. \rho >0.\right. \label{6aa}\end{aligned}$$Here the notation as follows. $N$ is the *NS nonlinear operator* $$N\mathbf{V}=\frac{D}{Dt}\mathbf{V}-\mathbf{F}_{H}, \label{NS
operator}$$with $\frac{D}{Dt}\mathbf{V}$ and** **$\mathbf{F}_{H}$ denoting respectively the *Lagrangian fluid acceleration* and the *total force* *per unit mass* $$\begin{aligned}
&&\left. \frac{D}{Dt}\mathbf{V}=\frac{\partial }{\partial t}\mathbf{V}+%
\mathbf{V}\cdot \nabla \mathbf{V,}\right. \label{2a} \\
&&\left. \mathbf{F}_{H}\equiv \mathbf{-}\frac{1}{\rho _{o}}\nabla p+\frac{1}{%
\rho _{o}}\mathbf{f}+\upsilon \nabla ^{2}\mathbf{V,}\right.
\label{2c}\end{aligned}$$while $\rho _{o}$ and $\nu >0$ are the constant* mass density* and the constant *kinematic viscosity*. In particular, $\mathbf{f}$ is the *volume force density* acting on the fluid, namely which is assumed of the form$$\mathbf{f=-\nabla }\phi (\mathbf{r},t)+\mathbf{f}_{R},$$$\phi (\mathbf{r},t)$ being a suitable scalar potential, so that the first two force terms \[in Eq.(\[2c\])\] can be represented as $-\nabla p+\mathbf{f%
}$ $=-\nabla p_{r}+\mathbf{f}_{R},$ with $$p_{r}(\mathbf{r},t)=p(\mathbf{r},t)-\phi (\mathbf{r},t),$$denoting the *reduced fluid pressure*. As a consequence of Eqs.([1b]{}) and (\[1ba\]) it follows that the fluid pressure necessarily satisfies the *Poisson equation*$$\nabla ^{2}p=S, \label{Poisson}$$where the source term $S$ reads $$S=-\rho _{o}\nabla \cdot \left( \mathbf{V}\cdot \nabla
\mathbf{V}\right) +\nabla \cdot \mathbf{f}.$$Here we shall assume, furthermore, that the fluid fields $\left\{
Z\right\} , $ together with the volume force density $\mathbf{f}$ and the initial and boundary conditions $\left\{
Z_{o}(\mathbf{r})\right\} ,$ $\left\{ \left.
Z_{w}(\mathbf{r,}t)\right\vert _{\partial \Omega }\right\} $[ are all stochastic functions (see Appendix) of the form ]{}$$\begin{aligned}
&&\left. Z=Z(\mathbf{r},t,\mathbf{\alpha }),\right. \notag \\
&&\left. \mathbf{f}\mathbf{=f}(\mathbf{r},t,\mathbf{\alpha
})\right. \notag
\\
&&\left. Z_{o}=Z_{o}(\mathbf{r},\mathbf{\alpha })\right. \\
&&\left. Z_{w}=\left. Z_{w}(\mathbf{r,}t)\right\vert _{\partial
\Omega }\right. \notag\end{aligned}$$where $\mathbf{\alpha }\in V_{\mathbf{\alpha }}$ are stochastic variables assumed independent of $(\mathbf{r},t$). Eqs.(\[1b\])-(\[1c\]) then denote the[ initial-boundary value problem for the ]{}stochastic incompressible Navier-Stokes equations[ (*strong stochastic INSE problem*). ]{}
3 - The IKT-statistical model
=============================
A fundamental aspect of fluid dynamics lays in the construction of statistical models for the fluid equations [@Tessarotto2009c].* *In this connection a possible viewpoint is represented by the construction of the so-called *IKT-statistical model *able to yield as moments of the PDF the *whole set of fluid fields* $\left\{ Z\right\} $ which determine the fluid state and in which the same PDF satisfies a Liouville equation.* *Despite previous attempts (Vishik and Fursikov, 1988 [@Vishik1988] and Ruelle, 1989 [@Ruelle1989]) the existence of such a dynamical system has remained for a long time an unsolved problem.
This type of approach has actually been achieved for incompressible NS fluids [@Tessarotto2004]. Its applicationsand extensions are wide-ranging and concern in particular: incompressible thermofluids [Tessarotto2008-2]{}, quantum hydrodynamic equations (see [Tessarotto2007a,Tessarotto2008-4]{}), phase-space Lagrangian dynamics [Tessarotto2008-5]{}, tracer-particle dynamics for thermofluids [Tessarotto2008-6,Tessarotto2009b]{}, the evolution of the fluid pressure in incompressible fluids [@Tessarotto2008-3], turbulence theory in Navier-Stokes fluids [@Tessarotto2008-4; @Tessarotto2008-7] and magnetofluids [@Tessarotto2009] and applications of IKT to lattice-Boltzmann methods [@Tessarotto2008-8].
The IKT-statistical model is based on the introduction of a PDF depending on the local configuration-space vector $\mathbf{r,}$ $f_{1}(t)\equiv f_{1}(%
\mathbf{r,v,}t,\mathbf{\alpha })$ (*1-point velocity PDF*) defined on the phase-space $\Gamma _{1}=\Omega \times U$ \[with $U\equiv
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{3}$\] and identified with a strictly positive function such that the complete set of fluid fields $\left\{ Z\right\} $ associate to the strong stochastic INSE problem can be represented in terms of the functionals (*velocity moments*) $$\int\limits_{U}d\mathbf{v}Gf_{1}(t)=\left\{ 1,\mathbf{V}(\mathbf{r,}t,%
\mathbf{\alpha }),p_{1}(\mathbf{r,}t,\mathbf{\alpha })\right\} ,
\label{MOMENTS-2}$$(*Requirement \#1; correspondence principle*). Here $G=\left\{ 1,%
\mathbf{v,}u^{2}/2\right\} $while $p_{1}(\mathbf{r,}t,\mathbf{\alpha })>0$ denotes the *kinetic pressure* $$p_{1}(\mathbf{r},t,\mathbf{\alpha })=p(\mathbf{r},t,\mathbf{\alpha }%
)+p_{0}(t,\mathbf{\alpha })-\phi (\mathbf{r},t,\mathbf{\alpha }),
\label{kinetic pressure}$$with $p_{0}(t,\mathbf{\alpha })>0$ (the *pseudo-pressure*) a strictly positive, smooth, i.e., at least $C^{(1)}(I)$, real function and $\phi (%
\mathbf{r},t,\mathbf{\alpha })$ a suitably defined potential. In addition $%
f_{1}(t)$ is assumed to obey the Liouville equation - or inverse kinetic equation (IKE) according to the notation of Ref.[@Ellero2005]) -$$L(\mathbf{r,v},t;f_{1}(t))f_{1}(t)=0 \label{LIOUVILLE EQ}$$(*Requirement \#2*) with $L(\mathbf{r,v},t;f_{1}(t))$ denoting the Liouville streaming operator $L(\mathbf{r,v},t;f_{1}(t))\cdot \equiv \frac{%
\partial }{\partial t}\cdot +\frac{\partial }{\partial \mathbf{x}}\cdot
\left\{ \mathbf{X}(\mathbf{x},t;f_{1}(t))\cdot \right\} $ and $\mathbf{F}%
(f_{1}(t))$ a suitable smooth vector field defined in such a way that the moment equations of (\[LIOUVILLE EQ\]) obtained for $G=\left\{ 1,\mathbf{%
v,\rho }_{o}u^{2}/3\right\} $ \[$\mathbf{u}$ denoting the relative velocity $%
\mathbf{u\equiv u}(\mathbf{r,}t,\mathbf{\alpha })=\mathbf{v-V(r},t,\mathbf{%
\alpha })$\] coincide respectively with Eqs.(\[1ba\]), (\[1bb\]) and again (\[1ba\]).* *This implies that the initial value problem associated to the vector field$$\mathbf{X}(\mathbf{x},t;f_{1}(t))=\left\{
\mathbf{v,F}(f_{1}(t))\right\} , \label{vector field}$$$$\left\{
\begin{array}{c}
\frac{d\mathbf{x}}{dt}=\mathbf{X}(\mathbf{x},t;f_{1}(t)), \\
\mathbf{x}(t_{o})=\mathbf{x}_{o}%
\end{array}%
\right. \label{Eq.1}$$necessarily defines a dynamical system. In particular if $\mathbf{x}(t)=%
\mathbf{\chi (x}_{o},t_{o},t\mathbf{)}$ is the general solution of ([Eq.1]{}), this is identified with the flow$$T_{t_{o},t}:\mathbf{x}_{o}\rightarrow \mathbf{x}(t)=T_{t_{o},t}\mathbf{x}%
_{o}\equiv \mathbf{\chi (x}_{o},t_{o},t\mathbf{)} \label{FLOW}$$*generated by* $\mathbf{X}(\mathbf{x},t;f_{1}(t))$*.* Furthermore it is assumed that Eq.(\[LIOUVILLE EQ\]) admits as a particular solution $f_{1}(t)$ the Gaussian PDF$$f_{M}(\mathbf{x},t,\mathbf{\alpha })=\frac{1}{\pi ^{3/2}v_{th}^{3}(\mathbf{r,%
}t,\mathbf{\alpha })}\exp \left\{ -\frac{u^{2}}{v_{th}^{2}(\mathbf{r,}t,%
\mathbf{\alpha })}\right\} , \label{GAUSSIAN}$$where $v_{th}^{2}(\mathbf{r,}t,\mathbf{\alpha })=2p_{1}(\mathbf{r,}t,\mathbf{%
\alpha })/\rho _{o}.$ More precisely it is assumed that $f_{M}(\mathbf{x},t,%
\mathbf{\alpha })$ is a particular solution of Eq.(\[LIOUVILLE EQ\]) if and only if the fluid fields $Z(\mathbf{r},t,\mathbf{\alpha })$ are solutions of the strong stochastic INSE problem (*Requirement \#3*).
In the following we intend to investigate, in particular, the consequences of requirements (\[MOMENTS-2\]), (\[LIOUVILLE EQ\]) and (\[GAUSSIAN\]) for the problem posed in this paper.
4 - The Navier-Stokes dynamical system
======================================
According to a certain misconception, dynamical systems for continuous fluids cannot be finite dimensional due to the fact that the fluid equations are PDEs for the relevant set of fluid fields $\left\{ Z\right\} $. However, it is easy to show that this is not the case even in the so-called Lagrangian description of fluid dynamics. For a NS fluid this is realized by parametrizing the fluid velocity $\mathbf{V}$ in terms of the Lagrangian path (LP) $\mathbf{r}(t).$ In the present notation this is solution of the problem $$\left\{
\begin{array}{l}
\frac{D\mathbf{r}(t)}{Dt}=\mathbf{V}(\mathbf{r}(t),t,\mathbf{\alpha }), \\
\mathbf{r}(t_{o})=\mathbf{r}_{o},%
\end{array}%
\right. \label{LP}$$where $\frac{D}{Dt}$ is the Lagrangian derivative (\[2a\]). As a consequence the NS equation becomes $$\frac{D\mathbf{V}(\mathbf{r}(t)\mathbf{,}t)}{Dt}=\mathbf{F}_{H}(\mathbf{r}(t)%
\mathbf{,}t),$$which \[with $\mathbf{F}_{H}(\mathbf{r}(t)\mathbf{,}t)$ considered prescribed\] can be treated as an ODE and integrated along a LP yielding $$\mathbf{V}(\mathbf{r}(t)\mathbf{,}t)=\mathbf{V}(\mathbf{r}_{o}\mathbf{,}%
t_{o})+\int\limits_{t_{o}}^{t}dt^{\prime }\mathbf{F}_{H}(\mathbf{r}%
(t^{\prime })\mathbf{,}t^{\prime }).$$This permits to determine, *for all* $\mathbf{r\equiv r}(t)$ *and* $\left( \mathbf{r}_{o}\mathbf{,}t_{o}\right) \in \Omega
\times I,$ the vector field $\mathbf{V}(\mathbf{r,}t)$. Therefore the *finite-dimensional dynamical system* $\left( \mathbf{r}_{o}\mathbf{,}%
t_{o}\right) \rightarrow $ $(\mathbf{r}(t),t)$ defined by Eq.(\[LP\]) actually generates the time evolution of $\mathbf{V}(\mathbf{r},t,\mathbf{%
\alpha }).$ Nevertheless, in this description the fluid pressure is actually not directly determined \[in fact this requires solving the Poisson equation (\[Poisson\])\].
We intend to show that a dynamical system advancing in time the complete set of fluid fields for a NS fluid is realized by the dynamical system $%
T_{t_{o},t}$ \[NS dynamical system; see Eq.(\[FLOW\])\]. Its precise definition depends manifestly on the vector field $\mathbf{F}(f_{1}(t))$. The task \[of defining $\mathbf{F}(f_{1}(t))]$ is achieved by the IKT approach developed in Ref.[@Ellero2005]. As a consequence it follows that $\mathbf{F}(f_{1}(t))$ can be non-uniquely determined [Ellero2005,Tessarotto2006]{} in terms of a smooth vector field which is at least $C^{(1)}(\Gamma _{1}\times I\times
V_{\mathbf{\alpha }})$.
In this case, as a fundamental mathematical result, we intend to prove here *the equivalence between the strong stochastic INSE problem and the NS dynamical system.* In other words the Liouville equation (\[LIOUVILLE EQ\]) can be shown to be equivalent to $$f_{1}(\mathbf{x},t,\mathbf{\alpha })=J(t,\mathbf{\alpha
})f_{1}(\mathbf{\chi (x},t,t_{o},\mathbf{\alpha
)},t_{o},\mathbf{\alpha }), \label{INTEGRAL LIOUVILLE EQ.}$$where $\mathbf{x}(t)=T_{t_{o},t}\mathbf{x}_{o}$ is the general solution of (\[Eq.1\]) for $\mathbf{F}\equiv \mathbf{F}(f_{1}(t))$ and where$$J(t,\mathbf{\alpha })=\exp \left\{ \int\limits_{t_{o}}^{t}dt^{\prime }\frac{%
\partial }{\partial \mathbf{v(t}^{\prime },\mathbf{\alpha })}\cdot \mathbf{F}%
(\mathbf{x}(t^{\prime },\mathbf{\alpha }),t^{\prime },\mathbf{\alpha ;}%
f_{1}(t^{\prime }))\right\} \equiv \left\vert \frac{\partial \mathbf{x}(t,%
\mathbf{\alpha })}{\partial \mathbf{x}_{o}}\right\vert$$ is the Jacobian of the flow $T_{t_{o},t}$ \[see Eq.(\[FLOW\])\]. Therefore, the NS dynamical system necessarily advances in time the PDF $f_{M}(\mathbf{x%
},t_{o},\mathbf{\alpha })$ so that it is *identically* a solution of the Liouville equation (\[LIOUVILLE EQ\]). The result* *can be established on general grounds, i.e., for an arbitrary vector field $\mathbf{%
F}((f_{1}(t))$ fulfilling Requirements \#1-\#3. The following result holds:
**THM.1 - Equivalence theorem**
[*In validity of Requirements $\#1-\#3$ the strong stochastic INSE problem is equivalent to the NS dynamical system (\[FLOW\])* ]{}.
PROOF
The proof is immediate. In fact, if $$\mathbf{x\equiv x}(t,\mathbf{\alpha })=\mathbf{\chi (x}_{o},t_{o},t,\mathbf{%
\alpha )}$$is the solution of Eq.(\[Eq.1\]) (which is assumed to exist and define at least a $C^{(2)}-$diffeomorphism, its inverse transformation is simply $$\mathbf{x}_{o}=\mathbf{\chi (x},t,t_{o},\mathbf{\alpha ).}$$Therefore by differentiating Eq.(\[INTEGRAL LIOUVILLE EQ.\]) it follows$$\frac{d}{dt}f_{1}(\mathbf{x},t,\mathbf{\alpha })-\frac{d}{dt}\left\{ J(t,%
\mathbf{\alpha })f_{1}(\mathbf{\chi (x},t,t_{o},\mathbf{\alpha )},t_{o},%
\mathbf{\alpha })\right\} =0,$$which recovers Eq.(\[INTEGRAL LIOUVILLE EQ.\]) and admits as a particular solution $f_{M}(t)\equiv
f_{M}(\mathbf{x},t,\mathbf{\alpha })$ when subject to the initial condition $f_{M}(\mathbf{x}_{o},t_{o},\mathbf{\alpha })$. Hence in terms of such an equation the NS dynamical system advances in time *the complete set of fluid fields.* Therefore, the fluid velocity and the kinetic pressure at time $t$, i.e.,$\mathbf{V}(t)\equiv \mathbf{V}(%
\mathbf{r},t,\mathbf{\alpha })$ and $p_{1}(t)\equiv p_{1}(\mathbf{r},t,%
\mathbf{\alpha }),$ follow from the moment equations (\[MOMENTS-2\]). Q.E.D.
5. Conclusions
==============
This work is motivated by the analogy between hydrodynamic description and the theory of classical dynamical systems. For greater generality the case of stochastic fluid equations has been considered. The problem of the equivalence between the initial-boundary value problem for incompressible Navier-Stokes equations and the Navier-Stokes dynamical system introduced in Ref.[@Ellero2005] has been investigated. Indeed, the theory here developed applies both to deterministic and stochastic fluid fields. In fact, in both cases the time evolution of $f_{1}$ is determined by a Liouville equation \[see Eq.(\[LIOUVILLE EQ\])\] which evolves in time also the complete set of fluid fields (all represented in terms of moments of the same PDF). Contrary to the misconception according to which the phase-space dynamical system characterizing the fluid fields $\left\{ Z\right\} $ of a continuous fluid system should be infinite dimensional, here we have proven that the finite-dimensional NS classical dynamical system advances in time *the complete set of fluid fields,* determined in terms of velocity moments of the 1-point PDF $f_{1}(\mathbf{x},t,\mathbf{\alpha })$. The theory here developed applies generally to stochastic fluid equations. As shown elsewhere [@Tessarotto2008-7; @Tessarotto2009; @Tessarotto2009c], this represents a convenient treatment for the statistical theory of turbulence, historically referred to the work of Kolmogorov (Kolmogorov, 1941 [Kolmogorov1941]{}) and Hopf (Hopf, 1950/51 [@Hopf1950/51]).
The theory has important consequences which concern fundamental aspects of fluid dynamics:
- determination of the NS dynamical system advancing in time the complete set of fluid fields of a turbulent NS fluid [@Tessarotto2009c];
- construction of the initial conditions for the 1-point PDF $f_{1}$ [@Tessarotto2010c]$;$
- determination of the time-evolution of passive scalar and tensor fields [@Tessarotto2008-3];
- construction of the exact equations of motion for ideal tracer-particle dynamics in a turbulent NS fluids [@Tessarotto2009b];
- construction of multi-point PDFs for turbulent NS fluids [Tessarotto2010e]{};
- statistical treatment of homogeneous, isotropic and stationary turbulence based on IKT [@Tessarotto2010d].
Appendix - ** **Stochastic variables
====================================
Let $(S,\Sigma ,P)$ be a probability space; a measurable function $\mathbf{%
\alpha :}S\longrightarrow V_{\mathbf{\alpha }}$, where $V_{\mathbf{\alpha }%
}\subseteq
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{k}$, is called *stochastic* (or *random*) *variable*.
A stochastic variable $\mathbf{\alpha }$ is called *continuous* if* *it is endowed with a *stochastic model* $\left\{ g_{%
\mathbf{\alpha }},V_{\mathbf{\alpha }}\right\} ,$* *namely a real function* *$g_{\mathbf{\alpha }}$ (called as *stochastic PDF*)* *defined on the set $V_{\mathbf{\alpha }}$ and such that:
1\) $g_{\mathbf{\alpha }}$ is measurable, non-negative, and of the form $$g_{\mathbf{\alpha }}=g_{\mathbf{\alpha
}}(\mathbf{r},t,\mathbf{\cdot }); \label{stochastic PDF}$$
2\) if $A\subseteq V_{\mathbf{\alpha }}$ is an arbitrary Borelian subset of $%
V_{\mathbf{\alpha }}$ (written $A\in \mathcal{B}(V_{\mathbf{\alpha
}})$), the integral $$P_{\mathbf{\alpha }}(A)=\int\limits_{A}d\mathbf{x}g_{\mathbf{\alpha }}(%
\mathbf{r},t,\mathbf{x}) \label{dist-of-alpha}$$exists and is the probability that $\mathbf{\alpha \in }A$; in particular, since $\mathbf{\alpha }\in V_{\mathbf{\alpha }}$, $g_{\mathbf{\alpha }}$ admits the normalization $$\int\limits_{V_{\mathbf{\alpha }}}d\mathbf{x}g_{\mathbf{\alpha }}(\mathbf{r}%
,t,\mathbf{x})=P_{\mathbf{\alpha }}(V_{\mathbf{\alpha }})=1.
\label{normalization}$$
The set function $P_{\mathbf{\alpha }}:\mathcal{B}(V_{\mathbf{\alpha }%
})\rightarrow \lbrack 0,1]$ defined by (\[dist-of-alpha\]) is a probability measure and is called distribution (or law) of $\mathbf{\alpha }$. Consequently, if a function $f\mathbf{:}V_{\mathbf{\alpha }%
}\longrightarrow V_{f}\subseteq
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{m}$ is measurable, $f$ is a stochastic variable too.
Finally define the *stochastic-averaging operator* $\left\langle
\cdot \right\rangle _{\mathbf{\alpha }}$(see also [@Tessarotto2009]) as* *$$\left\langle f\right\rangle _{\mathbf{\alpha }}=\left\langle f(\mathbf{y}%
,\cdot )\right\rangle _{\mathbf{\alpha }}\equiv \int\limits_{V_{\mathbf{%
\alpha }}}d\mathbf{x}g_{\mathbf{\alpha }}(\mathbf{r},t,\mathbf{x})f(\mathbf{y%
},\mathbf{x}), \label{stochastic averaging operator}$$for any $P_{\mathbf{\alpha }}$-integrable function $f(\mathbf{y},\cdot ):V_{%
\mathbf{\alpha }}\rightarrow
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$, where the vector $\mathbf{y}$ is some parameter.
Acknowledgments {#acknowledgments .unnumbered}
===============
Work developed in cooperation with the CMFD Team, Consortium for Magneto-fluid-dynamics (Trieste University, Trieste, Italy). Research partially performed in the framework of the GDRE (Groupe de Recherche Européen) GAMAS.
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|
---
abstract: 'Rather than an [*a priori*]{} arena in which events take place, space-time is a construction of our mind making possible a particular kind of ordering of events. As quantum entanglement is a property of states independent of classical distances, the notion of space and time has to be revised to represent the holistic interconnection of quanta. We also speculate about various forms of reprogramming, or reconfiguring, the propagation of information for multipartite statistics and in quantum field theory.'
author:
- Karl Svozil
title: Space and time in a quantized world
---
Intrinsic construction of space-time frames
===========================================
Physical space and time appear to be [*ordering events*]{} by quantifying top-bottom, left-right, front-back, as well as before-after. In that function, space-time relates to actual physical events, such as clicks in particle detectors. Without such events, space-time would be metaphysical at best, because there would be no operational basis that gave meaning to the aforementioned categories. Intrinsic space-time is tied to, or rather based upon, physical events; and is bound to operational means available to observers “located inside” the physical system.
In acknowledging this empirical foundation, Einstein’s centennial paper on space-time [@ein-05; @naber], and to a certain extent Poincare’s thoughts [@poincare02], introduced [*conventions and operational algorithmic procedures*]{} that allow the generation of space-time frames by relying on intrinsically feasible methods and techniques alone [@toffoli:79; @svozil-94]. This renders a space-time (in terms of clocks, scales and conventions for the definition of space-time frames, as well as their transformations) which is [*means relative*]{} [@Myrvold2011237] with respect to physical devices (such as clocks and scales), as well as to procedures and conventions (such as for [*defining*]{} simultaneity employing round-trip time, which is nowadays even used by [*Cristian’s Algorithm*]{} for computer networks). These unanimously executable measurements and “algorithmic” physical procedures need not rely upon any kind of absolute metaphysical knowledge (such as “absolute space or time”).
This approach is characterized by constructing operational, intrinsic space-time frames based on physical events alone; rather then by staging physical events in a Kantian [*a priori*]{} “space-time theatre.” One step in this direction is, for instance, the determination of the [*dimensionality*]{} of space and of space-time from empirical evidence [@sv2].
Consequently, space and time emerge as concepts that are not independent of the physical phenomena (as well as on assumptions or conventions) by which they are constructed. Therefore, it is quite legitimate to ask whether the space-time of classical physics can be carried over to quantum space-time [@Kreinovich-94; @Myrvold2002435].
Encoding information on single quanta
=====================================
So far, there is evidence that any kind of “will- and useful” classical or quantum information in terms of nonrandom bit(stream)s can be transferred from some space-time point $A$ to another space-time point $B$ only [*via*]{} individual quanta: single quanta are emitted at some space-time point $A$, and absorbed at another space-time point $B$. This is true, in particular, for quantum teleportation; that is, the entanglement assisted transmission of quantum information from one location to another. Thus we shall first concentrate on the generation of quantum space time – that is, on the construction of clocks and scales based upon quantum processes yielding space-time frames, as well as on their transformations by of a direct bit exchange. Issues often referred to as quantum “nonlocality” and entanglement are relegated to the next section.
Time scales
-----------
If indeed one takes seriously the idea that “quanta can be utilized to create space-time frames,” then we need to base space and time scales used in such frames on quantum mechanical entities, that is, on quantum clocks and on quantum scales.
Formally, by Cayley’s representation theorem the unitary quantum evolution can be represented by some subgroup of the symmetric group. One approach to quantum clocks and time might thus be to consider general distances and metrics on permutations, in particular, on the symmetric groups, thereby relating changes in quantum states to time.
Indeed, the current definition of the second in the International System (SI) of units is [*via*]{} 9 192 631 770 transitions between two orthogonal quantum states of a caesium 133 atom. That is, if we encode the two ground states by the subspaces spanned by the two orthogonal vectors $\vert \psi_0 \rangle \equiv (0,1)$ and $\vert \psi_1 \rangle \equiv (1,0)$, \[or, equivalently, by the projectors $\text{diag}(0,1)$ and $\text{diag}(1,0)$\] in two-dimensional Hilbert space, then the 9 192 631 770’th fraction of a second is delivered by the unitary operator that is known as the [*not gate*]{} [@mermin-07] $\textsf{\textbf{X}}
= \begin{pmatrix}0& 1\\1&0
\end{pmatrix}
$, representing a single permutation-transition $\textsf{\textbf{X}}\vert \psi_i \rangle $ between $\vert \psi_i \rangle \leftrightarrow \vert \psi_{i\oplus 1} \rangle $, $i\in \{ 0, 1\}$, of two orthogonal quantum states of a caesium 133 atom.
Space scales
------------
The current definition of spatial distances in the International System of units is in terms of the propagation of light quanta in vacuum. More specifically, the metre is the length of the path travelled by light in vacuum during a time interval of 1/299 792 458’th part of a second – or, equivalently, as light travels 299 792 458 metres per second, a duration in which 9 192 631 770 transitions between two orthogonal quantum states of a caesium 133 atom occur – during 9 192 631 770/299 792 458 $\approx 31$ transitions of two orthogonal quantum states of a caesium 133 atom.
More generally we may ask what, exactly, is a [*“spatial distance?”*]{} In particular, what quantum meaning can be ascribed to a “path travelled by light in vacuum?” First and foremost, any spatial distance seems to depend on two criteria: (i) separateness, or disconnectedness; as well as (ii) the capacity to (inter-)connect. The latter connection must, by quantum rules, be mediated [*via*]{} [*permutations.*]{} In the simplest sense, one could algorithmically model such a contact transmission by reversible [*cellular automata*]{} [@fredkin; @svozil-1996-time; @thooft-2013]; that is, by a tesselated, three-dimensional, discrete computation space [@zuse-67] constantly permuting itself.
Alexandrov-Zeemann theorem
--------------------------
In order to make operational sense without regress to absolute space-time frames, the SI definition of length implicitly assumes that the velocity of light in vacuum for all space-time frames is constant, regardless of the state of motion of that frame [@peres-84]. By these assumptions and other conventions, such as Einstein’s definition of simultaneity [@ein-05] and bijectivity of coordinate transformations, the Lorentz transformations are essentially (up to shift-translations and dilations with positive scalar constants) a consequence of the Alexandrov-Zeemann theorem of incidence geometry [@alex3; @zeeman; @lester; @naber]. Pointedly stated, if two observers “presiding over their reference frames agree” [@naber] that points connected by light rays can be interconnected, then linear transformations of space-time frames follow.
From a purely formal point of view, fixing the invariance (constancy) of the velocity of light with respect to changes of space-time frames appears to be purely conventional, and thus may be even considered as arbitrary and [*a priori*]{} unjustified, if not misleading. Any other velocity, both sub- as well as superluminal – even associated with no-signalling correlated events such as from phased arrays (see below) – would suffice for the construction of transformations between space-time frames.
The physical motivation for choosing light in vacuum is twofold: First, the [*form invariance*]{} of the equations of motion, such as Maxwell’s equation in vacuum, is a convenience. And secondly, all space-time frames correctly reflect the causality relative to the electromagnetic interaction.
It is thus suggested to “stay within a single type of interaction” when it comes to the construction of clocks and scales, and also to fix the invariance of the respective signals for the construction of space-time frames, as well as the transformation laws between them. The resulting space-time is defined means relative to (the causality induced by) this interaction [@svozil-relrel]. In this sense, the SI definition renders a space-time with is means relative to the electromagnetism.
Encoding information across quanta
==================================
Entanglement characteristics
----------------------------
At the time of conceptualizing special relativity theory, quantum mechanics was in its infancy, and quantum effects were therefore not considered for the definition of space-time scales. Alas, this has changed since Schrödinger pointed out the possibility of entangled quantum states of multipartite quantized systems; states that do not have any classical local counterpart. Entanglement is characterized by an encoding of (classical) information “across quanta” [@zeil-99; @zeil-Zuk-bruk-01; @svozil-2002-statepart-prl] that defy any kind of spatial apartness or locality, and yield experimental violations [@wjswz-98] of classical probabilities [@pitowsky]. These features alone suggest to reconsider quantum mechanical processes for the definition of space-time frames.
One of the characteristics of quantum entanglement is that information is not encoded in the single quanta which constitute an entangled system. Therefore, through context translation, any enquiry about the state of a single quantum is futile, because no such information is available prior to this “forced measurement.” The archetypical example of this situation is the Bell state $\vert \Psi_- \rangle = \left( 1/\sqrt{2} \right) \left(\vert +-\rangle - \vert -+\rangle \right)$. On the one hand, $\vert \Psi_- \rangle$ is totally and irreducible indeterminate about the states $\vert -\rangle$ or $\vert +\rangle$ of its individual two constituents. Indeed a “forced measurement” yields random outcomes [@svozil-qct]; and the concatenation of independent outcomes encoded as a binary sequence can, for instance, be expected to be Borel normal [@svozil-2006-ran; @PhysRevA.82.022102]; in particular, there is a 50:50 chance for $\vert -\rangle$ and $\vert +\rangle$, respectively. On the other hand, $\vert \Psi_- \rangle$ is totally determined by the joint correlations of the particles involved; in particular, by the two propositions [*“the spin states of the two particles along two orthogonal spatial directions are different”*]{} [@Zeilinger-97; @zeil-99; @svozil-2002-statepart-prl].
Alas, in this view, for the Bell state as well as for other nonlocalized multipartite entangled states, in which the constituents can be thought of as “torn apart” arbitrary spatial distances, there is no “spooky action at a distance” [@Nikolic] whatsoever, because the multiple constituents, if they become separated and “drift away” from their joint space-time preparation regions, do so at speeds not exceeding the velocity of light; with no further communication or information exchange between them.
Thereby, any greater-than-classical correlations and expectations these constituents carry are due to the particular type of quantum probabilities. Recall that the quantum probabilities are generalizations of classical probabilities: Due to Gleason’s theorem the Born rule can be derived from the noncontextual pasting of blocks of subalgebras (that is, maximal, co-measurable observables); whereas all classical probability distributions result from convex sums of two-valued states on the Boolean algebra of classical propositions.
Pointedly stated, the so-called “quantum nonlocality” is not non-local at all, because these correlations reside in the (entangled) quantum states which must be perceived holistically (as being one compound state) rather than as being constructed from separate single quantum states; regardless of the spatial separation of the constituent quanta forming such states. The measurements in spatially different regions (regardless of whether they are space-like separated or not) just recover this property encoded in the quantum states; thereby nothing needs to be exchanged, nor can information be gained in excess of the one encoded by the state preparation.
There exist even quasi-classical models (which are nonlocal as they require the exchange of one bit per particle pair) capable of realizing stronger-than-quantum correlations [@svozil-2004-brainteaser]. Claims that these larger-than-classical correlations expresses some kind of “spooky action at a distance” mistake correlation for causality. In this regard, the terminology “peaceful coexistence” [@shimony-78] between quantum theory and special relativity, suggesting or even implying some perceivable kind of inconsistency between them, is misleading, because there cannot occur any kind of “clash” or inconsistency between fundamental observables and processes and any entities, such as space-time, which are secondary constructions of the mind, based on the former observables and processes.
Quantum statistics
------------------
The remaining discussion is very speculative and should not be taken as claiming the existence of any faster-than-light signalling.
0.7mm
(175,50)(20,0) (100,20) (105,20)[(1,0)[50]{}]{} (95,20)[(-1,0)[50]{}]{} (120,30) (80,30) (122.5,30) (77.5,30) (125,30) (75,30) (127.5,30) (72.5,30) (130,30) (70,30) (132.5,30) (67.5,30) (32.25,20)[(20,20)\[l\]]{} (32.5,10)[(0,1)[20]{}]{} (27.5,20)[(0,0)\[cc\][$D$]{}]{} (45.5,20)[(-1,0)[.07]{}]{}(55.5,20)[(-1,0)[20]{}]{} (154.5,20)[(1,0)[.07]{}]{}(144.5,20)[(1,0)[20]{}]{} (96.388,16.608)(.036812183,.033614213)[197]{}[(1,0)[.036812183]{}]{} (103.503,16.44)(-.033615023,.033615023)[213]{}[(0,1)[.033615023]{}]{} (100.056,10.076)[(0,0)\[cc\][$S$]{}]{} (155,15)
------------------------------------------------------------------------
(170,20)[(0,0)\[cc\][$L$]{}]{} [(165,32)[(0,1)[.07]{}]{}(165,7)[(0,-1)[.07]{}]{}(165,7)[(0,1)[25]{}]{} ]{}
Suppose the constituent quanta of an entangled state are subjected to [*active*]{} stimulation rather than passive measurement. In particular, multi-partite quantum statistics can give rise to stimulated emission or absorption. For the sake of an attack [@svozil-slash] on local causality, consider the delayed choice of, say, either scattering a photon into a “box of identical photons” (or directing an electron into a region filled with other electrons occupying certain states attainable by the original electron), or passing this region without any other identical quanta, as depicted in Fig.\[2013-st1-dcsea\]. One might speculate that such a device might be used to communicate a message across the particle pair through controlling the outcome on one side, thereby [*spoiling outcome independence,*]{} because if some agent has free will to “induce” some state of one photon of a photon pair in an entangled singlet state, the other photon has no (random) choice any longer but to scatter into the corresponding state. One interesting way to argue against such a scenario is by pretending that the source “(en)forces” certain statistical properties of the single constituent particles – in particular their stochastic behaviour – of an entangled state even beyond the standard quantum predictions [@zeil-99; @svozil-2013-omelette].
Another possibility would be to transmit information across spatially extended quantum states of a large number of particles by affecting the statistical constraints on one side and observing the effects on the other end. For the sake of a concrete example consider a superconducting rod which is heated into the nonsuperconducting state (or otherwise “destroying it”) on one end of the rod, and observing the gap energy on the other end.
We will turn our attention now to “second quantization” effects on single (nonentangled) quanta; in particular, with regard to propagation. They are due to the presence of (spontaneous or controlled) many-partite excitations of the quantized fields involved.
Field theoretic models of signal propagation
============================================
When considering the propagation of light and other potential signals in vacuum [@einstein-aether; @dirac-aether], which will be considered as a [*signal carrier*]{}, there appear to exist at least two alternative conceptions. First, we could assume that light is “attenuated” by polarization and other (e.g., quantum statistical) effects. Without any such interactions such signals might travel arbitrarily fast. Thus, in order to increase signalling speeds, we must attempt to disentangle the signal from interacting with the vacuum. A somewhat related scenario is the hypothetical possibility to “shift gear” to another, less retarding, mode of propagation by (locally) changing the state of the signal carrier; for instance by supercavitation.
A second, entirely different, viewpoint may be that light needs a carrier for propagation; very much like a phonon needs, or rather subsumes, collective excitations of some carrier medium. In such scenarios, stronger couplings might result in higher signalling speeds.
If any such speculation will eventually yield superluminal communication and space travel is highly uncertain, but should not be outrightly excluded for the mere sake of orthodoxy. In what follows we briefly mention some possible directions of looking into these issues.
Multiple side hopping
---------------------
The capacity to transfer information can be modelled by some sort of interconnection between different spatial regions. One such microphysical model is the vibrating (linear) chain [@Henley-Thirring-EQFT Sec. 1.2] which requires some coupled (linearized) oscillators. The spatial signal carrier is modelled by an interconnected array of coupled oscillators. Thereby, (the energy of) an excitation is transferred from one oscillator to the next by the coupling between the two.
One possibility to change the resulting signal velocity would be to assume that any oscillator is coupled not only to its next neighbour, but to other oscillators which are farther apart but nevertheless topologically interconnected. In this way, by increasing the “hopping distance,” say, in a periodic medium, as depicted in Fig. \[2013-st1-msh\], faster modes of propagation (as compared to single side hopping) seem conceivable.
We suggest to employ [*phased array*]{} (radar) with faster-than-light synchronization, such as the one enumerated in Table \[2013-tablest1-msh\], of electrical signals for the exploration of multiple side hopping and the resulting higher order harmonics $2c, 3c,\ldots $ of the velocity of light $c$. Thereby, the signals generated by the phased array of electrical charges might resonate with the propagation modes of the field carrying those collective excitations. For random hopping distances, any such discretization cannot be expected.
In that way, one is not approaching any (supposedly impenetrable) speed-of-light barrier “from below” (i.e., with subluminal speeds) but attempts to induce carrier excitations at almost arbitrary velocities. We emphasize that the issue of whether or not the vacuum can actually carry such signals is a highly speculative suggestion that outrightly contradicts long-held beliefs, but remains empirically undecided and unknown.
0.5mm
(301,127.817)(0,0) [(0,10)[(1,0)[50]{}]{} (50,10)[(1,0)[50]{}]{} (99.962,10)[(1,0)[50]{}]{} (150,10)[(1,0)[50]{}]{} (200,10)[(1,0)[50]{}]{} (250,10)[(1,0)[50]{}]{} ]{} (0,10) (50,10) (100,10) (150,10) (200,10) (250,10) (300,10) [(100.067,15.977)[(3,-2)[.07]{}]{}(0,15.977)[(-3,-2)[.07]{}]{}(0.42,15.977)(50.875,49.613)(100.067,15.977) (200.135,15.977)[(3,-2)[.07]{}]{}(100.067,15.977)[(-3,-2)[.07]{}]{}(100.067,15.977)(150.942,49.613)(200.135,15.977) (300.202,15.977)[(3,-2)[.07]{}]{}(200.135,15.977)[(-3,-2)[.07]{}]{}(200.135,15.977)(251.009,49.613)(300.202,15.977)]{} [ (150.101,17.238)[(4,-3)[.07]{}]{}(.42,17.238)[(-4,-3)[.07]{}]{}(.42,17.238)(74.42,74.42)(150.101,17.238) (300.202,17.659)[(4,-3)[.07]{}]{}(150.521,17.659)[(-4,-3)[.07]{}]{}(150.521,17.659)(224.521,74.84)(300.202,17.659)]{} [ (200.135,19.341)[(4,-3)[.07]{}]{}(.42,19.341)[(-4,-3)[.07]{}]{}(.42,19.341)(99.857,91.658)(200.135,19.341)]{} [ (250.168,20.602)[(4,-3)[.07]{}]{}(.42,20.602)[(-3,-2)[.07]{}]{}(.42,20.602)(137.908,117.306)(250.168,20.602)]{} [ (298.94,22.704)[(4,-3)[.07]{}]{}(.42,22.704)[(-3,-2)[.07]{}]{}(.42,22.704)(152.203,127.817)(298.94,22.704)]{} (0,0)[(0,0)\[cc\][$1$]{}]{} (50,0)[(0,0)\[cc\][$2$]{}]{} (100,0)[(0,0)\[cc\][$3$]{}]{} (150,0)[(0,0)\[cc\][$4$]{}]{} (200,0)[(0,0)\[cc\][$5$]{}]{} (250,0)[(0,0)\[cc\][$6$]{}]{} (300,0)[(0,0)\[cc\][$7$]{}]{} (25,16)[(0,0)\[cc\][$a=2-1$]{}]{} (50,26)[(0,0)\[cc\][$2a$]{}]{} (75,40)[(0,0)\[cc\][$3a$]{}]{} (100,50.5)[(0,0)\[cc\][$4a$]{}]{} (135,63)[(0,0)\[cc\][$5a$]{}]{} (150,81)[(0,0)\[cc\][$6a$]{}]{}
------------------- --- --- --- --- --- --- --- ---------- -- -- --
array site 1 2 3 4 5 6 7 $\cdots$
[*versus*]{} time
c=1 1 0 0 0 0 0 0 $\cdots$
0 1 0 0 0 0 0 $\cdots$
0 0 1 0 0 0 0 $\cdots$
0 0 0 1 0 0 0 $\cdots$
0 0 0 0 1 0 0 $\cdots$
0 0 0 0 0 1 0 $\cdots$
0 0 0 0 0 0 1 $\cdots$
c=2 1 0 0 0 0 0 0 $\cdots$
0 0 1 0 0 0 0 $\cdots$
0 0 0 0 1 0 0 $\cdots$
0 0 0 0 0 0 1 $\cdots$
c=3 1 0 0 0 0 0 0 $\cdots$
0 0 0 1 0 0 0 $\cdots$
0 0 0 0 0 0 1 $\cdots$
c=4 1 0 0 0 0 0 0 $\cdots$
0 0 0 0 1 0 0 $\cdots$
c=5 1 0 0 0 0 0 0 $\cdots$
0 0 0 0 0 1 0 $\cdots$
c=6 1 0 0 0 0 0 0 $\cdots$
0 0 0 0 0 0 1 $\cdots$
------------------- --- --- --- --- --- --- --- ---------- -- -- --
: (Color online) Array synchronization for speculative multiple side hopping. In this discrete setup, the fundamental time unit is the time it takes for (the slowest) light signal in vacuum to propagate (“hop”) one fundamental spatial unit $a$. []{data-label="2013-tablest1-msh"}
Change of vacuum
----------------
Another possibility to change the propagation velocity of the signal carrier would be to alter its ability to carry a signal through attenuation and amplification of the processes responsible for sinalling. The most direct form would be to change the coupling between oscillators in the vibrating chain scheme mentioned earlier.
Another possibility would be to again use quantum statistical effects to reduce or increase the polarizability of the vacuum by placing bosons or fermions along the signalling path. A photon, for instance, seems to become accelerated if polarizability is reduced [@Scharnhorst-1998; @svozil-putz-sol].
Dimensionality
==============
One could speculate that the apparent three-dimensionality of physical configuration space is a reflection of the [*three-dimensional interconnection*]{} of the signal carrier of this universe on a very fundamental level. In this way, information is “permuted by point contact from one node to the other.” A discrete version of this would be a three-dimensional cellular automaton.
In another scenario the intrinsic, operational three-dimensionality is a (fractal) “shadow” on a higher dimensional signal carrier [@sv4]. In this view, if there is no “bending (yielding nontrivial topologies), folding or compactification” of the extra dimensions involved, information transfer might become even “slower” than in the lower dimensional case, since every extra dimension is nothing but an extra degree of freedom the bit can pursue, thereby even “getting lost” if, say, it travels a direction orthogonal to, or in other ways inaccessible for, physical three-space. On the other hand, if this fractal shadow constituting our accessible configuration space can be bent or even intersected by itself in topologically nontrivial ways, then information transfer, and thus signalling and space travel, from any point $A$ to any other point $B$ could in principle be obtained with arbitrary velocities.
Concluding remarks
==================
The short answer of the question of whether quantum space-time is different from classical space-time is this: since, according to the Alexandrov-Zeemann theorem, bijective space-time transformations are essentially determined by the [*causal ordering of events*]{}, any difference of classical [*versus*]{} quantum space-time can be reduced to the question of whether or not quantum events can be causally ordered differently than classical ones. Until now there is not the slightest indication that this is the case, so there is no evidence of any difference between classical and quantum space-time. However, there are [*caveats*]{} to this answer: certain processes, such as the ones discussed earlier, may give rise to a different quantum ordering, and thus to different space-times.
With respect to considerations regarding space-time as a construction based on empirical events, any attempt to unify gravity as a “geometrodynamic theory of curved space-time” [*on a par*]{} with the standard quantum field theories must inevitably fail: if space and time emerge as secondary “ordering” concepts based on our primary experience of quanta (e.g. detector clicks), they cannot be treated on an equal footing with these phenomena. Thus, if the equivalence principle “equating” inertial with gravitational mass is correct, one could speculate that the resulting geometrodynamic theory of gravity needs to be based upon some field theoretic effects accounting for this equivalence; such as “metrical elasticity” through vacuum quantum fluctuations [@Sakharov-67].
Beyond electromagnetic and gravitational interactions, other “fundamental” (strong, weak) interactions have been discovered, which, according to the standard unification model, propagate at the same speed as light, although no direct empirical evidence is available. In any case, [*a priori*]{}, different interactions need not always propagate with the same velocity, making necessary a sort of “relativized relativity” [@svozil-relrel] that has to cope with consistency issues, such as the “grandfather paradox.” The latter one is also resolved in “quantum time travelling” scenarios [@svozil-greenberger-2005].
Insofar multipartite and field theoretic considerations apply, it is prudent to distinguish on the one hand between the physical vacuum, which possesses some properties relevant for signal propagation; and, on the other hand, space-time frames, which are constructions based on and “tied to” some idealized physical properties of vacuum. One such typical assumption entering the formal derivation of the transformation properties of inertial space time frames is the constancy of the velocity of light in vacuum, regardless of the state of inertial motion of any observer.
This research has been partly supported by FP7-PEOPLE-2010-IRSES-269151-RANPHYS.
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|
---
abstract: 'The circumcircle of a planar convex polygon $P$ is a circle $C$ that passes through all vertices of $P$. If such a $C$ exists, then $P$ is said to be cyclic. Fix $C$ to have unit radius. While any two angles of a uniform cyclic triangle are negatively correlated, any two sides are independent. In contrast, for a uniform cyclic quadrilateral, any two sides are negatively correlated, whereas any two adjacent angles are uncorrelated yet dependent.'
author:
- Steven Finch
date: 'October 3, 2016'
title: Random Cyclic Quadrilaterals
---
To generate a cyclic triangle is easy: select three independent uniform points on the unit circle and connect them. To generate a cyclic quadrilateral is harder: select four such points and connect them in, say, a counterclockwise manner. Convexity follows immediately [@Pi-quadrilat], as does the fact that opposite angles are supplementary [@M1-quadrilat; @M2-quadrilat]. The inter-relationship of adjacent angles is more mysterious, as we shall soon see. Our initial focus, however, will be on adjacent sides, opposite sides and diagonals.
Let the four vertices be given by $\exp(i\,\theta_{k})$, where $i$ is the imaginary unit, $0\leq\theta_{1}<$ $\theta_{2}<\theta_{3}<\theta_{4}<2\pi$ are central angles relative to the horizontal axis, and $1\leq k\leq4$. Define $\theta_{0}=\theta_{4}-2\pi$ and $\theta_{5}=\theta_{1}+2\pi$ for convenience, then polygonal sides $s_{k}$ and polygonal angles $\alpha_{k}$ are given by$$\begin{array}
[c]{ccc}s_{k}=2\sin\left( \dfrac{\theta_{k}-\theta_{k-1}}{2}\right) , & &
\alpha_{k}=\dfrac{\theta_{k+1}-\theta_{k-1}}{2}.
\end{array}$$ Proof of the $s_{k}$ expression comes from the Law of Cosines and a half angle formula:$$\begin{aligned}
s_{k}^{2} & =1+1-2\cdot1\cdot1\cos(\theta_{k}-\theta_{k-1})=2\left[
1-\cos(\theta_{k}-\theta_{k-1})\right] \\
& =4\,\frac{1-\cos(\theta_{k}-\theta_{k-1})}{2}=4\sin^{2}\left(
\dfrac{\theta_{k}-\theta_{k-1}}{2}\right) .\end{aligned}$$ Proof of the $\alpha_{k}$ expression follows the fact that an inscribed angle is one-half the length of its intercepted circular arc. The polygonal diagonals $d_{k}$ clearly satisfy $d_{k}=2\sin(\alpha_{k})$. Let also $\omega$ denote the smaller of the two angles at the intersection point between the diagonals.
[quadrilat.eps]{}
Our labor draws upon the distribution of the order statistics $\theta_{1}$, $\theta_{2}$, $\theta_{3}$, $\theta_{4}$. We must be careful in summarizing the results because, while $s_{2}$, $s_{3}$, $s_{4}$ possess the same distribution, the one corresponding to $s_{1}$ is different. Hence, to make statements regarding arbitrary sides $s$, $t$, $u$, $v$ of the quadrilateral, we must use a $(3/4,1/4)$-mixture of densities. Likewise, $\alpha_{2}$ and $\alpha_{1}$ possess distinct distributions. Thus, to make statements regarding arbitrary adjacent angles $\alpha$, $\beta$ of the quadrilateral, we must use a $(1/2,1/2)$-mixture of densities.
A probabilistic analysis of the perimeter $s+t+u+v$ and area $2\sin
(\alpha)\sin(\beta)\sin(\omega)$ is beyond our current capabilities. Hopefully the groundwork established here will be a launching point for someone else’s research in the near future.
Sides
=====
Let $X_{1}<X_{2}<X_{3}<X_{4}$ denote the order statistics for a random sample of size $4$ from the uniform distribution on $[0,1]$. The density for $(X_{1},X_{2})=(x,y)$ is [@Gi-quadrilat; @HN-quadrilat]$$\left\{
\begin{array}
[c]{lll}12(1-y)^{2} & & \text{if }0<x<y<1,\\
0 & & \text{otherwise;}\end{array}
\right.$$ the density for $(X_{1},X_{3})=(x,y)$ is$$\left\{
\begin{array}
[c]{lll}24(y-x)(1-y) & & \text{if }0<x<y<1,\\
0 & & \text{otherwise;}\end{array}
\right.$$ the density for $(X_{1},X_{4})=(x,y)$ is$$\left\{
\begin{array}
[c]{lll}12(y-x)^{2} & & \text{if }0<x<y<1,\\
0 & & \text{otherwise;}\end{array}
\right.$$ the density for $(X_{2},X_{4})=(x,y)$ is$$\left\{
\begin{array}
[c]{lll}24x(y-x) & & \text{if }0<x<y<1,\\
0 & & \text{otherwise.}\end{array}
\right.$$ Consider the transformation $(x,y)\mapsto(y-x,y)=(u,v)$. Since this has Jacobian determinant $1$ and since $0<u<v<1$, it follows that the density for $X_{2}-X_{1}$ is$$12{\displaystyle\int\limits_{u}^{1}}
(1-v)^{2}dv=\left. -4(1-v)^{3}\right\vert _{u}^{1}=4(1-u)^{3};$$ the density for $X_{3}-X_{1}$ is$$24{\displaystyle\int\limits_{u}^{1}}
u(1-v)dv=\left. -12u(1-v)^{2}\right\vert _{u}^{1}=12u(1-u)^{2};$$ the density for $X_{4}-X_{1}$ is$$12{\displaystyle\int\limits_{u}^{1}}
u^{2}dv=\left. -12u^{2}(1-v)\right\vert _{u}^{1}=12u^{2}(1-u);$$ the density for $X_{4}-X_{2}$ is$$24{\displaystyle\int\limits_{u}^{1}}
u(v-u)dv=\left. 12u(v-u)^{2}\right\vert _{u}^{1}=12u(1-u)^{2}.$$ We disregard $X_{4}-X_{2}$ further since its distribution is the same as that for $X_{3}-X_{1}$. Consider the scaling $u\mapsto\pi\,u=x$. It follows that$$\begin{array}
[c]{ccc}\text{the density for }\dfrac{\theta_{2}-\theta_{1}}{2}\text{ is} & &
\dfrac{4}{\pi}\left( 1-\dfrac{x}{\pi}\right) ^{3};\\
\text{the density for }\dfrac{\theta_{3}-\theta_{1}}{2}\text{ is} & &
\dfrac{12}{\pi}\left( \dfrac{x}{\pi}\right) \left( 1-\dfrac{x}{\pi}\right)
^{2};\\
\text{the density for }\dfrac{\theta_{4}-\theta_{1}}{2}\text{ is} & &
\dfrac{12}{\pi}\left( \dfrac{x}{\pi}\right) ^{2}\left( 1-\dfrac{x}{\pi
}\right) .
\end{array}$$ Next, the function $x\mapsto\sin(x)=y$ possesses two preimages $\arcsin(y) $ and $\pi-\arcsin(y)$ in the interval $[0,\pi]$ and has derivative $\cos(x)=\sqrt{1-y^{2}}$. It follows that the three densities are [@Pa-quadrilat] $$\dfrac{4}{\pi\sqrt{1-y^{2}}}\left[ \left( 1-\dfrac{\arcsin(y)}{\pi}\right)
^{3}+\left( \dfrac{\arcsin(y)}{\pi}\right) ^{3}\right] ,$$$$\dfrac{12}{\pi\sqrt{1-y^{2}}}\left[ \left( \dfrac{\arcsin(y)}{\pi}\right)
\left( 1-\dfrac{\arcsin(y)}{\pi}\right) ^{2}+\left( 1-\dfrac{\arcsin
(y)}{\pi}\right) \left( \dfrac{\arcsin(y)}{\pi}\right) ^{2}\right] ,$$$$\dfrac{12}{\pi\sqrt{1-y^{2}}}\left[ \left( \dfrac{\arcsin(y)}{\pi}\right)
^{2}\left( 1-\dfrac{\arcsin(y)}{\pi}\right) +\left( 1-\dfrac{\arcsin
(y)}{\pi}\right) ^{2}\left( \dfrac{\arcsin(y)}{\pi}\right) \right]$$ respectively. The second and third expressions are identical. Finally, the scaling $y\mapsto2\,y=z$ and an algebraic expansion gives the density for $s_{2}$ as$$\dfrac{4}{\pi\sqrt{4-z^{2}}}\left[ 1-\frac{3\arcsin(\frac{z}{2})\left(
\pi-\arcsin(\frac{z}{2})\right) }{\pi^{2}}\right]$$ and the density for both $d_{2}$ and $s_{1}$ as$$\dfrac{4}{\pi\sqrt{4-z^{2}}}\left[ 0+\frac{3\arcsin(\frac{z}{2})\left(
\pi-\arcsin(\frac{z}{2})\right) }{\pi^{2}}\right] .$$ We omit details for $s_{3}$, $s_{4}$ (same as $s_{2}$) and $d_{1}$ (same as $d_{2}$).
Mixing the densities for $s_{2}$ (with weight $3/4$) and for $s_{1}$ (with weight $1/4$), the density for an arbitrary side $0\leq s\leq2$ emerges:$$\dfrac{3}{\pi\sqrt{4-s^{2}}}\left[ 1-\frac{2\arcsin(\frac{s}{2})\left(
\pi-\arcsin(\frac{s}{2})\right) }{\pi^{2}}\right]$$ which implies that$$\begin{array}
[c]{ccc}\operatorname*{E}\left( s\right) =\dfrac{6}{\pi}-\dfrac{24}{\pi^{3}}, & &
\operatorname*{E}\left( s^{2}\right) =2-\dfrac{3}{\pi^{2}}.
\end{array}$$ The corresponding moments for a diagonal $0\leq d\leq2$ are $48/\pi^{3}$ and $2+6/\pi^{2}$. Joint moments are available via the joint density of $\theta_{1}$, $\theta_{2}$, $\theta_{3}$, $\theta_{4}$:$$\left\{
\begin{array}
[c]{lll}\dfrac{4!}{(2\pi)^{4}} & & \text{if }0\leq\theta_{1}<\theta_{2}<\theta
_{3}<\theta_{4}<2\pi,\\
0 & & \text{otherwise.}\end{array}
\right.$$ For example,$$\begin{aligned}
\operatorname*{E}\left( s_{2}s_{3}\right) & =\frac{3}{2\pi^{4}}{\displaystyle\int\limits_{0}^{2\pi}}
{\displaystyle\int\limits_{\theta_{1}}^{2\pi}}
{\displaystyle\int\limits_{\theta_{2}}^{2\pi}}
{\displaystyle\int\limits_{\theta_{3}}^{2\pi}}
\left[ 2\sin\left( \dfrac{\theta_{2}-\theta_{1}}{2}\right) \right] \left[
2\sin\left( \dfrac{\theta_{3}-\theta_{2}}{2}\right) \right] d\theta
_{4}d\theta_{3}d\theta_{2}d\theta_{1}\\
& =\dfrac{48}{\pi^{2}}-\dfrac{384}{\pi^{4}}$$ (same for $\operatorname*{E}\left( s_{3}s_{4}\right) $) and $$\begin{aligned}
\operatorname*{E}\left( s_{1}s_{2}\right) & =\frac{3}{2\pi^{4}}{\displaystyle\int\limits_{0}^{2\pi}}
{\displaystyle\int\limits_{\theta_{1}}^{2\pi}}
{\displaystyle\int\limits_{\theta_{2}}^{2\pi}}
{\displaystyle\int\limits_{\theta_{3}}^{2\pi}}
\left[ 2\sin\left( \dfrac{\theta_{4}-\theta_{1}}{2}\right) \right] \left[
2\sin\left( \dfrac{\theta_{2}-\theta_{1}}{2}\right) \right] d\theta
_{4}d\theta_{3}d\theta_{2}d\theta_{1}\\
& =-\dfrac{24}{\pi^{2}}+\dfrac{384}{\pi^{4}}$$ (same for $\operatorname*{E}\left( s_{4}s_{1}\right) $) imply that, for arbitrary adjacent sides $s$ and $t$,$$\begin{array}
[c]{ccc}\operatorname*{E}\left( s\,t\right) =\dfrac{12}{\pi^{2}}, & &
\rho(s,t)\approx-0.183.
\end{array}$$ We used $$\dfrac{\theta_{1}-\theta_{0}}{2}=\dfrac{\theta_{1}-\left( \theta_{4}-2\pi\right) }{2}=\dfrac{2\pi-\left( \theta_{4}-\theta_{1}\right) }{2}=\pi-\dfrac{\theta_{4}-\theta_{1}}{2}$$ and $\sin(\pi-z)=\sin(z)$ in writing the preceding integral. The same value $12/\pi^{2}$ is also obtained for the expected product of arbitrary opposite sides $s$ and $t$. The proximity of quadrilateral sides is (evidently) immaterial when assessing their correlation.
[sides.eps]{}
[diagonals.eps]{}
Angles
======
By our work starting with $X_{3}-X_{1}$ and $X_{4}-X_{2}$, it is clear that $\alpha_{2}$ and $\alpha_{3}$ are identically distributed. Since $\alpha
_{1}=\pi-\alpha_{3}$, the density of $\alpha_{2}$ is $12x(\pi-x)^{2}/\pi^{4}$ while the density of $\alpha_{1}$ is $12x^{2}(\pi-x)/\pi^{4}$. Mixing the densities for $\alpha_{2}$ and for $\alpha_{1}$ with equal weighting, the marginal density for an arbitrary angle $0\leq\alpha\leq\pi$ becomes $6x(\pi-x)/\pi^{3}$.
We need, however, to find the joint distribution for arbitrary adjacent angles $\alpha$ and $\beta$. A fresh approach for obtaining this involves the Dirichlet$(1,1,1;1)$ distribution on a $3$-dimensional simplex [@Ra-quadrilat; @PC-quadrilat; @NT-quadrilat]:$$\left\{
\begin{array}
[c]{lll}6 & & \text{if }0<\xi_{1}<1,\text{ }0<\xi_{2}<1,\text{ }0<\xi_{3}<1\text{ and
}\xi_{1}+\xi_{2}+\xi_{3}<1,\\
0 & & \text{otherwise}\end{array}
\right.$$ and calculation of the joint density for $\eta_{1}=\xi_{1}+\xi_{2}$, $\eta
_{2}=\xi_{2}+\xi_{3}$. The list $\xi_{1}$, $\xi_{2}$, $\xi_{3}$ can be thought of as duplicating any one of the eight lists given in Table 1, each weighted with probability $1/8$. In words, up to the preservation of adjacency of angles $\pi\,\eta_{1}$, $\pi\,\eta_{2}$, any implicit ordering within $\xi_{1}$, $\xi_{2}$, $\xi_{3}$ has been removed. This formulation will simplify our work, removing the need to mix distributions (like before) as a concluding step.
Table 1. *Eight possibilities for* $\xi_{1}$, $\xi_{2}$, $\xi_{3}$*.*$$\begin{tabular}
[c]{|c|c|}\hline
Candidate Lists & Resulting Angles\\\hline
$\begin{array}
[c]{ccccc}\dfrac{\theta_{2}-\theta_{1}}{2\pi}, & & \dfrac{\theta_{3}-\theta_{2}}{2\pi
}, & & \dfrac{\theta_{4}-\theta_{3}}{2\pi}\end{array}
$ & $\begin{array}
[c]{ccc}\pi\,\eta_{1}=\alpha_{2}, & & \pi\,\eta_{2}=\alpha_{3}\end{array}
$\\\hline
$\begin{array}
[c]{ccccc}\dfrac{\theta_{4}-\theta_{3}}{2\pi}, & & \dfrac{\theta_{3}-\theta_{2}}{2\pi
}, & & \dfrac{\theta_{2}-\theta_{1}}{2\pi}\end{array}
$ & $\begin{array}
[c]{ccc}\pi\,\eta_{1}=\alpha_{3}, & & \pi\,\eta_{2}=\alpha_{2}\end{array}
$\\\hline
$\begin{array}
[c]{ccccc}\dfrac{\theta_{3}-\theta_{2}}{2\pi}, & & \dfrac{\theta_{4}-\theta_{3}}{2\pi
}, & & \dfrac{\theta_{5}-\theta_{4}}{2\pi}\end{array}
$ & $\begin{array}
[c]{ccc}\pi\,\eta_{1}=\alpha_{3}, & & \pi\,\eta_{2}=\alpha_{4}\end{array}
$\\\hline
$\begin{array}
[c]{ccccc}\dfrac{\theta_{5}-\theta_{4}}{2\pi}, & & \dfrac{\theta_{4}-\theta_{3}}{2\pi
}, & & \dfrac{\theta_{3}-\theta_{2}}{2\pi}\end{array}
$ & $\begin{array}
[c]{ccc}\pi\,\eta_{1}=\alpha_{4}, & & \pi\,\eta_{2}=\alpha_{3}\end{array}
$\\\hline
$\begin{array}
[c]{ccccc}\dfrac{\theta_{4}-\theta_{3}}{2\pi}, & & \dfrac{\theta_{5}-\theta_{4}}{2\pi
}, & & \dfrac{\theta_{2}-\theta_{1}}{2\pi}\end{array}
$ & $\begin{array}
[c]{ccc}\pi\,\eta_{1}=\alpha_{4}, & & \pi\,\eta_{2}=\alpha_{1}\end{array}
$\\\hline
$\begin{array}
[c]{ccccc}\dfrac{\theta_{2}-\theta_{1}}{2\pi}, & & \dfrac{\theta_{5}-\theta_{4}}{2\pi
}, & & \dfrac{\theta_{4}-\theta_{3}}{2\pi}\end{array}
$ & $\begin{array}
[c]{ccc}\pi\,\eta_{1}=\alpha_{1}, & & \pi\,\eta_{2}=\alpha_{4}\end{array}
$\\\hline
$\begin{array}
[c]{ccccc}\dfrac{\theta_{5}-\theta_{4}}{2\pi}, & & \dfrac{\theta_{2}-\theta_{1}}{2\pi
}, & & \dfrac{\theta_{3}-\theta_{2}}{2\pi}\end{array}
$ & $\begin{array}
[c]{ccc}\pi\,\eta_{1}=\alpha_{1}, & & \pi\,\eta_{2}=\alpha_{2}\end{array}
$\\\hline
$\begin{array}
[c]{ccccc}\dfrac{\theta_{3}-\theta_{2}}{2\pi}, & & \dfrac{\theta_{2}-\theta_{1}}{2\pi
}, & & \dfrac{\theta_{5}-\theta_{4}}{2\pi}\end{array}
$ & $\begin{array}
[c]{ccc}\pi\,\eta_{1}=\alpha_{2}, & & \pi\,\eta_{2}=\alpha_{1}\end{array}
$\\\hline
\end{tabular}$$
Introducing $\eta_{3}=\xi_{3}$, we have$$\begin{array}
[c]{l}\xi_{1}=\eta_{1}-\eta_{2}+\eta_{3},\\
\xi_{2}=\eta_{2}-\eta_{3},\\
\xi_{3}=\eta_{3}\end{array}$$ and calculate the Jacobian determinant to be equal to $1$. From$$\begin{array}
[c]{l}0<\eta_{1}-\eta_{2}+\eta_{3}<1,\\
0<\eta_{2}-\eta_{3}<1,\\
0<\eta_{3}<1,\\
0<\eta_{1}+\eta_{3}<1
\end{array}$$ it follows that$$\begin{array}
[c]{l}-\eta_{1}+\eta_{2}<\eta_{3}<1-\eta_{1}+\eta_{2},\\
-1+\eta_{2}<\eta_{3}<\eta_{2},\\
0<\eta_{3}<1,\\
-\eta_{1}<\eta_{3}<1-\eta_{1}\end{array}$$ hence $\max\{-\eta_{1}+\eta_{2},0\}<\eta_{3}<\min\{\eta_{2},1-\eta_{1}\}$. There are four cases:
1. If $1-\eta_{2}<\eta_{1}<\eta_{2}$, then $-\eta_{1}+\eta_{2}<\eta_{3}<1-\eta_{1}$
2. If $\eta_{1}<\eta_{2}<1-\eta_{1}$, then $-\eta_{1}+\eta_{2}<\eta_{3}<\eta_{2}$
3. If $1-\eta_{1}<\eta_{2}<\eta_{1}$, then $0<\eta_{3}<1-\eta_{1}$
4. If $\eta_{2}<\eta_{1}<1-\eta_{2}$, then $0<\eta_{3}<\eta_{2} $
giving rise to$$\begin{array}
[c]{cc}{\displaystyle\int\limits_{-\eta_{1}+\eta_{2}}^{1-\eta_{1}}}
6\,d\eta_{3}=6\left( 1-\eta_{2}\right) , &
{\displaystyle\int\limits_{-\eta_{1}+\eta_{2}}^{\eta_{2}}}
6\,d\eta_{3}=6\eta_{1},\\{\displaystyle\int\limits_{0}^{1-\eta_{1}}}
6\,d\eta_{3}=6\left( 1-\eta_{1}\right) , &
{\displaystyle\int\limits_{0}^{\eta_{2}}}
6\,d\eta_{3}=6\eta_{2}\end{array}$$ and thus the joint density for $\eta_{1}$, $\eta_{2}$ is$$\left\{
\begin{array}
[c]{lll}6\left( 1-\eta_{2}\right) & & \text{if }1-\eta_{2}<\eta_{1}<\eta_{2}\text{
and }1/2<\eta_{2}<1\text{,}\\
6\eta_{1} & & \text{if }\eta_{1}<\eta_{2}<1-\eta_{1}\text{ and }0<\eta
_{1}<1/2\text{,}\\
6\left( 1-\eta_{1}\right) & & \text{if }1-\eta_{1}<\eta_{2}<\eta_{1}\text{
and }1/2<\eta_{1}<1\text{,}\\
6\eta_{2} & & \text{if }\eta_{2}<\eta_{1}<1-\eta_{2}\text{ and }0<\eta
_{2}<1/2\text{.}\end{array}
\right.$$ The sought-after joint density for $\alpha$, $\beta$ is therefore$$\left\{
\begin{array}
[c]{lll}6\left( \pi-\beta\right) /\pi^{3} & & \text{if }\pi-\beta<\alpha
<\beta\text{ and }\pi/2<\beta<\pi\text{,}\\
6\alpha/\pi^{3} & & \text{if }\alpha<\beta<\pi-\alpha\text{ and }0<\alpha
<\pi/2\text{,}\\
6\left( \pi-\alpha\right) /\pi^{3} & & \text{if }\pi-\alpha<\beta
<\alpha\text{ and }\pi/2<\alpha<\pi\text{,}\\
6\beta/\pi^{3} & & \text{if }\beta<\alpha<\pi-\beta\text{ and }0<\beta<\pi/2
\end{array}
\right.$$ and we call this the bivariate tent distribution (as opposed to pyramid distribution, which already means something else [@Ke-quadrilat]). It is clear that $\rho(\alpha,\beta)=0$ yet $\alpha$ and $\beta$ are dependent.
[angles.eps]{}
[tent.eps]{}
Looking Back
============
Given a uniform cyclic triangle, the joint density for two arbitrary angles $\alpha$, $\beta$ is [@Ra-quadrilat; @Mil-quadrilat; @Mre-quadrilat] $$\left\{
\begin{array}
[c]{lll}2/\pi^{2} & & \text{if }0<\alpha<\pi\text{, }0<\beta<\pi\text{ and }\alpha+\beta<\pi,\\
0 & & \text{otherwise}\end{array}
\right.$$ and trivially $\rho(\alpha,\beta)=-1/2$. Let $\Delta$ denote the isosceles triangular support of this distribution. Let $a$ denote the side opposite $\alpha$ and $b$ denote the side opposite $\beta$. $\ $From$$\left(
\begin{array}
[c]{c}\alpha\\
\beta
\end{array}
\right) \mapsto\left(
\begin{array}
[c]{c}2\sin(\alpha)\\
2\sin(\beta)
\end{array}
\right) =\left(
\begin{array}
[c]{c}a\\
b
\end{array}
\right) ,$$ we have Jacobian determinant $4\cos(\alpha)\cos(\beta)$ and preimages$$\begin{array}
[c]{ccc}\left(
\begin{array}
[c]{c}\arcsin\left( \tfrac{a}{2}\right) \\
\arcsin\left( \tfrac{b}{2}\right)
\end{array}
\right) , & & \left(
\begin{array}
[c]{c}\pi-\arcsin\left( \tfrac{a}{2}\right) \\
\arcsin\left( \tfrac{b}{2}\right)
\end{array}
\right)
\end{array}$$ if $b<a$ and$$\begin{array}
[c]{ccc}\left(
\begin{array}
[c]{c}\arcsin\left( \tfrac{a}{2}\right) \\
\arcsin\left( \tfrac{b}{2}\right)
\end{array}
\right) , & & \left(
\begin{array}
[c]{c}\arcsin\left( \tfrac{a}{2}\right) \\
\pi-\arcsin\left( \tfrac{b}{2}\right)
\end{array}
\right)
\end{array}$$ if $a<b$. Reason: if $b<a$, then $\arcsin(b/2)<\arcsin(a/2)$ and hence both preimages fall in $\Delta$ because $\left[ \pi-\arcsin(a/2)\right]
+\arcsin(b/2)<\pi$. No other preimages exist when $b<a$ because $\arcsin(a/2)+\left[ \pi-\arcsin(b/2)\right] >\pi$ and $\left[ \pi
-\arcsin(a/2)\right] +\left[ \pi-\arcsin(b/2)\right] >\pi$. Likewise for $a<b$.
The joint density for $a$ and $b$ is thus $$\left\{
\begin{array}
[c]{lll}\dfrac{4}{\pi^{2}}\dfrac{1}{\sqrt{4-a^{2}}}\dfrac{1}{\sqrt{4-b^{2}}} & &
\text{if }0<a<2\text{ and }0<b<2\text{,}\\
0 & & \text{otherwise}\end{array}
\right.$$ which implies that sides $a$, $b$ are independent even though they are related so easily (via the sine function) to the dependent angles $\alpha$, $\beta$. As far as is known, this observation is new. We mention that the remaining side $c$ satisfies$$c=\left\{
\begin{array}
[c]{lll}\frac{1}{2}\left( a\sqrt{4-b^{2}}+b\sqrt{4-a^{2}}\right) & & \text{with
probability }1/2,\\
\frac{1}{2}\left\vert a\sqrt{4-b^{2}}-b\sqrt{4-a^{2}}\right\vert & &
\text{with probability }1/2
\end{array}
\right.$$ for completeness’ sake.
Looking Forward
===============
The polygonal angles $\alpha$, $\beta$, $\gamma$, $\delta$ associated with a uniform cyclic $5$-gon can be studied via the Dirichlet$(1,1,1,1;1)$ distribution on a $4$-dimensional simplex [@Ra-quadrilat; @PC-quadrilat; @NT-quadrilat]:$$\left\{
\begin{array}
[c]{lll}24 & & \text{if }0<\xi_{1}<1,\text{ }0<\xi_{2}<1,\text{ }0<\xi_{3}<1,\text{
}0<\xi_{4}<1\text{ and }\xi_{1}+\xi_{2}+\xi_{3}+\xi_{4}<1,\\
0 & & \text{otherwise}\end{array}
\right.$$ and calculation of the joint density for $\eta_{1}=\xi_{1}+\xi_{2}+\xi_{3}$, $\eta_{2}=\xi_{2}+\xi_{3}+\xi_{4}$, $\eta_{3}=1-\xi_{1}-\xi_{2} $, $\eta
_{4}=1-\xi_{2}-\xi_{3}$. Omitting elaborate details, we obtain the density to be $24$ when$$\max\{1-\eta_{1},1-\eta_{2}\}<\eta_{4}<\min\{2-\eta_{1}-\eta_{2},2-\eta
_{1}-\eta_{3}\}\text{ and }1<\eta_{1}+\eta_{3}<2$$ and $0$ otherwise. It follows that$$\begin{array}
[c]{ccccccc}\rho(\alpha,\beta)=1/6, & & \rho(\alpha,\gamma)=-2/3, & & \rho(\alpha
,\delta)=-2/3, & & \rho(\alpha,\varphi)=1/6
\end{array}$$ where $\varphi=3\pi-\alpha-\beta-\gamma-\delta$. In particular, adjacent angles are positively correlated and non-adjacent angles are negatively correlated.
For a uniform cyclic $6$-gon, we conjecture that$$\begin{array}
[c]{lllll}\rho(\alpha,\beta)=1/4, & & \rho(\alpha,\gamma)=-1/2, & & \rho(\alpha
,\delta)=-1/2,\\
\rho(\alpha,\varphi)=-1/2, & & \rho(\alpha,\psi)=1/4 & &
\end{array}$$ where $\varphi=2\pi-\alpha-\gamma$ and $\psi=2\pi-\beta-\delta$. Again, adjacent angles are positively correlated and non-adjacent angles are negatively correlated. The fact that $\delta$ is opposite $\alpha$ seems not to affect its correlation with $\alpha$, relative to either $\gamma$ or $\varphi$.
Area
====
Given a uniform cyclic triangle, moments of area $2\sin(\alpha)\sin(\beta
)\sin(\alpha+\beta)$ are computed by use of the joint angle density: $$\frac{2}{\pi^{2}}{\displaystyle\int\limits_{0}^{\pi}}
{\displaystyle\int\limits_{0}^{\pi-\beta}}
2\sin(\alpha)\sin(\beta)\sin(\alpha+\beta)\,d\alpha\,d\beta=\frac{3}{2\pi},$$$$\frac{2}{\pi^{2}}{\displaystyle\int\limits_{0}^{\pi}}
{\displaystyle\int\limits_{0}^{\pi-\beta}}
4\sin^{2}(\alpha)\sin^{2}(\beta)\sin^{2}(\alpha+\beta)\,d\alpha\,d\beta
=\frac{3}{8}.$$ The density for area itself is $8xK\left( 4x^{2}\right) $, where$$\begin{aligned}
K(y) & =\frac{1}{4\pi^{3}}\frac{1}{\sqrt{y}}\left\{ \Gamma\left( \frac
{1}{3}\right) ^{3}\left( \frac{4y}{27}\right) ^{-1/6}\,_{2}F_{1}\left(
\dfrac{1}{3},\dfrac{1}{3},\dfrac{2}{3},\frac{4y}{27}\right) -\right. \\
& \ \ \ \left. 3\Gamma\left( \frac{2}{3}\right) ^{3}\left( \frac{4y}{27}\right) ^{1/6}\,_{2}F_{1}\left( \dfrac{2}{3},\dfrac{2}{3},\dfrac{4}{3},\frac{4y}{27}\right) \right\} ,\end{aligned}$$ $_{2}F_{1}$ is the Gauss hypergeometric function and $0<y<27/4$. This formula corrects that which appears in Case III of [@MT-quadrilat].
Given a uniform cyclic quadrilateral, we conjecture that the joint density for angles $\alpha$, $\beta$, $\omega$ is$$f(\alpha,\beta,\omega)=\left\{
\begin{array}
[c]{lll}3/\pi^{3} & & \text{if }\alpha+\beta>\omega\text{, }\alpha+\omega
>\beta\text{, }\beta+\omega>\alpha\text{ and }\alpha+\beta+\omega<2\pi,\\
0 & & \text{otherwise.}\end{array}
\right.$$ It can be shown that, assuming the formula for $f$ is valid, any two angles from the list $\alpha$, $\beta$, $\omega$ are distributed according to the bivariate tent density. Our conjecture is consistent with computer simulation, but a rigorous proof is open. From this, we obtain area moments$${\displaystyle\int\limits_{0}^{\pi}}
{\displaystyle\int\limits_{0}^{\pi}}
{\displaystyle\int\limits_{0}^{\pi}}
2\sin(\alpha)\sin(\beta)\sin(\omega)f(\alpha,\beta,\omega)\,d\alpha
\,d\beta\,d\omega=\frac{3}{\pi},$$$${\displaystyle\int\limits_{0}^{\pi}}
{\displaystyle\int\limits_{0}^{\pi}}
{\displaystyle\int\limits_{0}^{\pi}}
4\sin^{2}(\alpha)\sin^{2}(\beta)\sin^{2}(\omega)f(\alpha,\beta,\omega
)\,d\alpha\,d\beta\,d\omega=\frac{1}{2}+\frac{105}{16\pi^{2}}$$ which again is consistent with experiment. The mean area for quadrilaterals is twice that for triangles. No formula for the density of area itself is known.
[tetra0.eps]{}
The problem with angles is that we do not know a suitable way of relating $\omega$ with parameters $\theta_{1}$, $\theta_{2}$, $\theta_{3}$, $\theta
_{4}$. For a cyclic quadrilateral with successive sides $a$, $b$, $c $, $d$, formulas like [@DR-quadrilat] $$\begin{array}
[c]{ccc}\tan\left( \dfrac{\alpha}{2}\right) =\sqrt{\dfrac{(-a+b+c+d)(a-b+c+d)}{(a+b-c+d)(a+b+c-d)}} & & \text{where }\alpha\text{ is angle between }a\text{
and }b\text{,}\end{array}$$$$\begin{array}
[c]{ccc}\tan\left( \dfrac{\beta}{2}\right) =\sqrt{\dfrac{(a-b+c+d)(a+b-c+d)}{(-a+b+c+d)(a+b+c-d)}} & & \text{where }\beta\text{ is angle between }b\text{
and }c\text{,}\end{array}$$$$\begin{array}
[c]{ccc}\tan\left( \dfrac{\omega}{2}\right) =\sqrt{\dfrac{(a-b+c+d)(a+b+c-d)}{(-a+b+c+d)(a+b-c+d)}} & & \text{where }\omega\text{ is angle between
diagonals}\end{array}$$ suggest an alternative approach to solution, but the path seems very complicated.
Acknowledgements
================
I am indebted to Chi Zhang for her hand calculations in Sections 2 and 4 (specifically, those involving $\xi$s and $\eta$s). I am also grateful to Guo-Liang Tian, Serge Provost and Paul Kettler for helpful discussions.
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\[c\][lll]{} & Steven Finch &\
& MIT Sloan School of Management &\
& Cambridge, MA, USA &\
& *steven\[email protected]* &
|
---
abstract: 'In this paper we present a new verification theorem for optimal stopping problems for Hunt processes. The approach is based on the Fukushima-Dynkin formula [@F] and its advantage is that it allows us to verify that a given function is the value function without using the viscosity solution argument. Our verification theorem works in any dimension. We illustrate our results with some examples of optimal stopping of reflected diffusions and absorbed diffusions.'
author:
- 'Achref Bachouch$^{1,2}$, Olfa Draouil$^{1,2}$ and Bernt Øksendal$^{1,2}$'
date: 30 August 2019
title: A new approach to optimal stopping for Hunt processes
---
#### MSC(2010):
60H05; 60H07; 60H40; 60G57; 91B70; 93E20.
#### Keywords:
Hunt processes, Dynkin-Fukushima formula, variational inequalities, optimal stopping
Introduction
============
Usually when solving optimal stopping problems it is assumed that the value function is twice continuously differentiable ($C^2$), and in order to find it we use the high contact principle. But in reality this is a strong condition and sometimes the high contact principle is not valid. Then one must use the viscosity solution approach to verify that a given function is indeed the value function. This is done in for example Dai & Menoukeu Pamen [@DM], where the authors use a viscosity solution approach to study optimal stopping of some processes with reflection. More precisely, they prove that the value function is the unique viscosity solution of the HJB equation associated with the optimal stopping problem of reflected Feller processes.\
In our paper we treat a more general case by considering the Hunt processes, i.e. strong Markov and quasi left continuous processes with respect a filtration $\{\mathcal{F}_t\}_{t\geq 0}$ the natural filtration of the Hunt process. We come back later in the next section with more details about Hunt processes. Using the Fukushima-Dynkin formula [@F], we obtain a variational inequality verification theorem for optimal stopping of Hunt processes. Note that in this theorem we do not need the use of high contact principle, nor do we need the viscosity approach. Moreover, our theorem works for multidimensional case, i.e. our Hunt process $X$ can take value in $\mathbb{R}^d$ for any $d\geq 1$.Therefore our work can be regarded as a generalisation of the result in the Mordecki & Salminen paper [@Salminen], which is based on methods applicable only in the 1-dimensional case.\
This paper is organised as follow:\
In Section 2 we define the Hunt process, we present the Fukushima-Dynkin formula [@F] then we make a connection between Fukushima-Dynkin formula and the Dynkin formula in Øksendal-Sulem formula [@SO].\
In Section 3 we introduce the optimal stopping problem for Hunt processes. We prove a verification theorem for this problem.\
Finally in Section 4 we illustrate our method by applying it to some optimal stopping problems for reflected or absorbed Hunt processes.
Our main result:\
Variational inequalities for optimal stopping
=============================================
In the following we let $\{X_t\}_{0\leq t\leq T}$ be a given Hunt process with values in a closed subset $K$ of $\mathbb{R}^d$. Our method and results are valid in any dimension, but for simplicity of notation we will in the following assume that $d=1.$ We let $\mathcal{T}$ be the set of stopping times $\tau \leq T$ with respect to the filtration $\mathbb{F}=\{\mathcal{F}_t\}_{t\geq 0}$ generated by $X$. We refer to the book by Fukushima [@F] for more information about Hunt processes and their associated Dirichlet forms and calculus.
We consider the following optimal stopping problem:
Given functions $f$ and $g$ and a constant $\alpha > 0$, find $\Phi_{\alpha}(x)$ and $\tau^* \in \mathcal{T}$ such that $$\Phi_{\alpha}(x)=\sup_{\tau\in \mathcal{T}}J_{\alpha}^{\tau}(x)=J_{\alpha}^{\tau^*}(x)$$ where $$J_{\alpha}^{\tau}(x)=E_x[\int_0^{\tau}e^{-\alpha t}f(X_t)dt+e^{-\alpha \tau}g(X_{\tau})]; \quad x \in K.$$
To study this problem we introduce the *resolvent* of $X$, defined by $$R_{\alpha}\varphi(x) = E_{x}[\int_0^{\infty} e^{-\alpha t} \varphi(X(t)) dt]$$ for all functions $\varphi$ such that the integral converges.
Recall that the *Fukushima-Dynkin formula* (see (4.2.6) p. 97 in [@F]), can be written $$E_x[e^{-\alpha \tau} R_{\alpha} \psi (X_{\tau)}]=R_{\alpha}\psi(x)-E_x[\int_0^{\tau} e^{-\alpha t}\psi(X_t)dt];\quad \text{ for all stopping times } \tau.$$
Using this, we obtain the following result, which is our main result. It can be regarded as a weak analogue of the variational inequality (4) of [@P]. Note that a viscosity solution interpretation is not needed here:
(Variational inequalities for optimal stopping)
Suppose there exists a measurable function $\psi:K\mapsto \mathbb{R}$ such that thew following hold:\
(i) $R_{\alpha}\psi(x) \geq g(x)$ for all $x\in K,$\
(ii) $\psi(x) \geq f(x)$ for all $x\in K,$\
(iii) Define $D=\{ x\in K, R_{\alpha}\psi(x)>g(x)\}$,\
and put\
$\tau_{D}=\inf\{ t>0, X_t\notin D\}.$\
Assume that\
(iv) $\psi(x)=f(x)$ on $D$,\
(v) the family $\{R_{\alpha}\psi(X_{\tau}), \tau \leq \tau_D \}$ is uniformly integrable with respect to $P_x$ for all $x\in K$.\
Then $$R_{\alpha}\psi(x)=\Phi_{\alpha}(x)=\sup_{\tau }J_{\alpha}(x), \forall x\in K$$ and $$\tau^*=\tau_D$$ is an optimal stopping time.
[[Proof.]{} ]{}First we remark that for $\tau=0$ we have $$\label{tau0}
J_{\alpha}^0(x)=g(x)\leq \Phi_{\alpha}(x).$$ Using (i),(ii) and the Fukushima-Dynkin formula we get that $$\begin{aligned}
R_{\alpha}\psi(x)&=E_x[\int_0^{\tau} e^{-\alpha t}\psi(X_t)dt]+E_x[e^{-\alpha \tau} R_{\alpha} \psi (X_{\tau)}]\nonumber\\
&\geq E_x[\int_0^{\tau}e^{-\alpha t}f(X_t)dt]+E_x[e^{-\alpha \tau}g(X_{\tau})]=J_{\alpha}^{\tau}(x), \forall x\in X.\end{aligned}$$ Since this inequality holds for arbitrary $\tau\leq T$ then we get $$\label{ineq1}
R_{\alpha}\psi(x)\geq \Phi_{\alpha}(x),\forall x\in K.$$ To prove the reverse inequality we consider two cases
- Suppose that $x\notin D$ then $$\label{xnotinD}
R_{\alpha}\psi(x)=g(x)\leq \Phi_{\alpha}(x).$$ Note that the last inequality of equation comes from . Combining and we conclude that $$J_{\alpha}^0(x)=R_{\alpha}\psi(x)=\Phi_{\alpha}(x),\forall x\notin D \text{ and } \tau^*=\tau^*(x,\omega)=0.$$
- Suppose $x\in D$. Let $\{D_k\}_{k\geq 1}$ be a sequence of increasing open sets of $D_k$ such that $\bar{D_k}\subset D$, $\bar{D_k}$ is compact and $D=\cup_{k=1}^{\infty}D_k$.\
Let $\tau_k=\inf\{ t>0, X_t\notin D_k\}$.\
Choose $x\in D_k$. Then by the Fukushima-Dynkin formula and (iv) we have $$\begin{aligned}
R_{\alpha}\psi(x)&=E_x[\int_0^{\tau_k} e^{-\alpha t}\psi(X_t)dt+e^{-\alpha \tau_k} R_{\alpha} \psi (X_{\tau_k)}]\nonumber\\
&=E_x[\int_0^{\tau_k} e^{-\alpha t}f(X_t)dt+e^{-\alpha \tau_k} R_{\alpha} \psi (X_{\tau_k)}].\end{aligned}$$ Using the uniform integrability of $R_{\alpha}\psi(X_{\tau_k})$, $\tau_k<\tau_D$, quasi left-continuity and the fact that $R_{\alpha}\psi(X_{\tau_D})=g(X_{\tau_D})$ we get $$\begin{aligned}
\label{xinD}
R_{\alpha}\psi(x)&=\lim_{k\rightarrow +\infty} E_x[\int_0^{\tau_k} e^{-\alpha t}f(X_t)dt+e^{-\alpha \tau_k} R_{\alpha} \psi (X_{\tau_k)}]\\
&=E_x[\int_0^{\tau_D} e^{-\alpha t}f(X_t)dt+e^{-\alpha \tau_D} R_{\alpha} \psi (X_{\tau_D)}]\\
&=E_x[\int_0^{\tau_D} e^{-\alpha t}f(X_t)dt+e^{-\alpha \tau_D} g(X_{\tau_D)}]=J_{\alpha}^{\tau_D}(x)\leq \Phi_{\alpha}(x).\end{aligned}$$ Combining and we get $$\Phi_{\alpha}(x)\leq R_{\alpha}\psi(x)=J_{\alpha}^{\tau_D}(x)\leq \Phi_{\alpha}(x).$$ Then we conclude that $$R_{\alpha}\psi(x)=\Phi_{\alpha}(x) \text{ and } \tau^*=\tau^*(x,\omega)=\tau_D, \forall x\in D.$$
[$\square$ ]{}
Examples
========
To illustrate our main result, we study some examples:
Consider the following optimal stopping problem:\
Find $\Phi_{\alpha}(x)$ and $\tau^*$ such that $$\Phi_{\alpha}(x)= \sup_{\tau} E[\int_0^{\tau} e^{-\alpha t}B(t)dt]=E[\int_0^{\tau*}e^{-\alpha t}B(t)dt].$$
We want to solve this problem in two ways:\
(i) By using the classical variational inequality theorem approach in the SDE book\
(ii) By using Theorem 0.1 above.
(i)\
Gessing $D=\{(t,x), x> x_0\}, x_0<0$.\
The function $\phi_{\alpha}$ should verify the following PDE $$\begin{cases}
\frac{\partial \phi_{\alpha}}{\partial t}(t,x)+\frac{1}{2}\frac{\partial^2 \phi_{\alpha}}{\partial x^2}(t,x)+xe^{-\alpha t}=0 \text{ on } D\\
\phi_{\alpha}(t,x)=0,\quad x\notin D.
\end{cases}$$ Put $\phi_{\alpha}(t,x)=\phi_0(x) e^{-\alpha t}$ then $\phi_0$ verifies the following second order differential equation $$\label{secondordereq}
\begin{cases}
\frac{1}{2}\frac{\partial^2 \phi_{0}}{\partial x^2}(x)-\alpha \phi_{0}(x) +x=0 \text{ on } D\\
\phi_{0}(t,x)=0,\quad x\notin D.
\end{cases}$$ The general solution of the equation is given by $$\phi_0(x)=C_1e^{\sqrt{2\alpha}x}+C_2e^{-\sqrt{2\alpha}x}
+\frac{1}{\alpha}x$$ where $C_1$ and $C_2$ are constants.\
Since we have $$E_x[\int_0^{\tau} e^{-\alpha t} B(t)dt] = \frac{1}{\alpha}(1-e^{-\alpha \tau})x + E_0[\int_0^{\tau} e^{-\alpha t} B(t)dt]$$
then only the first term depends on $x$ and it grows at most linearly as $x$ goes to $+\infty$. Hence $C_1=0$.
Using the continuity of $\phi_{\alpha}$ at $x=x_0$ we have $\phi_0(x_0)=0$ then $$C_2e^{-\sqrt{2\alpha}x_0}+\frac{1}{\alpha}x_0=0.$$
Hence $C_2=-\frac{1}{\alpha}x_0 e^{\sqrt{2\alpha}x_0}$.
Then $$\phi_{\alpha}(t,x)=e^{-\alpha t}(-\frac{1}{\alpha}x_0 e^{\sqrt{2\alpha}x_0}e^{-\sqrt{2\alpha}x}
+\frac{1}{\alpha}x).$$
Using now the high contact equation i.e, $\phi_{\alpha}$ is $C^1$ at $x=x_0$ we get the following equation $$-\frac{1}{\alpha}e^{\sqrt{2\alpha}x_0}e^{-\sqrt{2\alpha}x_0}(-\sqrt{2\alpha})+\frac{1}{\alpha}=0$$ Then we deduce that $$x_0 =-\frac{1}{\sqrt{2\alpha}}.$$
(ii)We now solve the problem using our approach. By condition (iii) of Theorem 2.2 we have $\psi(x)=f(x)$ on $D=]x_0,+\infty[$. i:e $\psi(x)=x$. $$\begin{aligned}
\label{ex1R}
R_{\alpha}\psi(x)&= \int_{x_0}^{+\infty}\psi(y)R_{\alpha}(x,dy)\\
&=\int_{x_0}^{+\infty}\psi(y)E_x[\int_{0}^{+\infty}e^{-\alpha t}1_{dy}(B_t)dt]\\
&=\int_{0}^{+\infty}e^{-\alpha t}\int_{x_0}^{+\infty}yE_x[1_{dy}(B_t)]dt\\
&=\int_{0}^{+\infty}e^{-\alpha t}\int_{x_0}^{+\infty}yP_x(B_t\in dy)dt\label{eq2.31}\\
&=\int_{0}^{+\infty}e^{-\alpha t}\int_{x_0}^{+\infty}y\frac{e^{-\frac{(y-x)^2}{2t}}}{\sqrt{2\pi t}}dy dt\\
&=\int_{0}^{+\infty}e^{-\alpha t}\int_{x_0-x}^{+\infty}(z+x)\frac{e^{-\frac{z^2}{2t}}}{\sqrt{2\pi t}}dz dt\\
&=\int_{0}^{+\infty}e^{-\alpha t}\int_{x_0-x}^{+\infty}z\frac{e^{-\frac{z^2}{2t}}}{\sqrt{2\pi t}}dz dt+x\int_{0}^{+\infty}e^{-\alpha t}\int_{x_0-x}^{+\infty}\frac{e^{-\frac{z^2}{2t}}}{\sqrt{2\pi t}}dz dt \\
&=\int_{0}^{+\infty}e^{-\alpha t}t\frac{e^{-\frac{(x_0-x)^2}{2t}}}{\sqrt{2\pi t}}dt+
x\int_{0}^{+\infty}e^{-\alpha t}\int_{x_0-x}^{+\infty}\frac{e^{-\frac{z^2}{2t}}}{\sqrt{2\pi t}}dz dt\label{eq3.17}\end{aligned}$$
We distinguish two cases:
- 1\) If $x>x_0$ then $x_0-x<0$. In this case we have from equation that $$\begin{aligned}
R_{\alpha}\psi(x)&=\int_{0}^{+\infty}e^{-\alpha t}t\frac{e^{-\frac{(x_0-x)^2}{2t}}}{\sqrt{2\pi t}}dt+
x\int_{0}^{+\infty}e^{-\alpha t}\int_{x_0-x}^{+\infty}\frac{e^{-\frac{z^2}{2t}}}{\sqrt{2\pi t}}dz dt\\
&=\int_{0}^{+\infty}e^{-\alpha t}t\frac{e^{-\frac{(x_0-x)^2}{2t}}}{\sqrt{2\pi t}}dt+
x\int_{0}^{+\infty}e^{-\alpha t}\int_{x_0-x}^{0}\frac{e^{-\frac{z^2}{2t}}}{\sqrt{2\pi t}}dz dt+
x\int_{0}^{+\infty}e^{-\alpha t}\int_{0}^{+\infty}\frac{e^{-\frac{z^2}{2t}}}{\sqrt{2\pi t}}dz dt\\
&=\int_{0}^{+\infty}e^{-\alpha t}t\frac{e^{-\frac{(x_0-x)^2}{2t}}}{\sqrt{2\pi t}}dt+x\int_{0}^{+\infty}e^{-\alpha t}\int_0^{x-x_0}\frac{e^{-\frac{z^2}{2t}}}{\sqrt{2\pi t}}dz dt+
\frac{x}{2}\int_{0}^{+\infty}e^{-\alpha t}dt\\
&=\int_{0}^{+\infty}e^{-\alpha t}t\frac{e^{-\frac{(x_0-x)^2}{2t}}}{\sqrt{2\pi t}}dt+x\int_{0}^{+\infty}e^{-\alpha t}\int_0^{\frac{x-x_0}{\sqrt{2t}}}\frac{e^{-z^2}}{\sqrt{\pi }}dz dt+
\frac{x}{2}\int_{0}^{+\infty}e^{-\alpha t}dt\\
&=\int_{0}^{+\infty}e^{-\alpha t}t\frac{e^{-\frac{(x_0-x)^2}{2t}}}{\sqrt{2\pi t}}dt+\frac{x}{2}\int_{0}^{+\infty}e^{-\alpha t}(\int_0^{\frac{x-x_0}{\sqrt{2t}}}\frac{e^{-z^2}}{\sqrt{\pi }}dz+1)dt
\\
&=\int_{0}^{+\infty}e^{-\alpha t}t\frac{e^{-\frac{(x_0-x)^2}{2t}}}{\sqrt{2\pi t}}dt+\frac{x}{2}\int_{0}^{+\infty}e^{-\alpha t}(\int_0^{\frac{x-x_0}{\sqrt{2t}}}\frac{2e^{-z^2}}{\sqrt{\pi }}dz-1)dt+x\int_{0}^{+\infty}e^{-\alpha t}dt\\
&=\int_{0}^{+\infty}e^{-\alpha t}t\frac{e^{-\frac{(x_0-x)^2}{2t}}}{\sqrt{2\pi t}}dt+\frac{x}{\alpha}+\frac{x}{2}\int_{0}^{+\infty}e^{-\alpha t}(\int_0^{\frac{x-x_0}{\sqrt{2t}}}\frac{2e^{-z^2}}{\sqrt{\pi }}dz-1)dt\label{eq3.29}\\
&=\int_{0}^{+\infty}e^{-\alpha t}t\frac{e^{-\frac{(x_0-x)^2}{2t}}}{\sqrt{2\pi t}}dt+\frac{x}{\alpha}-\frac{x}{2}\int_{0}^{+\infty}e^{-\alpha t}(1-H(\frac{x-x_0}{\sqrt{2t}}))dt\end{aligned}$$ where $$H(x)=\int_0^{x}\frac{2e^{-z^2}}{\sqrt{\pi }}dz.$$ By [@GR] we have the following result $$\int_0^{+\infty}(1-H(\frac{q}{2\sqrt{t}}))e^{-\alpha t} dt=\frac{1}{\alpha}e^{-q\sqrt{\alpha}}, \quad Re\text{ } \alpha>0, |arg \text{ }q|<\frac{\pi}{4}.$$ Then, for $q=\sqrt{2}(x-x_0)$ we get $$\label{eq3.33}
\int_{0}^{+\infty}e^{-\alpha t}(1-H(\frac{x-x_0}{\sqrt{2t}}))dt=\frac{1}{\alpha}e^{-\sqrt{2\alpha}(x-x_0)}$$ Therefore $$\label{eq3.34}
R_{\alpha}\psi(x) =\int_{0}^{+\infty}e^{-\alpha t}t\frac{e^{-\frac{(x_0-x)^2}{2t}}}{\sqrt{2\pi t}}dt+\frac{x}{\alpha}-\frac{x}{2\alpha}e^{-\sqrt{2\alpha}(x-x_0)}.$$ From we have $$\label{eq3.35}
\int_{0}^{+\infty}e^{-\alpha t}(1-\int_0^{\frac{x-x_0}{\sqrt{2t}}}\frac{2e^{-z^2}}{\sqrt{\pi }}dz)dt=\frac{1}{\alpha}e^{-\sqrt{2\alpha}(x-x_0)}.$$ Derive equation with respect to $x$ we find: $$\label{eq3.36}
2\int_{0}^{+\infty}e^{-\alpha t}\frac{e^{-\frac{(x-x_0)^2}{2t}}}{\sqrt{2\pi t}}dt=\sqrt{\frac{2}{\alpha}}e^{-\sqrt{2\alpha}(x-x_0)}.$$ Derive equation with respect to $\alpha$ we get $$\label{eq3.37}
\int_{0}^{+\infty}te^{-\alpha t}\frac{e^{-\frac{(x-x_0)^2}{2t}}}{\sqrt{2\pi t}}dt=e^{-\sqrt{2\alpha}(x-x_0)}(\frac{1}{\alpha\sqrt{2\alpha}}+\frac{x}{2\alpha}).$$ Replacing equation in we get $$\label{eq3.38b}
R_{\alpha}\psi(x)=\frac{1}{\alpha\sqrt{2\alpha}}e^{-\sqrt{2\alpha}(x-x_0)}+\frac{x}{\alpha}>0, \quad x>x_0.$$
- 2\) If $x\leq x_0$ then $x_0-x\geq 0$, then equation becomes $$\begin{aligned}
&R_{\alpha}\psi(x)=\int_{0}^{+\infty}e^{-\alpha t}t\frac{e^{-\frac{(x_0-x)^2}{2t}}}{\sqrt{2\pi t}}dt+
x\int_{0}^{+\infty}e^{-\alpha t}\int_{x_0-x}^{+\infty}\frac{e^{-\frac{z^2}{2t}}}{\sqrt{2\pi t}}dz dt\\
&=\int_{0}^{+\infty}e^{-\alpha t}t\frac{e^{-\frac{(x_0-x)^2}{2t}}}{\sqrt{2\pi t}}dt+
x\int_{0}^{+\infty}e^{-\alpha t}(\int_{0}^{+\infty}\frac{e^{-\frac{z^2}{2t}}}{\sqrt{2\pi t}}dz -\int_0^{x_0-x}\frac{e^{-\frac{z^2}{2t}}}{\sqrt{2\pi t}}dz)dt\\
&=\int_{0}^{+\infty}e^{-\alpha t}t\frac{e^{-\frac{(x_0-x)^2}{2t}}}{\sqrt{2\pi t}}dt+
\frac{x}{2}\int_{0}^{+\infty}e^{-\alpha t}(1-\int_0^{\frac{x_0-x}{\sqrt{2t}}}\frac{2e^{-z^2}}{\sqrt{\pi}}dz)dt.\end{aligned}$$ Or we have
$$\label{eq2.50}
\int_{0}^{+\infty}e^{-\alpha t}(1-\int_0^{\frac{x_0-x}{\sqrt{2t}}}\frac{2 e^{-z^2}}{\sqrt{\pi}}dz)dt=\frac{1}{\alpha}e^{-\sqrt{2\alpha}(x_0-x)}$$
Then $$\label{eq2.51}
R_{\alpha}\psi(x)=\int_{0}^{+\infty}e^{-\alpha t}t\frac{e^{-\frac{(x_0-x)^2}{2t}}}{\sqrt{2\pi t}}dt+\frac{x}{2\alpha}e^{-\sqrt{2\alpha}(x_0-x)}.$$ Let us derive with respect to $x$, then we get $$\label{eq2.52}
2\int_{0}^{+\infty}e^{-\alpha t}\frac{e^{-\frac{(x_0-x)^2}{2t}}}{\sqrt{2\pi t}}dt=\frac{\sqrt{2}}{\sqrt{\alpha}}e^{-\sqrt{2\alpha}(x_0-x)}.$$ Let us now derive with respect to $\alpha$ then we get: $$-2\int_{0}^{+\infty}te^{-\alpha t}\frac{e^{-\frac{(x_0-x)^2}{2t}}}{\sqrt{2\pi t}}dt=
e^{-\sqrt{2\alpha}(x_0-x)}\sqrt{2}(-\frac{1}{2\alpha\sqrt{\alpha}}-\frac{\sqrt{2}(x_0-x)}{2\alpha}).$$ Then $$\label{eq3.46b}
\int_{0}^{+\infty}te^{-\alpha t}\frac{e^{-\frac{(x_0-x)^2}{2t}}}{\sqrt{2\pi t}}dt=
e^{-\sqrt{2\alpha}(x_0-x)}(\frac{1}{2\alpha\sqrt{2\alpha}}+\frac{x_0-x}{2\alpha}).$$ Substituting into we get $$\label{eq3.47b}
R_{\alpha}\psi(x)=e^{-\sqrt{2\alpha}(x_0-x)}(\frac{1}{2\alpha\sqrt{2\alpha}}+\frac{x_0}{2\alpha}).$$ By continuity of $R_{\alpha}\psi$ at $x=x_0$ and combining with at $x=x_0$ we get $$x_0=-\frac{1}{\sqrt{2\alpha}}.$$ Replacing the value of $x_0$ in we get that $$R_{\alpha}\psi(x)=0, \quad x\leq x_0.$$
With $x_0=-\frac{1}{\sqrt{2\alpha}}$ it is easy to check that $R_{\alpha}\psi(x)>0$ for $x\in D=\{x>x_0\}$.\
Next we consider the following example:
Find $\Phi_{\alpha}(x)$ and $\tau^{*}$ such that $$\Phi_{\alpha}(x)= \sup_{\tau} E[e^{-\alpha \tau} B(\tau)] = E[e^{-\alpha \tau^{*}} B(\tau^*)].$$ Again we want to solve this problem in two ways:\
(i) By using the classical variational inequality theorem approach in the SDE book\
(ii) By using Theorem 0.1 above.
By waiting long enough we can always get a payoff which is positive, because the exponential goes to 0 faster than the Brownian motion grows, eventually. Therefore it does not make sense to stop while $B(t) < 0$. So the continuation region should be of the form: $$D=\{x<x_0\}, x_0>0.$$
The function $\phi_{\alpha}$ should verify the following PDE $$\begin{cases}
\frac{\partial \phi_{\alpha}}{\partial t}(t,x)+\frac{1}{2}\frac{\partial^2 \phi_{\alpha}}{\partial x^2}(t,x)=0 \text{ on } D\\
\phi_{\alpha}(t,x)=e^{-\alpha t} x,\quad x\notin D.
\end{cases}$$ Put $\phi_{\alpha}(t,x)=\phi_0(x) e^{-\alpha t}$ then $\phi_0$ verifies the following second order differential equation $$\label{secondordereq1}
\begin{cases}
\frac{1}{2}\frac{\partial^2 \phi_{0}}{\partial x^2}(x)-\alpha \phi_{0}(x)=0 \text{ on } D\\
\phi_{0}(x)=x,\quad x\notin D.
\end{cases}$$ The general solution of the equation is given by $$\phi_0(x)=C_1e^{\sqrt{2\alpha}x}+C_2e^{-\sqrt{2\alpha}x},$$ where $C_1$ and $C_2$ are constants.\
Since $\phi_{\alpha}$ is bounded as $x$ goes to $-\infty$ so we must have $C_2=0$.
Using the continuity of $\phi_{\alpha}$ at $x=x_0$ we have $\phi_0(x_0)=x_0$ then $$C_1e^{\sqrt{2\alpha}x_0}=x_0.$$
Hence $C_1=x_0 e^{-\sqrt{2\alpha}x_0}$.
Then $$\phi_{\alpha}(t,x)=e^{-\alpha t}x_0 e^{-\sqrt{2\alpha}x_0}e^{\sqrt{2\alpha}x} .$$
Using now the high contact equation i.e, $\phi_{\alpha}$ is $C^1$ at $x=x_0$ we get the following equation $$x_0 e^{-\sqrt{2\alpha}x_0}e^{\sqrt{2\alpha}x_0}(\sqrt{2\alpha})=1.$$ Then we deduce that $$x_0 =\frac{1}{\sqrt{2\alpha}}.$$
In $D^c=\{x\geq x_0\}, x_0\geq0$ we have $L\phi_{\alpha}+f=L\phi_{\alpha}=-\alpha e^{-\alpha t} x\leq 0.$\
(ii)By condition (iii) of Theorem 2.2 we have $\psi(x)=f(x)$ on $D$. i:e $\psi(x)=0$ on $D$. If $x\geq x_0$ we have $(\alpha-A)\phi(x)=\alpha x$. Then $\psi$ gets the following expression $$\psi(x)=\begin{cases}
0 , \quad x<x_0,\\
\alpha x, \quad x\geq x_0.
\end{cases}$$ Then $$\begin{aligned}
R_{\alpha}\psi(x)&= \int_{x_0}^{\infty}\psi(y)R_{\alpha}(x,dy)\nonumber\\
&=\int_{x_0}^{\infty}\psi(y)\int_0^{\infty} e^{-\alpha t}P_x(B_t\in dy)dt\nonumber\\
&=\alpha\int_0^{\infty} e^{-\alpha t}(\int_{x_0}^{\infty}y P_x(B_t\in dy)dt.\end{aligned}$$
Using the same computation as in we get that $$R_{\alpha}\psi(x)=\alpha\int_0^{+\infty}e^{-\alpha t}t\frac{e^{-\frac{(x_0-x)^2}{2t}}}{\sqrt{2\pi t}}dt+\alpha x \int_0^{+\infty}e^{-\alpha t}\int_{x_0-x}^{+\infty}\frac{e^{-\frac{z^2}{2t}}}{\sqrt{2\pi t}}dz dt.$$ We distinguish two cases:
- 1- $x<x_0$, in this case $x_0-x>0$.\
Using the same calculus as in the first example we get that $$\label{eq3.59}
R_{\alpha}\psi(x)=\alpha\int_0^{+\infty}e^{-\alpha t}t\frac{e^{-\frac{(x_0-x)^2}{2t}}}{\sqrt{2\pi t}}dt+\frac{\alpha x}{2}\int_{0}^{+\infty}e^{-\alpha t}(1-\int_0^{\frac{x_0-x}{\sqrt{2t}}}\frac{2e^{-z^2}}{\sqrt{\pi}}dz)dt.$$
Or we have $$\label{eq3.60}
\int_{0}^{+\infty}e^{-\alpha t}(1-\int_0^{\frac{x_0-x}{\sqrt{2t}}}\frac{2e^{-z^2}}{\sqrt{\pi}}dz)dt=\frac{1}{\alpha}e^{-\sqrt{2\alpha}(x_0-x)}$$ we derive equation first with respect to $x$ then $\alpha$ then we get $$\label{eq3.61}
\int_0^{+\infty}e^{-\alpha t}t\frac{e^{-\frac{(x_0-x)^2}{2t}}}{\sqrt{2\pi t}}dt=e^{-\sqrt{2\alpha}(x_0-x)}(\frac{1}{2\alpha\sqrt{2\alpha}}+\frac{x_0-x}{2\alpha}).$$ Replacing and in equation we get $$\label{eq3.62}
R_{\alpha}\psi(x)=e^{-\sqrt{2\alpha}(x_0-x)}(\frac{1}{2\sqrt{2\alpha}}+\frac{x_0}{2}).$$
- 2- By Theorem 2.2 we have that outside D i.e $$\label{eq3.63}
x\geq x_0 ,\quad R_{\alpha}\psi(x)=x.$$
Then by the continuity of $R_{\alpha}\psi$ at $x=x_0$ and combining and at $x=x_0$, we get $$x_0=\frac{1}{\sqrt{2\alpha}}.$$
Find $\Phi_{\alpha}(x)$ and $\tau^{*}$ such that $$\Phi_{\alpha}(x)= \sup_{\tau} E_{x}[e^{-\alpha \tau} B(\tau)] = E_{x}[e^{-\alpha \tau^{*}} B(\tau^*)].$$ where $B(t)$ is Brownian motion reflected upwards when $B(t)=0$. (i)The function $\phi_{\alpha}$ should verify the following PDE $$\begin{cases}
\frac{\partial \phi_{\alpha}}{\partial t}(t,x)+\frac{1}{2}\frac{\partial^2 \phi_{\alpha}}{\partial x^2}(t,x)=0 \text{ on } D,\\
\phi_{\alpha}(t,x)=e^{-\alpha t} x,\quad x\notin D,\\
\frac{\partial \phi_{\alpha}}{\partial x}(t,0)=0.
\end{cases}$$ Put $\phi_{\alpha}(t,x)=\phi_0(x) e^{-\alpha t}$ then $\phi_0$ verifies the following second order differential equation $$\label{secondreflectedbis}
\begin{cases}
\frac{1}{2}\frac{\partial^2 \phi_{0}}{\partial x^2}(x)-\alpha \phi_{0}(x)=0 \text{ on } D\\
\phi_{0}(x)=x,\quad x\notin D.\\
\phi_{0}'(0)=0
\end{cases}$$ The general solution of the equation is given by $$\phi_0(x)=C_1e^{\sqrt{2\alpha}x}+C_2e^{-\sqrt{2\alpha}x},$$ where $C_1$ and $C_2$ are constants.\
Since we have $\phi_{0}'(0)=0$ then $C_1=C_2$ and $$\phi_0(x)=C_1(e^{\sqrt{2\alpha}x}+e^{-\sqrt{2\alpha}x}).$$ Using the continuity of $\phi_{\alpha}$ at $x=x_0$ we have $\phi_0(x_0)=x_0$ then $$C_1=\frac{x_0}{e^{\sqrt{2\alpha}x_0}+e^{-\sqrt{2\alpha}x_0}}.$$ Then the solution of is given by $$\label{eqrefappbis}
\phi_0(x)=\frac{x_0}{e^{\sqrt{2\alpha}x_0}+e^{-\sqrt{2\alpha}x_0}}(e^{\sqrt{2\alpha}x}+e^{-\sqrt{2\alpha}x}).$$ Using the high contact condition at $x=x_0$ we get that $$\frac{\sqrt{2\alpha}x_0}{e^{\sqrt{2\alpha}x_0}+e^{-\sqrt{2\alpha}x_0}}(e^{\sqrt{2\alpha}x_0}-e^{-\sqrt{2\alpha}x_0})=1.$$ The $x_0$ is a solution of the following equation $$\tanh(\sqrt{2\alpha}x)=\frac{1}{\sqrt{2\alpha}x}.$$ This equation has two solutions one negative and one positive.\
For $x_0>0$ we have $D=\{ 0\leq x<x_0\}$.
(ii)By Theorem 2.2 we have that outside of $D$ i.e $$\label{express1bis}
x\geq x_0,\quad R_{\alpha}\psi(x)=x.$$
We have $\psi(x)=f(x)=0$ for $x<x_0$ and for $x\geq x_0, \psi(x)=(\alpha-A)\phi(x)=\alpha x$. Then we get that for all $x\in \mathbb{R}_+$
$$\begin{aligned}
R_{\alpha}\psi(x)&=\int_0^{\infty}e^{-\alpha t}\int_{x_0}^{\infty}\psi(y)P_x(|B_t|\in dy)dt\\
&=\alpha \int_0^{\infty} e^{-\alpha t}\int_{x_0}^{\infty}y\frac{e^{-\frac{(y-x)^2}{2t}}+e^{-\frac{(y+x)^2}{2t}}}{\sqrt{2\pi t}}dy dt\\\label{eqfireqbis}
&=\alpha \int_0^{\infty} \frac{e^{-\alpha t}}{\sqrt{2\pi t}}\int_{x_0}^{\infty}\{ye^{-\frac{(y-x)^2}{2t}}+ye^{-\frac{(y+x)^2}{2t}}\}dy dt\\
&=\alpha \int_0^{\infty} \frac{e^{-\alpha t}}{\sqrt{2\pi
t}}\{-t[e^{-\frac{(y-x)^2}{2t}}]_{x_0}^{\infty}+x\int_{x_0}^{\infty}e^{-\frac{(y-x)^2}{2t}}dy-t[e^{-\frac{(y+x)^2}{2t}}]_{x_0}^{\infty}-x\int_{x_0}^{\infty}e^{-\frac{(y+x)^2}{2t}}dy\}dt\\
&=\alpha \int_0^{\infty} \frac{e^{-\alpha t}}{\sqrt{2\pi
t}}\{te^{-\frac{(x_0-x)^2}{2t}}+x\int_{x_0}^{\infty}e^{-\frac{(y-x)^2}{2t}}dy+te^{-\frac{(x_0+x)^2}{2t}}-x\int_{x_0}^{\infty}e^{-\frac{(y+x)^2}{2t}}dy\}dt\\
&=\alpha \int_0^{\infty} \frac{e^{-\alpha t}}{\sqrt{2\pi
t}}\{te^{-\frac{(x_0-x)^2}{2t}}+te^{-\frac{(x_0+x)^2}{2t}}+x\int_{x_0}^{\infty}e^{-\frac{(y-x)^2}{2t}}dy-x\int_{x_0}^{\infty}e^{-\frac{(y+x)^2}{2t}}dy\}dt.\label{express2bis}\end{aligned}$$
We study now the sum of the two integrals with respect to $dy$ in the previous equation.\
We have $$\begin{aligned}
&\alpha x\int_0^{+\infty}e^{-\alpha t}\int_{x_0}^{+\infty}\frac{e^{-\frac{(y-x)^2}{2t}}}{\sqrt{2\pi t}}dy dt-\alpha x\int_0^{+\infty}e^{-\alpha t}\int_{x_0}^{+\infty}\frac{e^{-\frac{(y+x)^2}{2t}}}{\sqrt{2\pi t}}dy dt\nonumber\\
&=\alpha x\int_0^{+\infty}e^{-\alpha t}\int_{\frac{x_0-x}{\sqrt{2t}}}^{+\infty}\frac{e^{-z^2}}{\sqrt{\pi }}dz dt-\alpha x\int_0^{+\infty}e^{-\alpha t}\int_{\frac{x_0+x}{\sqrt{2t}}}^{+\infty}\frac{e^{-z^2}}{\sqrt{\pi }}dz dt\nonumber\\
&=\alpha\frac{x}{2}\int_0^{+\infty}e^{-\alpha t}\Big(\int_{\frac{x_0-x}{\sqrt{2t}}}^{+\infty}2\frac{e^{-z^2}}{\sqrt{\pi }}dz-\int_{\frac{x_0+x}{\sqrt{2t}}}^{+\infty}2\frac{e^{-z^2}}{\sqrt{\pi} }dz\Big)dt\end{aligned}$$ For $0\leq x<x_0$, we have that the last equation is equal to $$\begin{aligned}
&\alpha\frac{x}{2}\int_0^{+\infty}e^{-\alpha t}( 1-H(\frac{\sqrt{2}(x_0-x)}{2\sqrt{t}}))dt-\alpha\frac{x}{2}\int_0^{+\infty}e^{-\alpha t}( 1-H(\frac{\sqrt{2}(x_0+x)}{2\sqrt{t}}))dt\nonumber\\
&=\frac{x}{2}e^{-\sqrt{2\alpha}(x_0-x)}-\frac{x}{2}e^{-\sqrt{2\alpha}(x_0+x)},\end{aligned}$$ where $$H(u)=\int_0^u\frac{2e^{-z^2}}{\sqrt{\pi}}dz, \quad \forall u>0.$$ Then we deduce that for $0\leq x<x_0$ $$\label{eqcalc1bis}
R_{\alpha}\psi(x)=\alpha \int_0^{+\infty}\frac{e^{-\alpha t}}{\sqrt{2\pi t}}t(e^{-\frac{(x_0-x)^2}{2t}}+e^{-\frac{(x_0+x)^2}{2t}})dt+\frac{x}{2}(e^{-\sqrt{2\alpha}(x_0-x)}-e^{-\sqrt{2\alpha}(x_0+x)}).$$ Or we have $$\int_0^{+\infty}e^{-\alpha t}\Big(\int_{\frac{x_0-x}{\sqrt{2t}}}^{+\infty}\frac{e^{-z^2}}{\sqrt{\pi }}dz-\int_{\frac{x_0+x}{\sqrt{2t}}}^{+\infty}\frac{e^{-z^2}}{\sqrt{\pi} }dz\Big)dt=\frac{1}{2\alpha}(e^{-\sqrt{2\alpha}(x_0-x)}-e^{-\sqrt{2\alpha}(x_0+x)}).$$ We derive the previous equation with respect to $x$, we get $$\int_0^{+\infty}\frac{e^{-\alpha t}}{\sqrt{2\pi t}}\Big(e^{-\frac{(x_0-x)^2}{2t}}+e^{-\frac{(x_0+x)^2}{2t}}\Big)dt=\frac{1}{\sqrt{2\alpha}}(e^{-\sqrt{2\alpha}(x_0-x)}+e^{-\sqrt{2\alpha}(x_0+x)}).$$ We derive now with respect to $\alpha$ we get $$\begin{aligned}
\label{eqcalcbis}
&\int_0^{+\infty}t\frac{e^{-\alpha t}}{\sqrt{2\pi t}}\Big(e^{-\frac{(x_0-x)^2}{2t}}+e^{-\frac{(x_0+x)^2}{2t}}\Big)dt=\frac{1}{2\alpha\sqrt{2\alpha}}(e^{-\sqrt{2\alpha}(x_0-x)}+e^{-\sqrt{2\alpha}(x_0+x)})\nonumber\\
&+
\frac{1}{2\alpha}(x_0-x)e^{-\sqrt{2\alpha}(x_0-x)}+\frac{1}{2\alpha}(x_0+x)e^{-\sqrt{2\alpha}(x_0+x)}.\end{aligned}$$
Substituting in we get that $$\label{eqcalc3bis}
R_{\alpha}\psi(x)=(\frac{1}{2\sqrt{2\alpha}}+\frac{1}{2}x_0)e^{-\sqrt{2\alpha}x_0}(e^{\sqrt{2\alpha}x}+e^{-\sqrt{2\alpha}x}), \quad 0\leq x<x_0.$$ Let us now study $R_{\alpha}\psi(x_0)$.\
Using the continuity of $R_{\alpha}\psi$ at $x_0$ i.e combining the two expressions and of $R_{\alpha}\psi$ at $x=x_0$ we get $$\label{eqkhbis}
(\frac{1}{2\sqrt{2\alpha}}+\frac{1}{2}x_0)e^{-\sqrt{2\alpha}x_0}(e^{\sqrt{2\alpha}x_0}+e^{-\sqrt{2\alpha}x_0})=x_0.$$ Form this equation we get that $$(\frac{1}{2\sqrt{2\alpha}}+\frac{1}{2}x_0)e^{-\sqrt{2\alpha}x_0}=\frac{x_0}{e^{\sqrt{2\alpha}x_0}+e^{-\sqrt{2\alpha}x_0}}.$$ Replacing this in we get $$\label{eqcalc3biss}
R_{\alpha}\psi(x)=\frac{x_0(e^{\sqrt{2\alpha}x}+e^{-\sqrt{2\alpha}x})}{e^{\sqrt{2\alpha}x_0}+e^{-\sqrt{2\alpha}x_0}}, \quad 0\leq x<x_0.$$ Developing equation then multiplying it by $e^{\sqrt{2\alpha}x_0}$, we get
$$\label{eqx01bis}
\frac{1}{\sqrt{2\alpha}x_0}(e^{\sqrt{2\alpha}x_0}+e^{-\sqrt{2\alpha}x_0})=e^{\sqrt{2\alpha}x_0}-e^{-\sqrt{2\alpha}x_0}.$$
Then we deduce that $x_0$ satisfies the following equation $$\label{eqx0bis}
th(\sqrt{2\alpha}x)=\frac{1}{\sqrt{2\alpha}x}.$$ To summarize, we have proved that $$R_{\alpha}\psi(x)=
\begin{cases}
\frac{x_0(e^{\sqrt{2\alpha}x}+e^{-\sqrt{2\alpha}x})}{e^{\sqrt{2\alpha}x_0}+e^{-\sqrt{2\alpha}x_0}}, \quad 0\leq x<x_0\\
x, \quad x\geq x_0,
\end{cases}$$ where $x_0$ satisfies the equation . The next step is to verify the assertion (ii) of Theorem 2.2 i.e that for all $x\in \mathbb{R}_+$ we have $R_{\alpha}\psi(x)\geq x$. In fact, consider the function $$f(x)=R_{\alpha}\psi(x)-x=\begin{cases}
\frac{x_0(e^{\sqrt{2\alpha}x}+e^{-\sqrt{2\alpha}x})}{e^{\sqrt{2\alpha}x_0}+e^{-\sqrt{2\alpha}x_0}}-x, \quad 0\leq x<x_0\\
0, \quad x\geq x_0,
\end{cases}$$ For $0\leq x<x_0$, we have $$\label{eqf'bis}
f'(x)=\frac{\sqrt{2\alpha}x_0(e^{\sqrt{2\alpha}x}-e^{-\sqrt{2\alpha}x})}{e^{\sqrt{2\alpha}x_0}+e^{-\sqrt{2\alpha}x_0}}-1.$$ Then $$f'(x)=0\Rightarrow e^{\sqrt{2\alpha}x}-e^{-\sqrt{2\alpha}x}=\frac{e^{\sqrt{2\alpha}x_0}+e^{-\sqrt{2\alpha}x_0}}{\sqrt{2\alpha}x_0}$$ From equation , we have that $x_0$ satisfies $$e^{\sqrt{2\alpha}x_0}-e^{-\sqrt{2\alpha}x_0}=\frac{e^{\sqrt{2\alpha}x_0}+e^{-\sqrt{2\alpha}x_0}}{\sqrt{2\alpha}x_0}.$$ Then $x_0$ is the unique positive solution of $f'(x)=0$.\
Or we have that $f$ is strictly decreasing on $[0,x_0[$.\
In fact one can write $f'(x)$ in as the following $$f'(x)=\frac{\sqrt{2\alpha}x_0 sh(\sqrt{2\alpha}x)}{ch(\sqrt{2\alpha}x_0)}-1,$$ then $$f''(x)=\frac{2\alpha ch(\sqrt{2\alpha}x)}{ch(\sqrt{2\alpha}x_0)}>0.$$ Then $f'$ is strictly increasing in $[0,x_0[$ with supremum $f'(x_0)=0$. Then we deduce that for $x\in[0,x_0[, f'(x)<0$. Then $f(x)=R_{\alpha}\psi(x)-x$ is decreasing in $[0,x_0[$ with infimum $f(x_0)=R_{\alpha}\psi(x_0)-x_0=0$. Therefore $f(x)>0$ for all $x\in [0,x_0[$ i.e $R_{\alpha}\psi(x)>x$ for all $x\in [0,x_0[$.\
In addition for all $x\geq x_0$ we have $R_{\alpha}\psi(x)=x$. Then we deduce that for all $x\in[0,+\infty[, R_{\alpha}\psi(x)\geq x$. Then the assertions of the verification Theorem 2.2 are well satisfied.
Find $\Phi_{\alpha}(x)$ and $\tau^{*}$ such that $$\Phi_{\alpha}(x)= \sup_{\tau} E_{x}[e^{-\alpha \tau} B(\tau)] = E_{x}[e^{-\alpha \tau^{*}} B(\tau^*)].$$ where $B(t)$ is Brownian motion trapped when B(t)=0. More precisely, we consider $$\tau(0)=\inf\{ t\geq 0,|B(t)=0\}.$$ The Brownian motion process trapped at 0 is defined by $$B_0(t)=B(t\wedge \tau_0).$$ From now on we denote $B_0(t)$ by $B(t)$.\
(i)The function $\phi_{\alpha}$ should verify the following PDE $$\begin{cases}
\frac{\partial \phi_{\alpha}}{\partial t}(t,x)+\frac{1}{2}\frac{\partial^2 \phi_{\alpha}}{\partial x^2}(t,x)=0 \text{ on } D\\
\phi_{\alpha}(t,x)=e^{-\alpha t} x,\quad x\notin D.\\
\frac{\partial^2 \phi_{\alpha}}{\partial x^2}(t,0)=0
\end{cases}$$ Put $\phi_{\alpha}(t,x)=\phi_0(x) e^{-\alpha t}$ then $\phi_0$ verifies the following second order differential equation $$\label{secondreflected}
\begin{cases}
\frac{1}{2}\frac{\partial^2 \phi_{0}}{\partial x^2}(x)-\alpha \phi_{0}(x)=0 \text{ on } D\\
\phi_{0}(x)=x,\quad x\notin D.\\
\phi_{0}''(0)=0
\end{cases}$$ The general solution of the equation is given by $$\phi_0(x)=C_1e^{\sqrt{2\alpha}x}+C_2e^{-\sqrt{2\alpha}x}$$ where $C_1$ and $C_2$ are constants.\
Since we have $\phi_{0}''(0)=0$ then $C_2=-C_1$ and $$\phi_0(x)=C_1(e^{\sqrt{2\alpha}x}-e^{-\sqrt{2\alpha}x})$$ Using the continuity of $\phi_{0}$ at $x=x_0$ we have $\phi_0(x_0)=x_0$ then $$C_1=\frac{x_0}{e^{\sqrt{2\alpha}x_0}-e^{-\sqrt{2\alpha}x_0}}.$$ Then the solution of is given by $$\phi_0(x)=\frac{x_0}{e^{\sqrt{2\alpha}x_0}-e^{-\sqrt{2\alpha}x_0}}(e^{\sqrt{2\alpha}x}-e^{-\sqrt{2\alpha}x}).$$ Using the high contact condition at $x=x_0$ we get that $$\frac{\sqrt{2\alpha}x_0}{e^{\sqrt{2\alpha}x_0}-e^{-\sqrt{2\alpha}x_0}}(e^{\sqrt{2\alpha}x_0}+e^{-\sqrt{2\alpha}x_0})=1.$$ The $x_0$ is a solution of the following equation $$\coth(\sqrt{2\alpha}x)=\frac{1}{\sqrt{2\alpha}x}.$$ This equation has two solutions one negative and one positive.\
For $x_0>0$ we have $D=\{ 0<x<x_0\}$.
(ii)By Theorem 2.2 we have that outside of $D$ i.e $$\label{express1}
x\geq x_0,\quad R_{\alpha}\psi(x)=x.$$
We have $\psi(x)=f(x)=0$ for $0<x<x_0$ and for $x\geq x_0, \psi(x)=(\alpha-A)\phi(x)=\alpha x$.\
In addition the density of Brownian motion trapped at 0 is given by $$P_x(B_t\in dy)=\frac{e^{-\frac{(y-x)^2}{2t}}-e^{-\frac{(y+x)^2}{2t}}}{\sqrt{2\pi t}}dy.$$ Then we get that for all $x\in \mathbb{R}_+^*$
$$\begin{aligned}
R_{\alpha}\psi(x)&=\int_0^{\infty}e^{-\alpha t}\int_{x_0}^{\infty}\psi(y)P_x(B_t\in dy)dt\\
&=\alpha \int_0^{\infty} e^{-\alpha t}\int_{x_0}^{\infty}y\frac{e^{-\frac{(y-x)^2}{2t}}-e^{-\frac{(y+x)^2}{2t}}}{\sqrt{2\pi t}}dy dt\\
&=\alpha \int_0^{\infty} \frac{e^{-\alpha t}}{\sqrt{2\pi t}}\int_{x_0}^{\infty}\{ye^{-\frac{(y-x)^2}{2t}}-ye^{-\frac{(y+x)^2}{2t}}\}dy dt\\
&=\alpha \int_0^{\infty} \frac{e^{-\alpha t}}{\sqrt{2\pi
t}}\{-t[e^{-\frac{(y-x)^2}{2t}}]_{x_0}^{\infty}+x\int_{x_0}^{\infty}e^{-\frac{(y-x)^2}{2t}}dy+t[e^{-\frac{(y+x)^2}{2t}}]_{x_0}^{\infty}+x\int_{x_0}^{\infty}e^{-\frac{(y+x)^2}{2t}}dy\}dt\\
&=\alpha \int_0^{\infty} \frac{e^{-\alpha t}}{\sqrt{2\pi
t}}\{te^{-\frac{(x_0-x)^2}{2t}}+x\int_{x_0}^{\infty}e^{-\frac{(y-x)^2}{2t}}dy-te^{-\frac{(x_0+x)^2}{2t}}+x\int_{x_0}^{\infty}e^{-\frac{(y+x)^2}{2t}}dy\}dt\\
&=\alpha \int_0^{\infty} \frac{e^{-\alpha t}}{\sqrt{2\pi
t}}\{te^{-\frac{(x_0-x)^2}{2t}}-te^{-\frac{(x_0+x)^2}{2t}}+x\int_{x_0}^{\infty}e^{-\frac{(y-x)^2}{2t}}dy+x\int_{x_0}^{\infty}e^{-\frac{(y+x)^2}{2t}}dy\}dt.\label{express2}\end{aligned}$$
We study now the sum of the two integrals with respect to $dy$ in the previous equation.\
We have $$\begin{aligned}
&\alpha x\int_0^{+\infty}e^{-\alpha t}\int_{x_0}^{+\infty}\frac{e^{-\frac{(y-x)^2}{2t}}}{\sqrt{2\pi t}}dy dt+\alpha x\int_0^{+\infty}e^{-\alpha t}\int_{x_0}^{+\infty}\frac{e^{-\frac{(y+x)^2}{2t}}}{\sqrt{2\pi t}}dy dt\nonumber\\
&=\alpha x\int_0^{+\infty}e^{-\alpha t}\int_{\frac{x_0-x}{\sqrt{2t}}}^{+\infty}\frac{e^{-z^2}}{\sqrt{\pi }}dz dt+\alpha x\int_0^{+\infty}e^{-\alpha t}\int_{\frac{x_0+x}{\sqrt{2t}}}^{+\infty}\frac{e^{-z^2}}{\sqrt{\pi }}dz dt\nonumber\\
&=\alpha\frac{x}{2}\int_0^{+\infty}e^{-\alpha t}\Big(\int_{\frac{x_0-x}{\sqrt{2t}}}^{+\infty}2\frac{e^{-z^2}}{\sqrt{\pi }}dz+\int_{\frac{x_0+x}{\sqrt{2t}}}^{+\infty}2\frac{e^{-z^2}}{\sqrt{\pi} }dz\Big)dt\end{aligned}$$ For $0<x<x_0$, we have that the last equation is equal to $$\begin{aligned}
&=\alpha\frac{x}{2}\int_0^{+\infty}e^{-\alpha t}( 1-H(\frac{\sqrt{2}(x_0-x)}{2\sqrt{t}}))dt+\alpha\frac{x}{2}\int_0^{+\infty}e^{-\alpha t}( 1-H(\frac{\sqrt{2}(x_0+x)}{2\sqrt{t}}))dt\nonumber\\
&=\frac{x}{2}e^{-\sqrt{2\alpha}(x_0-x)}+\frac{x}{2}e^{-\sqrt{2\alpha}(x_0+x)},\end{aligned}$$ where $$H(u)=\int_0^u\frac{2e^{-z^2}}{\sqrt{\pi}}dz, \quad \forall u>0.$$ Then we deduce that for $0< x<x_0$ $$\label{eqcalc1}
R_{\alpha}\psi(x)=\alpha \int_0^{+\infty}\frac{e^{-\alpha t}}{\sqrt{2\pi t}}t(e^{-\frac{(x_0-x)^2}{2t}}-e^{-\frac{(x_0+x)^2}{2t}})dt+\frac{x}{2}(e^{-\sqrt{2\alpha}(x_0-x)}+e^{-\sqrt{2\alpha}(x_0+x)}).$$ Or we have $$\int_0^{+\infty}e^{-\alpha t}\Big(\int_{\frac{x_0-x}{\sqrt{2t}}}^{+\infty}\frac{e^{-z^2}}{\sqrt{\pi }}dz+\int_{\frac{x_0+x}{\sqrt{2t}}}^{+\infty}\frac{e^{-z^2}}{\sqrt{\pi} }dz\Big)dt=\frac{1}{2\alpha}(e^{-\sqrt{2\alpha}(x_0-x)}+e^{-\sqrt{2\alpha}(x_0+x)}).$$ We derive the previous equation with respect to $x$, we get $$\int_0^{+\infty}\frac{e^{-\alpha t}}{\sqrt{2\pi t}}\Big(e^{-\frac{(x_0-x)^2}{2t}}+e^{-\frac{(x_0+x)^2}{2t}}\Big)dt=\frac{1}{\sqrt{2\alpha}}(e^{-\sqrt{2\alpha}(x_0-x)}+e^{-\sqrt{2\alpha}(x_0+x)}).$$ We derive now with respect to $\alpha$ we get $$\begin{aligned}
\label{eqcalc}
&\int_0^{+\infty}t\frac{e^{-\alpha t}}{\sqrt{2\pi t}}\Big(e^{-\frac{(x_0-x)^2}{2t}}-e^{-\frac{(x_0+x)^2}{2t}}\Big)dt=\frac{1}{2\alpha\sqrt{2\alpha}}(e^{-\sqrt{2\alpha}(x_0-x)}-e^{-\sqrt{2\alpha}(x_0+x)})\nonumber\\
&+
\frac{1}{2\alpha}(x_0-x)e^{-\sqrt{2\alpha}(x_0-x)}-\frac{1}{2\alpha}(x_0+x)e^{-\sqrt{2\alpha}(x_0+x)}.\end{aligned}$$
Substituting in we get that $$\label{eqcalc3}
R_{\alpha}\psi(x)=(\frac{1}{2\sqrt{2\alpha}}+\frac{1}{2}x_0)e^{-\sqrt{2\alpha}x_0}(e^{\sqrt{2\alpha}x}-e^{-\sqrt{2\alpha}x}), \quad 0<x<x_0.$$ Let us now study $R_{\alpha}\psi(x_0)$.\
Using the continuity of $R_{\alpha}\psi$ at $x_0$ i.e combining the two expressions and of $R_{\alpha}\psi$ at $x=x_0$ we get $$\label{eqkh}
(\frac{1}{2\sqrt{2\alpha}}+\frac{1}{2}x_0)e^{-\sqrt{2\alpha}x_0}(e^{\sqrt{2\alpha}x_0}-e^{-\sqrt{2\alpha}x_0})=x_0.$$ Form this equation we get that $$(\frac{1}{2\sqrt{2\alpha}}+\frac{1}{2}x_0)e^{-\sqrt{2\alpha}x_0}=\frac{x_0}{e^{\sqrt{2\alpha}x_0}-e^{-\sqrt{2\alpha}x_0}}.$$ Replacing this in we get $$\label{eqcalc3bisss }
R_{\alpha}\psi(x)=\frac{x_0(e^{\sqrt{2\alpha}x}-e^{-\sqrt{2\alpha}x})}{e^{\sqrt{2\alpha}x_0}-e^{-\sqrt{2\alpha}x_0}}, \quad 0<x<x_0.$$ Developing equation then multiplying it by $e^{\sqrt{2\alpha}x_0}$, we get
$$\label{eqx01}
\frac{1}{\sqrt{2\alpha}x_0}(e^{\sqrt{2\alpha}x_0}-e^{-\sqrt{2\alpha}x_0})=e^{\sqrt{2\alpha}x_0}+e^{-\sqrt{2\alpha}x_0}.$$
Then we deduce that $x_0$ satisfies the following equation $$\label{eqx0}
coth(\sqrt{2\alpha}x)=\frac{1}{\sqrt{2\alpha}x}.$$ To summarize, we have proved that $$R_{\alpha}\psi(x)=
\begin{cases}
\frac{x_0(e^{\sqrt{2\alpha}x}-e^{-\sqrt{2\alpha}x})}{e^{\sqrt{2\alpha}x_0}-e^{-\sqrt{2\alpha}x_0}}, \quad 0<x<x_0\\
x, \quad x\geq x_0,
\end{cases}$$ where $x_0$ satisfies the equation . The next step is to verify the assertion (ii) of Theorem 2.2 i.e that for all $x\in \mathbb{R}_+$ we have $R_{\alpha}\psi(x)\geq x$. In fact, consider the function $$f(x)=R_{\alpha}\psi(x)-x=\begin{cases}
\frac{x_0(e^{\sqrt{2\alpha}x}-e^{-\sqrt{2\alpha}x})}{e^{\sqrt{2\alpha}x_0}- e^{-\sqrt{2\alpha}x_0}}-x, \quad 0< x<x_0\\
0, \quad x\geq x_0,
\end{cases}$$ For $0< x<x_0$, we have $$\label{eqf'}
f'(x)=\frac{\sqrt{2\alpha}x_0(e^{\sqrt{2\alpha}x}+e^{-\sqrt{2\alpha}x})}{e^{\sqrt{2\alpha}x_0}-e^{-\sqrt{2\alpha}x_0}}-1.$$ Then $$f'(x)=0\Rightarrow e^{\sqrt{2\alpha}x}+
e^{-\sqrt{2\alpha}x}=\frac{e^{\sqrt{2\alpha}x_0}-e^{-\sqrt{2\alpha}x_0}}{\sqrt{2\alpha}x_0}$$
![[]{data-label=""}](ValueFctsxlarge.png){width="0.9\linewidth"}
![[]{data-label=""}](ValueFcts-xlessThan1.png){width=".7\linewidth"}
From equation , we have that $x_0$ satisfies $$e^{\sqrt{2\alpha}x_0}+
e^{-\sqrt{2\alpha}x_0}=\frac{e^{\sqrt{2\alpha}x_0}-e^{-\sqrt{2\alpha}x_0}}{\sqrt{2\alpha}x_0}.$$ Then $x_0$ is the unique positive solution of $f'(x)=0$.\
Or we have that $f$ is strictly decreasing on $]0,x_0[$.\
In fact one can write $f'(x)$ in as the following $$f'(x)=\frac{\sqrt{2\alpha}x_0 ch(\sqrt{2\alpha}x)}{sh(\sqrt{2\alpha}x_0)}-1,$$ then $$f''(x)=\frac{2\alpha sh(\sqrt{2\alpha}x)}{sh(\sqrt{2\alpha}x_0)}>0\quad \forall x\in]0,x_0[.$$ Then $f'$ is strictly increasing in $]0,x_0[$ with supremum $f'(x_0)=0$. Then we deduce that for $x\in]0,x_0[, f'(x)<0$. Then $f(x)=R_{\alpha}\psi(x)-x$ is strictly decreasing in $]0,x_0[$ with infimum $f(x_0)=R_{\alpha}\psi(x_0)-x_0=0$. Therefore $f(x)> 0$ for all $x\in [0,x_0[$ i.e $R_{\alpha}\psi(x)\geq x$ for all $x\in ]0,x_0[$.\
In addition for all $x\geq x_0$ we have $R_{\alpha}\psi(x)=x$. Then we deduce that for all $x\in]0,+\infty[, R_{\alpha}\psi(x)\geq x$. Then the assertions of the verification Theorem 2.2 are well satisfied.
We denote by $x_0^o,x_0^r$ and $x_0^a$ the values of $x_0$ associated to the optimal stopping barriers for the usual Brownian motion i.e Example 2.2, reflected Brownian motion at 0 i.e Example 2.3 and absorbed (trapped) Brownian motion i.e Example 2.4 , respectively. Comparing these three values we have $$x_0^a<x_0^o<x_0^r.$$
Indeed this result is expected because if $B(t)$ is absorbed at 0 then the payoff is 0, and this is the worst that can happen. Therefore one is afraid to wait for large value of $B(t)$ before stopping. For the reflected Brownian motion, however, there is no disaster if it hits 0 because it is just reflected back. Therefore one can wait for a large value of $B(t)$ before stopping.
[99]{}
Menoukeu Pamen, O.: Optimal stopping time problem for reflected Brownian motion. Manuscript 30 November 2016.
Dai, S and Menoukou-Pamen,O.: Viscosity solution for optimal stopping problems of Feller processes. arXiv:1803.03832v1 (2018).
Ernesto Mordecki, E. and Salminen, P.: Optimal stopping of Hunt and L\` evy processes(2006),Stochastics: An International Journal of Probability and Stochastic Processes.
Sulem, A. and Øksendal, B.: Applied stochastic control of jump diffusions.Springer-Verlag Berlin Heidelberg (2005).
Fukushima, M. : Dirichlet form and Markov processes. North-Holland Mathematical Library (1980).
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**On Subvarieties of Abelian Varieties with**
**degenerate Gauß Mapping**
**Rainer Weissauer**
Let $X$ be an abelian variety over an algebraically closed field $k$ of dimension $g$. For a closed irreducible subvariety $Y$ of dimension $d$ in $X$, we let $S$ denote a Zariski dense open subset of its regular locus $Y_{reg}$. For the conormal bundle $p_S: T_{S}^*(X) \to S$ let $\Lambda_Y$ denote the closure of $T_S^*(X)$ in the cotangent bundle $T^*(X)$ of $X$. The cotangent bundle is trivial for an abelian variety. With respect to the decomposition $$T^*(X) \cong X \times T^*_0(X)$$ its bundle morphism $p_X: T^*(X)\to X$ is given by the projection on the first factor. The corresponding structure morphism $p_Y: \Lambda_Y \to Y$ is defined by the upper horizontal map of the diagram $$\xymatrix{ \Lambda_Y \ar@{^{(}->}[r] & T^*(X) \ar[r]^-{p_X} & X \cr
\Lambda_S \ar@{^{(}->}[u]\ar[rr]^-{p_S} & & S \ar@{^{(}->}[u] \cr } ,$$ using that the image of $p_Y$ is contained in $Y$, since $Y$ is the Zariski closure of $S$ and $\Lambda_Y$ is the Zariski closure of $\Lambda_S$. Notice, $\dim(\Lambda_Y)=g$ implies $\dim(\Lambda_Y \setminus \Lambda_S) < g$. For $y\in S$ let $\Lambda_{S,y}$ denote the fiber $p_S^{-1}(y)$, which is the conormal space $N^*_y(Y)$ of $Y$ in $T^*_y(X)$ at the point $y$. By a translation, $T^*_y(X)$ will be identifed with its image in $T^*_0(X) = - y + T^*_y(X)$, so $$\Lambda_{S,y} \subset T^*_0(X)\ .$$ For $\Lambda_Y \subset T^*(X)$, now the projection $T^*(X)\to T^*_0(X)$ on the second factor induces the Gauß mapping $$\gamma: \Lambda_Y \to T^*_0(X)\ .$$ The image under the Gauß mapping of $\lambda = (y,\tau)$, with components $y\in Y$ and $\tau \in \Lambda_{Y,y}\subset T^*_0(X)$, is $\tau\in T^*_0(X)$, whereas the image under the structure morphism $p_Y: \Lambda_Y \to Y$ is the first component $y\in Y\subset X$. Since, for $y\in S$, the vector $\tau$ is a conormal vector $\tau\in T_{Y,y}^*(X) \subset T_y^*(X) \cong T^*_0(X)$ via translation, the linear form $\tau$ annihilates the tangent space $T_y(Y)$ of $Y$ at $y$.
The case $Y=X$ is exceptional; in this case the image $\gamma(\Lambda_Y)$ of the Gauß mapping is contained in $\{0\}$. For $Y\neq X$, we may remove both the closure of the zero section in $\Lambda_Y$ and the zero section in $T^*(X)$ to obtain a proper morphism of the associated projective conormal bundles $${{\mathbb P}}\gamma: {{\mathbb P}}\Lambda_S \to {{\mathbb P}}(T^*_0(X)) \ .$$ Obviously, for $Y\neq X$, the mapping ${{\mathbb P}}\gamma$ is dominant if and only if the mapping $\gamma$ is dominant. Therefore we may ignore the trivial vector $\tau=0$ in $T^*_0(X)$ in subsequent arguments. Furthermore, if $\gamma$ is not dominant, the image ${{\mathbb P}}(\Lambda_Y)$ is a closed subvariety of ${{\mathbb P}}(T^*_0(X))$, hence contained in a hypersurface defined as the zero locus of some nontrivial homogenous polynomial on $T^*_0(X)$.
Since $\dim(\Lambda_Y)=g$, hence for $Y\neq X$ the following assertions are equivalent
1. $\dim(\gamma(\Lambda)) < g $
2. The proper morphism ${{\mathbb P}}\gamma: {{\mathbb P}}\Lambda_S \to {{\mathbb P}}(T^*_0(X)) $ is not dominant.
3. The image $\gamma(\Lambda_S)$ of the Gauß mapping is contained in the zero locus of a nontrivial homogenous polynomial $F$ on $T^*_0(X)$.
4. $\gamma: \Lambda_Y \to T^*_0(X)$ is not dominant.
5. $\gamma: \Lambda_Y \to T^*_0(X)$ is not generically finite.
6. For any point $y$ of general position in $S$ there exists a curve $C$ in $S$ containing $y$, which is contracted by $\gamma$.
Similarly one defines the Gauß mapping $\gamma$ for reducible closed varieties $Y$. In this case the Gauß mapping is dominant if and only if the Gauß mapping of one of its irreducible components is dominant.
An irreducible variety $Y$ in $X$ will be called [*degenerate*]{}, if there exists a positive dimensional abelian subvariety $A\subset X$ such that $A+Y=Y$. (This notion differs from the one used in \[R\]). Our main result is
.
*For a closed irreducible subvariety $Y$ of $X$ the following assertions are equivalent*
- The Gauss mapping $\gamma: \Lambda_Y \to T^*_0(X)$ is not dominant.
- $Y$ is degenerate.
The mapping defined on $S$, which assigns to each point $y\in S$ the tangent space $T_y(Y)$ in $T_y(X)$, gives rise to another Gauß mapping $\Gamma$, now with image in a Graßmann variety $$\Gamma: S \to Gr(d,g) \ .$$ Again the tangent spaces $T_y(X)$ are considered in $T_0(X)$ via translation by $y\in X$; and $\Gamma$ and $\gamma$ can be identified in the cases $d=1$ and $d=g-1$.
The next theorem reproduces theorem 4 of \[A\]. See also \[R\], chapter II.
. [*For a closed irreducible subvariety $Y$ of $X$ the following holds: If the Gauss mapping $\Gamma: \Lambda_Y \to Gr(d,g)$ is not generically finite, then the Gauss mapping $\gamma: \Lambda_Y \to T^*_0(X)$ is not dominant; and hence $Y$ is degenerate in the sense above.*]{}
. If $\Gamma$ is not generically finite, then through any point $y$ of $S$ in general position there exists an algebraic curve $C$ containing $y$, which is contracted under $\Gamma$. So for $y'$ in $C$ we have $T_y(Y)=T_{y'}(Y)$ in $T_0(X)$ and hence $\Lambda_{S,y} = N^*_yY = N^*_{y'}Y = \Lambda_{S,y'}$. Thus $\gamma(y,v)=\gamma(y',v)$ for all $v \in \Lambda_{S,y} \subset T^*_0(X)$. Therefore $C \times \{ v \} \subset \Lambda_S$ is contracted by $\gamma$ for all $v\in N^*_y(Y) \subset T^*_0(X)$. Hence there exist points in general position contracted by $\gamma$. Thus the Gauß mapping $\gamma: \Lambda_S \to T^*_0(X)$ is not dominant, and by theorem 1 there exists an abelian subvariety $A$ of $dim(A)>0$ in $X$ such that $A+Y=Y$.
. To show that the first assertion implies the second, will cover the next sections. The converse is rather trivial: For $A\subset X$ of $\dim(A)>0$, consider the image $\tilde Y$ of $Y$ in the quotient $B=X/A$. Notice that $A+Y=Y$ implies $T_y(A)\subset T_y(Y)$, hence $\Lambda_{Y,y} \subset T^*_0(B)$. Therefore $\gamma(\Lambda_S) = \tilde\gamma(\Lambda_{\tilde Y}) \subset T^*_0(B)$ holds, for the corresponding Gauß mapping $\tilde \gamma$ of $\tilde Y$ in $B$. Thus $\dim(\gamma(\Lambda_S)) =
\dim(\tilde\gamma(\Lambda_{\tilde S})) \leq \dim(B) < \dim(X)=g$, and therefore the Gauß mapping $\gamma:\Lambda_Y \to T^*_0(X)$ is not dominant.
The converse assertion of theorem 1 will be proved by induction on the dimension $d$ of $Y$. The case $d=0$ is trivial. So in the following let us fix some $d>0$, and suppose that theorem 1 already holds for irreducible subvarieties $Y'\subset X'$ of dimension $\dim(Y')< d$ of arbitrary abelian varieties $X'$. This assumption will be maintained during the proof almost until the end of the paper. Furthermore, it is easy to see that for the proof, without restriction of generality, we may assume that $Y$ generates $X$ $$\langle Y \rangle = X \ .$$ Under these conditions, we will then show that the assertion of theorem 1 also holds for varieties $Y$ of dimension $d$.
Before we proceed with the proof, we quote from \[A\] the following
**Characterization of degenerate subvarieties**
For a reduced, irreducible subvariety $Y$ of an abelian variety $X$ define $$Z(Y)=\bigl\{ y\in Y \ \vert \ \exists X'\subset X, X' \mbox{ closed subgroup}, dim(X')>0, y+X' \subset Y\bigr\} \ .$$ Then according to loc. cit. the following assertions hold
. [ *Suppose $Y$ is Zariski closed in $X$, then $Z(Y)$ is Zariski closed in $Y$.*]{}
. [ *Suppose $Y$ is closed in $X$ and $Z(Y)=Y$ holds. Then $Y$ is degenerate.*]{}
In loc. cit. this is stated in the more general context of semiabelian varieties.
. Suppose $U\subset Y$ is a Zariski open dense subvariety, and $Y$ is closed in $X$. Then $Z(U)=U$ implies $Z(Y)=Y$.
. Indeed $Z(U) \subset Z(Y)$ by definition. Hence proposition 1 implies $Y=\overline U =\overline{Z(U)}
\subset Z(Y)$. Hence $Z=Z(Y)$, which proves the lemma.
. With lemma 1 in mind we often replace $Y$ by some Zariski dense open subset $U$ contained in the nonsingular locus $S=Y_{reg}$ of $Y$. We then tacitly write $U=S$ by abuse of notation.
. Suppose $Y$, or a Zariski open dense subset $U$ of $Y$, is a union of not necessarily finitely many subvarieties $F$. Then $Z(F)=F$ for all the $F$ implies $Z(U)=U$, hence $Z(Y)=Y$.
**Exact sequences of abelian varieties**
1\) For a nontrivial abelian subvariety $X' \subset X$ of dimension $<g$ we have the quotient mapping $$q:X\to \tilde X=X/ X' \ .$$ The image $\tilde Y$ of $Y$ is considered as a closed irreducible variety of $\tilde X$ endowed with the reduced subscheme structure $$\xymatrix{ 0 \ar[r] & X' \ar[r] & X\ar[r]^q & \tilde X \ar[r] & 0\cr
& & Y\ar@{->>}[r]^q\ar@{^{(}->}[u] & \tilde Y \ar@{^{(}->}[u] & \cr} \ .$$ $\tilde Y$ generates the nontrivial abelian variety $\tilde X$, by our assumption $\langle Y\rangle =X$. Hence $\dim(\tilde Y) >0$. Therefore the fibers $F_{\tilde y}$ of the morphism $q: Y\to \tilde Y$ have dimension $$\dim(F_{\tilde y}) < d=\dim(Y)\ .$$ For $\tilde y\in \tilde Y$, there exists $y\in Y$ so that $ F_{\tilde y} \ =\ q^{-1}(\tilde y) \ \subset \ y + X' $.
2\) Both concerning the assertions and the assumptions of theorem 1, we may replace $X$ by a finite etale covering and $Y$ by its inverse image. This allows to assume for the proof that $X$ splits (noncanonically) into a direct product $$X = X' \times \tilde X \ .$$ Therefore, in the following, we often tacitly assume that some splitting of the exact sequence exists, and has been chosen. Then $$T^*(X)=T^*(X')\times T^*(\tilde X)$$ and $$F_{\tilde y} = Y \cap (\tilde y + X') \ .$$
3\) For (regular) points $y_1,y_2$ in $Y$, such that $q(y_1)=q(y_2)$ holds in $\tilde X$, the fibers $\Lambda_{Y,y_1} = N^*_{y_1}(Y)$ and $\Lambda_{Y,y_2} = N^*_{y_2}(Y)$ usually do not coincide. Let $i: X'\to X$ be the inclusion or, more generally, any of its translates $i(x') = y+x'$. We claim
. [*There exists a Zariki dense open subset $U$ of the set of regular points of $\tilde S$ of $\tilde Y$, such that for regular points $y$ of $Y$ in $q^{-1}(U)$ there exists a canonical exact sequence of vectorspaces $$\xymatrix{0\ \ar[r] & \ \Lambda_{\tilde S,\tilde y}\ \ar[r] & \ \Lambda_{S,y}\ \ar[r]^-{T^*(i)} & \ \Lambda'_{F_{\tilde y},y}\ \ar[r] & 0 }\ .$$*]{}
Hence, for fixed $\tilde y=q(y)$ in $\tilde X$ with fiber $F_{\tilde y}$ in $X'$, the variation of the conormal spaces $\Lambda_{S,y}$ is controlled by the variation of the conormal spaces $\Lambda'_{F_{\tilde y},y}$. Notice, for a subvariety $Y'$ of a translate of $X'$, we can define $\Lambda_{Y'}$ in $T^*(X)$, and also $\Lambda'_{Y'}$ in $T^*(X')$. Hence the prime index will indicate that the ambient space is a translate of $X'$.
. Consider $0\to T_y(F_{\tilde y}) \to T_y(Y) \to T_{\tilde y}(\tilde Y) \to 0$. This exact sequence of tangent spaces, at a point $y$ where $q$ is a smooth morphism locally, maps to $0\to T(X')\to T(X)\to T(\tilde X)\to 0$. Hence by the snake lemma we get $0\to N'_y(F_{\tilde y}) \to N_y(Y) \to N_{\tilde y}(\tilde Y) \to 0$, and the exact sequence in our assertion is the dual sequence. This proves the claim.
For completeness, we record that there exists another naturally defined exact sequence $ 0 \to \Lambda_{S,y} \to \Lambda_{F_{\tilde y},y} \to T^*_{\tilde y}(\tilde S) \to 0 $.
4\) Since $\Lambda_S = \bigcup_{y\in S} \Lambda_{S,y}$, the image $\gamma(\Lambda_Y)$ of the Gauß mapping in $T^*_0(X)$ is the Zariski closure of the union $$\gamma(\Lambda_S) = \bigcup_{y\in S} \gamma(\Lambda_{S,y})\ .$$ Here, of course, $S$ could be replaced by any Zariski dense open subset. We conclude that the image of $\gamma(\Lambda_Y)$ under the linear mapping $$T^*(i): T^*_0(X)\to T^*_0(X')$$ is the Zariski closure of $$T^*(i)\bigl(\bigcup_{y\in S} \gamma(\Lambda_{S,y})\bigr) = \bigcup_{y\in S} T^*(i)\bigl(\gamma(\Lambda_{S,y})\bigr) = \bigcup_{y\in S} \gamma'(\Lambda'_{F_{\tilde y ,y}}) \ ,$$ where $\gamma': \Lambda'_{F_{\tilde y,y}} \to T^*_0(X')$ denotes the Gauß mapping for $X'$ instead of $X$. Indeed, we have a commutative diagram $$\xymatrix{ \Lambda_{S,y} \ar@/_9mm/[dd]_{\gamma}\ar[r]^-{T^*(i)}\ar@{^{(}->}[d] & \Lambda'_{F_{\tilde y},y} \ar@/^9mm/[dd]^{\gamma'}\ar@{^{(}->}[d]\cr
T^*(X) \ar@{->>}[d] & T^*(X') \ar@{->>}[d]\cr
T^*_0(X) \ar[r]^-{T^*(i)} & T_0^*(X') \cr}$$ possibly after replacing $S$ by some Zariski open dense subset (also denoted $S$ by abuse of notation). Thus $$T^*(i) \bigl(\gamma(\Lambda_Y)\bigr) \ = \ \overline{ \bigcup_{y\in S} \gamma'(\Lambda'_{F_{\tilde y},y} ) } \ .$$
5\) Now consider the case where the Gauß mapping $$\gamma: \Lambda_Y \to T^*_0(X)$$ is ; in addition we assume $Y\neq X$.
Then, for a homogenous polynomial $F\neq 0$ on $T^*_0(X)=T^*_0(X')\oplus T^*_0(\tilde X)$, the zero locus of $F$ contains the image of the Gauss mapping $\gamma$.
Suppose
- $\tau'\neq 0$ in $T^*_0(X')$ is some fixed vector of general position in $T^*_0(X')$.
- $\tau'$ is contained in the subvectorspace $ \Lambda'_{F_{\tilde y,y}}$ of $T^*_0(X')$, for $y\in U \subset Y$ in some fixed Zariski dense open subset $U$ of $S$.
Then, by lemma 2, there exists $\tilde\tau$ in $T^*_0(\tilde X)$ such that the vector $\tau=(\tau',\tilde \tau)$ in $T^*_0(X')\oplus T^*_0(\tilde X)=T^*_0(X)$ is contained in the linear subspace $\Lambda_{S,y}$ of $T^*_0(X)$; and furthermore $$(\tau',\tilde\tau) + \Lambda_{\tilde S,\tilde y} \ \subset \ \Lambda_{S,y} \ \subset \ \gamma(\Lambda_S) \subset T_0^*(X) \ .$$ Hence the polynomial $F$ vanishes on all the vectors $ (\tau',\tilde\tau) + \Lambda_{\tilde S,\tilde y}$.
Notice, $\Lambda_{\tilde S,\tilde y} = 0 \times W $ for some linear subspace $W$ of $V=T^*(\tilde X)$. For $v\in V$ there exists an expansion of $F(\tau',v)$ $$F(\tau',v) = F_{m,\tau'}(v) + F_{m-1,\tau'}(v) + ... + F_{0,\tau'}(v) \ ,$$ where the $F_{\nu,\tau'}(v)$ are homogenous polynomials of degree $\nu$ on $V$. We may suppose $\tilde F := F_{m,\tau'} \neq 0$, since otherwise $F$ would not depend on the variable $v\in V$; and this would give as a contradiction $F=0$, since $\tau'$ in $T^*(X')$ is of general position by our assumptions. For any $v\in V$ and any fixed vector $\tilde\tau \in V$ we have (symbolically) $$\tilde F(v) \ = \ \lim_{t \to \infty} \ t^{-m} \cdot F(\tau',\tilde\tau + t\cdot v ) \ .$$ Since $F(\tau',\tilde\tau + W)=0$ vanishes, we get: For every $v\in W\subset V$ and any $t\in k^*$ also $t^{-m}\cdot F(\tau',\tilde\tau+ t\cdot v)=0$ vanishes. We summarize this as follows:
- $\tilde F\neq 0$ on $V=T^*_0(\tilde X)$
- $\tilde F= 0$ on $W=\Lambda_{\tilde S,\tilde y} \subset T^*_0(\tilde X)$.
Notice, the polynomial $\tilde F$ does not depend on the particular point $y\in U$. It only depends on the fixed decomposition $T^*_0(X)=T^*_0(X')\oplus T^*_0(\tilde X)$ and on the point $\tau'$ in $T^*_0(X')$, where the latter is in general position by assumption. We obtain
. [*Suppose $Y$ is closed and irreducible in $X$ such that the Gauß mapping $\gamma: \Lambda_Y \to T^*_0(X)$ is not dominant. If $0\to X'\to X\to \tilde X\to 0$ with $0< dim(X')<g$ is given together with $\tau'$ in $\gamma'(\Lambda'_{F_{\tilde y},y})\subset T^*(i)(\gamma(\Lambda_S))$, with sufficantly general position in $T^*_0(X')$, then the Gauß mapping $$\tilde\gamma: \Lambda_{\tilde Y} \to T^*_0(\tilde X)$$ is not dominant for all $\tilde y$ in a Zariski dense open subset of $\tilde S \subset \tilde Y$*]{}.
. For the proof we may assume $Y\neq X$, since otherwise $\tilde Y=\tilde X$ and the assertion is trivial. So, for $\tilde y\in \tilde S$, we know $(\tau',\tilde\tau) + \Lambda_{\tilde S,\tilde y}
\subset \gamma(\Lambda_S)$ holds for some $\tilde \tau$, as explained in section 5 above. For $v\in
\Lambda_{\tilde S,\tilde y}$ therefore $\tilde F(v)=0$ holds with respect to a fixed nontrivial polynomial $\tilde F$ on $T^*_0(\tilde X)$, not depending on $\tilde y$. In particular, $\tilde \gamma$ cannot be dominant.
. [*For $Y$ closed and irreducible in $X$ suppose $$\gamma: \Lambda_Y \to T^*_0(X)$$ is not dominant. Furthermore, given $0\to X'\to X\to \tilde X\to 0$ for an abelian subvariety $0\neq X' \subsetneq X$, suppose $$\tilde \gamma: \Lambda_{\tilde Y} \to T^*_0(\tilde X)$$ is dominant. Then, for all $\tilde y$ in a Zariski dense open subset of $\tilde S \subset \tilde Y$, none of the Gauß mappings $$\gamma': \Lambda'_{F_{\tilde y}} \to T^*_0(X')$$ is dominant (the same for the irreducible components $Y'$ of these fibers $F_{\tilde y}$).* ]{}
. If $\gamma' : \Lambda'_{F_{\tilde y}} \to T^*_0(X') $ is dominant for some $\tilde y\in \tilde S \subset \tilde Y$ in general position, there exists a conormal vector $\tau'\neq 0$ of general position in $T^*_0(X')$ such that $$\tau' \ \in \ \gamma'(\Lambda'_{F_{\tilde y,y}})$$ holds for some point $y$ on the fiber $F_{\tilde y}\subset Y$ with $q(y)=\tilde y$. By corollary 1 therefore $\tilde\gamma: \Lambda_{\tilde Y} \to T^*_0(\tilde X)$ is not dominant, contradicting our assumptions.
By our induction assumptions, theorem 1 holds for varieties $Y'$ of dimension $<d$. Thus, in the situation of proposition 3, we can therefore apply theorem 1 to the irreducible components $Y'$ of the fibers $F_{\tilde y}$ in $Y$. These have dimension $\dim(Y')\leq \dim(Y) - \dim(\tilde Y)< d$ for generic $\tilde y$ in $\tilde Y$, since $\tilde Y$ has dimension $>0$ by the assumption $\langle Y\rangle = X$. In the situation of the proof of the induction step for theorem 1, after renaming $X'$ by $A$, therefore the last proposition implies
.
*Suppose $Y$ is closed and irreducible in $X$ of dimension $d=\dim(Y)$ and generates $X$. Suppose there exists an abelian subvariety $A \subsetneq X$ such that*
- $ \gamma: \Lambda_Y \to T^*_0(X) $ is not dominant.
- $\tilde \gamma : \Lambda_{\tilde Y} \to T^*_0(\tilde X)$ is dominant, for the image $\tilde Y$ of $Y$ in $\tilde X=X/A$.
Then $Y$ is degenerate.
. The assumptions on the Gauß mappings imply $A\neq 0$ and $\tilde Y \neq \tilde X$. By the other assumptions $\tilde X\neq 0$. Hence $\dim(\tilde Y)>0$ by $\langle \tilde Y\rangle =\tilde X$. Then, by proposition 3, all Gauß mappings $\gamma': \Lambda'_{Y'} \to T^*_0(X')$ for the irreducible components $Y'$ of the fibers $F_{\tilde y}$ (for $\tilde y$ in general position) are not dominant. By the general induction assumption, underlying the proof of theorem 1, we conclude that for all these $\tilde y$ the components $Y'$ are degenerate. Hence $Z(Y')=Y'$, and therefore $Z(U)=U$ for some Zariski dense open subset $U$ of $Y$ by remark 2. Thus $Y$ is degenerate, by proposition 1 and 2.
In the situation of the induction step for theorem 1 the proposition 3 also implies
. [*Suppose $Y$ is irreducible and closed in $X$ of dimension $d$ and generates $X$. Suppose $Y$ is not degenerate and $ \gamma: \Lambda_Y \to T^*_0(X) $ is not dominant. Then for any abelian subvariety $X' \subset X$, with quotient $\tilde X:=X/X'$ and image $\tilde Y$ of $Y$ in $\tilde X$, the Gauß mapping $\tilde \gamma : \Lambda_{\tilde Y} \to T^*_0(\tilde X)$ is not dominant.* ]{}
. We can assume $\dim(X')>0$, so that proposition 4 can be applied.
**Fibers of the Gauß mapping**
Suppose $Y\subsetneq X$ is irreducible and closed, and suppose the Gauß mapping $$\gamma: \Lambda_Y \to T^*_0(X)$$ is not dominant. In this section we furthermore assume $\langle Y\rangle =X$ and $Y\neq X$. In this setting we now use the following argument as in \[R\] or \[KrW\]: Under these assumptions all nonempty fibers of the Gauß mapping $\gamma$ $$Z_\tau = \gamma^{-1}(\tau) \subset \Lambda_Y$$ have dimension $$\dim(Z_\tau) \geq 1 \ .$$ This follows from the upper semicontinuity of fiber dimensions, since it holds for generic points $\tau$ in $\gamma(\Lambda_Y)$. Notice, the image $Y_\tau = p_Y(Z_\tau)$ in $Y$ $$\xymatrix{ Z_\tau \ar@{^{(}->>}[d]_{p_Y}\ar@{^{(}->}[r] & \Lambda_Y \ar[d]^{p_Y}\ar[r]^-\gamma & T^*_0(X) \cr
Y_\tau \ar@{^{(}->}[r] & Y & \cr }$$ has the property that $ p_Y: Z_\tau \to Y_\tau $ is a set theoretic bijection, since over any $y\in Y_\tau$ the points $z\in Z_\tau$ are uniquely determined by the condition $\gamma(z)=\tau$. Indeed, set theoretically, $$Z_\tau = Y_\tau \times \{ \tau \} \subset X \times T^*_0(X) = T^*(X) \ .$$ Now assume $\tau\neq 0$. Then $\tau$ defines a nontrivial linear form $$\tau: T_0(X) \to k$$ whose kernel contains all tangent vectors in $T_y(Y)$, considered as vectors in $T_0(X)$ via a translation by $y\in X$. In particular all tangent vectors in $T_y(Y_\tau)$, at regular points $y$ of $Y_\tau$, are contained in the kernel of the linear form $\tau$.
So, for given $\tau\neq 0$, let us fix some point $y=y_\tau$ in $Y_\tau$.
Then $Y_\tau - y_\tau$ contains zero, and the abelian subvariety $X'$ of $X$ generated by $Y_\tau - y_\tau$ is a nontrivial abelian subvariety $X'\subsetneq X$. Indeed $\dim(Y_\tau)\geq 1$ implies $X'\neq 0$, and $\tau(T_0(X'))=0$ implies $X'\neq X$. Furthermore by construction of $X'$ $$Z_\tau \subset y_\tau + X' \ .$$ In this situation, a priori, the abelian variety $X'=X'(\tau,y)$ may depend on the choice of $\tau\in \gamma(\Lambda)$ and also on the choice of $y=y_\tau$ in $Y_\tau$, so that $$\xymatrix{ Y_\tau \ar@{^{(}->}[r]\ar@{^{(}->}[d] & X \ar[r]\ar@{^{(}->}[d] & T^*_0(X) \ni \tau \cr
F_{\tilde y_\tau} = Y\cap \bigl(y_\tau + X'(\tau,y_\tau)\bigr) \ar@{^{(}->}[r] & X \cr}$$ and $$Y = \bigcup_{\tau \in \gamma(\Lambda_Y)} Y_\tau \ .$$
. [*For all $\tau$ in a Zariski open dense subset of $\gamma(\Lambda_Y)$ and all $y_\tau$ in a Zariski open dense subset of $Y_\tau$, the abelian variety $X'=X'(\tau,y_\tau)$ does not depend on the choice of $\tau$ and $y_\tau$.* ]{}
. There exist only countably many abelian subvarieties $X'$ in $X$, and $X'=X'(\tau,y_\tau)$ depends algebraically on $\tau$ and $y_\tau$. Replace $k$ by an uncountable extension field.
By the rigidity property we can therefore assume that $X'$ is a fixed nontrivial proper abelian subvariety attached to $Y\subset X$, such that for all $y$ in a Zariski dense open subset $U\subset Y$ $$F_{\tilde y} = Y \cap (\tilde y + X') = Y \cap (y_\tau + X'(\tau,y_\tau))$$ contains $Y_\tau$, hence is of positive dimension $ dim(F_{\tilde y}) \geq 1 ;$ and the image $\tilde Y$ of $Y$ in $\tilde X = X/X'$ is irreducible of dimension $$\dim(\tilde Y) < \dim(Y) \ .$$ To summarize; this shows
. [*To irreducible closed $Y\neq X$ with $\langle Y\rangle =X$ and non-dominant Gauß mapping $\gamma: \Lambda_Y \to T^*_0(X)$, there is an abelian subvariety $0\neq X' \subsetneq X$ such that $\dim(\tilde Y) < \dim(Y)$ holds for the image $\tilde Y$ of $Y$ in $\tilde X=X/X'$, with the fibers of the Gauß mapping $\gamma$ contained in translates of $X'$.*]{}
Indeed $\langle Y\rangle =X$ can be assumed without restriction of generality.
In the situation of lemma 3, let us now assume that $Y$ is not degenerate with a non-dominant Gauß mapping $\gamma$, and let us also assume that theorem 1 holds for varieties of dimension $< \dim(Y)$. Then proposition 4 can be applied; it shows that the induced Gauß mapping $$\tilde\gamma: \tilde Y \to T^*_0(\tilde X)$$ is not dominant. Then $\langle \tilde Y \rangle = \tilde X$ is inherited from $\langle Y \rangle = X$. So suppose $$\tilde Y \neq \tilde X \ .$$ Then, if $\tilde Y\neq \tilde X$, we can apply once again lemma 3, now for the pair $(\tilde Y,\tilde X)$, to construct an exact sequence $$0\to \tilde X' \to \tilde X \to \tilde{\tilde X} \to 0$$ such that
- $\langle \tilde{\tilde Y} \rangle = \tilde{\tilde X}$
- ${\tilde{\tilde \gamma}} : \tilde{\tilde Y} \to T^*_0(\tilde{\tilde X} )$ is not dominant.
Obviously, this construction can be iterated. It must terminate after finitely many steps, since $$\cdots < \dim(\tilde{\tilde Y}) < \dim(\tilde Y) < \dim(Y)\ ,$$ thus provides an abelian subvariety $AÊ\subsetneq X$ containing $X'$, such that the image of $Y$ in $B=X/A$ is equal to $B$.
A closed irreducible variety $Y$ in $X$ will be called [*codegenerate*]{} (with respect to $A$), if there exists an abelian subvariety $A\neq X$ in $X$, such that the image of $Y$ in $B=X/A$ is equal to $B$.
Using this notion, in the situation of the induction step for the proof of theorem 1, we have therefore shown that for $Y$ closed irreducible of dimension $\dim(Y)=d$ the following corollary holds.
.
*Suppose the Gauß mapping $$\gamma: \Lambda_Y \to T^*_0(X)$$ is not dominant, then either*
- $Y$ is degenerate, or
- $Y$ is codegenerate with respect to an abelian subvariety $A$, so that the fibers of the Gauß mapping $\gamma$ are contained in translates of $A$.
**End of the proof of theorem 1**
To finish the proof of the induction step (i.e. the proof of theorem 1) for $Y$ of dimension $d$ with non-dominant Gauß mapping, it remains to consider the codegenerate case $\tilde Y=B$ of the last corollary 3. Of course, without restriction of generality, we can in addition assume that $Y$ is not degenerate. That $Y$ is not degenerate implies (by the induction assumption of theorem 1 and proposition 1 and 2): For a Zariski dense open subset $U$ of $\tilde Y=B$, the Gauß mappings of the fibers $F_{\tilde y}, \tilde y\in U$ of the projection $q: Y \to \tilde Y=B$ are nondegenerate. Indeed, if this non-degeneracy holds for a single fiber $F_{\tilde y}$, where $\tilde y$ is supposed to be in general position, it holds for all fibers $F_{\tilde y}$ with $\tilde y$ in a Zariski dense open subset $U$ of $\tilde Y$ by a specialization argument.
Assuming all these conditions together, we in fact $$\mbox{\bf Claim: } Y \mbox{ is degenerate.}$$ This contradiction implies the induction step for the proof of theorem 1.
To prove the last claim, we use the induction assumption stating that theorem 1 holds for dimension $< d$. Recall that in the last section we found $$\xymatrix{ 0 \ar[r] & A \ar[r]^i & X \ar[r]^q & B \ar[r] & 0}$$ such that $B=\tilde Y$, and such that the fibers $Z_\tau $ of the Gauß mapping $\gamma: \Lambda_Y \to T^*_0(X)$ map bijectively to varieties $Y_\tau \subset Y$ that are contained in the fibers $$F_{\tilde y} = Y \cap (\tilde y + A) \subset Y$$ of the projection $q:Y \to \tilde Y=B$. Here, without restriction of generality, we assume that $B$ splits, so that $B$ can be considered as a subvariety of $X$ complementary to $A$. Since $Y=\bigcup_{\tilde y \in B} F_{\tilde y}$, the variety $Y$ is degenerate if all the $F_{\tilde y}$ are degenerate for all $\tilde y$ in some Zariski open dense subset of $\tilde Y$ (using proposition 1 and 2). Therefore, if $Y$ were not degenerate, by the induction assumption of theorem 1 we conclude that the Gauß mapping $$\gamma_A : \Lambda_{F_{\tilde y}}^A \to T^*_0(A)$$ is dominant for all points $\tilde y$ of general position in $\tilde Y$.
Fix some $\tau' \neq 0$ in $T^*_0(A)$ in general position; fix some $\tilde y$, now with $\tilde y\in U$, so that $\gamma_A: \Lambda_{F_{\tilde y}} \to T^*_0(A)$ is dominant. Since $\gamma_A$ is dominant and since $\tau$ has general position in $T^*_0(A)$, there exists $$y \in F_{\tilde y} \subset \tilde y + A \mbox{ so that }
\tau' \mbox{ is contained in } N^*_y(F_{\tilde y}) = \Lambda_{F_{\tilde y},y} \ .$$ Since $\tilde S$ is Zariski dense in $\tilde Y=B$, we conclude $$\Lambda_{\tilde S, \tilde y} = \Lambda_{B,\tilde y} = \{ \tilde y \} \times \{ 0\}$$ in $T^*(B) = B\times T^*_0(B)$. Therefore, as explained in lemma 2, we obtain
. [*For $\tilde y$ in a suitably chosen Zariski open dense subset $U$ of $\tilde Y=B$, our assumptions imply that $\tau'$ (chosen in general position) is in $ \Lambda'_{F_{\tilde y},y}$ such that $$\Lambda_{S,y} \cong \Lambda'_{F_{\tilde y},y} \ .$$*]{}
In other words: $\tau'\in T^*_0(A)$ can be uniquely lifted to a point in $\Lambda_{S,y}$, once the corresponding base point $y\in F_{\tilde y}$ over $\tilde y\in U$ has been specified.
For the conormal bundle in $T^*(A)$ we write $\Lambda^A_{F_{\tilde y}}$, instead of $\Lambda'_{F_{\tilde y}}$,. Notice, $\gamma_A: \Lambda^A_{F_{\tilde y}} \to T^*_0(A)$ is dominant, hence generically finite, for all $\tilde y$ a Zariski open subset $\tilde U$ of $\tilde Y$. We therefore obtain
. [*For fixed $\tau'\neq 0$ with general position in $T^*_0(A)$ and fixed $\tilde y$ with general position in $ B$, there exist only finitely many points $y\in Y$ mapping to $\tilde y$ (i.e. $y \in F_{\tilde y}$) for which $\tau'$ is contained in the conormal bundle $\Lambda_{F_{\tilde y},y}$ of $F_{\tilde y}$ at $y$.* ]{}
Combining lemma 4 and 5 we get
.
*We have the following diagram $$\xymatrix{ \mbox{dense open subset of } \gamma(\Lambda_S) \ar@{=}[dd] \ar@{^{(}->}[rr] & & T^*_0(X) \ar[dd]^{T^*(i)}\cr
& & \cr
\bigcup_{\tilde y \in U} \gamma'(\Lambda^A_{F_{\tilde y}}) \ar[rr] & & T^*_0(A) }$$ and the image under the lower horizontal morphism of each $\gamma'(\Lambda^A_{F_{\tilde y}})$ for $ \tilde y\in U$ is Zariski dense in $T^*_0(A)$.*
Furthermore, for $\tau'$ of sufficantly general position in $T^*_0(A)$, the points $\gamma(\lambda) \in T^*_0(X)$ in $\gamma(\Lambda_S)$, which map under $T^*(i): T^*_0(X)\to T^*_0(A)$ to the point $\tau'$, correspond to points $\lambda$ in $\Lambda_S$ $$\lambda = (y,\tau) \in \Lambda_S \subset X \times T^*_0(X)$$ for which the base point $$\tilde y = q(y) \in B =X/A$$ of $y$ can be arbitrarily prescribed within a dense open subset of $B$. Once this base point $\tilde y \in B$ is fixed, there exists at least one, but at most finitely many, choices for the point $y\in X$ such that $$\tilde y = q(y) \mbox{ and } \lambda = (y,\tau)=(y,\tau',\tilde\tau) \in \Lambda_S$$ holds for some $\tilde\tau$ in $T^*_0(B)$.
Thus there exists a Zariki dense open subset $V\subset \Lambda_S$, such that on $V$ the mapping $\varphi=(q \circ p_S, T^*(i)\circ \gamma)$ defines a (generically) finite and hence dominant morphism $$\xymatrix{ \Lambda_S \supset V \ar[r]^-\varphi & \ B \times T^*_0(A) \cr }$$ $$\lambda =(y,\tau) \mapsto (\tilde y, \tau') \ .$$
For a point $\tau\in \gamma(\Lambda_S)$ in general position consider the fiber $Z_\tau \subset \Lambda_S$. Notice, $V\cap Z_\tau$ is a Zariski dense open subset of $Z_\tau$ by the choice of $\tau$. Furthermore, for all points $\lambda$ in $V\cap Z_\tau$ by definition $\lambda=(y,\tau) $ holds; and the fixed $\tau$ maps to a fixed $\tau'$ of general position in $T^*(A)$. In other words: The second component $T^*(i)\circ \gamma$ of the morphism $\varphi$ is constant on $Z_\tau$. Since $\varphi$ is finite on $V$, therefore the first component $$q\circ p_S: V \cap Z_\tau \to B$$ of the morphism $\varphi$ is a finite morphism on $V\cap Z_\tau$, such that $$V \cap Z_\tau \ni (y,\tau) \mapsto \tilde y=q(y)\ .$$ Since $X' \subset A$, using the notations of the argument preceding corollary 3, we already know from the beginning of this section that $$Z_\tau \subset y_\tau + X' \subset y_\tau + A = \tilde y_\tau + A \ .$$ Here we assumed $X=A\oplus B$ without restriction of generality, and $\tilde y_\tau \in B$ is the image of $y_\tau$ under the projection $q:X\to B=X/A$. Recall that $y_\tau$ was some fixed chosen point in $Y_\tau$, and only depends on $\tau$. Hence the inclusion $ Z_\tau \subset \tilde y_\tau + A $ implies $$q\circ p_S(V\cap Z_\tau) = \tilde y_\tau \ .$$ Since $q\circ p_S$ is finite on $V\cap Z_\tau$, the intersection $V\cap Z_\tau$ therefore contains only finitely many point. But this contradicts $\dim(V\cap Z_\tau)=\dim(Z_\tau) \geq 1$, and finishes the proof of the induction step for theorem 1.
[XXXX]{} Abramovich D., [*Subvarieties of semiabelian varieties*]{}. Comp. Math. 90 (1994), 37–52 Ginsburg V., [*Characteristic varieties and vanishing cycles*]{}. Invent. math. 84, 327 - 402 (1986) Kashiwara M., Shapira P., [*Micro-hyperbolic systems*]{}. Acta Math. 142, 1-55 (1979) Krämer T.,Weissauer R., [*Vanishing theorems for constructible sheaves on abelian varieties*]{}. Preprint, Heidelberg (2011) Ran Z., [*On Subvarieties of Abelian Varieties*]{}. Invent. math. 62, 459 – 479 (1981)
|
---
abstract: 'We demonstrate an algorithm for learning a flexible color-magnitude diagram from noisy parallax and photometry measurements using a normalizing flow, a deep neural network capable of learning an arbitrary multi-dimensional probability distribution. We present a catalog of 640M photometric distance posteriors to nearby stars derived from this data-driven model using Gaia DR2 photometry and parallaxes. Dust estimation and dereddening is done iteratively inside the model and without prior distance information, using the Bayestar map. The signal-to-noise (precision) of distance measurements improves on average by more than 48% over the raw Gaia data, and we also demonstrate how the accuracy of distances have improved over other models, especially in the noisy-parallax regime. Applications are discussed, including significantly improved Milky Way disk separation and substructure detection. We conclude with a discussion of future work, which exploits the normalizing flow architecture to allow us to exactly marginalize over missing photometry, enabling the inclusion of many surveys without losing coverage.'
author:
- 'Miles D. Cranmer'
- Richard Galvez
- Lauren Anderson
- 'David N. Spergel'
- Shirley Ho
bibliography:
- 'main.bib'
title: |
Modeling the Gaia Color-Magnitude Diagram with Bayesian Neural Flows\
to Constrain Distance Estimates
---
Introduction {#sec:intro}
============
Gaia’s precise astrometry has impacted the astrophysics community in countless ways. While Gaia data is predominantly useful for mapping out the Milky Way and its bulk properties, for example in [@bovy_stellar_2017]; it has been used for calibration of standard candles like the Red Clump as in [@hawkins_red_2017; @huber_asteroseismology_2017], and vice versa — using the Red Clump to improve Gaia parallax calibration — [@hall_testing_2019]; and mapping out the interstellar medium to create dust maps of the Milky Way, as in [@green_3d_2019]. Gaia’s astrometric data has been incredibly useful for the study of substructures in the Milky Way and its halo, for example, stellar streams, as in [@price-whelan_off_2018; @malhan_ghostly_2018; @brown_gaia_2018; @koposov_piercing_2019]. The precise positional and proper motion estimates even allow for automated algorithms to be used to detect stellar streams, as described in [@malhan_streamfinder_2018]. Recently the astrometric data has been used to observe possible dynamical evidence for dark matter in stellar streams in [@bonaca_spur_2019], although this stream was at a distance where the parallax measurements couldn’t be reliably used besides a foreground filter. Producing more accurate astrometry for the Gaia DR2 dataset would impact many current and future use cases. Gaia parallaxes greatly degrade in quality at a distance starting at about $\sim\SI{1}{kpc}$, with only $\sim 72$ million sources having greater than 10 signal-to-noise (SNR, defined here as parallax over parallax uncertainty), hampering studies of structure in the galactic halo such as stellar streams.
We can make reasonable distance estimates from these uncertain parallaxes by relying on distance priors such as those derived in [@bailer-jones_estimating_2015; @bailer-jones_estimating_2018], or by exploiting patterns in stellar photometry, either using theoretical Color-Magnitude Diagrams (CMDs) for stars or learning them as in [@leistedt_hierarchical_2017; @anderson_improving_2018]. If we were to rely on purely theoretical models for estimating distances from photometry, we would be subject to systematic errors from the models themselves. Distance prior-derived distances are also heavily prior-dominated, and of limited use for substructure studies, also demonstrated in Section \[sec:applications\]. By denoising in a model-independent fashion — training a flexible machine learning algorithm to model the most common photometric measurements for stars, hence acting as a prior for distance estimates — we avoid difficulties from modeling the theoretical photometry, and naturally learn over top of the Gaia systematic errors. Using a very flexible model for a color-magnitude population density would reduce potential model-dependent bias from entering distance estimates.
One way of building very flexible models is by using machine learning models. Typically in astronomy, machine learning is thought of purely as an approach to classification or regression, which is predicting an output scalar or vector value based on a set of input parameters, such as the simplest example: creating a line of best fit. The extension of this is to fitting quadratic functions, and then higher-order polynomials, and other simple models, which can be generalized to multi-input, multi-output using additional parameters. One generalization of these simple machine learning models which are fit to data is a popular technique called deep learning, which is a flexible approach capable of fitting very complex surfaces over input data, relying typically on stochastic gradient descent to fit millions of parameters. Deep learning can also be applied to many additional optimization problems than just regression, such as density estimation, which we will use it for in this paper. Density estimation is the problem of fitting a function that models an unknown probability distribution and can be approached with deep learning using a model called a normalizing flow.
Deep learning can be thought of as a recursive generalized linear regression — you repeatedly compute linear regression (fitting a hyperplane) on an input, following each regression with an element-wise nonlinearity (such as converting negative values to zero). In the case of normalizing flows, as we will see, we also apply a mask over the linear regression weights to make the function have a triangular Jacobian matrix. We use such a normalizing flow in this paper to model photometric measurements of Gaia stars.
One common criticism of deep learning models targets their lack of interpretability and potential over-flexibility. Generally, it is recommended that deep learning should only be used when it gives you large performance gains or lets you model a relation that would be extremely difficult to represent with classical machine learning models. As discussed in this paper, due to the large size of the Gaia DR2 dataset, and the non-Gaussian contours of the density of stars on a CMD, a deep neural network is a useful model for its scalability and flexibility, so we choose it over traditional machine learning models.
Data {#sec:data_description}
====
Our dataset is a slice of the Gaia DR2 catalog. For technical papers describing the 2nd data release used in this paper, consult [@brown_gaia_2018; @lindegren_gaia_2018; @riello_gaia_2018]. We make use of the following features: `bp_rp`, `bp_g`, `phot_g_mean_mag`, `ra`, `dec`, `parallax`, along with their corresponding uncertainties, and features to calculate the renormalized unit weight error (RUWE): `astrometric_chi2_al` and `astrometric_n_good_obs_al`. We choose to not apply any filters based on parallax or parallax uncertainty, meaning we take all noisy measurements (although we do filter using RUWE, discussed later), including negative parallaxes. We exclude all stars below -30 deg declination, since the Bayestar dust map, which we use from [@green_galactic_2018] in the package [@dustmaps], does not extend there. We also exclude stars lying in holes of this dust map. We include all stars during training, even the low-SNR parallaxes. However, we make cuts during training such that stars in high-density regions of the sky are excluded to avoid stars with bad goodness-of-fit for parallax. We do this by requiring that the renormalized unit-weight error of measured parallaxes, discussed on <https://www.cosmos.esa.int/web/gaia/dr2-known-issues#AstrometryConsiderations>, is less than 1.4.
We apply a single global parallax offset of 0.029 mas from [@lindegren_gaia_2018] to all of the Gaia data. While the true parallax offset is conditional on several variables, we believe that training with a constant parallax offset makes it easier to apply different post-processing offsets to the finished model. A more complex parallax offset can also be used during evaluation.
Model {#sec:model}
=====
We wish to train a model that takes `bp_rp`, `bp_g`, `phot_g_mean_mag`, `ra`, `dec`, `parallax` from the Gaia DR2 catalog, along with their uncertainties, and produces a Bayesian posterior over distance. Our model is similar in strategy, but not in terms of the actual density model, to [@anderson_improving_2018]. [@anderson_improving_2018] built a model on Gaia DR1 parallaxes cross-matched with 2MASS photometry, and applies the “Extreme Deconvolution” algorithm from [@bovy_jo_extreme_2011] to build a Gaussian Mixture Model (GMM) over color-magnitude values. This GMM is then used as a prior for the same data, to build a final parallax probability distribution for every data point. The model described in this paper is similar to this GMM model but differs by using a normalizing flow to model density in photometric space. Various other changes are made, such as that we propagate uncertainty from the dust map, both into our training (down-weighting photometry from uncertain dust estimates) and during evaluation (sampling the dust map).
Normalizing flows are not yet popular for density estimation in astrophysics or the natural sciences in general, although they are being used for some likelihood-free inference applications derived from [@papamakarios_sequential_2018] in Cosmology and Particle Physics, for example in the “MadMiner” package for LHC data [@brehmer_madminer:_2019], and “PyDELFI” for Cosmology [@alsing_fast_2019]. This paper represents the first application that we are aware of for normalizing flows being applied to learning a stellar CMD.
Our model makes several assumptions which largely follow those made in [@anderson_improving_2018]:
- The fundamental assumption of the model is that for a given star, there will be other stars with similar photometry, as is assumed for any regression-type optimization problem.
- We model the contents of Gaia DR2, not the Milky Way. Since parallax measurements are noisier for stars in the halo, which also tend to be lower metallicity, the CMD model will bias to metallicities in the disk. However, we lessen the effect of this by training on all stars, rather than just high-SNR.
- The model assumes that the expected color-magnitude relation is unchanging with respect to location in the Milky Way, in that we do not include $\alpha, \delta$ (right ascension and declination) as a parameter in our CMD.
- This model makes no further physical assumptions about stellar structure or the color-magnitude relation. This is a trade-off because while it does avoid potential inaccuracies of physical models, it also misses the many successes of these models. Future work will attempt to combine physical models with data-driven to help regularize the training.
- We assume that the Bayestar dust map of [@green_galactic_2018] produces accurate dust posteriors along each line of sight. We further assume that for training, the median dust estimate will not bias the model during training. However, we incorporate the full dust posterior for evaluation.
- We make the assumption that the Gaia catalog has no parallax bias relative to any parameters (such as proper motion, or a specific sky coordinate such as a high-density region).
A potentially problematic assumption in our model is the fact that the dust map we use, from [@green_galactic_2018], makes use of stellar models to estimate dust, meaning that there are implicitly some stellar models being introduced in our distance estimates. In the future, it would be desirable to combine the learning of a dust map with the learning of a CMD so we could make completely model-independent estimates, but for now, this is our approach. The assumptions for the model described in our paper are different from those in [@anderson_improving_2018] in that we do not model the color-magnitude relation using 128 Gaussians; rather, we use a deep normalizing flow which has more flexibility in describing the joint prior. The path of stellar evolution has many sharp turns and discontinuities, so, while the spread of stars about a single isochrone may be Gaussian-like due to a Gaussian spread in ages and metallicities, the overall CMD is better modelled with a highly flexible model that can express sharp turns, which is where deep learning can be extremely helpful, as GMMs perform poorly at modeling the contours of the CMD. We also deal with a significantly larger dataset, using the majority of DR2 (640M stars) versus a selection of DR1 (1.4M stars). We also incorporate the entirety of the samples from [@green_galactic_2018] rather than median dust estimates during evaluation.
The crux of our model is a very flexible “normalizing flow” neural network. A good introduction on this family of model can be found at <http://akosiorek.github.io/ml/2018/04/03/norm_flows.html>. This architecture models the joint posterior density for Gaia magnitudes, as $$P(g, bp-rp, bp-g),$$ where $g$ is the absolute (dereddened) G-band mean magnitude from Gaia DR2, corresponding to column [`phot_g_mean_mag`]{}, and $bp$ and $rp$ similarly for the absolute integrated BP and RP mean magnitudes. This prior density models the dashed lines in the graphical model in \[fig:graphicalmodel\].
{width="60.00000%"}
This posterior models the density of stars in Gaia DR2 and weights stars observed in Gaia DR2 by $$\frac{1}{\sigma_P^2},$$ where we define $\sigma_P$ as $$\sigma_P^2 = \sigma_g^2 + \sigma_{bp-rp}^2 + \sigma_{bp-g}^2,$$ which is a metric for the signal-to-noise ratio of data points, much like when calculating a mean from uncertain measurements, one weights each measurement by the inverse variance. These $\sigma_i^2$ values are the variance of estimates in each of the quantities. These variances incorporate both the uncertainty of the dust map as well as the intrinsic uncertainty in the Gaia parallax and distance prior.
During training, we sample 32 parallaxes for each Gaia DR2 data point from the truncated normal distribution: $$\label{eqn:distance}
P(\varpi) \propto \left\{
\begin{array}{cc}
\exp\left(-\frac{(\varpi-\varpi_\text{obs})^2}{2 \sigma^2}\right),
& \varpi>0 \\
0, & \varpi\leq0,
\end{array}
\right.$$ using the DR2 parallax value for $\varpi_\text{obs}$, and $\sigma$ equal to the DR2 standard deviation in the parallax value estimate. As stated in [@hogg_likelihood_2018], a likelihood for Gaia parallaxes treats the individual measurements as drawn from $\mathcal{N}(\varpi_\text{true}, \sigma)$. Therefore this truncated distribution acts as a minimal distance prior on the Gaia parallax likelihood, which states that we can only have positive distances.
Model architecture {#sec:arch}
------------------
A normalizing flow can roughly be thought of as a Gaussian that has been parametrically warped by an invertible neural network. This is a smooth invertible mapping between probability distributions: $\mathbb{R}^n \rightarrow \mathbb{R}^n$. Our specific model relies on the “Masked Autoregressive Flow” variant, described in [@papamakarios_masked_2017] for density estimators, which uses the following mappings: $$\begin{aligned}
y_1 &= \mu_1 + \sigma_1 z_1,\\
\forall i>1: y_i &= \mu(y_{1:i-1}) + \sigma(y_{1:i-1}) z_i,\end{aligned}$$ for $y, z, \mu \in \mathbb{R}^n$ and $\sigma\in\mathbb{R}^n_{+}$. This equation is similar to the common matrix multiply followed by vector addition that is found in neural networks, but with a particular mask applied. This is done so that the Jacobian of this mapping is triangular — and hence the determinant is easy to calculate as the product of elements along the diagonal. Recall that if you would like to change variables from $\vec{x}$ to $\vec{y}$ via a smooth function $f$, for probability distributions one must write: $$p(y) = p(x) \abs{\frac{df}{dx}}^{-1}$$ This mapping is invertible, and the inverse also has a triangular Jacobian. The fact that the Jacobian is easy to calculate lets us normalize the transformation between probability distributions. The inverse is $$\begin{aligned}
\forall i>1: z_i &= \frac{y_i - \mu (y_{1:i-1})}{\sigma(y_{1:i-1})},\end{aligned}$$ which transforms from our data variables ($y$) to a latent space ($z$) where we set the Gaussian. This transform is what we compute to calculate the probability of given data (though we use a reparametrized version, so it doesn’t need to be done sequentially). This transform can be repeated to form a complex flow. Autoregressive models can also be used to exactly marginalize over inputs in the case of missing data (such as missing photometry), as discussed in Section \[sec:future\].
Our model uses a sequence of blocks of <span style="font-variant:small-caps;">MADE</span> $\rightarrow$ <span style="font-variant:small-caps;">BatchNorm</span> $\rightarrow$ <span style="font-variant:small-caps;">Reverse</span>, where <span style="font-variant:small-caps;">MADE</span> is the Masked Autoencoder for Distribution Estimation model defined in [@germain_made:_2015], <span style="font-variant:small-caps;">Reverse</span> is the reversing layer found in [@dinh_density_2016], and <span style="font-variant:small-caps;">BatchNorm</span> is a batch norm-like layer (typically used in convolutional neural networks) also defined for normalizing flow models in [@dinh_density_2016]. We use the PyTorch code found at <https://github.com/ikostrikov/pytorch-flows/blob/master/flows.py> as the template of our codebase, which we alter for our usecase.
The <span style="font-variant:small-caps;">MADE</span> model essentially applies three densely-connected neural network layers with a mask (hence, a asked utoencoder) applied to the weights at each layer to satisfy properties of the neural flow. It forms an autoencoder such that each of the output units only depends on the preceding input units ($i \sim 1:i-1$). One can think of this transform as parametrizing an arbitrary bijective vector field, where the vectors show the flow of points from distribution to distribution. The <span style="font-variant:small-caps;">BatchNorm</span> is the equivalent of a batch normalization for normalizing flows, and <span style="font-variant:small-caps;">Reverse</span> permutes the order of the probability variables since the <span style="font-variant:small-caps;">MADE</span>’s mask treats each slightly differently (e.g., $y_i$ depends on $z_{1:i-1}$, whereas $y_1$ only on $z_1$). Hence the <span style="font-variant:small-caps;">BatchNorm</span> and <span style="font-variant:small-caps;">Reverse</span> layers help regularize training. Defining one block as <span style="font-variant:small-caps;">MADE</span> $\rightarrow$ <span style="font-variant:small-caps;">BatchNorm</span> $\rightarrow$ <span style="font-variant:small-caps;">Reverse</span>, our model has a probability distribution with $3$ variables. We then specify a hidden dimension size in each <span style="font-variant:small-caps;">MADE</span> of $500$ and use $35$ sequential blocks.
Dust estimation
---------------
We build the Bayestar dust map from [@green_galactic_2018] into the model with an iterative approach using the software package `dustmap` from [@dustmaps]. We use extinction values based on a blackbody integrated over an $R_V=3.1$ dust model from [@odonnell_rnu-dependent_1994], using the extinction package from [@barbary_extinction_2016], which gives us conversions from the Bayestar dust map to Gaia DR2 bands: $$\begin{aligned}
A_G &= 2.71 E(g-r)\\
E(BP-RP) &= 0.85 E(g-r)\\
E(BP-G) &= 0.39 E(g-r).\end{aligned}$$
In the future, we would like to expand this graphical model with an estimate for the temperature and other stellar parameters of each star, and incorporate a map of $R_V$. We would also like to fit a separate graphical model which learns the best extinction to maximize likelihood over the data.
Training and evaluation take different approaches due to the computational expense. During training, we calculate a distance by dividing each of the 32 parallax samples from \[eqn:distance\]. We calculate the mean of these distance samples to get the current best-estimate for a distance. For every best-estimate distance, we query Bayestar at the center [`ra`]{} and [`dec`]{} position. We convert this reddening into each of the Gaia bands, giving us estimates for $G, BP,$ and $RP,$ which gives us $bp-rp$ and $bp-g$. Then, using each of the 32 samples for parallax, we convert the $G$ estimate into 32 samples for $g$. Next, using the current model for $P(g, bp-rp, bp-g)$, we calculate the probability of each $(g, bp-rp, bp-g)$ tuple. These likelihood values are treated as weights, and the weights are used to calculate a new best-estimate for distance via a weighted sum: $$\label{eqn:weight_sum}
d_\text{best} =
\frac{\sum_i d_i P(g_i, bp-rp, bp-g)}
{\sum_i P(g_i, bp-rp, bp-g)},$$ where each of the $d_i$ is a distance sample. This $d_\text{best}$ is then fed back into the loop and a new reddening is found using Bayestar.
This iteration is repeated 5 times during training, due to the computational expense, but 10 times during evaluation. Once the final $d_\text{best}$ is given, we calculate a best-estimate value for the dust. We use this to get final estimates for the dereddened $G, bp-rp, bp-g$. We then calculate $g$ using the raw parallax samples and dereddened $G$. Note that we do not use $d_\text{best}$ to calculate the final $g$ since this could create a feedback loop for the probability density and create unphysical artifacts, which we experimentally observed. A final $g_\text{best}$ is then found by averaging the $g_i$. The standard deviation of the $g_i$ is used to calculate $\sigma_g$: $$\sigma_g^2 = \text{Var}(g_1, \ldots, g_{32}) + \sigma_{G, \text{Bayestar}}^2,$$ where $\sigma_{G, \text{Bayestar}}$ is the uncertainty in the dust map for $G$ band at the given distance and sky location. We can add these variances because $G$ and $g$ are linearly related. We could also choose to estimate $g_\text{best}$ by multiplying each of the weights in \[eqn:weight\_sum\] by $P(g, bp-rp, bp-g)$, though we found using it created artifacts in the density which did not go away with further training.
Using $d_\text{best}$ and the dust map we calculate the dereddened: $(bp-rp)_\text{best}$ and $(bp-g)_\text{best}$. We also calculate the uncertainty due to these colors: $$\begin{aligned}
\sigma_{bp-rp}^2 &= \sigma_{bp-rp, \text{Bayestar}}^2,\\
\sigma_{bp-g}^2 &= \sigma_{bp-g, \text{Bayestar}}^2.\end{aligned}$$ Finally, we are left with $(bp-rp)_\text{best}, (bp-g)_\text{best},$ and $g_\text{best}$, along with a measure of the combined uncertainty of the color-magnitude point $\sigma_P$. Our loss function for this point is then: $$-\frac{1}{\sigma_P^2} \log\left\{P\left(g, bp-rp, bp-g\right)_\text{best}\right\}.$$ We sum this over all stars in the DR2 catalog, after calculating the best-estimate color-magnitude points, and minimize.
This algorithm learns a very flexible prior on dereddened $(g, bp-rp, bp-g)$ tuples for stars in Gaia DR2, which can then be combined with a distance prior and raw parallax measurements to generate a Bayesian distance estimate for every star in Gaia. This process is done iteratively until the dust estimate converges.
Model Optimization
------------------
We conduct a hyperparameter search for the normalizing flow over 80 different models, finding the best model using Bayesian optimization with summed-log-likelihood as an optimization metric. The model is trained on the entirety of Gaia DR2 above , completing several passes over the data with a mini-batch size of 2048. We found this mini-batch size balances accuracy: much smaller batch sizes led to the creation of artifacts in the density map, and much larger batch sizes resulted in early convergence to less accurate models.
During model selection, we randomly initialize the model weights using Xavier initialization (see @glorot_understanding_2010) with a normal distribution, and train for one pass over Gaia DR2, recording the likelihood over each random one million-star subset. We explore the (learning rate, layers, hidden units) space such that the model fits in a 16 GB GPU, and record the likelihood as a function of these variables and fractional epochs. We then pass these measurements to a Gaussian Process with a radial basis function kernel. We select the next model architecture by maximizing likelihood plus the uncertainty in the likelihood. In total, we explore 80 different architectures and find that the best loss was for a model with 35 layers of 500 hidden units each. We also tried models that used mixtures of normalizing flows but found these underperformed. We also found that using the distance prior from [@bailer-jones_estimating_2018] during the estimate of $d_\text{best}$ gave the noisy measurements too much weight, $\frac{1}{\sigma_P^2},$ since for very noisy measurements, as $\sigma_\varpi\rightarrow \infty$, the distance prior and $P(x)$ will be equal, but $\sigma_P$ will be finite, so very noisy measurements will have a large effect on the CMD.
Results {#sec:results}
=======
We first demonstrate the ability of this model to estimate the true color-magnitude diagram of noisy Gaia data on simulated data in Section \[sec:simulation\]. Next, in Section \[sec:products\] onwards, we present the results on real data. The trained normalizing flow is visualized in \[fig:cmd\], which shows a clear main sequence and giant branch, with some color in the log plot indicating it has learned weight at the white dwarf part of the CMD as well, unprecedented in previous attempts using full CMD GMM models. We will release the catalog, which is described in Table \[tbl:data\_summary\], and code from links on <https://github.com/MilesCranmer/public_CMD_normalizing_flow>.
Simulation {#sec:simulation}
----------
We simulate a basic Gaia dataset of 30M stars drawn from simple analytic distributions in distance and color space. We apply a known noise-free dust map that is a 2D Gaussian over $(\alpha, \delta)$ that is uniform along each line of sight. We distribute the stars uniformly over each line of sight, $(\alpha, \delta)$ (though the distances follow a specific prior, see below). We fit a line in $(g, bp-rp, bp-g)$ space on the high-SNR Gaia data and use this to sample colors. We randomly sample $g$ values from the high-SNR Gaia data and project onto the line to get a $bp-rp$ and $bp-g$ value. We then randomly perturb these colors using a Gaussian to create the truth CMD shown in \[fig:true\_fake\_cmd\]. The distances are randomly sampled from the Bailer-Jones distance prior with $L=\SI{1}{kpc}$: $$P(d) \propto d^2 \exp{-d/L}.$$ These are used to create the $G$, $BP$, and $RP$ observed bands at Earth, followed by reddening according to $d \times \text{dust}(\alpha, \delta)$, for our dust map, and mapped to fixed extinction conversions for each band. Next, we use the formula: $$\sigma_\varpi = \frac{d}{\SI{10}{kpc}} \si{mas}$$ to create a parallax measurement error for all the stars. We then sample the observed parallax, $\varpi_\text{obs}$, for all stars from a Gaussian distributed with $\sigma_\varpi$ as the standard deviation. This transformation results in the adjusted parallax distribution shown in \[fig:fake\_cmd\_parallax\]. Using $\frac{1}{\varpi_\text{obs}}$ as a simple distance estimate, we can visualize the noisy reddened CMD in \[fig:noisy\_fake\_cmd\].
[0.375]{} {width="\textwidth"}
[0.375]{} {width="\textwidth"}
[0.375]{} {width="\textwidth"}
[0.375]{} {width="90.00000%"}
We train a model on this with $8$ blocks of $256$ hidden nodes with the same block scheme (MADE, BatchNorm, Reverse) as our real Gaia model for a few epochs, allowing the model to use the true dust map as part of its iterative dust estimation scheme (though recall it still needs to estimate accurate distances to calculate the true dust). We then numerically integrate this CMD over the $bp-g$ dimension and visualize it in the same space as \[fig:true\_fake\_cmd\] in \[fig:recon\_fake\_cmd\]. As can be seen, the reconstructed CMD, without any hyperparameter tuning or extensive training, and noisy reddened data, is very close to the original.
Data Products {#sec:products}
-------------
After training the model on the Gaia dataset, we apply it back to the data to generate a catalog. We choose not to use the Bailer-Jones distance priors, since we found they harmed the accuracy of stellar distance estimates in the halo as they are very prior dominated (which can be seen in \[fig:gd1\_example\_prior\]). Instead, we use a constant distance prior over non-negative distances. The catalog contains the mean and standard deviation in the Bayesian posterior estimate for each of the DR2 sources. We plan on adding another catalog that includes 100 quantiles of the posteriors. Alternatively, one can run our code, which will be added to <https://github.com/MilesCranmer/public_CMD_normalizing_flow>, to sample from the entire posteriors in our trained model. These will be described in the data documentation.
Number Gaia Value (if relevant) Model Value
---------------------------------------------------------------------------------- -------------------------- -------------
Total Catalog Entries 640,875,169
Fraction of Stars with $d \in (1 \text{ pc},10 \text{ pc}]$
Fraction of Stars with $d \in (10 \text{ pc},100 \text{ pc}]$
Fraction of Stars with $d \in (100 \text{ pc},1 \text{ kpc}]$
Fraction of Stars with $d \in (1 \text{ kpc},10 \text{ kpc}]$
Fraction of Stars with $d \in (10 \text{ kpc},100 \text{ kpc}]$
Fraction of Stars with $d \in (100 \text{ kpc},1000 \text{ kpc}]$
SNR $\in (-\infty, 0.1]$
SNR $\in (0.1, 1.0]$
SNR $\in (1 , 10]$
SNR $\in (10 , 100]$
SNR $\in (100 , 1000]$
Mean SNR Improvement of the points with ${\varpi_{\text{obs}}}>0$ 48.6%
Negative Fraction of Parallaxes 0
Mean SNR of the points with ${\varpi_{\text{obs}}}< 0$ 1.388
Average Distance of the points with ${\varpi_{\text{obs}}}< 0$ 5.376 kpc
Average Distance (weighted by SNR) of the points with ${\varpi_{\text{obs}}}< 0$ 5.845 kpc
A visualization of the normalizing flow marginalized over $bp-g$ can be seen in terms of probability and log-probability density in \[fig:cmd\].
[0.475]{} ![The trained normalized flow, representing a color-magnitude diagram, is visualized here as a probability density in the space of $bp-rp$ and $g$, marginalized over $bp-g$. Plot (a) shows probability and plot (b) shows log-probability.[]{data-label="fig:cmd"}](normal_power_57.png "fig:"){width="\textwidth"}
[0.475]{} ![The trained normalized flow, representing a color-magnitude diagram, is visualized here as a probability density in the space of $bp-rp$ and $g$, marginalized over $bp-g$. Plot (a) shows probability and plot (b) shows log-probability.[]{data-label="fig:cmd"}](log_power_57.png "fig:"){width="\textwidth"}
The tightening of the CMD from applying this prior to the data can be seen in \[fig:parallaxes\] through \[fig:50\_parallaxes\]. The tightening is used as a visual metric for the improvement in the distance estimates. As seen in Table \[tbl:data\_summary\], more stars have higher signal-to-noise ratios, with only 18.9% of stars having SNR less than 1.0 versus the Gaia catalog which has 46.1%. The mean SNR improvement of those stars which do not have negative parallaxes is 48.6%.
[0.8123]{} {width="\textwidth"}
[0.8123]{} {width="\textwidth"}
[0.8123]{} {width="\textwidth"}
Application Examples {#sec:applications}
--------------------
Here we describe several potential applications of this dataset: obtaining an estimate of the distance to M67, filtering foreground stars, and substructure detection in three spatial dimensions. First, we visualize the improved distance estimates to stars in the GD-1 stream. We make use of the reduction from [@price-whelan_off_2018] to select the stars in a strip of the DR2 sky, and then compare the distance estimates from Gaia parallaxes alone with distance estimates from Gaia parallaxes sampled using the Bailer-Jones distance prior, and finally our model distance. As seen in \[fig:gd1\_example\_model\] compared to \[fig:gd1\_example\_dold\] (raw Gaia) and \[fig:gd1\_example\_prior\] (distance priors), distance estimates to stars in GD-1 are greatly improved with this photometric model. By using the distance estimates from our model, one can perform kinematic searches for substructure farther than $\SI{1}{kpc}$ without requiring associated stars be part of an isochrone with the same metallicity and age. In other words, this enables generic filtering of foreground stars in Gaia data, and also adds a third distance dimension for clustering stellar substructures, such as with the `STREAMFINDER` algorithm in [@malhan_streamfinder_2018].
{width="80.00000%"}
{width="80.00000%"}
{width="80.00000%"}
Finally, we give an example of estimating the distance to the cluster M67, using full Bayesian posteriors from our model. The cluster M67 is at $\sim$ (@yakut_close_2009) from Earth and was also used as a demonstration cluster in [@anderson_improving_2018]. We do this with our model without distance priors (a flat prior over non-negative distances). First, we filter Gaia DR2 sources to the cone centered about $\alpha = 08^\text{h} 51.3^\text{m}$, $\delta$=, with radius . Since the estimate of inverted Gaia parallaxes is reasonable by itself (excluding negative parallaxes), we filter down to only parallaxes with SNR less than some bound, shown as the columns in \[fig:M67\]. to demonstrate that our model improves the precision and accuracy of noisy parallaxes. The top row of \[fig:M67\] shows the inverse parallaxes. The middle row in \[fig:M67\] has plots for the distribution of distances from Gaia parallaxes sampled using the [@bailer-jones_estimating_2018] distance prior, and then in the bottom row of \[fig:M67\], we show the distribution of the model-derived distances presented in this paper.
{width="\textwidth"}
We then fit a two-component Bayesian GMM to every distance distribution, modeling the cluster distance with one component and the background stars with the other. These are overplotted in \[fig:M67\]. The model estimates for each SNR are shown in \[tbl:M67\].
Model M67 Center Estimate (kpc) Spread (kpc)
------------------------------------------------------------------- --------------------------- --------------
Inverse Parallax with SNR $< 15$ (and ${\varpi_{\text{obs}}}>0$) $0.77$ $0.07$
Inverse Parallax with SNR $ < 50$ (and ${\varpi_{\text{obs}}}>0$) $0.80$ $0.04$
Bailer-Jones Prior with SNR $< 15$ $0.87$ $0.07$
Bailer-Jones Prior with SNR $< 50$ $0.85$ $0.01$
Normalizing Flow with SNR $< 15$ $0.83$ $0.06$
Normalizing Flow with SNR $< 50$ $0.845$ $0.006$
As is evident in \[tbl:M67\], our normalizing flow model improves precision and accuracy to the M67 cluster of the very low-quality parallaxes: not only is the spread in the Gaussian component almost half of that with the Bailer-Jones distance priors, but the estimate derived from SNR $<15$ parallaxes is much closer to the final estimate of $0.84$ kpc using our model. Also, since the distance priors rely on other surveys to estimate stellar density in every direction, they implicitly contain models for this particular cluster, meaning it is easier for the distance prior model in this example. The distance prior strategy fails in the example in \[fig:gd1\_example\_prior\], with distance estimates to stars in the GD-1 stellar stream: the estimates are completely off the accepted value range. When there are anomalous stars, the prior-derived distance estimates become much less useful, and a photometry model is necessary, as shown giving much more reasonable distances in \[fig:gd1\_example\_model\].
Future work {#sec:future}
===========
The Masked Autoencoder architecture from [@germain_made:_2015] and [@papamakarios_masked_2017] which we use in our technique models a joint posterior in $\mathbb{R}^n$ using conditional probabilities: $$\begin{aligned}
p(x_1, \ldots, x_n) &\sim p(x_1) p(x_2|x_1) p(x_3| x_1, x_2)\cdots p(x_n| x_1, \ldots x_{n-1}).\end{aligned}$$ What this means is that we can exactly marginalize over $x_{n-m}$ through $x_n$ for some $m$ without additional computation by excluding them from the joint posterior, as long as we fix a hierarchy of $x_i$ that maximizes. In other words, the marginalized probability is: $$\begin{aligned}
p(x_1, \ldots, x_{n-m}) &\sim p(x_1) p(x_2|x_1) p(x_3| x_1, x_2) \cdots p(x_{n-m}| x_1, \ldots x_{n-m-1}),\end{aligned}$$ which uses the same normalizing flow model. While we cannot marginalize over an arbitrary $x_i$ without using an ensemble model, if we have photometric bands with different coverage, we can fit a color-magnitude density that simultaneously models many different survey bands, and still train and evaluate the color-magnitude probability with limited information. E.g., if we were to model Gaia and AllWISE bands simultaneously in a color-magnitude density, and we order Gaia $>$ AllWISE, then we can exactly evaluate the color-magnitude density for a given Gaia photometry marginalized over AllWISE bands, but not vice versa. This technique, also used in [@alsing_fast_2019], is powerful and would be useful to exploit for future distance estimates to maximize coverage while using all available surveys. In the future, we would also like to make use of maps for $R_V$, as well as simultaneously model stellar parameters, rather than use extinction conversions based on a simple blackbody.
Conclusion {#sec:discussion}
==========
We have demonstrated an algorithm for learning a flexible probability distribution in Gaia color-magnitude space from noisy parallax and photometry measurements using a normalizing flow. These deep neural networks, capable of learning arbitrary multi-dimensional probability distributions, have been shown in this paper to be capable of modeling CMDs well, and work at predicting CMDs accurately in an iterative dust estimation scheme. We have also presented a catalog of 640M photometric distance posteriors derived from this data-driven model using Gaia DR2 photometry and parallaxes to learn a prior in Gaia color space. Overall, the signal-to-noise of distance measurements in this catalog improves on average by 48% over the raw Gaia data, including only the non-negative Gaia parallaxes, and we also demonstrate how the accuracy of distances have improved over other models. Applications are discussed for this catalog, including significantly improved Milky Way disk separation and substructure detection.
We also will maintain a GitHub repository at <https://github.com/MilesCranmer/public_CMD_normalizing_flow> where we will post links to the distance catalog along with future versions, host code of the normalizing flow for photometry data, and answer questions about the paper and implementing the algorithm.
Acknowledgments
===============
Miles Cranmer would like to thank David W. Hogg for the initial project idea of improving Gaia distances with a generative neural network model, Johann Brehmer and Thomas Kipf for the idea of creating this model with a normalizing flow, Wolfgang Kerzendorf for feedback on calculating extinction conversions, Gregory Green for help with his `dustmap` package, George Papamakarios for the idea of marginalizing over input to the normalizing flow using [@papamakarios_masked_2017], and Iain Murray for comments on an early draft. This work made use of Astropy (@robitaille_astropy:_2013), PyTorch (@paszke2017automatic), Scikit-Learn (@scikit-learn), among other scientific packages mentioned in the paper.
|
---
author:
- |
E.G. Drukarev and M.G. Ryskin\
Petersburg Nuclear Physics Institute\
Gatchina, St.Petersburg 188350, Russia
title: QCD sum rules as a tool for investigation of the baryon properties at finite densities
---
-1cm .2cm 0.25cm
Speaking about the properties of nucleons in nuclear matter, we have in mind, e.g.:
1. Potential energy of a nucleon in the medium.
2. Neutron-proton mass splitting in isotope-symmetric matter.
3. Parameters which describe the interaction of nucleons with external fields: axial coupling constant $g_A$ and magnetic moments $\mu_N$.
4. Structure functions of deep inelastic scattering.\
Turning to the strange baryons we can add.
5. Properties of a strange baryon in nuclear matter.
6. The system of strange baryons (“strange matter”).
Approach of traditional nuclear physics to description of properties of baryons in nuclear matter is based on conception of $NN$ interactions. The problem is that small internucleon distances, where the nucleons cannot be considered as structureless point particles appear to be of crucial importance. Thus the whole approach becomes complicated and not well defined.
However, while $NN$ interactions are complicated at small distances, the strong interactions are not. Indeed, due to asymptotic freedom of QCD, the latter are the perturbative interactions between quarks and gluons. The peculiarity of QCD is the finite value of the vacuum condensates of quark and gluon fields $\langle0|\bar qq|0\rangle$, $\langle0|\frac{\alpha_s}\pi G^2|0\rangle$, etc. This means that in the ground state of QCD there are finite densities of quark–antiquark and gluon fields.
Shifman et al. [@1] suggested the QCD sum rules (SR) for the description of characteristics of free mesons. The method was based on the features of QCD, mentioned above. Later it was expanded by Ioffe [@2] to the case of free baryons. Characteristics of free nucleons where expressed through the values of QCD condensates.
In 1988 Drukarev and Levin [@3] used the SR method for investigation of the properties of nucleons in nuclear matter. In [@3] the first steps were made to express the potential energy of the nucleon through in-medium values of QCD condensates. This paper was followed by a number of works of Petersburg (Leningrad) Nuclear Physics Institute — (PNPI) group [@4]–[@9]. In 1991 the Maryland University group joined this field of investigations [@10]. Also a number of papers on meson properties in nuclear matter was published later.\
They are based on dispersion relation $$\Pi_0(q^2)\ =\ \frac1\pi\int \frac{\mbox{Im }\Pi_0(k^2)dk^2}{k^2-q^2}$$ for the function $\Pi_0(q^2)$ which describes the propagation of the system carrying the quantum numbers of the nucleon (proton). Equation (1) is considered at $q^2\to-\infty$ where the system can be treated just as three quarks with perturbative interactions between themselves and with quarks and gluons of vacuum. At $q^2\to-\infty$ $\Pi_0(q^2)$ can be presented as power series $$\Pi_0(q^2)\ =\ \sum^2_{n=0}a_nq^{2n}\ln q^2+\sum^\infty_{n=0} c_nq^{-2n}$$ known as operator expansion. The coefficients $a_n,c_n$ are related to expectation values of certain QCD operators. As to the right-hand side (r.h.s.) of Eq.(1), the spectral density Im$\,\Pi_0(k^2)$ is related to observable spectrum of the system. The usual approach is to single out the lowest laying state (proton), approximating the higher states by continuum: $$\mbox{Im }\Pi_0(k^2)\ =\ \lambda^2\delta(k^2-m^2)+\theta(k^2-W^2)\Delta
\Pi_0(-k^2)\ .$$ This is known as “pole + continuum” model. Parameters $m$ and $\lambda^2$ which describe the position of the lowest laying pole and the residue are characteristics of proton. Continuum threshold $W^2$ is the parameter of the model: the cut with physical threshold and unknown spectral density is replaced by that with unknown value of the threshold $W^2$ and fixed spectral density $\Delta\Pi_0(-k^2)$. The special mathematical ansatz, the Borel transform (inversed Laplace transform) increases the role of lower laying states. A function of $q^2$ transforms into the one of Borel mass $M^2$, e.g. $$\widehat B\ \frac1{k^2+q^2}\ =\ \exp\left(-\frac{k^2}{M^2}\right)\ .$$
The model for the left-hand side (l.h.s.) of Eq.(1) becomes increasingly true at large values of $M^2$. The one for r.h.s. works better at small $M^2$. The basic assumption of the method is that there is certain region of the values of $M^2$ in which both r.h.s. and l.h.s. of Eq.(1) approximate the true function $\Pi_0(q^2)$ well enough. Then the parameters $m,\lambda^2$ and $W^2$ can be expressed through the values of QCD condensates. Ioffe [@2] found that the value of $m$ depends mainly on the condensate $\langle0|\bar qq|0\rangle$.
Thus the picture of formation of the proton mass turned out to be very simple. It appears due to the exchange by quarks between our probe system and the quark-antiquark pairs of QCD vacuum.
The generalization of the SR method to the case of finite densities is not straightforward. The spectrum of the function $\Pi(q)$ is more complicated now. One should single-out the singularities connected with the baryon but not with the medium itself. This can be done by the special choice of variables. Neglecting the Fermi motion of the nucleons of the matter, we can fix the pair energies $S$ of our probe hadron and that of the matter. Presenting the QCD SR for the function $\Pi(q)=\Pi(q^2,s)$ we can single-out the singularities connected with the probe hadron until we limit ourselves to its pair interactions with the nucleons of the matter. In this approach “pole+continuum” model, employed for vacuum can be used. This choice of variables insures the condition $q_0\to\infty$ at $q^2\to-\infty$ which is necessary for the operator expansion of the function $\Pi(q)$. Another problem comes since each term of operator expansion corresponds, in the general case, to infinite number of condensates. Due to the presence of logarithmic loops, several lowest order terms of operator expansion contain, however, finite number of the condensates.
In the papers [@3; @4; @6] QCD SR in nuclear matter were presented as Borel transformed dispersion relations for the difference of the functions $\Pi(q)$ in matter and in vacuum. The shifts of the parameters $m,\lambda^2$ and $W^2$ caused by interaction with the matter $(\Delta\,m$, $\Delta\lambda^2$ and $\Delta W^2$) were expressed through in-medium values of QCD condensates.
On the other hand the shift of the position of the nucleon pole in external field is $\Delta m=U$ with $U$ standing for the potential energy of the nucleon. It was found that the value of $U$ is determined mainly by the averaged values of the quark operators $\bar q\gamma_0q$ and $\bar qq$. The condensate $\langle M|\bar q\gamma_0q|M\rangle$, which vanishes in vacuum, is just the density of baryon number in the system. The expectation value $\langle M|\bar qq|M\rangle$ is the density of quark-antiquark pairs. Thus we come to a simple picture of formation of the potential energy $U$. It comes from exchange by quarks between our probe three quark system and the matter. The latter can contribute by its valence quarks $\langle M|\bar q\gamma_0q|M\rangle$ and by modification of its sea of quark-antiquark pairs $\langle
M|\bar qq|M\rangle-\langle0|\bar qq|0\rangle$.
One can immediately calculate the condensate $$\langle M|\bar q\gamma_0q|M\rangle\ =\ \sum_i n_{q_i}\rho_i$$ with $n_q$ being the number of $q$ quarks in a nucleon of the matter ($i$ denotes proton or neutron), $\rho_i$ stands for the density. The SR analysis provides the contribution $\Delta_vm\approx+200$ MeV caused by this condensate (at $\rho_n=\rho_p=\rho/2$, $\rho=0.17$ Fm$^{-3}$). The scalar condensate can be presented as $$\langle M|\bar qq|M\rangle-\langle0|\bar qq|0\rangle\ =\ \rho\langle
N|\bar qq|N\rangle+F(\rho)$$ with the first term in r.h.s. of Eq.(6) standing for the gas approximation while $F(\rho)$ describes the contribution of the meson cloud. Fortunately the first term can be expressed through observables since $$\langle N|\bar qq|N\rangle\ =\ \frac{2\sigma}{m_u+m_d}\ .$$ Here $\sigma$ denotes pion-nucleon sigma term which can be extracted from experimental data on low energy $\pi N$ scattering. The gas approximation provides the contribution $\Delta_sm\approx-300$ MeV to the potential energy. Several steps beyond the gas approximation have been made also. The function $F(\rho)$ is determined mainly by the relatively large distances of the order of inversed Fermi momenta or larger ones. This makes some approximate calculations available. They lead to the saturation curve with reasonable values of the equilibrium density and of the binding energy.
One can generalize the approach for the case of neutron-proton mass splitting in symmetric matter. In QCD language this effect is caused by finite values of the difference of quark masses $m_d-m_u$ and by the non-vanishing value of the operator $\bar dd-\bar uu$. Both contributions were included explicitly into QCD SR analysis [@7]. Neutron was found to be bound stronger than the proton with reasonable value of the mass difference.
As to parameters of interaction of the nucleons with external fields, the application of the method at finite densities is the straightforward generalization of this method in vacuum [@11]. In the left-hand side of SR the quark system interacts with external field while in the right-hand side the corresponding parameter of nucleon enters the equation. The first approach to the calculation of renormalization of axial coupling constant was made in [@5].
In the same way the method was applied to the calculation of the deep inelastic structure functions of nuclei. In this case the system interacts with the hard virtual photon. In our paper [@9] we calculated the deviations of the structure function $F_2$ from that of a system of free nucleons at intermediate values of Bjorken variable $x$. The calculated values followed typical EMC behaviour. As the next steps of application of the approach to this problem we plan to investigate cumulative aspects of the process. The method can be applied also to investigation of gluon structure function. Another interesting object is the structure function of a polarized nucleon.
One can see that the method can be applied for description of a strange baryon in the matter. All the problems considered in this section and in the previous one can be approached in the same way. Also behaviour of a baryon in the system of strange ones can be described in terms of the condensates of $u,d$ and $s$ quarks.
We made first steps in solving the problem of expressing the characteristics of baryons at finite densities through the in-medium values of the condensates. Potential energy of a nucleon in nuclear matter was expressed as the sum of the terms proportional to vector and scalar condensates. The former is positive while the latter is negative. Hence, the structure of potential energy reproduces that of quantum hadrodynamics. The saturation of the matter in our approach is provided by non-linear contribution to the scalar condensate $\langle M|\bar qq|M\rangle$. We obtained at least qualitative description of neutron-proton mass splitting in nuclear matter. We described also the influence of medium on nucleon structure functions.
Note that this approach does not describe quark effects only. It describes the hadron effects, expressing them through certain quark effects. For example exchange by mesons (pairs of strongly correlated quarks) in the r.h.s. of the sum rules is expressed through exchange by pairs of uncorrelated quarks in l.h.s.
We obtained some new knowledge. We show the scalar forces to be related to $\pi N$ sigma term. In the case of isotope-breaking forces we show the scalar channel to be as important as the vector one. Thus, the method provides guide-likes for traditional nuclear physics.
All the results, described above, were obtained without fitting parameters. We did not use a controversial conception of $NN$ interaction.
Note one more point. The QCD SR method provides a unique approach to the problems, listed in the beginning of this paper. In framework of traditional nuclear physics they require different knowledge and different skill. Thus, usually they attract attention of different communities of the explorers.
The method should be improved by inclusion of more complicated in-medium condensates. Also the role of higher order terms of expansion in powers of Fermi momentum should be clarified.
Note that investigation of QCD SR stimulated other directions of research. Say, the first analysis of the function $\langle M|\bar
qq|M\rangle$ carried out in [@3] was followed by more than a dozen works on the subject.
One of us (E.G.D.) is indebted to the Organizing Committee of the Conference and to the Russian Fund for Fundamental Research (grant \#97-02-27083) for the support. This activity is supported by the Russian Fund for Fundamental Research (grant\#95-02-03752-a).
[99]{} M.A. Shifman, A.I. Vainshtein, V.I. Zakharov, Nucl.Phys. [**B147**]{} (1979) 385. B.L. Ioffe, Nucl.Phys. [**B188**]{} (1981) 317. E.G. Drukarev and E.M.Levin, Sov.Phys. JETP Lett. [**48**]{} (1988) 338. E.G. Drukarev and E.M. Levin, Sov.Phys. JETP [**68**]{} (1989) 680;\
Nucl.Phys.A [**511**]{} (1990) 679; Progr.in Part and Nucl.Phys. [**27**]{} (1991) 77. E.G. Drukarev and E.M. Levin, Nucl.Phys.A [**532**]{} (1991) 695. E.G. Drukarev and M.G. Ryskin, Nucl.Phys.A [**578**]{} (1994) 333. E.G. Drukarev and M.G. Ryskin, Nucl.Phys.A [**572**]{} (1994) 560; [**577**]{} (1994) 375. E.G. Drukarev, M.G. Ryskin and V.A. Sadovnikova, Z.Phys.A [**353**]{} (1996) 455. E.G. Drukarev and M.G. Ryskin, Z.Phys.A [**356**]{} (1997) 457. T.D. Cohen, R.J. Furnstahl and D.K. Griegel, Phys.Rev.Lett. [**67**]{} (1991) 961. B.L. Ioffe and A.V. Smilga, Nucl.Phys.B [**252**]{} (1984) 109.
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---
abstract: 'This article contains a detailed and rigorous proof of the construction of a geometric invariant for initial data sets for the Einstein vacuum field equations. This geometric invariant vanishes if and only if the initial data set corresponds to data for the Kerr spacetime, and thus, it characterises this type of data. The construction presented is valid for boosted and non-boosted initial data sets which are, in a sense, asymptotically Schwarzschildean. As a preliminary step to the construction of the geometric invariant, an analysis of a characterisation of the Kerr spacetime in terms of Killing spinors is carried out. A space spinor split of the (spacetime) Killing spinor equation is performed, to obtain a set of three conditions ensuring the existence of a Killing spinor of the development of the initial data set. In order to construct the geometric invariant, we introduce the notion of approximate Killing spinors. These spinors are symmetric valence 2 spinors intrinsic to the initial hypersurface and satisfy a certain second order elliptic equation —the approximate Killing spinor equation. This equation arises as the Euler-Lagrange equation of a non-negative integral functional. This functional constitutes part of our geometric invariant —however, the whole functional does not come from a variational principle. The asymptotic behaviour of solutions to the approximate Killing spinor equation is studied and an existence theorem is presented.'
author:
- |
[Thomas Bäckdahl]{} [^1]\
[^2]\
School of Mathematical Sciences,\
Queen Mary University of London,\
Mile End Road, London E1 4NS, UK.
title: '**On the construction of a geometric invariant measuring the deviation from Kerr data**'
---
Introduction
============
The Kerr spacetime is, undoubtedly, one of the most important exact solutions to the vacuum Einstein field equations [@Ker63]. It describes a rotating stationary asymptotically flat black hole parametrised by its mass $m$ and its specific angular momentum $a$. One of the outstanding challenges of contemporary General Relativity is to obtain a full understanding of the properties and the structure of the Kerr spacetime, and of its standing in the space of solutions to the Einstein field equations.
There are a number of difficult conjectures and partial results concerning the Kerr spacetime. In particular, it is widely expected to be the only rotating stationary asymptotically flat black hole. This conjecture has been proved if the spacetime is assumed to be analytic ($C^\omega$) — see e.g. [@ChrCos08] and references within. Recently, there has been progress in the case where the spacetime is assumed to be only smooth ($C^\infty$) —see [@IonKla09a]. Moreover, it has been shown that a regular, non-extremal stationary black hole solution of the Einstein vacuum equations which is suitably close to a Kerr solution must be that Kerr solution —i.e. *perturbative stability* among the class of stationary solutions [@AleIonKla09].
Another of the conjectures concerning the Kerr spacetime is that it describes, in some sense, the late time behaviour of a spacetime with dynamical (that is, non-stationary) black holes —this is sometimes known as the *establishment point of view of black holes*, cfr. [@Pen73]. A step in this direction is to obtain a proof of the *non-linear stability* of the Kerr spacetime —this conjecture roughly states that the Cauchy problem for the vacuum Einstein field equations with initial data for a black hole which is suitably close to initial data for the Kerr spacetime gives rise to a spacetime with the same global structure as Kerr and with suitable pointwise decay. Numerical simulations support the conjectures described in this paragraph.
A common feature in the problems mentioned in the previous paragraphs is the need of having a precise formulation of what it means that a certain spacetime is *close* to the Kerr solution. Due to the coordinate freedom in General Relativity it is, in general, difficult to measure how much two spacetimes differ from each other. Statements made in a particular choice of coordinates can be deceiving. In the spirit of the geometrical nature of General Relativity, one would like to make statements which are coordinate and gauge independent. Invariant characterisations of spacetimes provide a way of bridging this difficulty.
Most analytical and numerical studies of the Einstein field equations make use of a 3+1 decomposition of the equations and the unknowns. In this context, the question of whether a given initial data set for the Einstein field equations corresponds to data for the Kerr spacetime arises naturally —an initial data set will be said to be data for the Kerr spacetime if its development is isometric to a portion (or all) of the Kerr spacetime. A related issue arises when discussing the (either analytical or numerical) 3+1 evolution of a spacetime: do the leaves of the foliation approach, as a result of the evolution, hypersurfaces of the Kerr spacetime? In order to address these issues it is important to have a geometric characterisation of the Kerr solution which is amenable to a 3+1 splitting.
A number of invariant characterisations are known in the literature. Each with their own advantages and disadvantages. For completeness we discuss some which bear connection to the analysis presented in this article:
**The Simon and Mars-Simon tensors.** A convenient way of studying stationary solutions to the Einstein field equations is through the quotient manifold of the orbits of the stationary Killing vector. The Schwarzschild spacetime is characterised among all stationary solutions by the vanishing of the Cotton tensor of the metric of this quotient manifold —see e.g. [@Fri04]. In [@Sim84] a suitable generalisation of the Cotton tensor of the quotient manifold was introduced —the *Simon tensor*. The vanishing of the Simon tensor together with asymptotic flatness and non-vanishing of the mass characterises the Kerr solution in the class of stationary solutions. In [@Mar99; @Mar00] a spacetime version of the Simon tensor was introduced —the so-called *Mars-Simon tensor*. The construction of this tensor requires the *a priori* existence of a Killing vector in the spacetime. Accordingly, it is tailored for the problem of the uniqueness of stationary black holes. The vanishing of the Mars-Simon tensor together with some global conditions (asymptotic flatness, non-zero mass, stationarity of the Killing vector) characterises the Kerr spacetime.
**Characterisations using concomitants of the Weyl tensor.** A concomitant of the Weyl tensor is an object constructed from tensorial operations on the Weyl tensor and its covariant derivatives. An invariant characterisation of the Kerr spacetime in terms of concomitants of the Weyl tensor has been obtained in [@FerSae09]. This result generalises a similar result for the Schwarzschild spacetime given in [@FerSae98]. These characterisations consist of a set of conditions on concomitants of the Weyl tensor, which if satisfied, characterise locally the Kerr/Schwarzschild spacetime. An interesting feature of the characterisation is that it provides expressions for the stationary and axial Killing vectors of the spacetime in terms of concomitants of the Weyl tensor. Unfortunately, the concomitants used in the characterisation are complicated, and thus, produce very involved expressions when performing a 3+1 split.
**Characterisations by means of generalised symmetries.** Generalised symmetries (sometimes also known as hidden symmetries) are generalisations of the Killing vector equation —like the Killing tensors and conformal Killing-Yano tensors. These tensors arise naturally in the discussion of the so-called Carter constant of motion and in the separability of various types of linear equations on the Kerr spacetime —see e.g. [@Car68b; @KamMcL84; @PenRin86]. In particular, the existence of a conformal Killing-Yano tensor is equivalent to the existence of a valence-2 symmetric spinor satisfying the Killing spinor equation. An important property of the Schwarzschild and Kerr spacetimes is that they admit a Killing spinor. This Killing spinor generates, in a certain sense, the Killing vectors and Killing-Yano tensors of the exact solutions in question [@HugSom73b]. Moreover, as it will be discussed in the main part of this article, for a spacetime which is neither conformally flat nor of Petrov type N, the existence of a Killing spinor associated to a Killing-Yano tensor together with the requirement of asymptotic flatness renders a characterisation of the Kerr spacetime. To the best of our knowledge, this property has only been discussed in the literature —without proof— in [@FerSae07].
Although at first sight independent, the characterisations of the Schwarzschild and Kerr spacetimes described in the previous paragraphs are interconnected —sometimes in very subtle manners. This is not too surprising as all these characterisations make use in a direct or indirect manner of the fact that the Kerr spacetime is a vacuum spacetime of Petrov type D —see e.g. [@SKMHH] for a discussion of the Petrov classification. The art in producing a useful characterisation of the Kerr spacetime lies in finding further conditions on type D spacetimes which are natural and simple to use.
A characterisation of Kerr data {#a-characterisation-of-kerr-data .unnumbered}
-------------------------------
Characterisations of initial data sets for the Schwarzschild and Kerr spacetimes have been discussed in [@GarVal07; @GarVal08b; @Val05b]. These characterisations make use of a number of local and global ingredients. For example, in [@GarVal08b] it is necessary to assume the existence of a Killing vector on the development of the spacetime.
In this article we present a rigorous and detailed discussion of a geometric invariant characterising initial data for the Kerr spacetime. A restricted version of this construction has been presented in [@BaeVal10a].
The starting point of our construction is the observation that the existence of a Killing spinor in the Kerr spacetime is a key property. It allows to relate the Killing vectors of the spacetime with its curvature in a neat way. The reason for its importance can be explained in the following way: from a specific Killing spinor it is possible to obtain a Killing vector which in general will be complex. It turns out that for the Kerr spacetime this Killing vector is in fact real and coincides with the stationary Killing vector. It can be shown that the Kerr solution is the only asymptotically flat vacuum spacetime with these properties, if one assumes that there are no points where the Petrov type is either N or O.
Given the aforementioned spacetime characterisation of the Kerr solution, the question now is how to make use of it to produce a characterisation in terms of initial data sets. For this, one has to encode the existence of a Killing spinor at the level of the data. The way of doing this was first discussed in [@GarVal08a] and follows the spirit of the well-known discussion of how to encode Killing vectors on initial data —see e.g. [@BeiChr97b].
The conditions on the initial data that ensure the existence of a Killing spinor in its development are called the *Killing spinor initial data equations* and are, like the Killing initial data equations (KID equations), overdetermined. In [@Dai04c], a procedure was given on how to construct equations which generalise the KID equations for time symmetric data. These generalised equations have the property that for a particular behaviour at infinity they always admit a solution. If the spacetime admits Killing vectors, then the solutions to the generalised KID equations with the same asymptotic behaviour as the Killing vectors are, in fact, Killing vectors. Therefore, one calls the solutions to the generalised KID equations *approximate symmetries*. The total number of approximate symmetries is equal to the maximal number of possible Killing vectors on the spacetime. A peculiarity of this procedure is that if the spacetime is not stationary, the approximate Killing vector associated to a time translation does not have the same asymptotic behaviour as a time translation[^3].
The Killing spinor initial data equations consist of three conditions: one of them differential (the *spatial Killing spinor equation*)[^4] and two *algebraic conditions*. Following the spirit of [@Dai04c] we construct a generalisation of the spatial Killing spinor equation —*the approximate Killing spinor equation*. This equation is elliptic and of second order. This equation is the Euler-Lagrange equation of an integral functional —the $L^2$-norm of the exact spatial Killing spinor equation. For this equation it is possible to prove the following theorem:
For initial data sets to the Einstein field equations with suitable asymptotic behaviour, there exists a solution to the approximate Killing spinor equation with the same asymptotic behaviour as the Killing spinor of the Kerr spacetime.
A precise formulation will be given in the main text. In particular, it will be seen that the conditions on the asymptotic behaviour of the initial data are rather mild and amount to requiring the data to be, in a sense, asymptotically Kerr data. Contrasted with the results in [@Dai04c], this result is notable because, arguably, the most important approximate symmetry of [@Dai04c] does not share the same asymptotic behaviour as the exact symmetry. The precise version of this theorem generalises the one discussed in [@BaeVal10a] in that it allows for boosted data. This generalisation is only possible after a detailed analysis of the asymptotic solutions of the exact Killing spinor equation.
The approximate Killing spinor discussed in the previous paragraphs can be used to construct a geometric invariant for the initial data. This invariant is global and involves the $L^2$ norms of the Killing spinor initial data equations evaluated at the approximate Killing vector. It should be observed that only part of the invariant satisfies a variational principle —this is a further difference with respect to the construction of [@Dai04c]. As the initial data set is assumed to be asymptotically Euclidean, one expects its development to be asymptotically flat. This renders the desired characterisation of Kerr data and our main result.
Consider an initial data set for the vacuum Einstein field equations whose development in a small slab is neither of Petrov type N nor O at any point, and such that the $L^2$ norm of the Killing spinor initial data equations evaluated at the solution (with the same asymptotic behaviour as the Killing spinor of the Kerr spacetime) to the approximate Killing spinor equation vanishes. Then the initial data set is locally data for the Kerr spacetime. Furthermore, if the 3-manifold has the same topology as that of hypersurfaces of the Kerr spacetime, then the initial data set is data for the Kerr spacetime.
There are several advantages of this characterisation over previous ones given in the literature. Most notably, it allows to condense the non-Kerrness of an initial data set in a single number. That this invariant constitutes a good distance in the space of initial data sets will be discussed elsewhere. Furthermore, the way the invariant is constructed is fully amenable to a numerical implementation —the elliptic solvers that one would need to compute the solution to the approximate Killing spinor equation are, nowadays, standard technology.
Detailed outline of the article {#detailed-outline-of-the-article .unnumbered}
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The outline of the article is as follows: in Section \[Section:Basics\] we study Killing spinors, and their influence on the algebraic type of the spacetime. We relate the Killing spinors to Killing vectors and Killing-Yano tensors. Using these results together with a characterisation of the Kerr spacetime by Mars [@Mar00], we conclude that the Kerr spacetime can be characterised in terms of existence of a Killing spinor related to a real Killing vector. This has previously been overlooked in the literature, but it is a key element in our analysis.
Section \[Section:SpaceSpinors\] follows with an exposition of space spinors, which will be the main computational tool for the remainder of the paper. Following that, in Section \[Section:KSD\] we study a 3+1 splitting of the Killing spinor equation. A similar analysis was carried out in [@GarVal08a], but here we manage to condense the result into three simple equations, the *spatial Killing spinor equation* and two algebraic equations. We also present general equations for the spatial derivatives of a general valence 2 spinor, which is not necessarily a Killing spinor. These equations are also used in later parts of the paper.
In Section \[Section:ApproximateKS\] we introduce the new concept of *approximate Killing spinors*. These are introduced as solutions to an elliptic equation formed by composing the spatial Killing spinor operator with its formal adjoint. That this composed operator is indeed elliptic and formally self adjoint is proved. We also see that the approximate Killing spinor equation can be derived from a variational principle.
To get unique solutions to the approximate Killing spinor equation, we need to specify the asymptotic behaviour. For a rigorous treatment of this, we use weighted Sobolev spaces; these are described in Section \[Section:AsymptoticallyEuclideanData\]. Here we also study the asymptotics of a Killing spinor on a boosted slice of the Schwarzschild spacetime. In general, we study slices of an arbitrary spacetime with asymptotics similar to those of the Schwarzschild spacetime. Using these assumptions, we can then in Section \[Section:AB\] prove existence of spinors with the same asymptotics as the Killing spinor in the Schwarzschild spacetime. We later use these spinors as seeds for solutions to the approximate Killing spinor equation. In this way we get the desired asymptotic behaviour of our approximate Killing spinors.
In Section \[Section:ApproximateKSinAEM\] we study the approximate Killing spinor equation in our asymptotically Euclidean manifolds to gain existence and uniqueness of solutions with the desired asymptotics. This is done by means of the Fredholm alternative on weighted Sobolev spaces, transforming the existence problem into a study of the kernel of the Killing spinor operator. In this process we get the first part of the geometric invariant —the $L^2$ norm of the approximate Killing spinor. This norm is proved to be finite. The geometric invariant is constructed in Section \[Section:Invariant\], by adding the $L^2$ norms of the algebraic conditions. There follows our main theorem: the invariant vanishes if and only if the spacetime is the Kerr spacetime. The invariant is as a consequence of the construction proved to be finite and well defined.
We also include two appendices. The first describes an alternative proof of finiteness of a particular boundary integral in Section \[Section:ApproximateKSinAEM\]. The other contains tensor versions of the invariant —this can be useful in applications.
General notation and conventions {#general-notation-and-conventions .unnumbered}
--------------------------------
All throughout, $(\mathcal{M},g_{\mu\nu})$ will be an orientable and time orientable globally hyperbolic vacuum spacetime. It follows that the spacetime admits a spin structure —see [@Ger68; @Ger70c]. Here, and in what follows, $\mu,\,\nu,\cdots$ denote abstract 4-dimensional tensor indices. The metric $g_{\mu\nu}$ will be taken to have signature $(+,-,-,-)$. Let $\nabla_\mu$ denote the Levi-Civita connection of $g_{\mu\nu}$. The sign of the Riemann tensor will be given by the equation $$\nabla_\mu\nabla_\nu\xi_\zeta-\nabla_\nu\nabla_\mu\xi_\zeta=R_{\nu\mu\zeta}{}^\eta\xi_\eta.$$
The triple $(\mathcal{S}, h_{ab},K_{ab})$ will denote initial data on a hypersurface of the spacetime $(\mathcal{M},g_{\mu\nu})$. The symmetric tensors $h_{ab}$, $K_{ab}$ will denote, respectively, the 3-metric and the extrinsic curvature of the 3-manifold $\mathcal{S}$. The metric $h_{ab}$ will be taken to be negative definite —that is, of signature $(-,-,-)$. The indices $a,\,b,\ldots$ will denote abstract 3-dimensional tensor indices, while $i,\,j,\ldots$ will denote 3-dimensional tensor coordinate indices. Let $D_a$ denote the Levi-Civita covariant derivative of $h_{ab}$.
Spinors will be used systematically. We follow the conventions of [@PenRin84]. In particular, $A,\,B,\ldots$ will denote abstract spinorial indices, while $\mathbf{A}, \,\mathbf{B},\ldots$ will be indices with respect to a specific frame. Tensors and their spinorial counterparts are related by means of the solder form $\sigma_\mu{}^{AA'}$ satisfying $g_{\mu\nu}=\sigma_\mu^{AA'}\sigma_\nu^{BB'} \epsilon_{AB}\epsilon_{A'B'}$, where $\epsilon_{AB}$ is the antisymmetric spinor and $\epsilon_{A'B'}$ its complex conjugate copy. One has, for example, that $\xi_\mu =
\sigma_{\mu}{}^{AA'} \xi_{AA'}$. Let $\nabla_{AA'}$ denote the spinorial counterpart of the spacetime connection $\nabla_\mu$. Besides the connection $\nabla_{AA'}$, two other spinorial connections will be used: $D_{AB}$, the spinorial counterpart of the Levi-Civita connection $D_a$ and $\nabla_{AB}$, the Sen connection of $(\mathcal{M},g_{\mu\nu})$ —full details will be given in Section \[Section:SpaceSpinors\].
**The Kerr spacetime.** For the Kerr spacetime it will be understood the maximal analytic extension of the Kerr metric as described by Boyer & Lindquist [@BoyLin67] and Carter [@Car68a]. When regarding the Kerr spacetime as the development of Cauchy initial data, we will only consider its maximal globally hyperbolic development.
Killing spinors: general theory {#Section:Basics}
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As mentioned in the introduction, our point of departure will be a characterisation of the Kerr spacetime based on the existence in the spacetime of a valence-2 symmetric spinor satisfying the Killing spinor equation. To the best of our knowledge, this characterisation of the Kerr spacetime has not explicitly been discussed in the literature, save for a side remark in [@FerSae07]. In this section we provide a summary of this characterisation and fill in some technical details.
Killing spinors and Petrov type D spacetimes
--------------------------------------------
A valence-2 Killing spinor is a symmetric spinor $\kappa_{AB}=\kappa_{(AB)}$ satisfying the equation $$\label{KillingSpinorEquation} \nabla_{A'(A} \kappa_{BC)}=0.$$ Killing spinors offer a way of relating properties of the curvature to properties of the symmetries of the spacetime. Taking a further derivative of equation , antisymmetrising and commuting the covariant derivatives one finds the integrability condition $$\Psi_{(ABC}{}^F\kappa_{D)F}=0, \label{IntegrabilityCondition}$$ where $\Psi_{ABCD}$ denotes the self-dual Weyl spinor. The above integrability imposes strong restrictions on the algebraic type of the Weyl spinor. More precisely, it follows that if $\Psi_{ABCD}\neq
0$ and $\kappa_{AB}\neq 0$, then $$\Psi_{ABCD} = \psi \kappa_{(AB} \kappa_{CD)}, \label{SolutionIntegrabilityCondition}$$ where $\psi$ is a scalar. Thus, $\Psi_{ABCD}$ must be of Petrov type D or N —see e.g. [@GarVal08a; @Jef84]. The converse is also true [@HugPenSomWal72; @PenRin86; @WalPen70]. Summarising:
\[Theorem:TypeDhasalwaysaKS\] A vacuum spacetime admits a valence-2 Killing spinor if and only if it is of Petrov type D, N or O.
From it can also be seen that $\Psi_{ABCD}$ is of Petrov type N if and only if $\kappa_{AB}$ is algebraically special. That is, there exists a spinor $\alpha_A$ such that $\kappa_{AB}=\alpha_A \alpha_B$. Thus, an algebraically general Killing spinor $\kappa_{AB}=\alpha_{(A}\beta_{B)}$ is always associated to a vacuum spacetime of Petrov type D.
The Killing vector associated to a Killing spinor and the generalised Kerr-NUT metrics
--------------------------------------------------------------------------------------
Given a Killing spinor $\kappa_{AB}$, the concomitant $$\label{ComplexKillingVector}
\xi_{AA'}=\nabla^B{}_{A'} \kappa{}_{AB},$$ is a complex Killing vector of the spacetime: its real and imaginary parts are themselves Killing vectors of the spacetime [@HugSom73b]. In relation to this it should be pointed out that all vacuum Petrov type D spacetimes are known [@Kin69]. It follows from the analysis in the latter reference that all vacuum, Petrov type D spacetimes have a commuting pair of Killing vectors. A key property of the Kerr spacetime is the following (cfr. [@HugSom73b; @PenRin86]):
\[Proposition:KSrendersKV\] Let $(\mathcal{M},g_{\mu\nu})$ be a vacuum Petrov type D spacetime. The Killing vector $\xi_{AA'}$ given by is real in the case of the Kerr spacetime.
**Remark 1.** In what follows, the class of Petrov type D spacetimes for which $\xi_{AA'}$ is real will be called the *generalised Kerr-NUT class* —cfr. [@FerSae07]. This class can be alternatively characterised —see e.g. [@KamMcL84]— by the existence of a Killing-Yano tensor $$Y_{\mu\nu}=Y_{[\mu\nu]}, \quad \nabla_{(\mu} Y_{\nu)\lambda}=0.$$ The correspondence between the Killing spinor $\kappa_{AB}$ and the spinorial counterpart $Y_{AA'BB'}$ of the Killing-Yano tensor, $Y_{\mu\nu}$, is given by $$Y_{AA'BB'} \equiv\mbox{i} \left( \kappa_{AB}\epsilon_{A'B'} - \epsilon_{AB} \bar{\kappa}_{A'B'} \right),$$ where the overbar denotes the complex conjugate.
**Remark 2.** In terms of the Kinnersley list of type D metrics, the class of generalised Kerr-NUT metrics contains, in addition to the proper Kerr-NUT metrics (II.C), also the metrics II.E —see [@DebKamMcL84].
An important property of the generalised Kerr-NUT metrics involves the Killing form,$F_{AA'BB'}=-F_{BB'AA'}$, of a real Killing vector $\xi_{AA'}$ defined by $$\label{KillingForm}
F_{AA'BB'} \equiv \frac{1}{2}\left( \nabla_{AA'}\xi_{BB'} - \nabla_{BB'}\xi_{AA'} \right).$$ Let $$\label{SelfDualKillingForm}
\mathcal{F}_{AA'BB'} \equiv \frac{1}{2}\left(F_{AA'BB'} + \mbox{i}F^*_{AA'BB'} \right)$$ denote the corresponding *self-dual Killing form*, with $F^*_{AA'BB'}$ the Hodge dual of $F_{AA'BB'}$. Due to the symmetries of the Killing form one can write $$\mathcal{F}_{AA'BB'} = \mathcal{F}_{AB} \epsilon_{A'B'},$$ with $$\label{KFSpinor}
\mathcal{F}_{AB} \equiv \frac{1}{2} F_{AQ'B}{}^{Q'} = \mathcal{F}_{BA}.$$ One has the following result
\[Lemma:PrincipalDirections\] For generalised Kerr-NUT spacetimes one has that $$\mathcal{F}_{AB} = \varkappa \kappa_{AB},$$ where $\varkappa$ is a non-vanishing scalar function, so that the principal spinors of $\mathcal{F}_{AB}$ and $\Psi_{ABCD}$ are parallel. Equivalently, one has that $$\Psi_{ABPQ} \mathcal{F}^{PQ} = \varphi \mathcal{F}_{AB},$$ with $\varphi$ a non-vanishing scalar.
One proceeds by a direct computation. One notes that the expressions , and assume that the Killing vector $\xi_{AA'}$ is real. Using equations and and the vacuum commutators for $\nabla_{AA'}$ one finds that $$\mathcal{F}_{AB} = \frac{3}{4} \Psi_{ABPQ} \kappa^{PQ}.$$ As the spacetime is assumed to be of Petrov type D one has that $\kappa_{AB}=\alpha_{(A}\beta_{B)}$ with $\alpha_A \beta^A
=\varsigma$, where $\varsigma$ is a non-vanishing scalar. From equation one finds then that $\Psi_{ABCD}\negthinspace =\psi \alpha_{(A}\alpha_B \beta_C \beta_{D)}$, so that $$\Psi_{ABPQ} \kappa^{PQ} = - \frac{1}{3} \psi\varsigma^2 \kappa_{AB},$$ and finally that $$\mathcal{F}_{AB} = -\frac{1}{4} \psi \varsigma^2 \kappa_{AB},$$ from where the desired result follows.
The property that allows us to single out the Kerr spacetime out of the generalised Kerr-NUT class is given by the following result proved by Mars [@Mar99; @Mar00].
\[Theorem:MMars\] Let $(\mathcal{M},g_{\mu\nu})$ be a smooth vacuum spacetime with the following properties:
- $(\mathcal{M},g_{\mu\nu})$ admits a Killing vector $\xi_{AA'}$ such that, $\mathcal{F}_{AB}$, the spinorial counterpart of the Killing form of $\xi_{AA'}$ satisfies $$\Psi_{ABPQ} \mathcal{F}^{PQ} = \varphi \mathcal{F}_{AB},$$ with $\varphi$ a scalar;
- $(\mathcal{M},g_{\mu\nu})$ contains a stationary asymptotically flat 4-end, and $\xi_{AA'}$ tends to a time translation at infinity, and the Komar mass of the asymptotic end is non-zero.
Then $(\mathcal{M},g_{\mu\nu})$ is locally isometric to the Kerr spacetime.
**Remark.** A stationary asymptotically flat 4-end is an open submanifold $\mathcal{M}_\infty \subset \mathcal{M}$ diffeomorphic to $I\times ({\mbox{\SYM R}}^3 \setminus \mathcal{B}_R)$, where $I\subset {\mbox{\SYM R}}$ is an open interval and $\mathcal{B}_R$ a closed ball of radius $R$ such that in the local coordinates $(t,x^i)$ defined by the diffeomorphism, the metric $g_{\mu\nu}$ satisfies $$\begin{aligned}
&& |g_{\mu\nu}-\eta_{\mu\nu}| + |r\partial_i g_{\mu\nu}| \leq C
r^{-\alpha}, \\
&& \partial_t g_{\mu\nu} =0,\end{aligned}$$ with $C$, $\alpha$ constants, $\eta_{\mu\nu}$ is the Minkowski metric and $r=\sqrt{(x^1)^2 + (x^2)^2 + (x^3)^2}$. In particular $\alpha\geq 1$. The definition of the Komar mass is given in [@Kom58]. In this context it coincides with the ADM mass of the spacetime.
Non-degeneracy of the Petrov type of the Kerr spacetime
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Finally, we note the following result about the non-degeneracy of the Petrov type of the Kerr spacetime [@Mar00].
\[Proposition:Kerrdoesnotdegenerate\] The Petrov type of the Kerr spacetime is always D —there are no points where it degenerates to type N or O.
A characterisation of the Kerr spacetime using Killing spinors
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As a consequence of Theorem \[Theorem:TypeDhasalwaysaKS\] and propositions \[Proposition:KSrendersKV\], \[Proposition:Kerrdoesnotdegenerate\] one obtains the following invariant characterisation of the Kerr spacetimes. From this characterisation we will extract, in the sequel, a characterisation of asymptotically Euclidean Kerr data.
\[Theorem:SpacetimeCharacterisation\] Let $(\mathcal{M},g_{\mu\nu})$ be a smooth vacuum spacetime such that $$\Psi_{ABCD}\neq 0 ,\qquad
\Psi_{ABCD}\Psi^{ABCD}\neq 0$$ on $\mathcal{M}$. Then $(\mathcal{M},g_{\mu\nu})$ is locally isometric to the Kerr spacetime if and only if the following conditions are satisfied:
- there exists a Killing spinor, $\kappa_{AB}$, such that the associated Killing vector, $\xi_{AA'}$, is real;
- the spacetime $(\mathcal{M},g_{\mu\nu})$ has a stationary asymptotically flat 4-end with non-vanishing mass in which $\xi_{AA'}$ tends to a time translation.
Clearly, the conditions (i) and (ii) are necessary to obtain the Kerr spacetime. For the sufficiency, assume that (i) holds, that is, the spacetime has a Killing spinor $\kappa_{AB}$ such that the associated Killing vector $\xi_{AA'}$ is real. Accordingly, the spacetime must be of type $D$, $N$ or $O$. As $\Psi_{ABCD}\neq 0$ and $\Psi_{ABCD}\Psi^{ABCD}\neq 0$ by hypothesis, the spacetime cannot be of types $N$ or $O$. By the reality of $\xi_{AA'}$ it must be a generalised Kerr-NUT spacetime and the conclusion of Lemma \[Lemma:PrincipalDirections\] follows. Now, if (ii) holds then by Theorem \[Theorem:MMars\], the spacetime has to be locally the Kerr spacetime.
**Remark.** It is of interest to see whether the conditions $\Psi_{ABCD}\neq 0$ and $\Psi_{ABCD}\Psi^{ABCD}\neq 0$ can be removed. An analysis along what is done in the proof of Theorem \[Theorem:MMars\] may allow to do this. This will be discussed elsewhere.
Space spinors: general theory {#Section:SpaceSpinors}
=============================
As mentioned in the introduction, in this article we will make use of a space spinor formalism to project the longitudinal and transversal parts of the Killing spinor equation with respect to the timelike vector field $\tau^\mu$. The space spinor formalism was originally introduced in [@Som80]. Here we follow conventions and notations similar to those in [@GarVal08a]. For completeness, we introduce all the relevant notation here.
Basic definitions
-----------------
Let $\tau^\mu$ be a timelike vector field on $(\mathcal{M},g_{\mu\nu})$ with normalisation $\tau_\mu \tau^\mu=2$. Define the projector $$h_{\mu\nu}\equiv g_{\mu\nu} -\frac{1}{2} \tau_\mu\tau_\nu.$$ We also define the following tensors: $$\begin{aligned}
&& K_{\mu\nu} = -h_\mu{}^\lambda h_\nu{}^\rho \nabla_\lambda
\tau_\rho, \\
&& K^\mu = -\frac{1}{2} \tau^\nu \nabla_\nu \tau^\mu.\end{aligned}$$ Note that it is not being assumed that $\tau^\mu$ is hypersurface orthogonal. Thus, the tensor $K_{\mu\nu}$ as defined above is not necessarily the second fundamental form of a foliation of the spacetime $(\mathcal{M},g_{\mu\nu})$.
Let $\tau^{AA'}$ denote the spinorial counterpart of $\tau^\mu$. One has that $\tau^{AA'}\equiv
\sigma_{\mu}{}^{AA'}\tau^\mu$ so that $$\tau_{AA'}\tau^{AA'}=2, \quad \tau^A{}_{A'}\tau^{BA'}=\epsilon^{AB}.$$ The spinor $\tau^{AA'}$ allows to introduce the *spatial solder forms* $$\sigma_\mu{}^{AB}\equiv\sigma_\mu{}^{(A}{}_{A'}\tau^{B)A'}, \quad
\sigma^\mu{}_{AB} \equiv \tau_{(B}{}^{A'} \sigma^\mu{}_{A)A'},$$ so that one has $$\begin{aligned}
&& \sigma^\mu{}_{AB} \sigma_\nu{}^{AB}=h^\mu{}_\nu, \quad g_{\mu\nu}\sigma^\mu{}_{AB} \sigma^\nu{}_{CD}=h_{\mu\nu}\sigma^\mu{}_{AB} \sigma^\nu{}_{CD} = \frac{1}{2}( \epsilon_{AC}\epsilon_{BD} + \epsilon_{AD}\epsilon_{BC}), \\
&& \quad \tau_\mu\sigma^\mu{}_{AB}=0, \quad \epsilon_{AB}\epsilon_{A'B'} = \frac{1}{2}\tau_{AA'}\tau_{BB'} + h_{\mu\nu} \sigma^\mu{}_{AE} \sigma^\nu{}_{BF} \tau^E{}_{A'}\tau^F{}_{B'}.\end{aligned}$$
If $\tau^\mu$ is hypersurface orthogonal, then $h_{ab}$, $K_{ab}$, $K^a$, $\sigma_a{}^{\mathbf{AB}}$, $\sigma^a{}_{\mathbf{AB}}$ denote, respectively , the pull-backs to the hypersurfaces orthogonal to $\tau^\mu$ of $h_{\mu\nu}$, $K_{\mu\nu}$, $K^\mu$, $\sigma_\mu{}^{\mathbf{AB}}$, $\sigma^\mu{}_{\mathbf{AB}}$ —note that these objects are spatial, in the sense that their contraction with $\tau^\mu$ vanishes, and thus, their pull-backs are well defined. The relevant properties of these tensors apply to their pull-backs. Often we will begin with a spacelike hypersurface $\mathcal{S}$, and define $\tau^\mu$ as the normal to this hypersurface, we then automatically get the desired properties.
Space spinor splittings
-----------------------
The spinor $\tau^{AA'}$ can be used to construct a formalism consisting of unprimed indices. For example, given a spacetime spinor $\zeta_{AA'}$ one can write $$\label{SpaceSpinorSplit}
\zeta_{AA'} = \frac{1}{2}\tau_{AA'} \zeta-
\tau_{A'}{}^P \zeta_{PA},$$ with $$\zeta \equiv \tau^{PP'}\zeta_{PP'}, \quad \zeta_{AB} \equiv \tau_{(A}{}^{P'}\zeta_{B)P'}.$$ This decomposition can be extended in a direct manner to higher valence spinors. Any spatial tensor has a space-spinor counterpart. For example, if $T_\mu{}^\nu$ is a spatial tensor (i.e. $\tau^\mu T_\mu{}^\nu=0$ and $\tau_\nu T_\mu{}^\nu=0$), then its space spinor counterpart is given by $T_{AB}{}^{CD}=\sigma^\mu{}_{AB}\sigma_\nu{}^{CD}T_\mu{}^\nu$.
Spinorial covariant derivatives
-------------------------------
Applying formally the space spinor split given by to the spacetime spinorial covariant derivative $\nabla_{AA'}$ one obtains $$\nabla_{AA'}=\frac{1}{2}\tau_{AA'}\nabla-\tau_{A'}{}^B\nabla_{AB} ,$$ where we have introduced the differential operators $$\begin{aligned}
&& \nabla \equiv \tau^{AA'}\nabla_{AA'},\\
&& \nabla_{AB} \equiv \tau^{A'}{}_{(A}\nabla_{B)A'}=\sigma^\mu{}_{AB}\nabla_\mu.\end{aligned}$$ The latter is referred to as the *Sen connection*. Let $K_{ABCD}$ denote the space spinor counterpart of the tensor $K_{\mu\nu}$. One has that $$K_{ABCD}=\tau_D{}^{C'}\nabla_{AB}\tau_{CC'}, \quad K_{ABCD}=K_{(AB)(CD)}.$$ In the sequel, it will be convenient to write $K_{ABCD}$ in terms of its irreducible components. For this define $$\Omega_{ABCD}\equiv K_{(ABCD)}, \quad \Omega_{AB}\equiv
K_{(A}{}^C{}_{B)C}, \quad K\equiv{K^{AB}_{\phantom{AB}AB}},$$ so that one can write $$\label{KSplit}
K_{ABCD}=\Omega_{ABCD}-\frac{1}{2}\epsilon_{A(C}\Omega_{D)B}-\frac{1}{2}\epsilon_{B(C}\Omega_{D)A}-\frac{1}{3}\epsilon_{A(C}\epsilon_{D)B}K,$$ If $\tau^\mu$ is hypersurface orthogonal, then $\Omega_{AB}=0$, and thus $K_{\mu\nu}$ can be regarded as the extrinsic curvature of the leaves of a foliation of the spacetime $(\mathcal{M},g_{\mu\nu})$. Let $K_{AB}$ denote the spinorial counterpart of the acceleration $K_\mu$. It has the symmetry $K_{AB}=K_{(AB)}$ and satisfies $$K_{AB}=\tau_B{}^{A'}\nabla\tau_{AA'} .$$
If $\tau^\mu$ is hypersurface orthogonal then the pull-back, $D_a$, of $D_\mu \equiv
h^\nu{}_\mu \nabla_\nu$ corresponds to the Levi-Civita connection of the intrinsic metric of the leaves of the foliation of hypersurfaces orthogonal to $\tau^\mu$. Its spinorial counterpart is given by $D_{AB}=D_{(AB)}=\sigma^a{}_{AB}D_a$. The Sen connection, $\nabla_{AB}$, and the Levi-Civita connection, $D_{AB}$, are related to each other through the spinor $K_{ABCD}$. For example, for a valence 1 spinor $\pi_C$ one has that $$\nabla_{AB} \pi_C = D_{AB}\pi_C + \frac{1}{2} K_{ABC}{}^{D} \pi_D,$$ with the obvious generalisations for higher valence spinors.
Hermitian conjugation
---------------------
Given a spinor $\pi_{A}$, we define its *Hermitian conjugate* via $$\hat{\pi}_{A} \equiv \tau_{A}{}^{E'}\bar{\pi}_{E'}.$$ The Hermitian conjugate can be extended to higher valence symmetric spinors in the obvious way. The spinors $\nu_{AB}$ and $\xi_{ABCD}$ are said to be real if $$\hat{\nu}_{AB}=-\nu_{AB},\quad \hat{\xi}_{ABCD}=\xi_{ABCD}.$$ It can be verified that $\nu_{AB}
\hat{\nu}^{AB}, \; \xi_{ABCD} \hat{\xi}^{ABCD}\geq 0$. If the spinors are real, then there exist real spatial tensors $\nu_a$, $\xi_{ab}$ such that $\nu_{AB}$ and $\xi_{ABCD}$ are their spinorial counterparts.
Notice that the differential operator $D_{AB}$ is real in the sense that $$\widehat{D_{AB}\pi_C}=-D_{AB}\hat\pi_C.$$ Crucially, however, one has that $$\widehat{\nabla_{AB}\pi_C}=-\nabla_{AB}\hat\pi_C + \tfrac{1}{2}K_{ABC}{}^D\hat\pi_D.$$
Commutators
-----------
The analysis in the sequel will require intensive use of the commutators of the covariant derivative operators $\nabla$ and $\nabla_{AB}$. These can be derived from a space spinor splitting of the commutator of $\nabla_{AA'}$.
Define $$\square_{AB} \equiv \nabla_{C'(A}\nabla_{B)}{}^{C'}, \quad
\widehat{\square}_{AB} \equiv \tau_A{}^{A'} \tau_B{}^{B'}\square_{A'B'}=\tau_A{}^{A'} \tau_B{}^{B'}\nabla_{C(A'}\nabla_{B')}{}^{C}.$$ The action of these operators on a spinor $\pi_A$ is given by $$\square_{AB}\pi_C =\Psi_{ABCQ}\pi^Q + \tfrac{1}{2}\Lambda
\epsilon_{C(A}\pi_{B)}, \quad \widehat{\square}_{AB} \pi_C
=\tau_A{}^{A'}\tau_B{}^{B'} \Phi_{FCA'B'}\pi^F,$$ where $\Phi_{ABA'B'}$ and $\Lambda$ denote respectively, the spinor counterparts of the tracefree part of the Ricci tensor $R_{\mu\nu}$ and the Ricci scalar $R$ of the spacetime metric $g_{\mu\nu}$. Clearly, the above expressions simplify in the case of a vacuum spacetime, where we have $\Phi_{ABA'B'}=0$, $\Lambda=0$.
In terms of $\square_{AB}$ and $\widehat{\square}_{AB}$, the commutators of $\nabla$ and $\nabla_{AB}$ read
$$\begin{aligned}
[\nabla,\nabla_{AB}] ={}&
\widehat\square_{AB}-\square_{AB}-\tfrac{1}{2}K_{AB}\nabla+K^D{}_{(A}\nabla_{B)D}-K_{ABCD}\nabla^{CD},\label{commutator1}\\
[\nabla_{AB},\nabla_{CD}] ={}& \frac{1}{2}\left(
\epsilon_{A(C}\square_{D)B} + \epsilon_{B(C}\square_{D)A}
\right) +\frac{1}{2}\left(
\epsilon_{A(C}\widehat\square_{D)B}
+\epsilon_{B(C}\widehat\square_{D)A}
\right)\nonumber \\
&+\frac{1}{2}(K_{CDAB}\nabla-K_{ABCD}\nabla)
+K_{CDQ(A}\nabla_{B)}{}^Q-K_{ABQ(C}\nabla_{D)}{}^Q. \label{commutator2}\end{aligned}$$
Decomposition of the Weyl spinor
--------------------------------
The Hermitian conjugation can be used to decompose the Weyl spinor $\Psi$ in terms of its electric and magnetic parts via $$E_{ABCD} \equiv \frac{1}{2}\left( \Psi_{ABCD} +
\hat{\Psi}_{ABCD}\right), \quad B_{ABCD}\equiv
\frac{\mbox{i}}{2}\left(\hat{\Psi}_{ABCD} - \Psi_{ABCD} \right),$$ so that $$\Psi_{ABCD} = E_{ABCD} + \mbox{i}B_{ABCD}.$$ The spinorial Bianchi identity $\nabla^{AA'}\Psi_{ABCD}=0$ can be split using the space spinor formalism to render
$$\begin{aligned}
&& \nabla\Psi_{ABCD}=2\nabla^E{}_A\Psi_{BCDE}, \label{Bianchi1}\\
&& \nabla^{AB}\Psi_{ABCD}=0. \label{Bianchi2}\end{aligned}$$
Crucial for our applications is that the spinors $E_{ABCD}$ and $B_{ABCD}$ can be expressed in terms of quantities intrinsic to a hypersurface $\mathcal{S}$. More precisely, if $\Omega_{AB}=0$, one has that
$$\begin{aligned}
&& E_{ABCD}= -r_{(ABCD)} + \tfrac{1}{2}\Omega_{(AB}{}^{PQ}\Omega_{CD)PQ}
- \tfrac{1}{6}\Omega_{ABCD}K, \label{Weyl:Electric}\\
&& B_{ABCD}=-\mbox{i}\ D^Q{}_{(A}\Omega_{BCD)Q}, \label{Weyl:Magnetic}\end{aligned}$$
where $r_{ABCD}$ is the space spinor counterpart of the Ricci tensor of the intrinsic metric of the hypersurface $\mathcal{S}$.
Space spinor expressions in Cartesian coordinates
-------------------------------------------------
In some occasions it will be necessary to give spinorial expressions in terms of Cartesian or asymptotically Cartesian frames and coordinates. For this we make use of the spatial Infeld-van der Waerden symbols $\sigma^{i}{}_{\mathbf{A}\mathbf{B}}$, $\sigma_{i}{}^{\mathbf{A}\mathbf{B}}$. Given $x^{i}, \; \xi_{i}\in
{\mbox{\SYM R}}^3$ we shall follow the convention that $$x^{\mathbf{AB}} \equiv
\sigma_{i}{}^{\mathbf{A}\mathbf{B}} x^{i},
\quad \xi_{\mathbf{AB}} \equiv
\sigma^{i}{}_{\mathbf{A}\mathbf{B}} \xi_{i},$$ with $$x^{\mathbf{AB}}= \frac{1}{\sqrt{2}}
\left(
\begin{array}{cc}
-x^1 + \mbox{i}x^2 & x^3 \\
x^3 & x^1 +\mbox{i} x^2
\end{array}
\right),
\quad
\xi_{\mathbf{AB}}= \frac{1}{\sqrt{2}}
\left(
\begin{array}{cc}
-\xi_1 - \mbox{i}\xi_2 & \xi_3 \\
\xi_3 & \xi_1 -\mbox{i}\xi_2
\end{array}
\right).
\label{CartesianSpinor}$$
Killing spinor data {#Section:KSD}
===================
In this section we review some aspects of the space spinor decomposition of the Killing spinor equation . A first analysis along these lines was first carried out in [@GarVal08a]. The current presentation is geared towards the construction of geometric invariants.
General observations
--------------------
Given a symmetric spinor $\kappa_{AB}$ (not necessarily a Killing spinor), it will be convenient to define the following spinors:
$$\begin{aligned}
\xi &\equiv \nabla^{PQ}\kappa_{PQ},\label{xi_sen_1} \\
\xi_{BF} &\equiv \frac{3}{2}\nabla_{(F}{}^{D}\kappa_{B)D},\label{xi_sen_2}\\
\xi_{ABCD} &\equiv \nabla_{(AB}\kappa_{CD)}\label{xi_sen_3},\\
\xi_{AA'} &\equiv \nabla^B{}_{A'}\kappa_{AB},\\
H_{A'ABC} &\equiv 3 \nabla_{A'(A}\kappa_{BC)},\\
S_{AA'BB'} &\equiv \nabla_{AA'}\xi_{BB'} + \nabla_{BB'}\xi_{AA'}.\end{aligned}$$
We will use this notation throughout the rest of the paper. Clearly, for a Killing spinor one has $$H_{A'ABC}=0, \quad S_{AA'BB'}=0.$$ The spinors $\xi$, $\xi_{AB}$ and $\xi_{ABCD}$ arise in the space spinor decomposition of the spinors $H_{A'ABC}$ and $\xi_{AA'}$. To see this, let $\tau^{AA'}$ denote, as in section \[Section:SpaceSpinors\], the spinorial counterpart of a timelike vector with normalisation $\tau_{AA'}\tau^{AA'}=2$. Some manipulations show that
$$\begin{aligned}
&&
\xi_{AA'}=\tfrac{1}{2}\tau_{AA'}\xi-\tfrac{2}{3}\tau^B{}_{A'}\xi_{AB}+\tfrac{1}{2}\tau^B{}_{A'}\nabla\kappa_{AB},
\label{Split:xi}\\
&&
H_{A'ABC}=\tau_{A'(A}\xi_{BC)}+\tfrac{3}{2}\tau_{A'(A}\nabla\kappa_{BC)}-3\tau_{A'}{}^D\xi_{ABCD}. \label{Split:H}\end{aligned}$$
Furthermore, the spinors $\xi$, $\xi_{AB}$ and $\xi_{ABCD}$ correspond to the irreducible components of $\nabla_{AB}\kappa_{CD}$ so that one can write: $$\label{SenDiffKappaSplit}
\nabla_{AB}\kappa_{CD}=\xi_{ABCD}-\tfrac{1}{3}\epsilon_{A(C}\xi_{D)B}-\tfrac{1}{3}\epsilon_{B(C}\xi_{D)A}-\tfrac{1}{3}\epsilon_{A(C}\epsilon_{D)B}\xi.$$
Using the commutator for vacuum one can obtain equations for the derivatives of $\xi$ and $\xi_{AB}$ —these will be used systematically in the sequel. The irreducible components of the derivative $\nabla_{AB}\xi_{CD}$ are given by:
$$\begin{aligned}
\nabla^{AB}\xi_{AB}={}&
-\tfrac{1}{2}K\xi+\tfrac{3}{4}\Omega^{ABCD}\xi_{ABCD}
+\tfrac{1}{2}\Omega^{AB}\xi_{AB}
-\tfrac{3}{4}\Omega^{AB}\nabla\kappa_{AB}, \label{Dxi1}\\
\nabla^C{}_{(A}\xi_{B)C}={}&
\nabla_{AB}\xi
+\tfrac{3}{2}\Psi_{ABCD}\kappa^{CD}
-\tfrac{2}{3}K\xi_{AB}
-\tfrac{1}{2}\Omega_{ABCD}\xi^{CD}
-\tfrac{3}{2}\xi_{(A}{}^{CDF}\Omega_{B)CDF}\nonumber\\
&-\tfrac{3}{2}\nabla^{CD}\xi_{ABCD}
-\tfrac{1}{2}\Omega_{AB}\xi
+\tfrac{1}{2}\Omega_{(A}{}^C\xi_{B)C}
+\tfrac{3}{4}\Omega^{CD}\xi_{ABCD}
-\tfrac{3}{2}\Omega_{(A}{}^C\nabla\kappa_{B)C},
\label{Dxi2}\\
\nabla_{(AB}\xi_{CD)}={}&
3\Psi_{F(ABC}\kappa_{D)}{}^F
+K\xi_{ABCD}
-\tfrac{1}{2}\Omega_{ABCD}\xi
+\Omega_{(ABC}{}^F\xi_{D)F}
-\tfrac{3}{2}\Omega^{PQ}{}_{(AB}\xi_{CD)PQ} \nonumber \\
&+3\nabla^Q{}_{(A}\xi_{BCD)Q}
+\tfrac{1}{2}\Omega_{(AB}\xi_{CD)}
-\tfrac{3}{2}\Omega^F{}_{(A}\xi_{BCD)F}
+\tfrac{3}{2}\Omega_{(AB}\nabla\kappa_{CD)}.
\label{Dxi3}\end{aligned}$$
We note the appearance of the term $\nabla_{AB}\xi$ in . Thus, there is no independent equation for the derivative of $\xi$.
Finally, we consider the equations for the second order derivatives of $\xi$. For the sake of simplicity, we restrict our attention to the case when $\Omega_{AB}=0$ so that $K_{ABCD}=K_{CDAB}$. For notational purposes we define $\Omega_{ABCDEF}\equiv\nabla_{(AB}\Omega_{CDEF)}$. One finds:
$$\begin{aligned}
\nabla^{AB} \nabla_{AB} \xi ={}&
-\tfrac{1}{6}K^2\xi
-\tfrac{1}{2}\Omega^{ABCD}\Omega_{ABCD}\xi
+3\Psi_A{}^{CDF}\Omega_{BCDF}\kappa^{AB}
+\xi_{AB}\nabla^{AB}K\nonumber\\
&+\tfrac{3}{4}\hat\Psi^{ABCD}\xi_{ABCD}
-\tfrac{9}{4}\Psi^{ABCD}\xi_{ABCD}
+2 K \Omega^{ABCD}\xi_{ABCD}\nonumber\\
&-\tfrac{15}{4}\Omega^{ABFH}\Omega^{CD}{}_{FH}\xi_{ABCD}
+\tfrac{9}{2}\Omega^{ABCD}\nabla^F{}_D\xi_{ABCF} \nonumber \\
&+\tfrac{3}{2}\nabla^{AB}\nabla^{CD}\xi_{ABCD},\label{nabla2xi0a}\\
\nabla^C{}_{(A}\nabla_{B)C}\xi ={}
&\tfrac{1}{2}\Omega_{ABCD}\nabla^{CD}\xi
-\tfrac{1}{3}K \nabla_{AB}\xi, \label{nabla2xi0b}\\
\nabla_{(AB}\nabla_{CD)}\xi={}&
-4 K \Psi_{(ABC}{}^E \kappa_{D)E}
+\tfrac{1}{2} \hat\Psi_{ABCD}\xi
-\tfrac{5}{2} \Psi_{ABCD}\xi
-\tfrac{2}{3} \hat\Psi_{(ABC}{}^E \xi_{D)E} \nonumber\\
& -\tfrac{10}{3} \Psi_{(ABC}{}^E \xi_{D)E}
+\Omega_{ABCDEL} \xi^{EL}
+\tfrac{4}{3} K{}^2 \xi_{ABCD}
+3 \Omega_{EFL(ABC}\xi_{D)}{}^{ELF}\nonumber\\
&+3 \Psi_{(AB}{}^{EL} \xi_{CD)EL}
-\tfrac{3}{2}\xi_{(A}{}^{ELF}\Omega_{BCD)}{}^H\Omega_{ELFH}
-3 \Psi_{EL(A}{}^F \kappa ^{EL} \Omega_{BCD)F}\nonumber\\
&-\xi^{EL} \Omega_{ELF(A} \Omega_{BCD)}{}^F
+\tfrac{2}{3}K\xi_{(A}{}^E\Omega_{BCD)E}
+\tfrac{1}{2} \xi ^{ELFH} \Omega_{EL(AB} \Omega_{CD)FH}\nonumber\\
&-3 \Psi_{E(B}{}^{LF} \kappa_A{}^E \Omega_{CD)LF}
-3 \Psi_{E(AB}{}^F \kappa^{EL}\Omega_{CD)LF}
-\Omega_{ELF(B}\xi_A{}^E \Omega_{CD)}{}^{LF}\nonumber\\
&-4 K \xi_{(AB}{}^{EL} \Omega_{CD)EL}
-\tfrac{1}{2}\xi\Omega_{(AB}{}^{EL}\Omega_{CD)EL}
+\tfrac{3}{2} \xi ^{ELFH} \Omega_{E(ABC} \Omega_{D)LFH}\nonumber\\
&-2 \Omega_{E(BC}{}^H \xi_A{}^{ELF} \Omega_{D)LFH}
+\tfrac{1}{4}\xi^{ELFH}\Omega_{ABCD}\Omega_{ELFH}
-\tfrac{1}{3} K \xi \Omega_{ABCD}\nonumber \\
&+\tfrac{1}{2}\xi_{(AB}{}^{EL}\Omega_{CD)}{}^{FH}\Omega_{ELFH}
+\tfrac{2}{5} \xi_{(CD} \nabla _{AB)}K +\tfrac{12}{5} \xi_{E(BCD} \nabla_{A)}{}^EK \nonumber\\
&-3 \Omega_{E(BCD} \nabla_{A)}{}^E\xi
-\tfrac{3}{2}\Omega_{(A}{}^{ELF}\nabla_{CD}\xi_{B)ELF}
-\tfrac{3}{2} \Omega_{F(A}{}^{EL} \nabla_D{}^F\xi_{BC)EL}\nonumber\\
&-\tfrac{9}{2} \Omega_{(AB}{}^{EL} \nabla_D{}^F\xi_{C)ELF}
-\tfrac{9}{2} \nabla_{L(D}\nabla_C{}^E\xi_{AB)E}{}^L
-\tfrac{3}{2} \nabla_{L(D}\nabla^{EL}\xi_{ABC)E}\nonumber\\
&-6 K \nabla_{E(D}\xi_{ABC)}{}^E
+3 \Omega_{L(AB}{}^E \nabla^{LF}\xi_{CD)EF}
-3 \Omega_{(ABC}{}^E \nabla^{LF}\xi_{D)ELF}\nonumber\\
&-3\kappa^{EL}\nabla_{L(D}\Psi_{ABC)E}
+3\kappa_{(A}{}^E \nabla_D{}^L\Psi_{BC)EL}.\label{nabla2xi0c}\end{aligned}$$
The equations presented in this section have been deduced using the tensor algebra suite [xAct]{} for [Mathematica]{} —see [@xAct].
Propagation of the Killing spinor equation
------------------------------------------
A straightforward consequence of the Killing spinor equation in a vacuum spacetime is that: $$\label{boxkappa}
\square\kappa_{AB}=-\Psi_{ABCD}\kappa^{CD},$$ where $\square \equiv \nabla^{AA'}\nabla_{AA'}$. The latter equation is obtained by applying the differential operator $\nabla^{AA'}$ to equation and then using the vacuum commutator relation for the spacetime Levi-Civita connection.
The wave equation plays a role in the discussion of the *propagation* of the Killing spinor equation. More precisely, one has the following result —cfr. [@GarVal08a] for further details.
Let $\kappa_{AB}$ be a solution to equation . Then the corresponding spinor fields $H_{A'ABC}$ and $S_{AA'BB'}$ will satisfy the system of wave equations
$$\begin{aligned}
&& \square H_{A'ABC}= 4\left(\Psi_{(AB}{}^{PQ} H_{C)PQA'} + \nabla_{(A}{}^{Q'}S_{BC)Q'A'}\right), \label{wave1} \\
&& \square S_{AA'BB'} = -\nabla_{AA'} \left( \Psi_B{}^{PQR}H_{B'PQR} \right)-\nabla_{BB'}\left( \Psi_A{}^{PQR}H_{A'PQR}\right) \nonumber \\
&& \hspace{4cm} + 2\Psi_{AB}{}^{PQ}S_{PA'QB'} + 2 \bar{\Psi}_{A'B'}{}^{P'Q'}S_{AP'BQ'}. \label{wave2}\end{aligned}$$
The crucial observation is that the right hand sides of equations and are homogeneous expressions of the unknowns and their first order derivatives. The hyperbolicity of equations and imply the following result —again, cfr. [@GarVal08a] for further details.
\[Proposition:KSDevelopment\] The development $(\mathcal{M},g_{\mu\nu})$ of an initial data set for the vacuum Einstein field equations, $(\mathcal{S},h_{ab},K_{ab})$, has a Killing spinor in the domain of dependence of $\mathcal{U}\subset\mathcal{S}$ if and only if the following equations are satisfied on $\mathcal{U}$.
$$\begin{aligned}
&& H_{A'ABC}=0, \label{old_kspd1}\\
&& \nabla H_{A'ABC}=0, \label{old_kspd2}\\
&& S_{AA'BB'}=0, \label{old_kspd3}\\
&& \nabla S_{AA'BB'}=0. \label{old_kspd4}\end{aligned}$$
The Killing spinor data equations
---------------------------------
The *Killing spinor data* conditions obtained in Proposition \[Proposition:KSDevelopment\] can be reexpressed in terms of conditions on the spinor $\kappa_{AB}$ which are intrinsic to the hypersurface $\mathcal{S}$. For this one uses the split of $\xi_{AA'}$ and $H_{A'ABC}$ given by equations -. Extensive computations using the [xAct]{} suite for [Mathematica]{} render the following result.
\[Theorem:KSData\] Let $(\mathcal{S},h_{ab},K_{ab})$ be an initial data set for the Einstein vacuum field equations, where $\mathcal{S}$ is a Cauchy hypersurface. Let $\mathcal{U}\subset\mathcal{S}$ be an open set. The development of the initial data set will then have a Killing spinor in the domain of dependence of $\mathcal{U}$ if and only if
$$\begin{aligned}
&& \xi_{ABCD}=0,\label{kspd1}\\
&& \Psi_{(ABC}{}^F\kappa_{D)F}=0, \label{kspd2}\\
&& 3\kappa_{(A}{}^E\nabla_B{}^F\Psi_{CD)EF}+\Psi_{(ABC}{}^F\xi_{D)F}=0,\label{kspd3}\end{aligned}$$
are satisfied on $\mathcal{U}$. The Killing spinor is obtained by evolving with initial data satisfying conditions - and $$\nabla\kappa_{AB}=-\tfrac{2}{3}\xi_{AB} \label{kspd4}$$ on $\mathcal{U}$.
**Remark 1.** Conditions - are intrinsic to $\mathcal{U}\subset \mathcal{S}$ and will be referred to as the *Killing spinor initial data equations*. In particular, equation , which can be written as $$\label{SpatialKillingSpinorEquation}
\nabla_{(AB}\kappa_{CD)}=0,$$ will be called the *spatial Killing spinor equation*, whereas and will be known as the *algebraic conditions*.
**Remark 2.** Theorem \[Theorem:KSData\] is an improvement on Proposition 6 of [@GarVal08a] where the interdependence of the equations implied by - was not analysed.
The proof of Theorem \[Theorem:KSData\] consists of a space spinor decomposition of the conditions - and of an analysis of the dependencies of the resulting conditions. All calculations are made on $\mathcal{U}\subset\mathcal{S}$.
- *Decomposition of equation .* Splitting $\tau_F{}^{A'}H_{A'ABC}$ into irreducible parts gives that is equivalent to
$$\begin{aligned}
&&\xi_{ABCD}=0,\label{xi4vanishing}\\
&&\nabla\kappa_{AB}=-\tfrac{2}{3}\xi_{AB}\label{nablakappa}.\end{aligned}$$
- *Decomposition of equation .* It follows that $$\tau_D{}^{A'}\nabla H_{A'ABC}=\nabla(\tau_D{}^{A'}
H_{A'ABC})+H_{A'ABC}K_{DF}\tau^{FA'}.$$ Hence, under the condition , the irreducible parts of $\tau_D{}^{A'}\nabla H_{A'ABC}$ are given by
$$\begin{aligned}
&& \nabla\xi_{ABCD}=0,\label{nablaxi4}\\
&& \nabla^2\kappa_{AB}=-\tfrac{2}{3}\nabla\xi_{AB}.\label{nabla2kappa}\end{aligned}$$
From the commutator together with and we get $$\begin{aligned}
\nabla\xi_{ABCD}={}&\nabla\nabla_{(AB}\kappa_{CD)}\\
={}&2\Psi_{(ABC}{}^F\kappa_{D)F}-\tfrac{1}{3}\Omega_{(AB}\xi_{CD)}-\tfrac{1}{3}\Omega_{ABCD}\xi+\tfrac{2}{3}\Omega_{(ABC}{}^F\xi_{D)F}-\tfrac{2}{3}\nabla_{(AB}\xi_{CD)}.\end{aligned}$$ Equation and again and then yield $$\nabla\xi_{ABCD}=4\Psi_{(ABC}{}^F\kappa_{D)F}.$$ Using the commutator one obtains that
$$\begin{aligned}
\nabla\xi={}&
\nabla_{AB}\nabla\kappa^{AB}
-\tfrac{1}{3}K\xi
+\tfrac{2}{3}K^{AB}\xi_{AB}
+\tfrac{2}{3}\Omega^{AB}\xi_{AB}
-\Omega^{ABCD}\xi_{ABCD}
-\tfrac{1}{2}K^{AB}\nabla\kappa_{AB}
\label{nablaxi0a}\\
\nabla\xi_{AB}={}&
\tfrac{3}{2}\Psi_{ABCD}\kappa^{CD}
-\tfrac{1}{2}K_{AB}\xi
-\tfrac{1}{3}K\xi_{AB}
+\tfrac{1}{2}K^C{}_{(A}\xi_{B)C}
+\tfrac{3}{4}K^{CD}\xi_{ABCD}
-\tfrac{1}{2}\xi\Omega_{AB} \nonumber \\
&-\tfrac{1}{2}\xi^C{}_{(A}\Omega_{B)C}
+\tfrac{3}{4}\Omega^{CD}\xi_{ABCD}
+\tfrac{3}{2}\xi_{(A}{}^{CDF}\Omega_{B)CDF}
+\tfrac{1}{2}\xi^{CD}\Omega_{ABCD} \nonumber \\
&-\tfrac{3}{4}K^C{}_{(A}\nabla\kappa_{B)C}
+\nabla_{C(A}\nabla\kappa_{B)}{}^C\label{nablaxi2a}\end{aligned}$$
In terms of the normal derivative and the Sen connection, equation reads $$\begin{aligned}
\nabla^2\kappa_{AB}={}&
-2\Psi_{ABCD}\kappa^{CD}
-K\nabla\kappa_{AB}
-\tfrac{2}{3}\nabla_{AB}\xi
-\tfrac{4}{3}\nabla_{C(A}\xi^C{}_{B)}
-2\nabla^{CD}\xi_{ABCD} \nonumber \\
&+\tfrac{1}{3}K_{AB}\xi -\tfrac{2}{3}K^C{}_{(A}\xi_{B)C}
+K^{CD}\xi_{ABCD}
+\tfrac{2}{3}\Omega_{AB}\xi
+\tfrac{4}{3}\xi^C{}_{(A}\Omega_{B)C} \nonumber \\
&+2\xi_{ABCD}\Omega^{CD}.\label{boxkappasplit}\end{aligned}$$ It is worth stressing that equations , and are valid not only on $\mathcal{U}$, but on the spacetime. Hence, it makes sense taking normal derivatives of these equations. Using , and , the wave equation is seen to imply $$\begin{aligned}
\nabla^2\kappa_{AB}+\tfrac{2}{3}\nabla\xi_{AB}={}&
-\Psi_{ABCD}\kappa^{CD}
+\tfrac{4}{9}K\xi_{AB}
+\tfrac{1}{3}\Omega_{AB}\xi
+\xi^C{}_{(A}\Omega_{B)C}\nonumber \\
&+\tfrac{1}{3}\Omega_{ABCD}\xi^{CD}
-\tfrac{2}{3}\nabla_{AB}\xi
-\tfrac{2}{3}\nabla_{C(A}\xi^C{}_{B)}.\end{aligned}$$ Using equations , , , the latter equation reduces to . This far we have that for all solutions to , the system , is equivalent to the system , , .
- *Decomposition of equation .* Splitting $\tau_C{}^{A'}\tau_D{}^{B'}S_{AA'BB'}$ into irreducible parts yields
$$\begin{aligned}
&\nabla_{(AB}\nabla\kappa_{CD)}-\Omega_{ABCD}\xi+\tfrac{4}{3}K_{(ABC}{}^F\xi_{D)F}-K_{(ABC}{}^F\nabla\kappa_{D)F}-\tfrac{4}{3}\nabla_{(AB}\xi_{CD)}=0,\label{S0Full1}\\
&2\nabla\xi-\tfrac{4}{3}K^{AB}\xi_{AB}+K^{AB}\nabla\kappa_{AB}=0, \label{S0Full2}\\
&\tfrac{4}{3}\nabla_{AB}\xi^{AB}+K\xi-\tfrac{4}{3}\Omega^{AB}\xi_{AB}+\Omega^{AB}\nabla\kappa_{AB}-\nabla_{AB}\nabla\kappa^{AB}=0, \label{S0Full3}
\\
&\tfrac{1}{2}K_{BD}\xi
-\tfrac{2}{3}K^A{}_{(B}\xi_{D)A}
+\tfrac{1}{2}K^A{}_{(B}\nabla\kappa_{D)A}
-\tfrac{2}{3}K_{BDAC}\xi^{AC}
+\tfrac{1}{2}K_{BDAC}\nabla\kappa^{AC}\nonumber\\
&+\tfrac{2}{3}\nabla\xi_{BD}
-\tfrac{1}{2}\nabla^2\kappa_{BD}
+\nabla_{BD}\xi =0. \label{S0Full4}\end{aligned}$$
Using equations , , , and , one sees that equations - simplify to
$$\begin{aligned}
&& \Psi^F{}_{(ABC}\kappa_{D)F}=0,\\
&& \nabla\xi=K^{AB}\xi_{AB},\label{nablaxi0}\\
&& \nabla\xi_{BD}=-\tfrac{1}{2}K_{BD}\xi
+K^A{}_{(B}\xi_{D)A}
+K_{BDAC}\xi^{AC}
-\nabla_{BD}\xi,\label{nablaxi2}\end{aligned}$$
while equation is seen to be satisfied identically. Furthermore, employing equations , , , , and one obtains equation and . Hence, they are a consequence of the commutators, and . One concludes that for all solutions to , the equations , together with are equivalent to , , .
- *Decomposition of equation* . A straightforward computation shows that $$\begin{aligned}
&& \tau_C{}^{A'}\tau_D{}^{B'}\nabla S_{AA'BB'} \\
&& \hspace{1cm} =\nabla(\tau_C{}^{A'}\tau_D{}^{B'}S_{AA'BB'})+K_{CF}S_{AA'BB'}\tau_D{}^{B'}\tau^{FA'}+K_{DF}S_{AA'BB'}\tau_C{}^{A'}\tau^{FB'}.\end{aligned}$$ Hence, if condition holds, the irreducible parts of $\tau_C{}^{A'}\negthinspace\tau_D{}^{B'}\nabla S_{AA'BB'}$ are $\nabla$-derivatives of -. Using equation , these components become
$$\begin{aligned}
&&
\Omega_{ABCD}\nabla\xi
+\Omega_{(AB}\nabla\xi_{CD)}
-2\Omega^F{}_{(ABC}\nabla\xi_{D)F}
+\tfrac{2}{3}\xi_{(AB}\nabla\Omega_{CD)}\nonumber\\
&&\hspace{1cm}-\tfrac{1}{2}\nabla\kappa_{(AB}\nabla\Omega_{CD)}
+\xi\nabla\Omega_{ABCD}
+\tfrac{4}{3}\xi^F{}_{(A}\nabla\Omega_{BCD)F}\nonumber\\
&&\hspace{1cm}-(\nabla\kappa^F{}_{(A})\nabla\Omega_{BCD)F}
+\tfrac{4}{3}\nabla\nabla_{(AB}\xi_{CD)}
-\nabla\nabla_{(AB}\nabla\kappa_{CD)}=0, \label{nablaS0Full1}\\
&&2\nabla^2\xi-2K^{AB}\nabla\xi_{AB}+(\nabla K^{AB})\nabla\kappa_{AB}-\tfrac{4}{3}\xi^{AB}\nabla K_{AB}=0, \label{nablaS0Full2}\\
&&\xi\nabla K
+K\nabla\xi
-2\Omega^{AB}\nabla\xi_{AB}
-\tfrac{4}{3}\xi^{AB}\nabla\Omega_{AB}
+(\nabla\kappa^{AB})\nabla\Omega_{AB}\nonumber\\
&&\hspace{1cm}+\tfrac{4}{3}\nabla\nabla_{AB}\xi^{AB}
-\nabla\nabla_{AB}\nabla\kappa^{AB}=0, \label{nablaS0Full3}\\
&&\nabla^3\kappa_{BD}+\tfrac{2}{3}\nabla^2\xi_{BD}
=\tfrac{4}{3}\xi^A{}_{(B}\nabla\kappa_{D)A}
+\xi\nabla\kappa_{BD}
-\tfrac{4}{3}\xi^{AC}\nabla K_{BDAC} \nonumber \\
&& \hspace{1cm}+(\nabla K^A{}_{(B})\nabla\kappa_{D)A}+(\nabla K_{BDAC})\nabla\kappa^{AC}
+K_{BD}\nabla\xi -2 K^A{}_{(B}\nabla\xi_{D)A}\nonumber \\
&& \hspace{1cm}-2 K_{BDAC}\nabla^{AC}+2\nabla^2\xi_{BD}
+2\nabla\nabla_{BD}\xi. \label{nablaS0Full4}\end{aligned}$$
Now, using the commutator , and equations and it is easy so see that $$\label{nablaDdnablakappa}
\nabla\nabla_{AB}\nabla\kappa_{CD}=-\tfrac{2}{3}\nabla\nabla_{AB}\xi_{CD}.$$ Taking the normal derivative of the spacetime equations - and using the relations , , , , , and one gets $$\begin{aligned}
\nabla^2\xi={}&\xi^{AB}\nabla K_{AB}+K^{AB}\nabla\xi_{AB},\\
\nabla^2\xi_{AB}={}&
-\tfrac{1}{2}\xi\nabla K_{AB}
-\xi^C{}_{(A}\nabla K_{B)C}
+\tfrac{1}{3}\xi_{AB}\nabla K
-\tfrac{1}{2}K_{AB}\nabla\xi
+\tfrac{1}{3}K \nabla\xi_{AB} \nonumber \\
&+K^C{}_{(A}\nabla\xi_{B)C}-\Omega^C{}_{(A}\nabla\xi_{B)C}
+\Omega_{ABCD}\nabla\xi^{CD}
+\xi^C{}_{(A}\nabla\Omega_{B)C} \nonumber \\
&+\xi^{CD}\nabla\Omega_{ABCD}
-\nabla\nabla_{AB}\xi.\end{aligned}$$ Using these last two equations together with equations , , , , , and one finds that the system - reduces to
$$\begin{aligned}
&&
4\Psi^F{}_{(ABC}\xi_{D)F}+6\kappa^F{}_{(A}\nabla\Psi_{BCD)F}\label{TempEqnablaPsi}=0, \\
&&\nabla^3\kappa_{BD}+\tfrac{2}{3}\nabla^2\xi_{BD}=0.\label{nabla3kappa}\end{aligned}$$
Taking the normal derivative of equation and using equations , , , , and one gets equation . Finally, using the Bianchi equation , one has that equation reduces to $$3\kappa_{(A}{}^E\nabla_B{}^F\Psi_{CD)EF}
+\Psi_{(ABC}{}^F\xi_{D)F}=0$$
This completes the proof.
**Remark.** Note that the result is independent of $K_{AB}$ and $\Omega_{AB}$.
### The Killing spinor initial data conditions in terms of the Levi-Civita connection
It should be stressed that the Killing spinor equations - are truly intrinsic to the hypersurface $\mathcal{S}$. This can be more easily seen by expressing the Sen connection, $\nabla_{AB}$, in terms of the intrinsic (Levi-Civita) connection of the hypersurface, $D_{AB}$, and the second fundamental form $K_{ABCD}$. One obtains the following completely equivalent set of equations: $$\begin{aligned}
&&D_{(AB}\kappa_{CD)}+\Omega_{(ABC}{}^E\kappa_{D)E}=0,\\
&&\Psi_{(ABC}{}^F\kappa_{D)F}=0,\\
&&3\kappa_{(A}{}^E D_B{}^F\Psi_{CD)EF}
-\tfrac{3}{4}\Psi_{L(ABC}D^{HL}\kappa_{D)H}
-\tfrac{3}{4}\Psi_{L(ABC}D_{D)}{}^F\kappa^L{}_F\nonumber\\
&&\hspace{1cm}+\tfrac{3}{4}\Psi_{(ABC}{}^L\Omega_{D)FHL}\kappa^{FH}
+\tfrac{3}{2}\Psi_{(AB}{}^{HL}\kappa_{C}{}^{F}\Omega_{D)FHL}
-\tfrac{3}{2}\Psi_{FH(A}{}^{L}\Omega_{BCD)L}\kappa^{FH}\nonumber\\
&&\hspace{1cm}+\tfrac{3}{8}\Psi_{FH(AB}\kappa_{CD)}\Omega^{FH}
+\tfrac{3}{4}\Psi_{FH(AB}\Omega_{CD)}\kappa^{FH}=0,\end{aligned}$$ where the last expression was simplified using the first algebraic condition, and the value of the Weyl spinor is expressed in terms of initial data quantities via formulae -.
The integrability conditions of the spatial Killing spinor equation
-------------------------------------------------------------------
For the rest of the paper we assume that the tensor $K_{ab}$ is symmetric —accordingly, $\Omega_{AB}=0$. The condition $\xi_{ABCD}\equiv\nabla_{(AB}\kappa_{CD)}=0$ does not immediately give information about the other irreducible components of $\nabla_{AB}\kappa_{CD}$, namely $\xi$ and $\xi_{AB}$. However, using $\xi_{ABCD}=0$ and $\Omega_{AB}=0$ in the relations - one finds that $\nabla_{AB}\xi_{CD}$ can be written in terms of $\nabla_{AB}\xi$ and lower order derivatives of $\kappa_{AB}$. Furthermore, using $\xi_{ABCD}=0$ in the relations -, we see that the second order derivatives of $\xi$ can be expressed in terms of lower order derivatives of $\kappa_{AB}$. This yields the following result which will play a role in the sequel:
\[Lemma:ThirdDerivative\] Assume that $\nabla_{(AB}\kappa_{CD)}=0$, then $$\nabla_{AB}\nabla_{CD} \nabla_{EF} \kappa_{GH} = H_{ABCDEFGH},$$ where $H_{ABCDEFGH}$ is a linear combination of $\kappa_{AB}$, $\nabla_{AB}\kappa_{CD}$ and $\nabla_{AB}\nabla_{CD} \kappa_{EF}$ with coefficients depending on $\Psi_{ABCD}$, $\hat{\Psi}_{ABCD}$ and $K_{ABCD}$.
**Remark.** It is important to point out that the assertion of the lemma is false if $\nabla_{(AB}\kappa_{CD)}\neq 0$.
The approximate Killing spinor equation {#Section:ApproximateKS}
=======================================
In what follows we will regard the spatial Killing spinor equation as the key condition of the Killing spinor initial data equations. Equation (\[kspd1\]) is an overdetermined condition for the 3 (complex) components of the spinor $\kappa_{AB}$: not every initial data set $(\mathcal{S},h_{ab},K_{ab})$ admits a solution. One would like to deduce a new equation which always has a solution and such that any solution to equation (\[kspd1\]) is also a solution to the new equation.
The approximate Killing spinor operator
---------------------------------------
Let $\mathfrak{S}_2$ and $\mathfrak{S}_4$ denote, respectively, the spaces of totally symmetric valence 2 and valence 4 spinors. Given $\zeta_{ABCD}, \; \chi_{ABCD}\in \mathfrak{S}_4$, we introduce an inner product in $\mathfrak{S}_4$ via: $$\langle \zeta_{ABCD}, \chi_{EFGH}\rangle = \int_{\mathcal{S}} \zeta_{ABCD} \hat{\chi}^{ABCD} \mbox{d}\mu,$$ where $\mbox{d}\mu$ denotes the volume form of the 3-metric $h_{ab}$. We introduce the spatial Killing spinor operator $\Phi$ via $$\Phi: \mathfrak{S}_2 \rightarrow \mathfrak{S}_4, \quad \Phi(\kappa)_{ABCD}= \nabla_{(AB}\kappa_{CD)}.$$ Now, consider the pairing $$\begin{aligned}
&& \langle \nabla_{(AB}\kappa_{CD)}, \zeta_{EFGH} \rangle = \int_{\mathcal{S}} \nabla_{(AB}\kappa_{CD)} \hat{\zeta}^{ABCD} \mbox{d}\mu \\
&& \phantom{\langle \nabla_{(AB}\kappa_{CD)}, \zeta_{EFGH} \rangle}= \int_{\mathcal{S}} \nabla_{AB}\kappa_{CD} \hat{\zeta}^{ABCD} \mbox{d}\mu.\end{aligned}$$ The formal adjoint, $\Phi^*$, of the spatial Killing operator can be obtained from the latter expression by integration by parts. To this end we note the identity: $$\begin{aligned}
&& \int_{\mathcal{U}} \nabla^{AB} \kappa^{CD} \hat{\zeta}_{ABCD} \mbox{d}\mu - \int_{\mathcal{U}} \kappa^{AB} \widehat{\nabla^{CD} \zeta_{ABCD}}\mbox{d}\mu + \int_{\mathcal{U}} 2\kappa^{AB} \Omega^{CDF}{}_A\hat\zeta_{BCDF}\mbox{d}\mu \nonumber \\
&& \hspace{2cm}= \int_{\partial \mathcal{U}} n^{AB} \kappa^{CD} \hat{\zeta}_{ABCD} \mbox{d}S, \label{IntegrationbyParts}\end{aligned}$$ with $\mathcal{U}\subset \mathcal{S}$, and where $\mbox{d}S$ denotes the area element of $\partial \mathcal{U}$, $n_{AB}$ is the spinorial counterpart of its outward pointing normal, and $\zeta_{ABCD}$ is a symmetric spinor. From it follows that $$\label{FormalAdjoint}
\Phi^*:\mathfrak{S}_4 \rightarrow \mathfrak{S}_2, \quad \Phi^*(\zeta)_{CD}=\nabla^{AB}\zeta_{ABCD}-2\Omega^{ABF}{}_{(C}\zeta_{D)ABF}.$$
The composition operator $L\equiv \Phi^*\circ \Phi: \mathfrak{S}_2\rightarrow \mathfrak{S}_2$ given by: $$L(\kappa_{CD}) \equiv \nabla^{AB} \nabla_{(AB} \kappa_{CD)}-\Omega^{ABF}{}_{(A}\nabla_{|DF|}\kappa_{B)C}-\Omega^{ABF}{}_{(A}\nabla_{B)F}\kappa_{CD}=0, \label{ApproximateKillingEquation}$$ will be called the *approximate Killing spinor operator*, and equation the *approximate Killing spinor equation*.
**Remark.** Note that every solution to the spatial Killing spinor equation is also a solution to equation .
Ellipticity of the approximate Killing spinor operator
------------------------------------------------------
As a prior step to the analysis of the solutions to the approximate Killing spinor equation , we look first at its ellipticity properties.
The operator $L$ defined by equation is a formally self-adjoint elliptic operator.
The operator is by construction formally self-adjoint as it is given by the composition of an operator and its formal adjoint. In order to verify ellipticity, it suffices to look at the operator $$L'(\kappa)_{\mathbf{CD}}\equiv \partial^{\mathbf{AB}} \partial_{(\mathbf{AB}} \kappa_{\mathbf{CD})},$$ corresponding to the principal part of $L$ in some Cartesian spin frame. In the corresponding Cartesian coordinates $(x^1,x^2,x^3)$ one has that $$\partial_{\mathbf{AB}}= \frac{1}{\sqrt{2}}
\left(
\begin{array}{cc}
-\partial_1 - \mbox{i}\partial_2 & \partial_3 \\
\partial_3 & \partial_1 -\mbox{i}\partial_2
\end{array}
\right),
\quad
\partial^{\mathbf{AB}}= \frac{1}{\sqrt{2}}
\left(
\begin{array}{cc}
-\partial_1 + \mbox{i}\partial_2 & \partial_3 \\
\partial_3 & \partial_1 +\mbox{i}\partial_2
\end{array}
\right).$$ In particular, $\partial^{\mathbf{AB}} \partial_{\mathbf{AB}}=\Delta\equiv \partial^2_1 +\partial^2_2 +\partial^2_3$, the flat Laplacian. One notes that $$\partial^{\mathbf{PQ}}\partial_{(\mathbf{PQ}} \kappa_{\mathbf{AB})} = \frac{1}{6} \partial^{\mathbf{PQ}}\partial_{\mathbf{PQ}} \kappa_{\mathbf{AB}}+ \frac{2}{3} \partial^{\mathbf{PQ}} \partial_{\mathbf{P}(\mathbf{A}} \kappa_{\mathbf{B})\mathbf{Q}} + \frac{1}{6} \partial^{\mathbf{PQ}} \partial_{\mathbf{AB}} \kappa_{\mathbf{PQ}}.$$ Now, writing $$\kappa_{0}\equiv\kappa_{00}, \quad \kappa_{1}\equiv\kappa_{01}, \quad \kappa_{2}\equiv\kappa_{11},$$ one has that $L'$ can be expressed in matricial form as $A^{ij} \partial_i \partial_j u$, where $$\label{matrixA}
A^{ij}\partial_i \partial_j \equiv
\frac{1}{12}
\left(
\begin{array}{cccccc}
7\Delta -\partial_3^2 & -2\partial_1 \partial_3 & \partial_2^2-\partial_1^2 & 0 & -2\partial_2 \partial_3 & -2\partial_1 \partial_2 \\
-\partial_1\partial_3 & 6\Delta+2\partial^2_3 & \partial_1 \partial_3 & \partial_2\partial_3 & 0 & \partial_2 \partial_3 \\
\partial_2^2-\partial_1^2 & 2\partial_1\partial_3 & 7\Delta-\partial^2_3 & 2 \partial_1\partial_2 & -2\partial_2\partial_3 & 0 \\
0 & 2\partial_2\partial_3 & 2\partial_1\partial_2 & 7\Delta-\partial_3^2 & -2\partial_1\partial_3 & \partial^2_2-\partial_1^2 \\
-\partial_2\partial_3 & 0 & -\partial_2\partial_3 & -\partial_1\partial_3 & 6\Delta+2\partial_3^2 & \partial_1\partial_3 \\
-2\partial_1\partial_2 & \partial_2\partial_3 & 0& \partial_2^2-\partial_1^2 & 2 \partial_1\partial_3 & 7\Delta-\partial_3^2
\end{array}
\right),$$ and $$\label{vectorU}
u\equiv
\left(
\begin{array}{c}
\mbox{Re}(\kappa_0) \\
\mbox{Re}(\kappa_1) \\
\mbox{Re}(\kappa_2) \\
\mbox{Im}(\kappa_0) \\
\mbox{Im}(\kappa_1) \\
\mbox{Im}(\kappa_2) \\
\end{array}
\right).$$ The symbol, $l(\xi_i)$, of the operator given by is then given by replacing $\partial_i$ with $\xi_i\in {\mbox{\SYM R}}^3$. One finds that $$\det l(\xi_i)=\frac{1}{36}\left(
(\xi_1)^2+(\xi_2)^2+(\xi_3)^2\right)^6,$$ so that $\det l(\xi_i)=0$ if and only if $\xi_i=0$. Accordingly, the operator $L=\Phi^*\circ\Phi$ is elliptic.
A variational formulation
-------------------------
We note that the approximate Killing spinor equation arises naturally from a variational principle.
The approximate Killing spinor equation is the Euler-Lagrange equation of the functional $$J = \int_{\mathcal{S}} \nabla_{(AB} \kappa_{CD)} \widehat{\nabla^{AB} \kappa^{CD}} \mbox{d}\mu. \label{functional}$$
This is a direct consequence of the identity .
Asymptotically Euclidean manifolds {#Section:AsymptoticallyEuclideanData}
==================================
After having studied some formal properties of the Killing spinor initial data equations -,, and the approximate Killing spinor equation , we proceed to analyse their solvability on asymptotically Euclidean manifolds. In order to do this we introduce some relevant terminology and ancillary results.
General assumptions
-------------------
In what follows, we will be concerned with vacuum spacetimes arising as the development of asymptotically Euclidean data sets. Let $(\mathcal{S},h_{ab},K_{ab})$, denote a smooth initial data set for the vacuum Einstein field equations. The pair $(h_{ab},K_{ab})$ satisfies on the 3-dimensional manifold $\mathcal{S}$ the vacuum constraint equations
$$\begin{aligned}
&& -2r - K^a{}_a K^b{}_b + K_{ab}K^{ab}=0, \label{Hamiltonian}\\
&& D^a K_{ab}- D_b K^a{}_a=0, \label{Momentum}\end{aligned}$$
where $r$ and $D$ denote, respectively, the Ricci scalar and the Levi-Civita connection of the negative definite 3-metric $h_{ab}$, while $K_{ab}$ corresponds to the extrinsic curvature of $\mathcal{S}$. The unusual coefficients in the formulae above come from our normalisation of $\tau^\mu$. For an *asymptotic end* it will be understood an open set diffeomorphic to the complement of a closed ball in ${\mbox{\SYM R}}^3$. In what follows, the 3-manifold $\mathcal{S}$ will be assumed to be the union of a compact set and two asymptotically Euclidean ends, $i_1, \;i_2$.
Weighted Sobolev norms
----------------------
In order to discuss the decays of the various fields on the 3-manifold $\mathcal{S}$ we make use of weighted Sobolev spaces. In what follows, we follow the ideas of [@Can81] written in terms of the conventions of [@Bar86]. Choose an arbitrary point $O\in \mathcal{S}$, and let $$\sigma(x) \equiv (1 +d(O,x)^2 )^{1/2},$$ where $d$ denotes the Riemannian distance function on $\mathcal{S}$. The function $\sigma$ is used to define the following weighted $L^2$ norm: $$\label{Definition:WeightedSobolev}
\Vert u\Vert_\delta \equiv \left(\int_{\mathcal{S}} |u|^2 \sigma^{-2\delta-3} \mbox{d}x\right)^{1/2},$$ for $\delta\in \mathbb{R}$. In particular, if $\delta=-3/2$ one recovers the usual $L^2$ norm. Different choices of origin give rise to equivalent weighted norms —as mentioned before, the convention of indices used in the definition of the norm follows the one of Bartnik [@Bar86]. The fall off conditions of the various fields will be expressed in terms of weighted Sobolev spaces $H^s_\delta$ consisting of functions for which the norm $$\Vert u\Vert_{s,\delta} \equiv \sum_{0\leq |\alpha| \leq s} \Vert
D^\alpha u\Vert_{\delta-|\alpha|} < \infty,$$ with $s$ a non-negative integer, and where $\alpha=(\alpha_1,\alpha_2,\alpha_3)$ is a multiindex, $|\alpha|=\alpha_1 +\alpha_2 +\alpha_3$. We say that $u\in
H^\infty_\delta$ if $u\in H^s_\delta$ for all $s$. We will say that a spinor or a tensor belongs to a function space if its norm does. For instance, the notation $\zeta_{AB}\in H^s_\delta$ is a short hand notation for $(\zeta_{AB}\hat\zeta^{AB}+\zeta_A{}^A\hat\zeta_B{}^B)^{1/2}\in
H^s_\delta$.
We will make use of the following result:
Let $u\in H^\infty_\delta$. Then $u$ is smooth (i.e. $C^\infty$) over $\mathcal{S}$ and has a fall off at infinity such that $D^l u = o(r^{\delta-|l|})$.
The smoothness of $u$ follows from the Sobolev embedding theorems. The proof of the behaviour at infinity of $u$ can be found in [@Bar86] —cfr. Theorem 1.2 (iv)— while the decay for the derivatives follows from the definition of the weighted Sobolev norms.
**Remark.** Here $r$ is a radial coordinate on the asymptotic end —see the next section for details.
We also note the following multiplication lemma —cfr. e.g. Theorem 5.6 in [@Can81].
\[Lemma:Multiplication\] Let $u\in H^\infty_{\delta_1}$, $v\in H^\infty_{\delta_2}$. Then $$u v \in H^\infty_{\delta_1+\delta_2+\varepsilon}, \quad \varepsilon>0.$$
**Notation.** We will often write $u=o_\infty(r^\delta)$ for $u\in H^\infty_\delta$ at an asymptotic end.
For the present applications we will require a somehow finer multiplication lemma concerning the behaviour at infinity. For this we exploit the fact that we are working with smooth functions. More precisely:
\[Lemma:FinerMultiplication\] Let $u=o_\infty(r^{\delta_1})$, $v=o_\infty(r^{\delta_2})$ and $w=O(r^\gamma)$. Then $$uv =o(r^{\delta_1+\delta_2}), \quad uw=o(r^{\delta_1+\gamma}).$$
Let $\partial \mathcal{S}_r$ denote the surfaces of constant $r$. For sufficiently large $r$ (so that one is in an asymptotic end), the surface $\partial \mathcal{S}_{r}$ has the topology of the 2-sphere. Now, the functions $u, \; v$ are continuous and the surfaces $\partial \mathcal{S}_{r}$ are compact. Therefore, for sufficiently large $r$ the functions $$f(r)\equiv \max_{\partial \mathcal{S}_r}|u r^{-\delta_1}|, \quad g(r)\equiv \max_{\partial \mathcal{S}_r}|v r^{-\delta_2}|,$$ are finite and well defined. Furthermore $r^{\delta_1}|u|\leq f(r)$, $r^{\delta_2}|v|\leq g(r)$. By construction, one has that $f(r)=o(1)$ and $g(r)=o(1)$ —that is, $f,\; g\rightarrow 0$ for $r\rightarrow
\infty$. One also has that $|w r^{-\gamma}|$ is bounded by a constant $C$. Hence, $$\begin{aligned}
&& |uv| \leq f(r)g(r) r^{\delta_1+\delta_2} = o(r^{\delta_1+\delta_2}), \\
&& |uw| \leq f(r) r^{\delta_1} |w| \leq Cf(r) r^{\delta_1+\gamma} =o(r^{\delta_1+\gamma}), \end{aligned}$$ from where the desired result follows.
**Remark.** The lemmas extend to symmetric spatial spinors with even number of indices by the Cauchy-Schwartz inequality.
Decay assumptions
-----------------
As mentioned before, our analysis will be restricted to initial data sets $(\mathcal{S},h_{ab},K_{ab})$ with 2 asymptotic ends. Without loss of generality one of the ends will be denoted by the subscript/superscript $+$ on the relevant objects, while those of the other end by $-$. Often, when no confusion arises the subscript/superscript will be dropped.
**Remark.** We do not need to assume any topological restriction apart from paracompactness, orientability and the requirement of 2 asymptotically flat ends. Hence, we can have an arbitrary number of handles. For black holes, this means that we can handle Misner-type data with several black holes [@Mis63].
The standard assumption for asymptotic flatness is that on each end it is possible to introduce asymptotically Cartesian coordinates $x^i_\pm$ with $r=((x^1_\pm)^2 +
(x^2_\pm)^2 + (x^3_\pm)^2)^{1/2}$, such that the intrinsic metric and extrinsic curvature of $\mathcal{S}$ satisfy
$$\begin{aligned}
&& h_{ij} = -\delta_{ij}+o_\infty(r^{-1/2}), \label{decay1} \\
&& K_{ij} = o_\infty(r^{-3/2}) \label{decay2}.\end{aligned}$$
Note that the decay conditions and allow for data containing non-vanishing linear and angular momentum. For the purposes of our analysis, it will be necessary to have a bit more information about the behaviour of leading terms in $h_{ij}$ and $K_{ij}$. More precisely, we will require the initial data to be *asymptotically Schwarzschildean* in some suitable sense. For example, in [@BaeVal10a] the assumptions
$$\begin{aligned}
&& h_{ij} = -\left(1+ 2m_\pm r^{-1}\right)\delta_{ij} + o_\infty(r^{-3/2}), \label{OldDecay1} \\
&& K_{ij} = o_\infty(r^{-5/2}), \label{OldDecay2}\end{aligned}$$
have been used. This class of data can be described as *asymptotically non-boosted Schwarzschildean*. Here, we consider a more general class of data which includes boosted Schwarzschild data. Following [@BeiOMu87; @Hua10] we assume
$$\begin{aligned}
&& h_{ij} = -\left(1+\frac{2A_\pm}{r}\right)\delta_{ij} - \frac{\alpha_\pm}{r}\left( \frac{2x_ix_j}{r^2}-\delta_{ij}\right)+o_\infty(r^{-3/2}), \label{BoostedDecay1} \\
&& K_{ij} = \frac{\beta_\pm}{r^2}\left( \frac{2x_ix_j}{r^2}-\delta_{ij} \right) + o_\infty(r^{-5/2}), \label{BoostedDecay2}\end{aligned}$$
where $\alpha_\pm$ and $\beta_\pm$ are smooth functions on the 2-sphere and $A_\pm$ denotes a constant. The functions $\alpha$ and $\beta$ are related to each other via the vacuum constraint equations and . We will later need to be more specific about their particular form. The decay assumption for the metric, equation and hence also , is included in the analysis of [@Can81].
Important for our analysis is that boosted Schwarzschild data is of this form —see [@BeiOMu87]. It is noticed that a second fundamental form of the type given by (\[BoostedDecay2\]) is, in general, not trace-free: $$K_i{}^i = \frac{\beta_\pm}{r^2} + o_\infty(r^{-5/2}).$$
Henceforth, we drop the superscripts/subscripts $\pm$ for ease of presentation. If $\pm$ appears in any formula, $+$ is assumed for the $(+)$-end, $-$ for the $(-)$-end. For the $\mp$ sign we assume the opposite.
ADM mass and momentum
---------------------
The ADM energy, $E$, and momentum, $p_i$, at each end are given by the integrals: $$\begin{aligned}
&& E= \frac{1}{16\pi} \int_{\partial\mathcal{S}_\infty} \delta^{ij} \left(\partial_i h_{jk}- \partial_k h_{ij} \right)\frac{x^k}{r} \mbox{d}S, \\
&& p_i = \frac{1}{8\pi} \int_{\partial\mathcal{S}_\infty} \left(K_{ij} - K h_{ij} \right)\frac{x^j}{r} \mbox{d}S,\end{aligned}$$ so that the ADM 4-momentum covector is given by $p_\mu=(E,p_i)$. In what follows it will be assumed that $p_\mu$ is timelike —that is, $p_\mu p^\mu>0$. The need of this assumption will become clear in the sequel. From the ADM 4-momentum, we define the constants $$m\equiv\sqrt{p^\nu p_\nu}, \quad p^2\equiv E^2-m^2.$$
Asymptotically Schwarzschildean data
------------------------------------
Boosted Schwarzschild data sets—i.e. initial data for the Schwarzschild spacetime for which $p_i\neq 0$ satisfy the decay assumptions -. This type of data satisfies: $$\begin{aligned}
&& A=\frac{m}{\sqrt{1-v^2}},\\
&& \alpha = 2m\left(1+ 2\frac{(n\cdot v)^2}{1-v^2}\right)\left(1+\frac{(n\cdot v)^2}{1-v^2} \right)^{-1/2}-\frac{2m}{\sqrt{1-v^2}}, \\
&& \beta = 2m \frac{n\cdot v}{1-v^2}\left(\frac{3}{2}+\frac{(n\cdot v)^2}{1-v^2}\right)\left(1+\frac{(n\cdot v)^2}{1-v^2}\right)^{-3/2},\end{aligned}$$ where $n^i\equiv x^i/r$, $n\cdot v\equiv n^i v_i$, $v^2\equiv\delta^{ij}v_i v_j$, $v_i$ is a constant 3-covector —cfr. [@BeiOMu87], and $m_\pm=m$. Note that if $v_i=0$ then - reduce to -. It can be checked that $$E= \frac{m}{\sqrt{1-v^2}}, \quad p_i=\frac{mv_i}{\sqrt{1-v^2}}.$$ Rewriting this in terms of $(E,p_i)$, we get $$\label{alphabeta}
A=E,\qquad
\alpha = \frac{2m^2+4(n\cdot p)^2}{\sqrt{m^2+(n\cdot p)^2}}-2E, \qquad
\beta = \frac{(n\cdot p) E (3m^2+2(n\cdot p)^2)}{(m^2+(n\cdot p)^2)^{3/2}},$$ where $n\cdot p=n^i p_i=r^{-1}x^ip_i$.
**Assumption.** In the sequel, we will restrict our analysis to initial data sets which are asymptotically Schwarzschildean to the order given by -. For any asymptotically flat data that admits ADM 4-momentum, one can compute $(E,p_i)$, and then try to find coordinates that cast the metric and extrinsic curvature into the form - with $(A,\alpha,\beta)$ given by with $m=m_\pm$. If this is possible, we will say that the data is *asymptotically Schwarzschildean*. We expect this to be the case for a large class of data. The initial data sets excluded by this assumption will be deemed pathological. Examples of such pathological cases can be found in [@Hua10]. We stress that all data of the form - is included in our more general analysis.
The need to restrict our analysis to asymptotically Schwarzschildean data as defined in the previous paragraph will become evident in the sequel, where we need to find an asymptotic solution to the spatial Killing spinor equation.
Asymptotic behaviour of the spatial Killing spinors {#Section:AB}
===================================================
In this section we discuss in some detail the asymptotic behaviour of solutions to the spatial Killing spinor equation on an asymptotically Euclidean manifold. We begin by studying the asymptotic behaviour of the appropriate Killing spinor in the Kerr spacetime. Then, we will impose the same asymptotics on the approximate Killing spinor on a slice of a much more general spacetime. In what follows, we concentrate our discussion on a particular asymptotic end.
Asymptotic form of the stationary Killing vector
------------------------------------------------
As seen in section \[Section:Basics\], the Killing spinor of the Kerr spacetime gives rise to its stationary Killing vector $\xi^\mu$. It will be assumed that the spacetime is such that $p_\mu=(E, p_i)$ is timelike at each asymptotic end. If this is the case, then $p^\mu/\sqrt{p^\nu p_\nu}$ gives the asymptotic direction of the stationary Killing vector at each end —see e.g.[@BeiChr96]. Let $$m\equiv\sqrt{p^\nu p_\nu}, \quad p^2\equiv E^2-m^2.$$ Recall now, that $\xi$ and $\xi_{AB}$ denote the lapse and shift of the spinorial counterpart, $\xi^{AA'}$, of the Killing vector $\xi^\mu$. One finds that for non-boosted initial data sets of the form -, one has in terms of the asymptotic Cartesian coordinates and spin frame, that $$\xi= \pm\sqrt{2} +o_\infty(r^{-1/2}), \quad \xi_{\mathbf{AB}}= o_\infty(r^{-1/2}).$$ The factor of $\sqrt{2}$ arises due to the particular normalisations used in the space spinor formalism. This particular form of the asymptotic behaviour of the Killing vector has been discussed in [@BaeVal10a].
Now consider the more general case given by -. Again, adopting asymptotically Cartesian coordinates, we extend $p_i$ to a constant covector field on the asymptotic end. In terms of the associated asymptotically Cartesian spin frame, we then define $p_{\mathbf{AB}}
\equiv \sigma^i{}_{\mathbf{AB}} p_i$. One finds that $$\xi= \pm\frac{\sqrt{2}E}{m}+o_\infty(r^{-1/2}), \quad \xi_{\mathbf{AB}}=\pm\frac{\sqrt{2}p_{\mathbf{AB}}}{m}+o_\infty(r^{-1/2}).\label{xi_first_leading}$$
We see that the conditions are well defined even if we do not have a Killing vector in the spacetime. Hence, for the general case when the metric satisfies - and the ADM 4-momentum is well defined, we can still impose the asymptotics for our approximate Killing spinor. We will however need to assume that the functions in the metric are given by . Otherwise we will not be able to assume $\xi_{ABCD}\in H^\infty_{-3/2}$, as we will do in the next section. We will later see that this condition is important for the solvability of the elliptic equation .
Asymptotic form of the spatial Killing spinor {#Section:DecayKappa}
---------------------------------------------
In the sequel, given an initial data set $(\mathcal{S},h_{ab},K_{ab})$ satisfying the decay conditions - with $A$, $\alpha$ and $\beta$ given by with $m=m_\pm$, it will be necessary to show that it is always possible to solve the equation $$\label{AsymptoticSpatialKillingSpinorEquation}
\nabla_{(AB}\kappa_{CD)}=o_\infty(r^{-3/2}),$$ order by order without making any further assumptions on the data. A direct calculation allows us to verify that:
Let $(\mathcal{S},h_{ab},K_{ab})$ denote an initial data set for the vacuum Einstein field equations satisfying at each asymptotic end the decay conditions - with $A$, $\alpha$ and $\beta$ given by and $m$ the ADM mass of the respective end. Then $$\begin{aligned}
&& \kappa_{\mathbf{AB}} =
\mp\frac{\sqrt{2}E}{3m}\left (1+\frac{2E}{r}\right)x_{\mathbf{AB}} \nonumber \\
&&\hspace{2cm}\pm\frac{2\sqrt{2}}{3m}\left(1
+\frac{4E}{r}
-\frac{m^2+2(n\cdot p)^2}
{\sqrt{m^2+(n\cdot p)^2}r}
\right)p_{\mathbf{Q}(\mathbf{A}}x_{\mathbf{B})}{}^{\mathbf{Q}}
+o_\infty(r^{-1/2}), \label{KillingSpinorLeading}\end{aligned}$$ with $x_{\mathbf{AB}}$ as in , and $n\cdot
p=r^{-1}x^{\mathbf{AB}} p_{\mathbf{AB}}$ satisfies equation .
**Remark.** Formula implies the following expansions for $\xi$ and $\xi_{\mathbf{AB}}$:
$$\begin{aligned}
&&\xi=\pm\frac{\sqrt{2}E}{m}\mp\frac{\sqrt{2}E(m^2+2(n\cdot p)^2)}{m\sqrt{m^2+(n\cdot p)^2}}r^{-1}+o_\infty(r^{-3/2}),\label{xi0Leading}\\
&& \xi_{\mathbf{AB}}=
\pm\Biggl(-\frac{2\sqrt{2}E}{m}
+\frac{\sqrt{2}(E^2+4(n\cdot p)^2)}{m\sqrt{m^2+(n\cdot p)^2}}
+\frac{mE^2}{\sqrt{2}(m^2+(n\cdot p)^2)^{3/2}}
\Biggr)(n\cdot p)r^{-2}x_{\mathbf{AB}}\nonumber\\
&&\hspace{2cm}\pm\Biggl(\frac{\sqrt{2}}{m}
+\frac{2\sqrt{2}E}{mr}
-\frac{2\sqrt{2}(m^2+2(n\cdot p)^2)}{m\sqrt{m^2+(n\cdot p)^2}r}
\Biggr)p_{\mathbf{AB}}
+o_\infty(r^{-3/2}).\label{xi2Leading}\end{aligned}$$
In the case of non-boosted data the expansions , and reduce to $$\begin{aligned}
&& \kappa_{\mathbf{AB}} =
\mp\frac{\sqrt{2}}{3}\left (1+\frac{2m}{r}\right)x_{\mathbf{AB}}
+o_\infty(r^{-1/2}),\\
&& \xi=\pm\sqrt{2}\mp\sqrt{2}mr^{-1}+o_\infty(r^{-3/2}),\\
&& \xi_{\mathbf{AB}}=o_\infty(r^{-3/2}),\end{aligned}$$ as discussed in [@BaeVal10a].
Existence and uniqueness of spinors with Killing spinor asymptotics
-------------------------------------------------------------------
In this section we prove that given a spinor $\kappa_{AB}$ satisfying equation and , then the asymptotic expansion is unique up to a translation.
\[ExistensKillingSpinorAsymptotics\] Assume that on an asymptotic end of the slice $\mathcal{S}$, one has an asymptotically Cartesian coordinate system such that - hold. Then there exists $$\kappa_{\mathbf{AB}}=o_\infty(r^{3/2}),\label{AsymptoticAssumptions1}$$ such that $$\xi_{\mathbf{ABCD}}=o_\infty(r^{-3/2}), \quad
\xi_{\mathbf{AB}}=\pm\frac{\sqrt{2}p_{\mathbf{AB}}}{m}+o_\infty(r^{-1/2}), \quad
\xi=\pm\frac{\sqrt{2}E}{m}+o_\infty(r^{-1/2}).
\label{AsymptoticAssumptions2}$$ The spinor $\kappa_{\mathbf{AB}}$ is unique up to order $o_\infty(r^{-1/2})$, apart from a (complex) constant term.
**Remark 1.** The complex constant term arising in Theorem \[ExistensKillingSpinorAsymptotics\] contains 6 real parameters. In the sequel, given a particular choice of asymptotically Cartesian coordinates and frame, we will set this constant term to zero. Note that a change of asymptotically Cartesian coordinates would introduce a similar term containing only 3 real parameters —which by construction could be removed by a suitable choice of gauge. In what follows, we will use coordinate independent expressions, and therefore, this translational ambiguity will not affect the result.
**Remark 2.** Note that $\xi_{\mathbf{ABCD}}=o_\infty(r^{-3/2})$ implies $\xi_{ABCD}\in L^2$. The conditions in Theorem \[ExistensKillingSpinorAsymptotics\] are coordinate independent.
A direct calculation shows that the expansion yields , and $\xi_{\mathbf{ABCD}}=o_\infty(r^{-3/2})$. Hence, gives a solution of the desired form. In order to prove uniqueness we make use of the linearity of the integrability conditions - and -. Note that the translational freedom gives an ambiguity of a constant term in $\kappa_{\mathbf{AB}}$. Let $$\mathring\kappa_{\mathbf{AB}} \equiv
\mp\frac{\sqrt{2}E}{3m}\left (1+\frac{2E}{r}\right)x_{\mathbf{AB}}
\pm\frac{2\sqrt{2}}{3m}\left(1
+\frac{4E}{r}
-\frac{m^2+2(n\cdot p)^2}
{\sqrt{m^2+(n\cdot p)^2}r}
\right)p_{\mathbf{Q}(\mathbf{A}}x_{\mathbf{B})}{}^{\mathbf{Q}}.
\label{KillingSpinorLeadingTerms}$$ Let $\breve\kappa_{\mathbf{AB}}$, be an arbitrary solution to the system , . Furthermore, let $\kappa_{\mathbf{AB}}\equiv\breve\kappa_{\mathbf{AB}}-\mathring\kappa_{\mathbf{AB}}$. We then have $$\xi_{ABCD}=o_\infty(r^{-3/2}), \quad \xi_{AB}=o_\infty(r^{-1/2}), \quad \xi=o_\infty(r^{-1/2}), \quad \kappa_{AB}=o_\infty(r^{3/2}).$$ To obtain the desired conclusion we only need to prove that $\kappa_{\mathbf{AB}}=C_{\mathbf{AB}}+o_\infty(r^{-1/2})$, where $C_{\mathbf{AB}}$ is a constant. This is equivalent to $D_{AB}\kappa_{CD}=o_\infty(r^{-3/2})$. Note that we now have coordinate independent statements to prove.
We note that from - it follows that $$K_{ABCD}=o_\infty(r^{-2+\varepsilon}), \quad \Psi_{ABCD}=o_\infty(r^{-3+\varepsilon}),$$ with $\varepsilon>0$. From and Lemma \[Lemma:Multiplication\] we have $$\begin{aligned}
&& D_{AB}\kappa_{CD}=\xi_{ABCD}-\tfrac{1}{3}\epsilon_{A(C}\xi_{D)B}-\tfrac{1}{3}\epsilon_{B(C}\xi_{D)A}-\tfrac{1}{3}\epsilon_{A(C}\epsilon_{D)B}\xi-K_{AB(C}{}^E\kappa_{D)E} \\
&& \phantom{ D_{AB}\kappa_{CD}}=o_\infty(r^{-1/2+\varepsilon}).\end{aligned}$$ Integrating the latter yields $$\kappa_{AB}=o_\infty(r^{1/2+\varepsilon}).$$ The constant of integration is incorporated in the remainder term. Repeating this procedure allows to gain an $\varepsilon$ in the decay so that $$D_{AB}\kappa_{CD}=o_\infty(r^{-1/2}), \quad \kappa_{AB}=o_\infty(r^{1/2}).$$ Estimating all terms in , and gives
$$\begin{aligned}
\nabla^{AB}\nabla_{AB}\xi={}&
\xi_{AB}\nabla^{AB}K +o_\infty(r^{-7/2}) \nonumber \\
={}&o_\infty(r^{-7/2+\varepsilon}),\label{estnabla2xi0a}\\
\nabla^C{}_{(A}\nabla_{B)C}\xi
={}&\tfrac{1}{2}\Omega_{ABCD}\nabla^{CD}\xi -\tfrac{1}{3}K \nabla_{AB}\xi \nonumber \\ ={}&o_\infty(r^{-7/2+\varepsilon}),\label{estnabla2xi0b}\\
\nabla_{(AB}\nabla_{CD)}\xi
={}&
+\tfrac{1}{2} \hat\Psi_{ABCD}\xi -\tfrac{5}{2} \Psi_{ABCD}\xi -\tfrac{2}{3} \hat\Psi_{(ABC}{}^E \xi_{D)E} -\tfrac{10}{3} \Psi_{(ABC}{}^E \xi_{D)E}\nonumber\\ &+\Omega_{ABCDEL} \xi^{EL} +\tfrac{2}{5} \xi_{(CD} \nabla _{AB)}K-3 \Omega_{E(BCD} \nabla_{A)}{}^E\xi\nonumber\\ &-3\kappa^{EL}\nabla_{L(D}\Psi_{ABC)E}+3\kappa_{(A}{}^E \nabla_D{}^L\Psi_{BC)EL}+o_\infty(r^{-7/2}) \nonumber \\
={}& o_\infty(r^{-7/2+\varepsilon}). \label{estnabla2xi0c}\end{aligned}$$
Hence, $\nabla_{AB} \nabla_{CD}\xi=o_\infty(r^{-7/2+\varepsilon})$, and therefore $D_{AB}
D_{CD}\xi=o_\infty(r^{-7/2+\varepsilon})$. Integrating this yields $D_{AB}\xi=o_\infty(r^{-5/2+\varepsilon})$. In this step the constants of integration are forced to vanish by the condition $D_{AB}\xi=o_\infty(r^{-3/2})$, which is a consequence of $\xi=o_\infty(r^{-1/2})$. Integrating $D_{AB}\xi=o_\infty(r^{-5/2+\varepsilon})$ and using $\xi=o_\infty(r^{-1/2})$ to remove the constants of integration yields $$\xi=o_\infty(r^{-3/2+\varepsilon}).$$ Estimating all terms in , and yields
$$\begin{aligned}
&& \nabla^{AB}\xi_{AB}
= o_\infty(r^{-7/2+\varepsilon}),\label{estDxi1}\\
&& \nabla^C{}_{(A}\xi_{B)C}
=\tfrac{3}{2}\Psi_{ABCD}\kappa^{CD} -\tfrac{2}{3}K\xi_{AB} -\tfrac{1}{2}\Omega_{ABCD}\xi^{CD} +\nabla_{AB}\xi +o_\infty(r^{-5/2}) \nonumber \\
&& \phantom{\nabla^C{}_{(A}\xi_{B)C}}= o_\infty(r^{-5/2+\varepsilon}),\label{estDxi2}\\
&& \nabla_{(AB}\xi_{CD)}
=3\Psi_{E(ABC}\kappa_{D)}{}^E -\Omega_{E(ABC}\xi_{D)}{}^E +o_\infty(r^{-5/2}) \nonumber \\
&& \phantom{\nabla_{(AB}\xi_{CD)}}=o_\infty(r^{-5/2+\varepsilon}).\label{estDxi3}\end{aligned}$$
Hence, $\nabla_{AB} \xi_{CD}=o_\infty(r^{-5/2+\varepsilon})$, and therefore $D_{AB} \xi_{CD}=o_\infty(r^{-5/2+\varepsilon})$. Integrating and using $\xi_{AB}=o_\infty(r^{-1/2})$ to remove the constants of integration yields $$\xi_{AB}=o_\infty(r^{-3/2+\varepsilon}).$$ Now, $$\begin{aligned}
&& D_{AB}\kappa_{CD}=\xi_{ABCD}-\tfrac{1}{3}\epsilon_{A(C}\xi_{D)B}-\tfrac{1}{3}\epsilon_{B(C}\xi_{D)A}-\tfrac{1}{3}\epsilon_{A(C}\epsilon_{D)B}\xi-K_{AB(C}{}^E\kappa_{D)E} \nonumber \\
&& \phantom{D_{AB}\kappa_{CD}} =o_\infty(r^{-3/2+\varepsilon}).\end{aligned}$$ Integrating the latter we get $$\kappa_{AB}=C_{AB}+o_\infty(r^{-1/2+\varepsilon}),$$ where $C_{AB}$ is a constant in some frame. To get a frame independent statement one can still use the estimate $\kappa_{AB}=o_\infty(r^{\varepsilon})$. Reevaluating the estimates , and yields $$\begin{aligned}
&& \nabla^{AB}\nabla_{AB}\xi=
o_\infty(r^{-7/2}),\\
&& \nabla^C{}_{(A}\nabla_{B)C}\xi
=o_\infty(r^{-9/2+\varepsilon}),\\
&& \nabla_{(AB}\nabla_{CD)}\xi
=o_\infty(r^{-7/2}).\end{aligned}$$ Hence, one obtains $$\nabla_{AB}\nabla_{CD}\xi=o_\infty(r^{-7/2}).$$ Integrating as before, we get $$\xi=o_\infty(r^{-3/2}).$$ Finally, we can reevaluate the estimates and , to get $$\begin{aligned}
&& \nabla^C{}_{(A}\xi_{B)C}
=o_\infty(r^{-5/2}),\\
&& \nabla_{(AB}\xi_{CD)}
= o_\infty(r^{-5/2}).\end{aligned}$$ Combining this with , we obtain $$\nabla_{AB}\xi_{CD}=o_\infty(r^{-5/2}).$$ Integrating as before, we get $$\xi_{AB}=o_\infty(r^{-3/2}).$$ Hence, $$D_{AB}\kappa_{CD}=\xi_{ABCD}-\tfrac{1}{3}\epsilon_{A(C}\xi_{D)B}-\tfrac{1}{3}\epsilon_{B(C}\xi_{D)A}-\tfrac{1}{3}\epsilon_{A(C}\epsilon_{D)B}\xi-K_{AB(C}{}^E\kappa_{D)E}=o_\infty(r^{-3/2}),$$ from where the result follows.
From the asymptotic solutions we can obtain a globally defined spinor $\mathring{\kappa}_{AB}$ on $\mathcal{S}$ that will act as a seed for our approximate Killing spinor.
\[corkapparing\] There are spinors $\mathring{\kappa}_{AB}$, defined everywhere on $\mathcal{S}$, such that the asymptotics at each end is given by , where opposite signs are used at each end. Different choices of $\mathring{\kappa}_{AB}$ can only differ by a spinor in $H^\infty_{-1/2}$.
**Remark.** The opposite signs at each end are motivated by looking at the explicit example of standard Kerr data.
Theorem \[ExistensKillingSpinorAsymptotics\] gives the existence at each end. Smoothly cut off these functions, and paste them together. This gives a smooth spinor $\mathring{\kappa}_{AB}$ defined everywhere on $\mathcal{S}$. Furthermore $\nabla_{(AB}\mathring{\kappa}_{CD)}\in H^\infty_{-3/2}$.
The approximate Killing spinor equation in asymptotically Euclidean manifolds {#Section:ApproximateKSinAEM}
=============================================================================
In this section we study the invertibility properties of the approximate Killing spinor operator $L:\mathfrak{S}_2 \rightarrow
\mathfrak{S}_2$ given by equation on a manifold $\mathcal{S}$ which is asymptotically Euclidean in the sense discussed in section \[Section:AsymptoticallyEuclideanData\]. In order to do so, we first present some adaptations to our context of results for elliptic equations that can be found in [@Can81; @ChrOMu81; @Loc81].
Ancillary results of the theory of elliptic equations on asymptotically Euclidean manifolds
-------------------------------------------------------------------------------------------
### Asymptotic homogeneity of $L$
Let $u$ be the vector given by equation . The elliptic operator defined by can be written matricially in the form $$(A^{ij}+a^{ij}_2)D_i D_j u + a^i_1 D_i u + a_0u =0,$$ where $A^{ij}$ corresponds to the matrix associated to the elliptic operator with constant coefficients $L'$ given by equation , and $a^{ij}_2$, $a^j_1$, $a_0$ are matrix valued functions such that $$a^{ij}_2 \in H^\infty_{-1/2}, \quad a^j_1 \in H^\infty_{-3/2}, \quad a_0 \in H^\infty_{-5/2}.$$ Using the terminology of [@Can81; @Loc81] we say that $L$ is an *asymptotically homogeneous elliptic operator*[^5]. This is the standard assumption on elliptic operators on asymptotically Euclidean manifolds. It follows from [@Can81], Theorem 6.3 that:
The elliptic operator $$L: H^{2}_{\delta} \rightarrow H^{0}_{\delta-2},$$ with $\delta$ is not a non-negative integer is a linear bounded operator with finite dimensional Kernel and closed range.
### The Kernel of $L$
We investigate some relevant properties of the Kernel of $L$. This, in turn, requires an analysis of the Kernel of the operator of the Killing spinor equation .
The following is an adaptation to the smooth spinorial setting of an ancillary result from [@ChrOMu81][^6].
\[Lemma:ChrOMu81\] Let $\nu_{A_1B_1\cdots A_pB_p}$ be a $C^\infty$ spinorial field over $\mathcal{S}$ such that $$\nabla_{E_{m+1}F_{m+1}}\cdots \nabla_{E_1 F_1} \nu_{A_1B_1 \cdots A_pB_p} = H_{E_{m+1}F_{m+1}\cdots E_1 F_1 A_1 B_1 \cdots A_p B_p}$$ with $m,\;p$ non-negative integers, and where $H_{E_{m+1}F_{m+1}\cdots
E_1 F_1 A_1 B_1 \cdots A_p B_p}$ is a linear combination of $\nu_{A_1B_1 \cdots A_pB_p}$, $\nabla_{E_1F_1}\nu_{A_1B_1 \cdots
A_pB_p}$, $\ldots$, $\nabla_{E_mF_m}\cdots \nabla_{E_1F_1}\nu_{A_1B_1
\cdots A_pB_p}$ with coefficients $b_k$ where $k$ denotes the order of the derivative the coefficient is associated to. If $b_k\in
H^\infty_{\delta_k}$ with $$k-m-1 > \delta_k, \quad 0\leq k \leq m$$ and $\nu_{A_1B_1\cdots A_pB_p}\in H^\infty_\beta$, $\beta<0$, then $$\nu_{A_1B_1\cdots A_pB_p}=0 \quad { on } \;\;\mathcal{S}.$$
This last result, together with Lemma \[Lemma:ThirdDerivative\] allows to show that there are no non-trivial Killing spinor candidates that go to zero at infinity —in [@ChrOMu81] an analogous result has been proved for Killing vectors. More precisely,
\[Proposition:TrivialityKernel\] Let $\nu_{AB}\in H^\infty_{-1/2}$ such that $\nabla_{(AB}\nu_{CD)}=0$. Then $\nu_{AB}=0$ on $\mathcal{S}$.
From Lemma \[Lemma:ThirdDerivative\] it follows that $\nabla_{AB} \nabla_{CD} \nabla_{EF} \nu_{GH}$ can be expressed as a linear combination of lower order derivatives with smooth coefficients with the proper decay. Thus, Theorem \[Lemma:ChrOMu81\] applies with $m=2$ and one obtains the desired result.
We are now in the position to discuss the Kernel of the approximate Killing spinor operator in the case of spinor fields that go to zero at infinity. The following is the main result of this section.
\[EllipticKernel\] Let $\nu_{AB}\in H^\infty_{-1/2}$. If $L(\nu_{AB})=0$, then $\nu_{AB}=0$.
Using the identity with $\zeta_{ABCD}= \nabla_{(AB} \nu_{CD)}$ and assuming that $L(\nu_{CD})=0$, one obtains $$\int_{\mathcal{S}} \nabla^{AB}\nu^{CD} \widehat{\nabla_{(AB}
\nu_{CD)}} \mbox{d}\mu = \int_{\partial\mathcal{S}_\infty}
n^{AB}\nu^{CD} \widehat{\nabla_{(AB}\nu_{CD)}} \mbox{d}S,$$ where $\partial S_\infty$ denotes the sphere at infinity. Assume now, that $\nu_{AB}\in H^\infty_{-1/2}$. It follows that $\nabla_{(AB} \nu_{CD)}
\in H^\infty_{-3/2}$ and furthermore, using Lemma \[Lemma:FinerMultiplication\] that $$n^{AB} \nu^{CD}
\widehat{\nabla_{(AB}\nu_{CD)}} = o(r^{-2}).$$ The integration of the latter over a finite sphere of sufficiently large radius is of type $o(1)$. Thus one has that $$\int_{\partial\mathcal{S}_\infty}
n^{AB}\nu^{CD} \widehat{\nabla_{(AB}\nu_{CD)}} \mbox{d}S=0,$$ from where $$\int_{\mathcal{S}} \nabla^{AB}\nu^{CD} \widehat{\nabla_{(AB} \nu_{CD)}} \mbox{d}\mu =0.$$ Therefore, one concludes that $$\nabla_{(AB} \nu_{CD)}=0.$$ That is, $\nu_{AB}$ has to be a spatial Killing spinor. Using Proposition \[Proposition:TrivialityKernel\] it follows that $\nu_{AB}= 0$ on $\mathcal{S}$.
### The Fredholm alternative and elliptic regularity
We will make use of the following adaptation of the Fredholm alternative for second order asymptotically homogeneous elliptic operators on asymptotically Euclidean manifolds —cfr. [@Can81].
\[Theorem:FredholmAlternative\] Let $A$ be an asymptotically homogeneous elliptic operator of order 2 with smooth coefficients. Given $\delta<0$, the equation $$A(\zeta_{AB}) = f_{AB}, \quad f_{AB}\in H^0_{\delta-2},$$ has a solution $\zeta_{AB}\in H^2_{\delta}$ if $$\int_{\mathcal{S}} f_{AB} \hat{\nu}^{AB} \mbox{d}\mu=0$$ for all $\nu_{AB}$ satisfying $$\nu_{AB} \in H^0_{-1-\delta}, \quad A^*(\nu_{AB})=0,$$ where $A^*$ denotes the formal adjoint of $A$.
In order to assert the regularity of solutions, we will need the following elliptic estimate —see expression (62) in the proof of Theorem 6.3 of [@Can81].
\[Lemma:Regularity\] Let $A$ be an asymptotically homogeneous elliptic operator of order 2 with smooth coefficients. Then for any $\delta\in {\mbox{\SYM R}}$ and any $s\geq
2$ there exists a constant $C$ such that for every $\zeta_{AB} \in
H^s_{loc} \cap H^0_\delta$, the following inequality holds $$\Vert \zeta_{AB} \Vert_{H^s_\delta} \leq C \left( \Vert A(\zeta_{AB})
\Vert_{H^{s-2}_{\delta-2}} + \Vert \zeta_{AB} \Vert_{H^{s-2}_\delta} \right).$$
**Notation.** $H^s_{loc}$ denotes the local Sobolev space. That is, $u \in H^s_{loc}$ if for an arbitrary smooth function $v$ with compact support, $uv \in H^s$.
**Remark.** If $A$ has smooth coefficients, and $A(\zeta_{AB})=0$ then it follows that all the $H^s_\delta$ norms of $\zeta_{AB}$ are bounded by the $H^0_\delta$ norm. Thus, it follows that if a solution to $ A(\zeta_{AB})=0$ exists, it must be smooth —*elliptic regularity*.
Existence of approximate Killing spinors
----------------------------------------
We are now in the position of providing an existence proof to solutions to equation with the asymptotic behaviour discussed in section \[Section:DecayKappa\].
\[Theorem:ExistenceKS\] Given an asymptotically Euclidean initial data set $(\mathcal{S},h_{ab},K_{ab})$ satisfying the asymptotic conditions - and , there exists a smooth unique solution to equation with asymptotic behaviour at each end given by .
We consider the Ansatz $$\kappa_{AB} = \mathring{\kappa}_{AB} + \theta_{AB}, \quad \theta_{AB} \in H^2_{-1/2},$$ with $\mathring{\kappa}$ given by Corollary \[corkapparing\]. Substitution into equation renders the following equation for the spinor $\theta_{AB}$: $$\label{elliptic:general}
L(\theta_{CD}) = -L(\mathring{\kappa}_{CD}).$$ By construction it follows that $$\nabla_{(AB} \mathring{\kappa}_{CD)}\in
H^\infty_{-3/2},$$ so that $$F_{CD}\equiv-L(\mathring{\kappa}_{CD})\in H^\infty_{-5/2}.$$ Using Theorem \[Theorem:FredholmAlternative\] with $\delta=-1/2$, one concludes that equation has a unique solution if $F_{AB}$ is orthogonal to all $\nu_{AB}\in H^0_{-1/2}$ in the Kernel of $L^*=L$. Proposition \[Proposition:TrivialityKernel\] states that this Kernel is trivial. Thus, there are no restrictions on $F_{AB}$ and equation has a unique solution as desired. Due to elliptic regularity, any $H^2_{-1/2}$ solution to the previous equation is in fact a $H^\infty_{-1/2}$ solution —cfr. Lemma \[Lemma:Regularity\]. Thus, $\theta_{AB}$ is smooth. To see that $\kappa_{AB}$ does not depend on the particular choice of $\mathring\kappa_{AB}$, let $\mathring\kappa'_{AB}$, be another choice. Let $\kappa'_{AB}$ be the corresponding solution to . Due to Corollary \[corkapparing\], we have $\mathring\kappa_{AB}-\mathring\kappa'_{AB} \in H^\infty_{-1/2}$. Hence, we have $\kappa_{AB}-\kappa'_{AB}\in H^\infty_{-1/2}$ and $L(\kappa_{AB}-\kappa'_{AB})=0$. According to Proposition \[EllipticKernel\], $\kappa_{AB}-\kappa'_{AB}=0$, and the proof is complete.
The following is a direct consequence of Theorem \[Theorem:ExistenceKS\], and will be crucial for obtaining an invariant characterisation of Kerr data.
\[Corollary:Boundedness\] A solution, $\kappa_{AB}$, to equation with asymptotic behaviour given by satisfies $J<\infty$ where $J$ is the functional given by equation .
The functional $J$ given by equation is the $L^2$ norm of $\nabla_{(AB}\kappa_{CD)}$. Now, if $\kappa_{AB}$ is the solution given by Theorem \[Theorem:ExistenceKS\], one has that $\nabla_{(AB}\kappa_{CD)}\in H^0_{-3/2}$. In Bartnik’s conventions one has that $$\Vert\nabla_{(AB}\kappa_{CD)}\Vert_{L^2} =\Vert\nabla_{(AB}\kappa_{CD)}\Vert_{H^0_{-3/2}}<\infty.$$ The result follows.
**Remark.** Again, let $\kappa_{AB}$ be the solution to equation given by Theorem \[Theorem:ExistenceKS\]. Using the identity with $\zeta_{ABCD}=\nabla_{(AB}\kappa_{CD)}$ one obtains that $$J = \int_{\partial \mathcal{S}_\infty} n^{AB}\kappa^{CD}
\widehat{\nabla_{(AB}\kappa_{CD)}} \mbox{d}S <\infty.$$ Thus, the invariant $J$ evaluated at the solution $\kappa_{AB}$ given by Theorem \[Theorem:ExistenceKS\] can be expressed as a boundary integral at infinity. A crude estimation of the integrand of the boundary integral does not allow directly to establish its boundedness. This follows, however, from Corollary \[Corollary:Boundedness\]. Hence, the leading order terms of $n_{AB}\kappa_{CD}$ and $\nabla_{(AB}\kappa_{CD)}$ are orthogonal.
For an independent proof of this fact, see appendix \[AlternativeEstimate\].
The geometric invariant {#Section:Invariant}
=======================
In this section we show how to use the functional (\[functional\]) and the algebraic conditions (\[kspd2\]) and (\[kspd3\]) to construct the desired geometric invariant measuring the deviation of $(\mathcal{S},h_{ab},K_{ab})$ from Kerr initial data. To this end, let $\kappa_{AB}$ be a solution to equation as given by Theorem \[Theorem:ExistenceKS\]. Furthermore, let $\xi_{AB} \equiv
\tfrac{3}{2}\nabla^{P}{}_{(A}\kappa_{B)P}$. Define
$$\begin{aligned}
&& I_1 \equiv \int_{\mathcal{S}} \Psi_{(ABC}{}^{F}\kappa_{D)F}
\hat{\Psi}^{ABCG}\hat{\kappa}^D{}_G \mbox{d}\mu, \label{I1} \\
&& I_2 \equiv{} \int_{\mathcal{S}} \left(3\kappa_{(A}{}^{E}\nabla_{B}{}^{F}\Psi_{CD)EF}+\Psi_{(ABC}{}^{F}\xi_{D)F}\right) \nonumber \\
&& \hspace{3cm}\times \left(3\hat\kappa^{AP}\widehat{\nabla^{BQ}\Psi^{CD}{}_{PQ}}+\hat\Psi^{ABCP}\hat\xi^D{}_P\right){} \mbox{d}\mu. \label{I2}\end{aligned}$$
The geometric invariant is then defined by $$\begin{aligned}
I \equiv J + I_1 + I_2. \label{geometric:invariant}\end{aligned}$$
**Remark.** It should be stressed that by construction $I$ is coordinate independent and that $I\geq 0$. We also have the following lemma.
The geometric invariant given by is finite for an initial data set $(\mathcal{S},h_{ab},K_{ab})$ satisfying the decay conditions -.
From Corollary \[Corollary:Boundedness\] we already have $J<\infty$. From the form of the decay assumptions - we have $\Psi_{ABCD}\in
H^\infty_{-3+\varepsilon}$, $\varepsilon>0$. By Lemma \[Lemma:Multiplication\] and $\kappa_{AB}\in
H^\infty_{1+\varepsilon}$ we have $$\Psi_{(ABC}{}^{F}\kappa_{D)F} \in H^\infty_{-3/2}.$$ Thus, again one finds that $I_1<\infty$. A similar argument shows that $$3\kappa_{(A}{}^{E}\nabla_{B}{}^{F}\Psi_{CD)EF}+\Psi_{(ABC}{}^{F}\xi_{D)F}\in H^\infty_{-3/2},$$ from where it follows that $I_2 <\infty$. Hence, the invariant (\[geometric:invariant\]) is finite and well defined.
Finally, we are in the position of stating the main result of this article. It combines all the results in the sections 2 to 7.
Let $(\mathcal{S},h_{ab},K_{ab})$ be an asymptotically Euclidean initial data set for the Einstein vacuum field equations satisfying on each of its two asymptotic ends the decay conditions - and with a timelike ADM 4-momentum. Furthermore, assume that $\Psi_{ABCD}\neq
0$ and $\Psi_{ABCD}\Psi^{ABCD}\neq 0$ everywhere on $\mathcal{S}$. Let $I$ be the invariant defined by equations , , and , where $\kappa_{AB}$ is given as the only solution to equation with asymptotic behaviour on each end given by . The invariant $I$ vanishes if and only if $(\mathcal{S},h_{ab},K_{ab})$ is locally an initial data set for the Kerr spacetime.
Due to our smoothness assumptions, if $I=0$ it follows that equations - are satisfied on the whole of $\mathcal{S}$. Thus, the development of $(\mathcal{S},h_{ab},K_{ab})$ will have, at least in a slab, a Killing spinor. Accordingly, it must be of Petrov type D, N or O on the slab —see Theorem \[Theorem:TypeDhasalwaysaKS\]. The types N and O are excluded by the assumptions $\Psi_{ABCD}\neq 0$ and $\Psi_{ABCD}\Psi^{ABCD}\neq 0$ on $\mathcal{S}$ —by continuity, these conditions will also hold in a suitably small slab. Thus the development of the data can only be of Petrov type D —at least on a suitably small slab.
Now, from the general theory on Killing spinors, we know that $\xi_{AA'}=\nabla_{A'}{}^Q \kappa_{AQ}$ will be, in general, a complex Killing vector. In particular, both the real and imaginary parts of $\xi_{AA'}$ will be real Killing vectors. The Killing initial data for $\xi_{AA'}$ on $\mathcal{S}$ consists of the fields $\xi$ and $\xi_{AB}$ on $\mathcal{S}$ calculated from $\kappa_{AB}$ using the expressions and . It can be verified that $$\xi-\hat{\xi}=o_\infty(r^{-1/2}), \quad \xi_{AB}+\hat{\xi}_{AB}=o_\infty(r^{-1/2}).$$ The latter corresponds to the Killing initial data for the imaginary part of $\xi_{AA'}$. It follows that the imaginary part of $\xi_{AA'}$ goes to zero at infinity. However, there are no non-trivial Killing vectors of this type [@BeiChr96; @ChrOMu81]. Thus, $\xi_{AA'}$ is a real Killing vector. This means that the spacetime belongs, at least in a suitably small slab of $\mathcal{S}$, to the generalised Kerr-NUT class. By construction, it tends to a time translation at infinity so that, in fact, it is a stationary Killing vector. By virtue of the decay assumptions - the development of the initial data will be asymptotically flat, and it can be verified that the Komar mass of each end coincides with the corresponding ADM mass —these are non-zero by assumption. Hence, Theorem \[Theorem:SpacetimeCharacterisation\] applies and the slab of $\mathcal{S}$ is locally isometric to the Kerr spacetime.
If furthermore, the slice $\mathcal{S}$ is assumed, a priori, to have the same topology as a slice of the Kerr spacetime one has that the invariant $I$ vanishes if and only if $(\mathcal{S},h_{ab},K_{ab})$ is an initial data set for the Kerr spacetime.
This follows from the uniqueness of the maximal globally hyperbolic development of Cauchy data —see [@ChoGer69].
**Remark 1.** A improvement of Theorem \[Theorem:SpacetimeCharacterisation\] in which no *a priori* restrictions on the Petrov type of the spacetime are made —see the remark after Theorem \[Theorem:SpacetimeCharacterisation\]— would allow to remove the conditions $\Psi_{ABCD}\neq 0$ and $\Psi_{ABCD}\Psi^{ABCD}\neq 0$, and thus obtain a stronger characterisation of Kerr data.
**Remark 2.** It is of interest to analyse whether the same conclusion of the corollary can be obtained without making *a priori* assumptions on the topology of the 3-manifold.
Future prospects
================
We have seen that one can construct a geometric invariant for a slice with two asymptotically flat ends. A natural extension of this work would be to also allow asymptotically hyperboloidal and asymptotically cylindrical slices. Furthermore, one would like to analyse parts of manifolds in the same way. In this case we need to find appropriate conditions that can be imposed on $\kappa_{AB}$ on the boundary of the region we would like to study. A typical scenario would be to study the domain of outer communication for a black hole, or the exterior of a star.
Another natural question to be asked is how the geometric invariant behaves under time evolution. A great part of this problem is to obtain a time evolution of $\kappa_{AB}$ such that it satisfies on every leaf of the foliation. If the geometric invariant is small, one could instead use as an approximate evolution equation for the approximate Killing spinor. In this case the system , could be used to gain control over the evolution.
If some type of constancy or monotonicity property could be established for the geometric invariant, this would be a useful tool for studying non-linear stability of the Kerr spacetime and also in the numerical evolutions of black hole spacetimes.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank A García-Parrado and J M Martín-García for their help with computer algebra calculations in the suite [xAct]{} [@xAct], and M Mars and N Kamran for valuable comments. TB is funded by a scholarship of the Wenner-Gren foundations. JAVK is funded by an EPSRC Advanced Research fellowship.
An alternative estimation of the boundary integral {#AlternativeEstimate}
==================================================
In this section we present an alternative argument to show that the boundary integral $$\int_{\partial \mathcal{S}_r} n^{AB} \kappa^{CD} \widehat{\nabla_{(AB}\kappa_{CD)}} \mbox{d}S,$$ is finite as $r\rightarrow \infty$ —cfr. the remark after Corollary \[Corollary:Boundedness\]. For simplicity, we only consider the non-boosted case, so we have $$\kappa_{AB}=\pm \frac{\sqrt{2}}{3}r n_{AB} +O(1).$$ A similar, but much lengthier argument can be implemented in the boosted case. It is only necessary to consider the finiteness of the integral $$\label{BoundaryIntegral}
r \int_{\partial \mathcal{S}_r} n^{AB} n^{CD} \widehat{\nabla_{(AB}\kappa_{CD)}} \mbox{d}S \quad \mbox{ as } r\rightarrow \infty.$$
We begin by investigating the multipole structure of $\xi_{ABCD}\equiv
\nabla_{(AB}\kappa_{CD)}$ in an asymptotically flat end $\mathcal{U}\subset\mathcal{S} $. The equation satisfied by $\xi_{ABCD}$ is $$\label{Divergence:xi}
\nabla^{AB} \xi_{ABCD} - 2\Omega^{ABF}{}_{(C} \xi_{D)ABF}=0,$$ —see equation . As $\mathcal{U}\approx (r_0,\infty)\times {\mbox{\SYM S}}^2$, with $r_0\in{\mbox{\SYM R}}$, it will be convenient to work in spherical coordinates. For simplicity, we adopt the point of view that all the angular dependence of the various functions involved is expressed in terms of (spin-weighted) spherical harmonics. Accordingly, we use the differential operators $\eth, \;\bar{\eth} \in
\mbox{T}{\mbox{\SYM S}}^2$—see e.g. [@PenRin84]. Let $\omega_+,\;\omega_-\in \mbox{T}^*{\mbox{\SYM S}}^2$ denote the 1-forms dual to $\eth$ and $\bar{\eth}$: $$\langle \eth, \omega_+ \rangle =1, \quad \langle \bar{\eth},
\omega_-\rangle =1.$$ In addition, we consider $\partial_r\in \mbox{T}\mathcal{U}$. The operators $\eth,\;\bar{\eth}$ are extended into $\mbox{T} \mathcal{U}$ by requiring that $$[\eth,\partial_r]=[\bar{\eth},\partial_r]=0.$$ Again, let $\mbox{d}r\in\mbox{T}^*\mathcal{U}$ denote the form dual to $\partial_r$. One has that $$\delta_{ij} \mbox{d} x^i \otimes \mbox{d}x^j = \mbox{d}r \otimes \mbox{d} r
+ r^2 \left( \omega_+ \otimes \omega_- + \omega_-\otimes \omega_+ \right).$$ Now, recalling that $$h_{ij} = -\left(1+\frac{2m}{r}\right) \delta_{ij} + o_{\infty}(r^{-3/2}),$$ we introduce the following frame and coframe: $$\begin{aligned}
&& e_{01}= \left( 1-\frac{m}{r} \right)\partial_r +
o_{\infty}(r^{-3/2}), \quad \sigma^{01}= \left( 1+\frac{m}{r} \right)\mbox{d}r +
o_{\infty}(r^{-3/2}) \\
&& e_{00}= \left( 1-\frac{m}{r} \right)\frac{1}{r}\eth +
o_{\infty}(r^{-5/2}), \quad \sigma^{00} = \left( 1+\frac{m}{r}
\right)r \omega_+ + o_\infty(r^{-1/2})\\
&& e_{11}= \left( 1-\frac{m}{r} \right)\frac{1}{r}\bar{\eth} +
o_{\infty}(r^{-5/2}), \quad \sigma^{11}= \left( 1+\frac{m}{r} \right)r
\omega_- + o_\infty(r^{-1/2}).\end{aligned}$$ The fields $e_{AB}$ and $\sigma^{AB}$ satisfy $$\langle e_{AB} , \sigma^{CD} \rangle = h_{AB}{}^{CD}, \quad h =
h_{ABCD} \sigma^{AB} \otimes \sigma^{CD}.$$ where $h_{ABCD} \equiv - \epsilon_{A(C}\epsilon_{D)B}$. Let $\mu_{AB}$ denote a smooth spinorial field. Its covariant derivative $D_{EF}\mu_{AB}$ can be computed using $$D_{EF}\mu_{AB}= e_{EF}(\mu_{AB}) -\Gamma_{EF}{}^Q{}_A \mu_{QB}-\Gamma_{EF}{}^Q{}_B \mu_{AQ},$$ where $\Gamma_{EF}{}^Q{}_A$ denote the spin coefficients of the frame $e_{AB}$.
The components of the spinor field $\xi_{ABCD}$ with respect to the frame $e_{AB}$ can be written as $$\xi_{ABCD} = \xi_0 \epsilon^0_{ABCD}+ \xi_1 \epsilon^1_{ABCD} +\xi_2
\epsilon^2_{ABCD}+ \xi_3 \epsilon^3_{ABCD}+ \xi_4 \epsilon^4_{ABCD},$$ where $$\epsilon^k_{ABCD} \equiv \epsilon_{(A}{}^{(E} \epsilon_B{}^F \epsilon_C{}^G \epsilon_{D)}{}^{H)_k},$$ where ${}^{(EFGH)_k}$ means that after symmetrisation, $k$ indices are set to $1$. In terms of this formalism, equation is given by $$\label{DivergenceFrame}
\epsilon^{AP}\epsilon^{BQ} e_{PQ}(\xi_{ABCD}) - 4 \Gamma^{ABQ}{}_{(A}
\xi_{BCD)Q} + 2 K^{ABQ}{}_{(A} \xi_{BCD)Q} - 2\Omega^{ABQ}{}_{(C}\xi_{D)ABQ}=0.$$ Recalling that by assumption $\xi_{ABCD}=o_\infty(r^{-3/2})$, a lengthy but straightforward calculation shows that implies the equations
$$\begin{aligned}
&& \partial_r \xi_1 -\frac{1}{r}\bar{\eth}\xi_0 + \frac{1}{6}\frac{1}{r} \eth \xi_2 + \frac{3}{r}\left( 1+\frac{m}{r} \right) \xi_1 = o_\infty(r^{-5}), \label{Frame1}\\
&&\partial_r \xi_2 + \frac{3}{2}\frac{1}{r} \bar{\eth} \xi_1 +
\frac{3}{2}\frac{1}{r} \eth\xi_3 +
\frac{3}{r}\left( 1+\frac{m}{r} \right) \xi_2 = o_\infty(r^{-5}), \label{Frame2}\\
&& \partial_r \xi_3 + \frac{1}{r} \eth \xi_4 - \frac{1}{6}\frac{1}{r} \bar{\eth} \xi_2 + \frac{3}{r}\left( 1+\frac{m}{r} \right)\xi_3 =
o_\infty(r^{-5}). \label{Frame3}\end{aligned}$$
A computation shows that $$n_{(AB} n_{CD)}= \epsilon^2_{ABCD},$$ so that the boundary integral involves only the component $\xi_2$. Furthermore, only the harmonic $Y_{0,0}$ (monopole) contributes to the integral as $\epsilon^2_{ABCD}$ is a constant spinor in our frame. From the equations -, it follows that the coefficient $\xi_{2;0}$ of $\xi_2$ associated to the harmonic $Y_{0,0}$ satisfies the ordinary differential equation $$\left(1-\frac{m}{r}\right)\partial_r \xi_{2;0} + \frac{3}{r} \xi_{2;0}
=f(r), \quad f(r)=o_\infty(r^{-5}).$$ Consequently one has that $$\xi_{2;0} = \frac{\alpha}{(r-m)^3} + \frac{1}{(r-m)^3}\int r(r-m)^2
f(r) \mbox{d}r, \quad \alpha\in {\mbox{\SYM C}}.$$ It follows that $$\xi_{2;0} = \frac{\alpha}{r^3} +o_\infty(r^{-4}).$$ Using this last expression in the integral and recalling that $\mbox{d}S=O(r^2)$, it follows that $$r \int_{\partial \mathcal{S}_r} n^{AB} n^{CD}
\widehat{\nabla_{(AB}\kappa_{CD)}} \mbox{d}S = 4\pi\alpha<\infty.$$ It is worth noting that the constant $\alpha$ contains information of global nature and it is only known after one has solved the approximate Killing spinor equation.
Tensor expressions
==================
For many applications, it is useful to have tensor expressions for the invariants. To this end, define the following tensors on $\mathcal{S}$: $$\begin{aligned}
\kappa_a&\equiv \sigma_a{}^{AB}\kappa_{AB},&
\zeta&\equiv \xi,\\
\zeta_a&\equiv \sigma_a{}^{AB}\xi_{AB},&
\zeta_{ab} &\equiv \sigma_a{}^{AB}\sigma_b{}^{CD}\xi_{ABCD},\\
C_{ac} &\equiv E_{ac}+\mbox{i}B_{ac}.\end{aligned}$$ Here $\epsilon_{abc}$, $E_{ac}$ and $B_{ac}$ are the pull-backs of $\tfrac{1}{\sqrt{2}}\tau^\mu\epsilon_{\mu\alpha\beta\gamma}$, $\tfrac{1}{2}\tau^\gamma \tau^\delta C_{\alpha\beta\gamma\delta}$ and $\tfrac{1}{4}\epsilon_{\mu\nu\gamma\delta}\tau^\beta \tau^\delta C_{\alpha\beta}{}^{\mu\nu}$ respectively. Observe that we are using a negative definite metric. In this section we assume $K_{ab}=K_{ba}$.
The tensorial versions of the equations , , then read
$$\begin{aligned}
\zeta &=D^a\kappa_a,\\
\zeta_a &=\tfrac{3}{2\sqrt{2}}\mbox{i}\epsilon_{abc} D^c\kappa^b
-\tfrac{3}{4}K_{ab}\kappa^b+\tfrac{3}{4}K_b{}^b\kappa_a,\\
\zeta_{ab}&=D_{(a}\kappa_{b)}-\tfrac{1}{3} h_{ab}D^c\kappa_c-
\tfrac{1}{\sqrt{2}}\mbox{i}\epsilon_{cd(a}K_{b)}{}^d \kappa^c.\end{aligned}$$
Note that the spatial Killing spinor equation $\zeta_{ab}=0$ reduces to the conformal Killing vector equation in the time symmetric case ($K_{ab}=0$).
Expressed in terms of these tensors the elliptic equation takes the form $$\label{elliptictensor}
D^b\zeta_{ab}-\tfrac{1}{\sqrt{2}}\mbox{i}\epsilon_{acd}K^{bc}\zeta_b{}^d=0.$$
Let $\kappa_a\in H^\infty_{3/2}$ be the solution to with the asymptotics $$\begin{aligned}
\kappa_i ={}&
\mp\frac{\sqrt{2}E}{3m}\left (1+\frac{2E}{r}\right)x_i
\pm\frac{2\mbox{i}}{3m}\left(1
+\frac{4E}{r}
-\frac{m^2+2(n\cdot p)^2}
{\sqrt{m^2+(n\cdot p)^2}r}
\right)\epsilon_i{}^{jk}p_j x_k
+o_\infty(r^{-1/2}),\end{aligned}$$ at each end, where $p_\mu=(E, p_i)$ is the ADM-4 momentum, $m\equiv\sqrt{p^\mu p_\mu}$, and $n \cdot p=r^{-1}x^i p_i$. The metric and extrinsic curvature are assumed to have the asymptotics and respectively.
The integrand in is $$\mathfrak{J}\equiv\xi_{ABCD}\hat\xi^{ABCD}=\zeta_{ab}\bar\zeta^{ab}.$$
From the equation $$\begin{aligned}
\sigma_a{}^{AB}\sigma_b{}^{CD}\Psi_{(ABC}{}^F\kappa_{D)F}&=
\tfrac{1}{\sqrt{2}}\mbox{i}\epsilon_{cd(a}C_{b)}{}^d\kappa^c.\end{aligned}$$ we get the integrand for the $I_1$ part of the invariant $$\mathfrak{I}_1\equiv\Psi_{(ABC}{}^F\kappa_{D)F}\hat\Psi^{ABCP}\hat\kappa^D{}_{P}=
-\tfrac{1}{2}C^{bc}\bar C_{bc}\kappa^a\bar\kappa_a
+\tfrac{1}{2}C_b{}^c\bar C_{ac}\kappa^a\bar\kappa^b
+\tfrac{1}{4}C_a{}^c\bar C_{bc}\kappa^a\bar\kappa^b.$$
In order to discuss the integrand of $I_2$ we introduce the spinor $\Sigma_{ABCD}\equiv \nabla_{(A}{}^F\Psi_{BCD)F}$, and its tensor equivalent $\Sigma_{ab}=\sigma_a{}^{AB}\sigma_b{}^{CD}\Sigma_{ABCD}$. One finds that $$\begin{aligned}
0&=\sigma_a{}^{AB}\nabla^{CD}\Psi_{ABCD}=D^bC_{ab}-\tfrac{\mbox{i}}{\sqrt{2}}\epsilon_{acd}C^{bc}K_b{}^d, \\
\Sigma_{ab}&=\tfrac{\mbox{i}}{\sqrt{2}}\epsilon_{df(a}D^fC^d{}_{b)}
+\tfrac{1}{2}C^{cd}K_{cd}h_{ab}+C_{ab}K^f{}_f
-\tfrac{3}{2}C^c{}_{(a}K_{b)c}.\end{aligned}$$ The integrand for $I_2$ is given by $$\begin{aligned}
\mathfrak{I}_2={}&(3\kappa_{(A}{}^F\Sigma_{BCD)F}+\Psi_{(ABC}{}^F\xi_{D)F})
(3\hat\kappa^{AP}\hat\Sigma^{BCD}{}_P+\hat\Psi^{ABCP}\hat\xi^D{}_P)
\nonumber\\
={}&-\tfrac{9}{2}\Sigma^{bc}\bar\Sigma_{bc}\kappa^a\bar\kappa_a
+\tfrac{9}{2}\Sigma_b{}^c\bar\Sigma_{ac}\kappa^a\bar\kappa^b
+\tfrac{9}{4}\Sigma_a{}^c\bar\Sigma_{bc}\kappa^a\bar\kappa^b
+\tfrac{3}{2}\bar\Sigma_{bc} C^{bc}\bar\kappa^a\zeta_a
-\tfrac{3}{4}\bar\Sigma_{ac} C_b{}^c\bar\kappa^a\zeta^b \nonumber \\
&-\tfrac{3}{2}\bar\Sigma_{bc} C_a{}^c\bar\kappa^a\zeta^b
+\tfrac{3}{2}\Sigma_{bc}\bar C^{bc}\kappa^a\bar\zeta_a
-\tfrac{3}{4}\Sigma_{ac}\bar C_b{}^c\kappa^a\bar\zeta^b
-\tfrac{3}{2}\Sigma_{bc}\bar C_a{}^c\kappa^a\bar\zeta^b
+\tfrac{1}{2}C^{bc}\bar C_{bc}\zeta^a\bar\zeta_a \nonumber \\
&+\tfrac{1}{2}C_b{}^c\bar C_{ac}\zeta^a\bar\zeta^b
+\tfrac{1}{4}C_a{}^c\bar C_{bc}\zeta^a\bar\zeta^b.\end{aligned}$$ The complete invariant is given by $$I=\int_\mathcal{S}(\mathfrak{J}+\mathfrak{I}_1+\mathfrak{I}_2)\mbox{d}\mu.$$
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[^1]: E-mail address: [[email protected]]{}
[^2]: E-mail address: [[email protected]]{}
[^3]: Here and in what follows, for a time translation it is understood a Killing vector which in some asymptotically Cartesian coordinate system has a leading term of the form $\partial_t$.
[^4]: The idea of using the spatial part of spinorial equations to characterise slices of particular spacetimes is not new. In [@Tod84] the spatial twistor equation has been used to characterise slices of conformally flat spacetimes. See also [@BeiSza97].
[^5]: The sharp conditions for a second order elliptic operator to be asymptotically homogeneous are that $$a_2^{ij} \in H^\infty_\delta, \quad a_1^i \in H^\infty_{\delta-1},
\quad a_0 \in H^\infty_{\delta-2},$$ for $\delta <0$. As one sees, our operator $L$ satisfies these conditions with a margin.
[^6]: The hypotheses in [@ChrOMu81] are much weaker than the ones presented here. The adaptation to the smooth setting has been chosen for simplicity.
|
---
abstract: 'We report a novel insulator-insulator transition arising from the internal charge degrees of freedom in the two-dimensional quarter-filled organic salt $\beta$-([*meso*]{}-DMBEDT-TTF)$_2$PF$_6$. The optical conductivity spectra above $T_c = 70$ K display a prominent feature of the dimer-Mott insulator, characterized by a substantial growth of a dimer peak near 0.6 eV with decreasing temperature. The dimer-peak growth is rapidly quenched as soon as a peak of the charge order shows up below $T_c$, indicating a competition between the two insulating phases. Our infrared imaging spectroscopy has further revealed a spatially competitive electronic phases far below $T_c$, suggesting a nature of quantum phase transition driven by material-parameter variations.'
author:
- 'Ryuji Okazaki$^{1,\ast}$'
- Yuka Ikemoto$^2$
- Taro Moriwaki$^2$
- 'Takahisa Shikama$^{3}$'
- 'Kazuyuki Takahashi$^{3,4}$'
- 'Hatsumi Mori$^{3}$'
- 'Hideki Nakaya$^{5}$'
- 'Takahiko Sasaki$^{5}$'
- 'Yukio Yasui$^{1,6}$'
- 'Ichiro Terasaki$^{1}$'
title: 'Optical Conductivity Measurement of a Dimer-Mott to Charge-Order Phase Transition in a Two-Dimensional Quarter-Filled Organic Salt Compound'
---
=10000
Organic molecular conductors exhibit complex electronic phase diagram owing to their unique internal degrees of freedom coupled with correlation effects. Among them, the quasi two-dimensional (2D) quarter-filled salts $R_2X$, where $R_2$ is a dimer organic molecule and $X$ is a monovalent anion, are attracting much interest [@Seo06; @Hotta12]. In the weakly dimerized materials, the correlated carrier is localized on the molecular site due to long-range nature of Coulomb interaction, leading to charge order as seen in $\theta$-(ET)$_2$RbZn(SCN)$_4$ [@Miyagawa00] and $\alpha$-(ET)$_2$I$_3$ [@Takano01] \[ET=bis(ethylenedithio)-tetrathiafulvalene\]. On the other hand, under strong dimerization, the hole is localized on the dimer to act as a Mott insulator (the dimer-Mott insulator), as realized in $\kappa$-(ET)$_2X$, where $X$=Cu$_2$(CN)$_3$, Cu\[N(CN)$_2$\]Cl.
Recently, intriguing behaviors have been found in materials located near the border of such insulating phases. In the dimer-Mott insulator $\kappa$-(ET)$_2$Cu$_2$(CN)$_3$, the localized hole behaves as a magnetic dipole but shows no magnetic order down to very low temperatures, leading to a novel spin-liquid state [@Shimizu03; @SYamashita08; @MYamashita09]. Interestingly, in sharply contrast to conventional Mott insulators, this dimer Mott insulator exhibits unusual temperature and frequency variations of dielectric constant [@Abdel]. This experimental fact strongly suggests that the ET dimer is *electrically* polarized owing to the intra-dimer charge degree of freedom, which Hotta called “dipolar liquid” [@Hotta10]. The dielectric anomaly is also observed in the dimer-Mott insulators $\kappa$-(ET)$_2$Cu\[N(CN)$_2$\]Cl [@kCl] and $\beta '$-(ET)$_2$ICl$_2$ [@bICl]. These results have revealed that the instability to electric dipole order (charge order) is hidden in the dimer-Mott insulators [@Naka10]. The collective mode of the electric dipole order is theoretically calculated to be gapless or extremely narrow-gapped [@gapless; @Naka12], and can affect the low-temperature specific heat and thermal conductivity [@SYamashita08; @MYamashita09]. On the other hand, in these dimer-Mott insulators, there is no direct spectroscopic evidence of charge disproportionation [@Shimizu06; @Sed12; @Tom12], which can be sensitively probed through optical conductivity [@rev04; @rev12] or nuclear magnetic resonance measurements [@Takahashi06]. These results indicate that the observed dielectric anomaly is not simply attributed to charge order, and then raise a crucial question whether the instability toward the charge order exists in such dimer-Mott phase. The aim of this paper is to explore the opposite case where the instability to the dimer-Mott phase is hidden in a well established charge-ordered dimer-type organic conductor. Here we show such a phenomenon in the dimer-type quarter-filled organic salt $\beta$-([*meso*]{}-DMBEDT-TTF)$_2$PF$_6$ through the infrared optical study including a spatial imaging measurement.
The polarized reflectivity spectra in a large area ($\sim 400 \times50$ $\mu$m$^2$) on the sample surface were measured for energies between 90 meV and 1.4 eV using a Fourier transform infrared spectrometer (FTIR) equipped with an infrared microscope. The details of measurement area will be shown later. The sample was fixed with a conductive carbon paste on the cold head of helium flow-type refrigerator. The sample cooling rate was about 1 K/min. We used a standard gold overcoating technique for measuring the reference spectrum at each temperature. The complex optical conductivity is obtained from the Kramers-Kronig (KK) analysis. Standard extrapolation of $\omega^{-4}$ dependence was employed above 1.4 eV. In low energies, we extrapolated the reflectivity using several methods, but the peak shapes and positions are negligibly affected. Infrared-imaging measurements using a synchrotron radiation (SR) light were performed at BL43IR, SPring-8, Japan [@HKimura04; @SKimura04]. We measured the positional dependence of the $c^*$-axis-polarized local reflectivity spectra on a crystal surface of $1400\times 100$ $\mu$m$^2$ by using an FTIR for energies between 0.1 eV and 0.8 eV. High spatial resolution of $\sim$ 10 $\mu$m was achieved with high-brilliance SR light (spot diameter of 10 $\mu$m) and an infrared microscope equipped with a precision [*xy*]{}-scanning stage.
![(Color online) (a) The two-dimensional (2D) molecular plane of $\beta$-([*meso*]{}-DMBEDT-TTF)$_2$PF$_6$, where DMBEDT-TTF = 2-(5,6-dihydro-1,3-dithiolo\[4,5-[*b*]{}\]\[1,4\]dithiin-2-ylidene)-5,6-dihydro-5,6-dimethyl-1,3-dithiolo\[4,5-[*b*]{}\]\[1,4\]dithiin. Hydrogen atoms are omitted for clarity. Two donor molecules circled by the dotted ellipsoid are dimerized. (b) A checkerboard-type charge-ordered state below $T_c = 70$ K. (c) A dimer-Mott insulating state above $T_c$. ](fig1.eps){width="1\linewidth"}
The single crystals of $\beta$-([*meso*]{}-DMBEDT-TTF)$_2$PF$_6$ were grown by the electrochemical method [@KimuraCC], where DMBEDT-TTF stands for 2-(5,6-dihydro-1,3-dithiolo\[4,5-[*b*]{}\]\[1,4\]dithiin-2-ylidene)-5,6-dihydro-5,6-dimethyl-1,3-dithiolo\[4,5-[*b*]{}\]\[1,4\]dithiin. The crystal structure is composed of the stacking of conducting donor layers separated by insulating anionic ones [@KimuraCC]. In the conducting layer, two donor molecules are weakly dimerized as shown in Fig. 1(a). The x-ray diffraction [@KimuraJACS] as well as the infrared and Raman studies [@Tanaka08] clearly resolve a charge disproportionation below $T_c = 70$ K. The charge ordering pattern is of checkerboard type \[Fig. 1(b)\] [@KimuraJACS], which can be regarded as an antiferro-type electric dipole order, contrast to stripe-type charge ordered states in other 2D quarter-filled salts [@Takahashi06].
![(Color) The real part of optical conductivity $\sigma_1$ of $\beta$-([*meso*]{}-DMBEDT-TTF)$_2$PF$_6$ single crystal measured in the large area on the sample surface. (a) Temperature-dependent $\sigma_1$ spectra measured with polarization parallel to the [*c*]{}\* axis. The dc conductivity $\sigma_1(\omega \to 0)$ is plotted with the circles. (b) Expanded view of a temperature-dependent vibrational molecular modes around 180 meV in $\sigma_1$ spectra measured with polarization parallel to the [*b*]{}\* axis.](fig2.eps){width="1\linewidth"}
Figure 2(a) displays the real part of the optical conductivity spectra $\sigma_1$ obtained from the KK transformation of the reflectivity spectra measured in the large area on the sample surface with polarization parallel to the conducting [*c*]{}\* axis. A low-energy spectrum below 0.2 eV is gradually enhanced with decreasing temperature above $T_c$ but is suddenly suppressed and replaced by a sharp peak structure at $\hbar\omega_{\rm CO} \simeq$ 0.2 eV at $T_c$. This indicates that the formation of charge order drastically modifies the electronic structure in a wide energy range and such spectroscopic feature has also been observed in other charge-ordered material [@Ivek11]. The vibrational molecular modes in $\sigma_1$ measured with polarization parallel to the insulating [*b*]{}\* axis provide evidence for charge order below $T_c$ \[Fig. 2(b)\], consistent with previous results [@Tanaka08]. As shown in Fig. 2(b), the $\nu_{14}$ mode peak (out-of-phase stretching mode of ring C=C) of DMBEDT-TTF$^{0.5+}$ at $\hbar\omega_{\rm v} \simeq181$ meV splits into two bands (charge-rich site at $\hbar\omega_{\rm v}^{\rm R} \simeq178$ meV and charge-poor site $\hbar\omega_{\rm v}^{\rm P} \simeq185$ meV) owing to the charge disproportionation below $T_c$.
Firstly we discuss the high-temperature phase above $T_c$. The distinct feature newly found in $\sigma_1$ is a pronounced peak structure at $\hbar\omega_{\rm dimer} \sim 0.6$ eV, which exhibits a strong enhancement with lowering temperature from 300 K down to $T_c$. This mid-infrared peak can be assigned to a dimer peak, a transition from bonding to anti-bonding orbitals of dimerized molecules, as observed in $\kappa$-type dimerized ET salts [@Faltermeier07]. Most importantly, the dimer-peak intensity is enhanced with decreasing temperature in the dimer-Mott insulating phase, while it is reduced in a correlated metallic phase [@Sasaki04]. Thus the observed enhancement of dimer-peak intensity down to $T_c$ strongly indicates that the high-temperature phase in this material should be regarded as a dimer-Mott insulating phase \[Fig. 1(c)\], rather than a conventional metal. Below 200 K, $\sigma_1$ seems to exhibit a Drude-like response below 0.3 eV, but it should show a peak structure centered at a finite energy to connect low $\sigma_{\rm DC}$ values plotted in Fig. 2(a). Such a low-energy response strikingly resembles those of the quarter-filled salt $\theta$-(ET)$_2$I$_3$, in which an incoherent transport is realized at high temperatures [@Takenaka05].
{width="1\linewidth"}
To shed further light on the unusual electronic state in high-temperature dimer-Mott phase of $\beta$-([*meso*]{}-DMBEDT-TTF)$_2$PF$_6$, we discuss the effective carrier number $N_{\rm eff}$ expressed by, $$N_{\rm eff}(\omega_c) = \frac{2m_0V}{\pi e^2}\int_0^{\omega_c}\sigma_1(\omega')d\omega',$$ where $m_0$ is the free electron mass, $e$ is the charge of an electron, and $V$ is the volume occupied by one formula unit of $\beta$-([*meso*]{}-DMBEDT-TTF)$_2$PF$_6$. To evaluate the low-energy spectral weight, the cut-off energy $\hbar\omega_c$ was adopted to be 0.3 eV at which the spectra exhibit a minimum above $T_c$ \[Fig. 2(a)\]. Figure 3 shows the temperature variations of $N_{\rm eff}$ obtained using several extrapolation methods for reflectivity spectra. Note that difference in the extrapolation methods is negligible in following discussion. Now two important features are noticed: firstly, $N_{\rm eff}$ is increased with lowering temperature above $T_c$. This indicates that a temperature-dependent transfer energy probably due to a shrinkage of the sample volume contributes to the conductive nature in this phase. This differs from the situation in conventional metals, in which temperature variation of resistivity is mostly governed by the reduced scattering rate. Secondly, $N_{\rm eff}$ is significantly smaller than unity, showing that the low-energy metallic weight is about 10% of what is expected in conventional metals. We stress that this $N_{\rm eff}$ involves a sizable contribution from high-energy transitions including the dimer peak as seen in Fig. 2(a), and therefore the contribution from conduction electrons should be smaller than the $N_{\rm eff}$ values shown in Fig. 3. Furthermore, the magnitudes of $\sigma_1$ and the resulting $N_{\rm eff}$ in $\beta$-(*meso*-DMBEDT-TTF)$_2$PF$_6$ are roughly one order smaller than those in $\theta$-(ET)$_2$I$_3$ [@Takenaka05] and such considerable suppression of low-energy spectral weight has been observed in the dimer-Mott insulator $\kappa$-(ET)$_2$Cu\[N(CN)$_2$\]Cl [@Dumm09] and the Mott insulating phase of NiS$_{2-x}$Se$_x$ [@Perucchi09]. These results capture an insulating nature at the high-temperature phase in $\beta$-(*meso*-DMBEDT-TTF)$_2$PF$_6$, and thus indicate that the charge order transition at $T_c$ is *a transition from dimer-Mott to charge-order phase*.
![(Color) Spatially-competing charge-order and dimer-Mott insulating states in $\beta$-([*meso*]{}-DMBEDT-TTF)$_2$PF$_6$. (a-d) The vibrational molecular modes near 180 meV in the local reflectivity spectra $R(\omega)$ measured with polarization parallel to the [*c*]{}\* axis. On each panel, 60 local reflectivity spectra, which were measured in the region surrounded by solid-line rectangular box (from $x_0$ to $x_1$ point) shown in (e), are displayed. These spectra are separated by 20 $\mu$m and span a total distance of 1.2 mm as represented with the vertical offset. (e-k) The reflectivity-ratio mapping on the crystal surface of $1400\times 100$ $\mu$m$^2$. The color scale shows the reflectivity ratio $R$(184.5 meV)$/R$(181 meV). The right-side white region is a gold-coated area for measuring the reference spectra. The dashed-line rectangular box in (e) shows a region in which the large-area reflectivity measurements were performed.](fig4.eps){width="1\linewidth"}
Next let us discuss the low-temperature charge-ordered phase. As seen in Fig. 2(a), the growth of dimer peak is completely quenched by the formation of charge order below $T_c$, indicating a competitive nature between dimer-Mott and charge-order phases. An intriguing question is how these insulating phases compete in real space. Here we show the positional dependence of local reflectivity spectrum. Figures 4(a-d) show the vibrational molecular modes near 180 meV in the local reflectivity $R(\omega)$ measured at several temperatures. Each panel displays 60 local reflectivity spectra measured at from $x_0$ to $x_1$ points shown in Fig. 4(e). As seen in Fig. 4(a), reflectivity spectra are spatially homogeneous at $T_c$: almost all spectra possess the single peak near 181 meV (red-color spectra) as expected in the dimer-Mott phase \[Fig. 2(b)\]. Charge ordering characterized by the split peak at 178.2 and 184.5 meV emerges at a tiny portion inside the crystal as shown by the blue-color spectra. Meanwhile, the spectra measured below $T_c$ \[Figs. 4(b-d)\] are obviously inhomogeneous and can be sorted into two groups, red-color spectra having one single peak originating from the dimer-Mott insulating state and blue-color spectra with split peaks from the charge-ordered state.
Here we evaluate the reflectivity ratio $R$(184.5 meV)$/R$(181 meV) at each point, which gives a relative strength between the charge-ordered state with 184.5-meV peak and the dimer-Mott state with 181-meV peak, and plot its spatial distribution in Figs. 4(e-k). The red- and blue-color areas indicate the dimer-Mott and the charge-ordered states, respectively. Note that the large-area reflectivity spectra shown in Figs. 2(a) and (b) were measured in the rectangular box surrounded by dashed line in Figs. 4(e), from which the charge order appears just below $T_c$. In contrast to spatially-homogeneous spectra above $T_c$, the low-temperature spectra are highly inhomogeneous. Note that there is negligible temperature gradient inside the crystal [@sup]. The observed inhomogeneity does not originate from the phase separation in the hysteresis region near first-order phase transitions since the resistive hysteresis loop is closed within 10 K near $T_c$ as shown in the inset of Fig. 3, while the observed inhomogeneity survives far below $T_c$. Indeed, the present spatial pattern does not depend on the past environment and shows no hysteresis [@sup]. An extrinsic stress effect is also excluded [@sup]. We note that our results are obtained at ambient pressure, in sharply contrast to spatial inhomogeneity observed only in pressure [@Tanaka08].
Below $T_c$, the charge-ordered state gradually invades the dimer-Mott insulating state with lowering temperature, significantly different from conventional phase transitions occurring at finite temperatures, at which a high-entropy phase at high temperatures is immediately replaced by a low-temperature low-entropy phase. In this material, most surprisingly, the dimer-Mott state survives even at 10 K far below $T_c$. This experimental fact strongly indicates that the phase transition in this material is not driven by entropy term in the free energy. But rather, this transition seems to occur, when the materials parameters reach a critical value through their temperature variation. We suggest that this type of transition is of quantum nature in the sense that the transition is driven by the materials parameters. In the present system, it has been found that the interdimer transfer integral is doubly increased from room temperature to 11.5 K [@unp]. Such a considerable change of interdimer integral drives a phase transition from dimer-Mott to antiferro-type electric dipole order [@Dayal]. Now the phase competition expands an inhomogeneous region near the border between those two insulating ground states in the parameter space [@Burgy01; @Dagotto05]. The present compound may locate near the border at low temperatures.
Here we stress that the observed inhomogeneity well explains previous results. While the peak splitting in the local optical spectroscopy is abrupt [@Tanaka08], the superlattice intensity of bulk x-ray measurement exhibits a gradual increase with temperature [@KimuraJACS]. Our results show that the gradual increase originates from the volume-fraction change of charge-ordered state. Below $T_c$, the magnetic susceptibility sets to a finite value [@unp], indicating the survived dimer-Mott state inside the crystal, well consistent with the present results.
Let us finally discuss the nucleation mechanism of charge order. In the previous samples, the superlattice peak intensity and the resistivity are gradually increased below 90 K [@KimuraCC; @KimuraJACS], while a recent sample of higher quality shows a sharp increase of resistivity at $T_c$ [@unp]. This difference may originate from the sample quality. Also, the seeding position of charge order depends on sample [@sup]. Thus we speculate that the seeding is unavoidable crystalline imperfections, as also indicated from other competing correlated system [@Kolb04]. Similar nucleation has also been proposed in a ferroelectric relaxor: a polar domain is created near the nano-sized chemically ordering regions [@Fu09]. After the nucleation, the charge-ordered state is gradually expanded into the whole region with temperature. In the present compound, $dT_c(P)/dP$ ($P$: pressure) is negative [@Tanaka08], indicating that the volume of charge-ordered state is larger than that of dimer-Mott state. Thus, the induced charge-ordered state pressurizes the dimer-Mott state near the boundary locally, leading to $T_c$ reduction in such a boundary region. This indicates a large energy to move the boundary, that may cause the observed macroscopic inhomogeneity.
In summary, the infrared measurements on $\beta$-([*meso*]{}-DMBEDT-TTF)$_2$PF$_6$ reveal anomalous phase transition phenomena from dimer-Mott to charge-order state. We suggest a quantum nature of this transition driven by variations of temperature-dependent materials parameters, which possibly induce a spatial inhomogeneity owing to competitive nature between two insulating states. This quantum nature might be ubiquitous among organic systems which exhibit phase transitions at high temperatures, where the materials parameters considerably change with temperature.
We thank Y. Nogami, V. Robert, H. Seo, Y. Suzumura, H. Taniguchi and M. Tsuchiizu for fruitful discussion. The imaging experiments using synchrotron radiation were performed at BL43IR in SPring-8 with the approvals of JASRI (No. 2011B1221, 2011B1232, 2012A1082, 2012A1141, 2012B1352, 2012B1223). This work was supported by a Grant-in-Aid for Scientific Research on Innovative Areas “Molecular Degrees of Freedom” and “Heavy Electrons” from MEXT, Japan.
Email: [email protected]
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$ $
Supplemental Material
=====================
Temperature homogeneity
-----------------------
Figures 5(a) and (b) show the positional dependence of the dimer-peak energy at 80 and 70 K, respectively, obtained from the same sample shown in the main manuscript. Figure 5(c) depicts the positional dependence of the dimer-peak energy in the dotted outlined area shown in Fig. 5(b). The dimer-peak energy is spatially homogeneous above 70 K. This clearly shows that the sample temperature is also spatially homogeneous above 70 K, because the dimer-peak energy exhibits strong temperature dependence at the high-temperature phase as seen in Fig. 2(a) in the main manuscript. In fact, the 80-K and 70-K results in Fig. 5 are explicitly distinguished with each other. If there is a temperature gradient inside this sample, a position-dependent dimer peak energy will be obtained as shown by the dashed lines in Fig. 5(c). On the other hand, our experimental results show no spatial variations of the dimer peak energy. This evidences that our sample has negligible temperature gradient.
Sample dependence
-----------------
In Fig. 6, we show the imaging data measured on other single crystals. In the sample \#2, the blue-color charge-order state emerges from the left side, indicating that the gold deposition for the reference measurement does not affect the observed inhomogeneity. In the sample \#3, the charge-order state appears from the left and right sides. These samples show spatial inhomogeneity below 70 K, indicating that the inhomogeneity is an intrinsic property in $\beta$-([*meso*]{}-DMBEDT-TTF)$_2$PF$_6$.
Extrinsic stress effect
-----------------------
In $\kappa$-(ET)$_2$Cu\[N(CN)$_2$\]Cl, grease coating induces a trace of superconductivity due to the extrinsic stress originating from the difference of expansion coefficients between samples and grease [@S_coat]. Here we fixed the sample on the cold head by using the carbon paste, which may affect the transition temperature to charge ordering because the external pressure reduces $T_c$ in $\beta$-([*meso*]{}-DMBEDT-TTF)$_2$PF$_6$ [@S_Tanaka08]. To clarify this extrinsic effect, we fixed a part of the crystal on the cryostat by using paste, and measured the positional variation of local reflectivity spectrum. Figure 7 shows the local reflectivity spectra measured with 50-$\mu$m interval on sample \#4. The charge order state with split peak was developed from the fixed area, clearly showing that the survived dimer-Mott state far below $T_c$ does not originate from the stress effect due to the carbon paste. Moreover, the charge order appears just below $T_c$ in all samples we measured, excluding such extrinsic stress effect.
Hysteresis
----------
The present spatial pattern does not depend on the past environment and shows no hysteresis. The experiment shown in the main text was done as follows: the sample was firstly cooled-down to 30 K, and then the mapping data was measured from 30 K to 80 K with heating. Then, the sample was cooled-down from 80 K to 50 K, and 50-K data was measured again. After that, 10-K data was measured. Figures 8(a) and (b) show the reflectivity-ratio mapping measured at 50 K in the first heating and second cooling processes, respectively. Similar pattern is found in both results. Figure 9 displays the reflectivity-ratio mapping measured on the same sample in the main paper with fast scanning rate (40 minutes per one imaging). The results are almost same as those measured with slow scanning rate (3 hours per one imaging) shown in Fig. 4 (main paper), indicating that the sample is essentially in equilibrium at each measurement temperature.
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abstract: 'I describe how superstring theory may violate spin-statistics in an experimentally observable manner. Reviewing the basics of superstring interactions and how to utilize these to produce a statistical phase, I then apply these ideas to two specific examples. The first is the case of heterotic worldsheet linkings, whereby one small closed string momentarily enlarges sufficiently to pass over another, producing such a statistical phase. The second is the braneworld model with noncommutative geometry, whereby matter composed of open strings may couple to a background in which spacetime coordinates do not commute, modifying the field (anti)commutator algebra. I conclude with ways to sharpen and experimentally test these exciting avenues to possibly verify superstring theory.'
---
Spin-Statistics Violations in Superstring Theory
**Mark G. Jackson**
Particle Astrophysics Center and Theory Group
Fermi National Accelerator Laboratory
Batavia, Illinois 60510
*[email protected]*
Introduction
============
A principle which has been very well-tested at low precision and energies is the Spin-Statistics Theorem (SST). This states that given the assumptions of locality, Lorentz invariance and the vacuum being the lowest-energy state for a unitary point-particle field theory in 3+1 dimensions, integral-spin particles must be in a completely symmetric (‘bosonic’) wavefunctions whereas half-integral spin particles must be in completely antisymmetric (‘fermionic’) wavefunctions. Despite many attempts at a simple proof (for an extensive review see [@Duck:1998cp]) there is none known, and so its validity is usually simply assumed when quantizing field modes. This is done by imposing different (anti)commutators for the creation/annihilation operators: $$\label{commut}
{\rm bosons:} \ [a_{\bf k},a^\dagger_{\bf p} ] = \delta_{{\bf k},{\bf p}}, \hspace{0.5in} {\rm fermions:} \ \{ b_{\bf k},b^\dagger_{\bf p} \} = \delta_{{\bf k},{\bf p}} .$$ There are a variety of ways these relationships could be modified, as described in a review by Greenberg [@Greenberg:2000zy], each of which requires relaxing at least one of the assumptions of locality, Lorentz invariance or the idea of a point-particle field theory altogether.
Such violations of spin-statistics are theoretically interesting and could have dramatic physical and even cosmological consequences [@Jackson:2007tn], but ideally they should be motivated by a UV-complete theory predicting such violations. The leading such model, superstring theory, is fundamentally based upon extended objects and so clearly has the potential to produce such violations. These could appear either at high energies or perhaps suppressed by some small but nonzero parameter in the theory.
In this article I will summarize two ways in which such violations might be produced in superstring theory. $\S$2 contains a review of superstring interactions and how this produces a statistical phase. In $\S$3 I present one specific way to possibly violate spin-statistics in the heterotic string theory through worldsheet linkings, and in $\S$4 a second method relying upon braneworlds and noncommutative geometry. In $\S$5 I will offer some concluding remarks.
Superstring Interactions
========================
Both types of potential string theory spin-statistics violations come about from interactions with a gauge field, so we will briefly review this important issue here.
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Point Particles, Aharonov-Bohm, and Anyons
------------------------------------------
Point particles naturally couple to the 1-form gauge field $A_\mu$ via an interaction on their worldline $X^\mu(l)$, $$S = q \int dl \ {\dot X}^\mu A_\mu.$$ This action is then combined with kinetic terms and inserted into a path integral over $X$ and $A$ to calculate scattering amplitudes, $$\mathcal A( \cdots) = \int \left[ \mathcal D A \right] \left[ \mathcal DX \right] e^{i S[A,X]} ( \cdots)$$ where the $\cdots$ indicate field insertions. Aharonov and Bohm first observed how such an interaction could be used to modify statistical phases in 2+1 dimensions [@Aharonov:1959fk]. Consider two interacting particles, where the second one sources $A$ to produce some localized magnetic flux equal to $\Phi$, which in a particular gauge can be written $$A_i = - \frac{\Phi {\epsilon}_{ij} X^j}{4 \pi |X|^2}, \hspace{0.5in} B_{12} = \Phi \delta^2(X) .$$ Now arrange for the first particle to perform a closed circuit around the second, as shown in Figure 1(a). Such a circuit surrounding the source is topologically well-defined in the sense that one could smoothly adjust the path in an arbitrary fashion and yield the same enclosed flux. Although the first particle is never in contact with the flux and so feels no force, it nonetheless induces a relative phase in the path integral $$\Delta \phi = q \int dl \ {\dot X}^i \left( - \frac{\Phi}{4 \pi} \epsilon_{ij} \partial^j \ln |X| \right) = q \Phi.$$ This phase will then modify the statistics (\[commut\]), effectively violating spin-statistics and generalizing bosons and fermions into ‘anyons’ [@Wilczek:1982wy]. In order to evade the SST we had to break Lorentz invariance in the dimensional reduction.
Superstrings and the Kalb-Ramond Field
--------------------------------------
Similarly, superstrings naturally couple to the 2-form Kalb-Ramond gauge field $B_{\mu \nu}$ via the worldsheet interaction [@Rohm:1985jv] $$\label{stringaction}
S = \int d^2 z \ \partial X^\mu {\bar \partial X}^\nu B_{\mu \nu}.$$ This is introduced into a path integral exactly the same as for a point particle and is also capable of producing a phase[^1].
In the case of the string, however, there are now two possibilities, depending on whether the string is closed or open. For a closed string, the topologically invariant quantity is not the amount of flux traversed in a particle’s circular orbit but rather the amount of flux which has passed through the string loop [@Aneziris:1990gm] [@Bergeron:1994ym], as shown in Figure 1(b); this is referred to as a ‘linking’ in the literature. One way this could happen is if a small string passed through a cosmic superstring [@Witten:1985fp] [@Copeland:2003bj] or other extended non-perturbative object [@Hartnoll:2006zb]. While observing such an effect would be impressive, this isn’t quite in the spirit of violating spin-statistics, and we would first have to find such a cosmically extended string! [@Polchinski:2004ia] The second way is to begin with two small closed strings and allow one to momentarily enlarge; this is the mechanism elaborated upon in the next section.
Now specializing to open strings, such a linking is not possible, but there is a phase induced nonetheless. In the case where $B$ is only defined at the string endpoints (as in $\S4$), the action and hence the phase will be equal to $$\Delta \phi = \int d \tau \left. B_{\mu \nu} {\dot X}^\mu {X}^{\prime \nu} \right|^{\pi}_{\sigma = 0} .$$ This is shown in Figure 1(c). Thus both types of string can produce statistical phases. I will now apply these two cases to specific mechanisms leading to possible violation of spin-statistics.
Method \#1: Heterotic Worldsheet Instantons
===========================================
Motivation
----------
Let us consider two strings in 3+1 dimensions, where one is kept at finite size and the other is approximated as pointlike. We have just seen that $B$-field flux passing through the string loop can produce a statistical phase, and so we desire that the second particle/string source this flux. This can be achieved for a particle charged under a gauge field $A$ by using the (topological) interaction term of the form $$\label{bf}
S_{BF} = \int d^4x \ \epsilon^{\mu \nu \rho \lambda} B_{\mu \nu} \partial_\rho A_\lambda$$ which arises naturally from anomaly cancellation in the heterotic string [@Gross:1984dd]. An instanton-like mechanism to utilize this fact was proposed by Harvey and Liu [@Harvey:1990wa], whereby one string will momentarily open up and pass over another string before collapsing again, as shown in Figure 2. The magnitude of this spin-statistics-violating effect was estimated to be of order $e^{-1/\alpha' E^2}$, assuming that one string must open up to at least the de Broglie wavelength of the other. Naively $1/\sqrt{\alpha'} \sim 10^{16}$ GeV and so this is prohibitively too small to be observed, but if $1/\sqrt{\alpha'} \sim$ TeV (as in some recent warped models [@Randall:1999ee] [@Kachru:2003sx] [@Curio:2000dw]) then perhaps this effect is observable at achievable energies and worth revisiting. Note that this intrinsically stringy effect would never show up in the low-energy effective action, which is a Taylor expansion in small $\alpha'$.
{width="1.45in"} {width="1.45in"} {width="1.45in"} {width="1.45in"}
Explicit Instanton Solutions
----------------------------
Since the $BF$ term arises at 1-loop in a perturbative expansion in the string coupling, we would also expect the spin-statistics violation to occur at this order[^2]. While a detailed analysis is currently underway [@hellermanjackson], one could try to estimate the magnitude of the effect by constructing instantonlike linking solutions as attempted in [@Jackson:2008bs]. The complete action for the first string, with momentum $k_1$ and coupled to the Kalb-Ramond 2-form $B$, is $$S_{1} = \frac{1}{2 \pi \alpha'} \int d^2 z \left[ \partial X^\mu {\bar \partial} X^\nu (\delta_{\mu \nu} + 2 \pi \alpha' B_{\mu \nu}) + 2 \pi \alpha' \delta^2(z, {\bar z}) k_1\cdot X \right].$$ Note again that the term containing $B$ is imaginary and thus produces a phase in the path integral, and that we are considering worldsheet instanton solutions so the momentum $k_1$ is real. The action for the second string (which we approximate as a particle) with momentum $k_2$ coupled with charge $q$ to the pseudoanomalous $U(1)$ gauge field $A$ is $$S_2 = \int dl \ \left[ \frac{1}{2 \alpha'} {\dot Y} \cdot {\dot Y} + {\dot Y} \cdot \left( iq A-k_2 \right) \right].$$ Again note that the term coupling to $A$ is imaginary. The spacetime action governing the gauge fields $F=dA$ and ${\tilde H}=dB-A \wedge dA$ is the usual kinetic terms plus the $BF$ coupling in (\[bf\]), $$S_{gauge} = \int d^4 x \left[ \frac{3 \alpha'}{32g^2} {\tilde H}^2 + \frac{1}{4g^2} F ^2 + \theta \epsilon^{\mu \nu \rho \lambda} B_{\mu \nu} \partial_\rho A_\lambda \right]$$ where $g^2$ and $\theta$ are the dimensionless effective 4D couplings after compactification. In the heterotic string theory with different compactifications we can get different values of $\theta = c/32 \pi^2$, where $c$ is determined by the massless fermion content of the theory. In the case of compactification on a Calabi-Yau manifold [@Candelas:1985en] we break $SO(32) \rightarrow SU(3) \times SO(26) \times U(1)$ and then embed the spin connection in the gauge group. This yields $c=-\frac{3}{2} \chi$, where $\chi$ is the Euler number of the Calabi-Yau, and the fermion charges are $q=\pm 1, \pm 2$. The worldsheet and worldline then produce, respectively, $F$ and ${\tilde H}$ flux tubes with width $\sim \sqrt{\alpha'} / \theta g^2$ (this is reversed from the usual case due to the $B F$ term). If we approximate these as infinitesimally thin by taking $\theta g^2 \rightarrow \infty$ we may neglect the gauge kinetic terms and integrate the fields out, resulting in the effective action equal to $$\begin{aligned}
S_{eff} &=& \frac{1}{2 \pi \alpha'} \int d^2 z\ | \partial (X - \alpha' k_1 \ln |z|)|^2 + \frac{1}{2 \alpha'} \int dl |\dot Y - \alpha' k_2|^2 + i \frac{qN}{\theta} \\
&=& \frac{1}{2 \pi \alpha'} \int d^2 z \ \left| \partial (X^\mu - \alpha' k_1^\mu \ln |z|) \right. \\
&& \left. \mp \ i \frac{ \pi qC\alpha' }{\theta} {\epsilon^\mu}_{ \nu \rho \lambda} \partial (X^\nu + \alpha' k_1^\nu \ln |z|) \int dY^\rho \partial^\lambda G(X-Y) \right|^2 \\
&+& \frac{1}{2 \alpha'} \int dl \ |\dot Y - \alpha' k_2|^2 + \frac{qN}{\theta} \left( i \pm C \right)\end{aligned}$$ where $N=\frac{\epsilon^{\mu \nu \rho \lambda }}{4 \pi^2} \int d\Sigma_{\mu \nu}(X) \int dY_\rho \frac{ (X-Y)_\lambda}{|X-Y|^4}$ is the linking number. The equation for $Y$ is trivial and yields $Y(l) = \alpha' k_2 l$, whereas that for $X$ is nontrivial and must first be transformed so that the derivative of $X$ is isolated on the LHS, $$\label{bpsx}
z \partial X^\mu = \alpha' \left( {\delta^\mu}_\nu + i \frac{ qC \alpha'}{4 \theta} {\epsilon^\mu}_{ \nu \rho \lambda} \frac{ X_\perp^\rho {\hat k}_2^\lambda}{ |X_\perp |^3} \right)^{-1} \left( {\delta^\nu}_\gamma - i \frac{ qC \alpha'}{4 \theta} {\epsilon^\nu}_{ \gamma \kappa \sigma} \frac{ X_\perp^\kappa {\hat k}_2^\sigma}{ |X_\perp |^3} \right) k_1^\gamma.$$ This can be shown to have no solutions except in the trivial case $X = \alpha' k_1 \ln |z|$, so that there exist no instanton solutions for the model above.
The reason for this is easy to understand: there is no force acting on the worldsheet to keep it open as it passes over the second string. The most natural way to produce such an interaction is to recall that the (left-moving component of the) first string may also carry a charge $Q$ under the pseudoanomalous $U(1)$ gauge field, $$\begin{aligned}
\label{ds1}
\Delta S_1 &=& \frac{1}{2 \pi} \int d^2 z \ J(z) A_\mu {\bar \partial} X^\mu \\
&\approx& i Q \int d \tau A_\mu {\dot X}^\mu
\end{aligned}$$ where $J$ is the holomorphic $U(1)$ current normalized so that $\oint dz \ J(z) = 2 \pi i Q$. Then the electrostatic repulsion between the two strings would expand the worldsheet to a radius $$R \sim \sqrt{ g^2 q Q \alpha' } .$$ The addition of (\[ds1\]) to the action for strings with $q Q > 0$ could then plausibly produce instanton solutions, and could be analyzed using techniques similar to those employed here. Unfortunately explicit solutions for this model are likely much more difficult to construct due to the necessity of finite coupling.
The Spacetime Effective Action
------------------------------
Since explicit solutions may be difficult to obtain, let us for the moment assume that such solutions with linking number $N$ do exist and are of the Bogomol’nyi-Prasad-Sommerfeld form conjectured above, with a radially-symmetric trajectory producing this linking and an action proportion to $N$, $$\begin{aligned}
\nonumber
X_N &=& \alpha' k_1 \ln |z| + f_N(|z|), \\
\label{sn}
S_N &=& \frac{q}{\theta} \left( i N+ C|N| \right) .\end{aligned}$$ How would such a string theory process actually produce spin-statistics violations from the viewpoint of an effective field theory? On one hand, such a violation is reasonable because spacetime spin-statistics only comes about in an indirect way in string theory, after Gliozzi-Scherk-Olive (GSO)-projection involving worldsheet spin-statistics (which are undisturbed even with spacetime background fields) [@GSW]. Since the entire notion of vertex operators/GSO projection relies fundamentally on the fact that the string is an extended object, it is then reasonable to imagine that it could slightly violate spacetime spin-statistics. On the other hand, string theory produces an effective action of Lorentz-invariant, local, point particle fields, which the SST demands obey usual spin-statistics. How do we resolve this apparent paradox?
To see how, consider the correlation function between two strings in the background described by (\[sn\]), $$\label{a12}
\mathcal A_{12} = \int d^2z \sum_N \ e^{-k_2 \cdot \left[ \alpha' k_1 \ln |z| + f_N(|z|) \right] + iN/\theta - |N|C/\theta}$$ where we have analytically continued back $X \rightarrow iX$ and summed over linking numbers. To see what relation this has to the spacetime propagator, recall that the string propagator $\Delta$ can be represented in terms of worldsheet Hamiltonian $H = (p^2-m^2)$ and momentum $P$, $$\Delta = \frac{1}{2 \pi} \int_0 ^\infty d \tau \ e^{-H\tau} \int _{-\pi} ^\pi d \sigma \ e^{i \sigma P} .$$ Thus the correlation (\[a12\]) represents states contracted via the effective propagator $$\Delta_{eff} = \frac{1}{2 \pi} \int_0 ^\infty d \tau \ e^{-H\tau} \sum_N e^{F_N(H,\tau) + iN/\theta - |N|C/\theta} \int _{-\pi} ^\pi d \sigma \ e^{i \sigma P}$$ where $z = e^{\tau + i \sigma}$ and $F_N$ is some function of both kinetic operators and worldsheet coordinates. As $\theta \rightarrow 0$, only the $N=0$ term contributes and we recover the usual propagator. Regardless of the details of the instantonlike solution, we see the propagator will necessarily be modified into something nonlocal, requiring an infinite number of derivatives. It is probably not coincidence that such solutions likely require coupling to a gauge field via (\[ds1\]), as happens in field theory [@Gulzari:2006sa], allowing one to evade the spin-statistics theorem [@daCruz:2004si]: $${\rm bosons:} \ \frac{1}{(p^2-m^2)^{1+\epsilon}}, \hspace{0.5in} {\rm fermions:} \ \frac{\displaystyle{\not} p+m}{(p^2-m^2)^{1+\epsilon}}, \hspace{0.5in} 0 < |\epsilon| \ll 1.$$ The introduction of nonlocal propagators in string theory has a precedent, but only on very unusual backgrounds [@Taylor:2003gn].
It is important to stress that this nonlocality is not the usual nonlocality on size $\Delta x \sim \sqrt{\alpha '}$ because one has integrated out massive string modes. This spin-statistics-violating nonlocality must be present at arbitrarily large distances, corresponding to adiabatically moving one particle around another[^3].
Experimental Constraints
------------------------
It is difficult to place experimental constraints on such instanton effects without an explicit solution, since it is not clear whether the effect would scale non-perturbatively with energy or some small parameter such as coupling. Therefore let us consider each:
- [**Energy scale**]{} If the instantons scale with energy, as originally believed, it is possible that the LHC might see them if the effective string tension is very low, $\alpha' \sim (10 \ {\rm TeV})^{-2}$. Also, some of the “Transplankian" literature has discussed whether field-theory modifications like this could be observed in inflationary perturbations, such as Kempf’s modified Heisenberg uncertainty [@Easther:2001fi]. This could in principle probe (very) high energy, but there is not a great amount of precision data yet.
- [**Coupling constant**]{} If the instantons instead scale with some small parameter such as a coupling constant, it is more likely that a precision experiment could see this. Ramberg and Snow were the first to precisely measure possible deviations in fermionic spin-statistics [@Ramberg:1988iu], at low energy but incredibly precise experiments detecting forbidden transitions. Their approach has been refined by the VIP (VIolations of the Pauli exclusion principle) Experiment [@VIP] which has thus far constrained the deviation away from Fermi statistics in terms of the Ignatiev-Kuzmin-Greenberg-Mohapatra $\beta$ parameter [@Ignatiev:1987cd] [@Greenberg:1988um] as $$\frac{\beta^2}{2} \leq 4.5 \times 10^{-28}.$$ This bound is expected to improve another 2 orders of magnitude over the next few years due to larger integrated currents. Though the energy scale is low at only 8 keV, the incredible precision means this might be a viable way of detecting superstring-motived violations.
Method \#2: Braneworlds and Noncommutative Geometry
===================================================
Motivation
----------
The second such scenario we will study is that of brane worlds. These are models in which our universe is represented as a D-brane whose worldvolume[^4] contains open strings representing Standard Model particles [@Blumenhagen:2005mu], as shown in Figure 3. The fact that the strings are open means that their boundary conditions are sensitive not just to the naive metric $g_{\mu \nu}$ but rather the metric and Kalb-Ramond $B$-field from before, $$\left. g_{\mu \nu} (\partial - {\bar \partial}) X^\nu + 2 \pi \alpha' B_{\mu \nu} ( \partial + {\bar \partial}) X^\nu \right|_{z={\bar z}} = 0.$$ This generally difficult set of boundary conditions can be simplified by identifying $\theta^{\mu \nu} = \left( B^{-1} \right)^{\mu \nu}$ while simultaneously taking the string tension $\alpha' \sim \sqrt{\epsilon} \rightarrow 0$ and the metric $g_{\mu \nu} \sim \epsilon \rightarrow 0$ [@Seiberg:1999vs]. This limit imposes noncommutative geometry in the sense that fields corresponding to these open strings are now multiplied by the Moyal star product $\star$ defined as $$\label{moyaldef}
\star \equiv e^{- \frac{i}{2} \theta^{\mu \nu} P_\mu P_\nu }$$ which for spatially-dependent fields means it acts as $$\phi(x) \star \Phi(y) = e^{\frac{i}{2} \theta^{\mu \nu} \frac{\partial}{\partial x^\mu} \frac{\partial}{\partial y^\nu} } \phi(x) \Phi(y) .$$ This algebra can be summarized by stating that spatial coordinates fail to commute by a constant $\theta$, $$[{\hat x}^\mu, {\hat x}^\nu] = i \theta^{\mu \nu}.$$ Since the noncommutative coordinates now produce nonlocal interactions it is reasonable that this might violate spin-statistics, especially given the close connection of noncommutative geometry and the Quantum Hall Effect [@Susskind:2001fb] which relies fundamentally on such violations.
{width="3in"}
Spin-Statistics Violations from Noncommutativity
------------------------------------------------
This idea has been studied in a series of a papers [@Chaichian:2002vw] [@Balachandran:2005eb] [@Tureanu:2006pb] which we will now summarize. To see that noncommutative geometry is non-local, consider the equal-time commutator matrix element between the vacuum and a 2-particle state in $d$-dimensions, which should vanish in a local theory since measurements outside the lightcone can’t influence each other: $$\begin{aligned}
&& \langle 0 | \left. [ : \phi(x) \star \phi(x) :, : \phi(y) \star \phi(y) : ] \right|_{x^0 = y^0} | p,p' \rangle \\
&=& - \frac{2i}{(2 \pi)^{2d}} \frac{1}{ \sqrt{\omega_p \omega_{p'} }} \left( e^{-i p'x - i py} + e^{-ipx - ip'y} \right) \int \frac{ d^3 k}{\omega_k} \sin \left[ k \cdot (x-y) \right] \cos \left( \frac{1}{2} k \cdot \theta \cdot p \right) \cos \left( \frac{1}{2} k \cdot \theta \cdot p' \right) .\end{aligned}$$ If $\theta^{i0}=0$ then the integrand is antisymmetric under $k^i \rightarrow -k^i$, and so the correlation vanishes upon integration over the spatial measure $d^3 k$. Thus only if $\theta^{i 0} \neq 0$ could such a nonlocal process occur. Unfortunately, such timelike noncommutative theories are found to violate unitarity, the reason being that they cannot be formulated as field theories decoupled from massive open string modes [@Gomis:2000zz]. This does not mean all hope is lost, however. There is a technical loophole in that *lightlike* noncommutativity, for which $\theta^{\mu \nu} \theta_{\mu \nu} = 0$, produces theories which are indeed unitarity! [@Aharony:2000gz] There has been relatively little work done in this but hopefully the motivation from spin-statistics violations will spark some interest.
One might instinctively guess that the mixing of the coordinates would produce spin-statistics violations, and so this may come as a rather counterintuitive conclusion. Let us verify the matter explicitly for only $\theta^{ij} \neq 0$. Consider a real scalar field $\phi$, $$\label{phi}
\phi(x) = \int \frac{d^3 k}{(2 \pi)^{3/2}} \left( a_{\bf k} e^{-i {\bf k} \cdot {\bf x}} + a^\dagger_{\bf k} e^{i {\bf k} \cdot {\bf x}} \right).$$
For the Moyal $\star$-product we can again choose the spatial representation $P_i \rightarrow - i \partial_i$, in which case it is simply the Fourier phases which are multiplied: $$\begin{aligned}
\nonumber
\phi(x) \star \phi(y) &=& \int d^3 k \ d^3 p \ {\tilde \phi}(k) {\tilde \phi} (p) \left( e^{-ikx} \star e^{-ipy} \right) \\
\label{moyal}
&=& \int d^3 k \ d^3 p \ {\tilde \phi}(k) {\tilde \phi}(p) e^{-ikx - ipy + {1 \over 2} k \theta p}\end{aligned}$$ and the raising and lowering operators will still obey the usual algebra $[a_{\bf k}, a^\dagger_{\bf p}] = \delta_{{\bf k},{\bf p}}$. So from this perspective we get a noncommutative theory but one which respects the usual spin-statistics relation.
In [@Balachandran:2005eb] it is claimed that one could make an alternate choice of the Moyal Star representation, since the Fourier modes ${\tilde \phi} (k)$ of a field $\phi(x)$ also furnish representations of the momentum generators $P^i$: $$P^i {\tilde \phi} (k) = k^i {\tilde \phi} (k) .$$ Given the mode expansion (\[phi\]) this can be interpreted as deformed operators $a_{\bf k}, a^\dagger_{\bf k}$ relative to the undeformed ones $c_{\bf k}, c^\dagger_{\bf k}$, $$a_{\bf k} = c_{\bf k} e^{- \frac{i}{2} p_\mu \theta^{\mu \nu} P_\nu }, \hspace{0.5in} a^\dagger_{\bf k} = e^{\frac{i}{2} p_\mu \theta^{\mu \nu} P_\nu } c^\dagger_{\bf k}$$ which will then produce the following deformed commutation relations $$\begin{aligned}
a_{\bf k} a_{\bf p} &=& e^{-i p \cdot \theta \cdot k} a_{\bf p} a_{\bf k}, \hspace{0.5in} a_{\bf k}^\dagger a_{\bf p}^\dagger = e^{-i p \cdot \theta \cdot k} a_{\bf p}^\dagger a_{\bf k}^\dagger, \\
a_{\bf k} a_{\bf p}^\dagger &=& e^{i p \cdot \theta \cdot k} a_{\bf p}^\dagger a_{\bf k} + 2 E_{\bf k} \delta^3 ({\bf p}-{\bf k}).\end{aligned}$$ A field quantized with these deformed commutation relations will undo the $\star$ operation in (\[moyal\]), and thus render the $S$-matrix identical to that for a standard (commuting geometry) field, suggesting that a noncommutative theory with usual spin-statistics could be interpreted as a commuting theory with modified spin-statistics. In fact this is not true, as detailed in [@Tureanu:2006pb]. Were we to include the Fourier components in the noncommutative field multiplication, there must now be *three* Moyal star multiplications required: $$\phi(x) \star \phi(y) = \int d^3 k \ d^3 p \ {\tilde \phi}(k) \star e^{-ikx} \star {\tilde \phi} (p) \star e^{-ipy} .$$ The first and third $\star$ operations are trivial, but the second will produce the identical result as that obtained in (\[moyal\]). Thus the theory is truly noncommutative, and obeys the standard spin-statistics relations.
Experimental Constraints
------------------------
For this model violations of spin-statistics are parameterized in terms of the noncommutativity parameter $\theta$. Besides the usual constraints on Lorentz violation[@Kostelecky:2002hh] [@Kostelecky:2003fs], there exist bounds on the spatial components as $|\theta^{XY}| \lsim (10^{14} \ {\rm GeV})^{-2}$ from QCD [@Mocioiu:2000ip] and $|\theta^{XY}| \lsim (10 \ {\rm TeV})^{-2}$ from QED [@Carroll:2001ws]. Constant $H = dB$ has also been studied [@Nastase:2006na], finding that it would behave as stiff matter $\rho_H \sim a^{-6}$ but with strange properties such as solitons moving arbitrarily faster than the speed of light, and so the amount of such flux must be limited. Finally, there has been so-called Transplanckian research specifically studying whether noncommutative geometry might be measurable in the cosmic microwave background power spectrum [@Chu:2000ww].
These constraints on the spatial components should provide some constraints on the lightlike components[^5]. However, it was noted in [@Aharony:2000gz] that for lightlike $\theta^{i-}$ there is apparently no meaningful way to parameterize such violation, since the Minkowski square of such a quantity will always be zero by definition! Thus it is difficult to say whether such a background exists, making the theoretical and experimental method to constrain lightlike noncommutativity an important challenge.
Conclusion, Future Directions and Open Questions
================================================
There are several interesting theoretical and experimental issues that need to be addressed in the context of superstring violations of spin-statistics:
1. Does quantum gravity manifest itself as violations of spin-statistics? If so, is there a simple way to encode this new physics? Greenberg [@Greenberg:2000zy] makes the interesting observation that such violations cannot be encoded into an effective action as a “statistics violating term," so perhaps this is one reason we have found it difficult to quantize gravity?
2. Is there a mechanism in string theory to produce such violations, and is it related to the Kalb-Ramond field $B$? Both mechanisms mentioned here directly involve the $B$ field because this is the simplest way to introduce a phase, but are there others? Would a greater understanding of why we don’t observe such $B$-quanta help? Note also that after compactification to 4 dimensions this field is actually an axion (since $d a = \ast d B$), so perhaps this is related to axion physics [@Svrcek:2006yi].
3. Are such violations way beyond any possible experiment, or are they within reach of current technology? (such as precise but low-energy experiments like VIP or less-precise but higher-energy experiments like the LHC). Do the violations scale with energy, fixed parameters, or a lightlike noncommutative parameter? If the latter, how would we parameterize it?
I believe that attempting to answer these questions will prove to be an important step in understanding the relationship between quantum gravity and possibly proving superstring theory.
Acknowledgments
===============
I would like to thank J. Gomis, S. Hartnoll, J. Harvey, S. Hellerman, C. Hogan, A. Kostelecky, J. Lykken, C. Petrascu and D. Tong for useful discussions and collaborations. This work was supported by the DOE at Fermilab.
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[^1]: While the action (\[stringaction\]) is purely imaginary, the string path integral is Euclidean in the sense that it is introduced as $e^{-S}$ and so this action produces a phase just as for point particles.
[^2]: I would like to thank S. Hellerman for discussions on this point.
[^3]: I am grateful to D. Tong for emphasizing this fact.
[^4]: We are ignoring the fact that in order for the matter to admit chiral representations of gauge symmetries, this D-brane must actually be the intersection of two higher-dimensional D-branes, but this is irrelevant for the present discussion.
[^5]: I would like to thank A. Kostelecky for private communication on this point.
|
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abstract: 'Let $f$ be a $C^1$-diffeomorphism and $\mu$ be a hyperbolic ergodic $f$-invariant Borel probability measure with positive measure-theoretic entropy. Assume that the Oseledec splitting $$T_xM=E_1(x) \oplus\cdots\oplus E_s(x) \oplus E_{s+1}(x) \oplus\cdots\oplus E_l(x)$$ is dominated on the Oseledec basin $\Gamma$. We give extensions of Katok’s Horseshoes construction. Moreover there is a dominated splitting corresponding to Oseledec subspace on horseshoes.'
address:
- 'School of mathematics, physics and statistics, Shanghai University of Engineering Science, Shanghai 201620, P.R. China'
- 'School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, P.R. China'
- 'Departament of Mathematics, Soochow University, Suzhou 215006, Jiangsu, P.R. China'
- 'Departament of Mathematics, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200062, P.R. China'
author:
- Juan Wang
- Rui Zou
- Yongluo Cao
bibliography:
- 'bib.bib'
title: 'The approximation of Lyapunov exponents by horseshoes for $C^1$-diffeomorphisms with dominated splitting'
---
[^1]
Introduction
============
Let $f$ be a $C^r$ $(r\geq1)$ diffeomorphism of a compact Riemannian manifold $M$. An $f$-invariant subset $\Lambda\subset M$ is called a *hyperbolic set* if there exists a continuous splitting of the tangent bundle $T_\Lambda M = E^{s}\oplus E^{u}$, and constants $c > 0,\ 0 < \tau < 1$ such that for every $x \in \Lambda$,
1. $d_xf(E^s(x)) = E^s(f(x)),\ d_xf(E^u(x)) = E^u(f(x))$;
2. for all $n \geq 0,\ \|d_xf^n(v)\|\leq c\tau ^n\|v\|$ if $v \in E^s(x)$, and $\|d_xf^{-n}(v)\|\leq c\tau^n\|v\|$ if $v \in E^u(x)$.
A hyperbolic set $\Lambda$ is called *locally maximal*, if there exists a neighbourhood $U$ of $\Lambda$ such that $\Lambda=\bigcap_{n\in\mathbb{Z}}f^n(U)$. Let $\mathcal{M}_f(\Lambda)$ be the space of all $f$-invariant Borel probability measures on $\Lambda$. Let $\mu$ be a hyperbolic ergodic $f$-invariant Borel probability measure on $M$. We say $\mu$ *hyperbolic* if it possesses at least one negative and one positive, and no zero Lyapunov exponents. Let $\Gamma$ be the Oseledec’s basin of $\mu$ (see Theorem \[oseledectheo\]). For $x\in\Gamma$, denote its distinct Lyapunov exponents by $$\lambda_1(\mu)<\cdots<\lambda_s(\mu)<0<\lambda_{s+1}(\mu)<\cdots<\lambda_l(\mu)$$ with multiplicities $n_1, n_2, \cdots, n_l\geq1$ and let $$T_xM=E_1(x)\oplus\cdots\oplus E_s(x)\oplus E_{s+1}(x)\oplus\cdots\oplus E_l(x)$$ be the corresponding decomposition of its tangent space. Denote $E^s=E_1\oplus\cdots\oplus E_s$ and $E^u=E_{s+1}\oplus\cdots\oplus E_l$. In this paper, we consider $r=1$ and the following assumption.
\[assumpdominated\] The splitting $$T_{\Gamma} M=E_1\oplus\cdots\oplus E_s\oplus E_{s+1}\oplus\cdots\oplus E_l$$ is dominated.
We state the main result of this paper:
\[maintheorem\] Let $f: M\rightarrow M$ be a $C^1$ diffeomorphism of a compact Riemannian manifold $M$ and $\mu$ be a hyperbolic ergodic $f$-invariant Borel probability measure on $M$ with positive measure-theoretic entropy $h_\mu(f)>0$. Under Assumption \[assumpdominated\], we have for every small $\varepsilon>0$, there exists a compact set $\Lambda^*\subseteq M$ and a positive integer $m$ satisfying
- $\Lambda^*$ is a locally maximal hyperbolic set and topologically mixing with respect to $f^m$.
- $|h_{top}(f|_\Lambda)-h_\mu(f)|<\varepsilon$ where $\Lambda=\Lambda^*\cup f(\Lambda^*)\cup\cdots\cup f^{m-1}(\Lambda^*)$.
- $\Lambda$ is contained in the $\varepsilon$-neighborhood of the support of $\mu$.
- $d(\nu,\mu)<\varepsilon$ for every $\nu\in\mathcal{M}_f(\Lambda)$, where $d$ is a metric that generates the weak\* topology.
- There is a dominated splitting $T_{\Lambda} M=\widetilde{E}_1\oplus_< \widetilde{E}_2\oplus_< \cdots\oplus_< \widetilde{E}_l$ on $\Lambda$ with $\dim \widetilde{E}_i=n_i$, and $$\label{lyexloup}
e^{[\lambda_{i}(\mu)-6\varepsilon]km}\|u\|\leq \|d_xf^{km}(u)\|\leq e^{[\lambda_{i}(\mu)+6\varepsilon]km}\|u\|$$ for every $x\in\Lambda$, $k\geq 1$ and $0\neq u\in \widetilde{E}_i(x)$, $i=1, \cdots, l$.
This paper is motived by Katok [@Ka80], Avila, Crovisier, Wilkinson [@ACW17] and Cao, Pesin, Zhao [@cpz17]. Katok [@Ka80] proved for a $C^2$ diffeomorphism $f$ preserving an ergodic hyperbolic measure with positive entropy, there exists a sequence of horseshoes, and the topology entropy of $f$ restricted to horseshoes can be arbitrarily close to the measure-theoretic entropy, implying an abundance of hyperbolic periodic points. Mendoza [@Me88] proved that an ergodic hyperbolic SRB measure of $C^2$ surface diffeomorphism can be approximated by a sequence of measures supported on horseshoes and the Hausdorff dimension for horseshoes on the unstable manifold converges to $1$. Katok and Mendoza also elaborated on the related results in the part of supplement of [@KH95]. Avila, Crovisier, Wilkinson [@ACW17] explicitly gave a dominated splitting $T_\Lambda M=E_1\oplus_<\cdots\oplus_<E_l$ on each horseshoe $\Lambda$ and the Lyapunov exponential approximation in each subbundle $E_i$ is obtained over $\Lambda$, for $i=1,2,\cdots,l$. For $C^r (r>1)$ maps, results related to Katok’s approximation were obtained by Chung [@Chung], Gelfert [@Gelfert10] and Yang [@Yang]. For every ergodic invariant measure $\mu$ with positive entropy for $C^{1+\alpha}$ nonconformal repellers, Cao, Pesin, Zhao [@cpz17] constructed a compact expanding invariant set with dominated splitting corresponding to Oseledec splitting of $\mu$, and for which entropy and Lyapunov exponents approximate to entropy and Lyapunov exponents for $\mu$. Then Cao, Pesin, Zhao[@cpz17] used this construction to show the continuity of sub-additive topological pressure and give a sharp estimate for the lower bound estimate of Hausdorff dimension of non-conformal repellers. Lian and Young extended the results of Katok [@Ka80] to mappings of Hilbert spaces [@ly11] and to semi-flows on Hilbert spaces [@ly12]. In the case of Banach quasi-compact cocycles for $C^r (r>1)$ diffeomorphism, for an ergodic hyperbolic measure with positive entropy, Zou and Cao [@zc17] constructed a Horseshoe with dominated splitting corresponding to Oseledec splitting of $\mu$ for cocycles, for which there are entropy and Lyapunov exponents’s approximations.
Gelfert [@Gelfert16] relaxed the smoothness to $C^1$. Gelfert’s results assert the following: let $f$ be a $C^1$ diffeomorphism of a smooth Riemannian manifold, and let $\mu$ be an ergodic hyperbolic $f$-invariant Borel probability measure whose support admits a dominated splitting $T_{{\operatorname{supp}}\mu}M=E^s\oplus_<E^u$. Assume that $(f,\mu)$ has positive measure-theoretic entropy. Then she proved the analogous results of Katok [@Ka80]. In this paper we assume the splitting $$T_{\Gamma} M=E_1\oplus\cdots\oplus E_s\oplus E_{s+1}\oplus\cdots\oplus E_l$$ is dominated. We also prove the existence of locally maximal hyperbolic sets (horseshoes) in a neighborhood of the support set of $\mu$ by using Katok’s technique. Moreover we use some properties of the dominated splitting to show the invariance of the cones. Then we obtain a dominated splitting $T_\Lambda M=\widetilde{E}_1\oplus_<\cdots\oplus_<\widetilde{E}_l$ on each horseshoe $\Lambda$. We also use the properties of $C^1$ nonuniform hyperbolic dynamical systems to prove the approximation of Lyapunov exponents on each subbundle $\widetilde{E}_i$ over $\Lambda$ (see (\[lyexloup\])). We didn’t use the Lyapunov charts and Lyapunov norm in the proof.
The remainder of this paper is organized as follows. In Section 2, we recall some related notations, properties and theorems. Section 3 provides the proof of the main result.
Preliminaries
=============
Let $M$ be a compact Riemannion manifold, and $f$ be a $C^1$ diffeomorphism from $M$ to itself. Let $\dim M$ be the dimension of M. For $x\in M$, we denote $$\| d_xf\|= \sup_{0\neq u \in T_xM}\displaystyle
\frac{\|d_xf(u)\|}{\|u\|},\ \ m(d_xf) = \inf_{0 \neq u \in
T_xM}\displaystyle \frac{\|d_xf(u)\|}{\|u\|}$$ which are respectively called the maximal norm and minimum norm of the differentiable operator $d_xf : T_xM \to T_{fx}M$, where $\|\cdot\|$ is the norm induced by the Riemannian metric $d$ on $M$. Let $n$ be a natural number we define a metric $d_n$ on $M$ by $d_n(x,y)=\max_{0\leq i\leq n-1}d(f^ix,f^iy)$. For any $\rho>0$ and $x\in M$, *an $(n,\rho)$ Bowen ball of $x$* is $$B_n(x,\rho)=\Big\{y\in M: d_n(x,y) < \rho\Big\}.$$ A subset $E$ of $M$ is said to be *$(n,\rho)$-separated with respect to $f$* if $x,y\in E$, $x\neq y$, implies $d_n(x,y)>\rho$.
Dominated splitting
-------------------
Ma$\tilde{\text{n}}\acute{\text{e}}$, Liao and Pliss introduced independently the concept of dominated splitting in order to prove that structurally stable systems satisfy a hyperbolic condition on the tangent map. We recall the definition and some properties of the dominated splitting(see Appendix B in [@BDV]).
\[dominated-splitting\] Let $K \subseteq M$ be an $f$-invariant subset. A $df$-invariant splitting $T_K M = E_1 \oplus E_2 \oplus \cdots \oplus E_k$ of the tangent bundle over $K$ is *dominated* if there exists $N\in \mathbb{N}$ such that for every $i<j$, every $x\in K$, and each pair of unit vectors $u\in E_i(x)$ and $v\in E_j(x)$, one has $$\frac{\|d_xf^N(u)\|}{\|d_xf^N(v)\|}\leq\frac{1}{2}$$ and the dimension of $E_i(x)$ is independent of $x\in K$ for every $i\in\{1,\cdots,k\}$. We denote $T_K M = E_1 \oplus_< E_2\oplus_<\cdots\oplus_<E_k$.
\[conds\]
- We can assume $N=1$ for a smooth change of metric on $M$.
- The dominated splitting $T_K M = E_1 \oplus_< E_2\oplus_<\cdots\oplus_<E_k$ can be extended to a dominated splitting $TM = E_1 \oplus_< E_2\oplus_<\cdots\oplus_<E_k$ over the closure of $K$. See [@BDV] for a proof.
- The dominated splitting $T_K M = E_1 \oplus_< E_2\oplus_<\cdots\oplus_<E_k$ can be extended to a continuous splitting $TM = E_1 \oplus E_2\oplus\cdots\oplus E_k$ in a neighborhood of $K$. See [@BDV] for a proof.
- The dominated splitting is unique if one fixes the dimensions of the subbundles.
- Every dominated splitting is continuous, i.e. the subspaces $E_i(x)$, $i=1$, $2,\cdots, k$, depend continuously on the point $x$.
Preliminaries of Lyapunov exponents
-----------------------------------
We review the Oseledec’s Theorem which contains the definitions of the Oseledec’s basin, the Oseledec’s splitting, the Lyapunov exponents and the multiplicities.
\[oseledectheo\]([@Ose68]) Let $f$ be a $C^1$ diffeomorphism of a compact Riemannian manifold $M$ preserving an ergodic $f$-invariant Borel probability measure $\mu$. Then there exist
- real numbers $\lambda_1(\mu)<\lambda_2(\mu)<\cdots<\lambda_l(\mu)(l\leq \dim M)$;
- positive intergers $n_1, n_2, \cdots, n_l$, satisfying $n_1+\cdots+n_l=\dim M$;
- a Borel set $\Gamma:=\Gamma(\mu)$, called the Oseledec’s basin of $\mu$, satisfying $f\big(\Gamma)=\Gamma$ and $\mu\big(\Gamma\big)=1$;
- a measurable splitting, called the Oseledec’s splitting, $T_xM=E_1(x)\oplus\cdots \oplus E_l(x)$ with $\dim E_i(x)=n_i$ and $d_xf(E_i(x))=E_i(f(x))$, such that $$\lim_{n\to\pm\infty}\frac{\log\|d_xf^n(v)\|}{n}=\lambda_i(\mu),$$ for any $x\in\Gamma$, $0\not=v\in E_i(x)$, $i=1,2,\cdots,l$.
The real numbers $\lambda_1(\mu),\lambda_2(\mu),\cdots,\lambda_l(\mu)$ are called the *Lyapunov exponents*, and $n_1,n_2,\cdots,n_l$ are called the *multiplicities*.
In the result to follow, we will review the Lyapunov exponents as a limit of Birkhoff sums in terms of $d_{(\cdot)} f^N$ for natural numbers $N$ large enough, which is computed in [@ABC11].
\[malyexoffN\](Lemma $8.4$ in [@ABC11]) Let $f$ be a $C^1$-diffeomorphism, $\mu$ be an ergodic invariant probability measure, and $E\subseteq T_{\text{supp}(\mu)}M$ be a $df$-invariant continuous subbundle defined over the support of $\mu$. Let $\lambda_E^+$ be the upper Lyapunov exponent in $E$ of the measure $\mu$. Then, for any $\varepsilon>0$, there exists an integer $N_1(\varepsilon)$ such that, for $\mu$ almost every point $x\in M$ and any $N\geq N_1(\varepsilon)$, the Birkhoff averages $$\frac{1}{kN}\sum_{l=0}^{k-1}\log\|d_{f^{lN}(x))}f^N|_{E(f^{lN}(x))}\|$$ converge towards a number contained in $[\lambda_E^+,\lambda_E^++\varepsilon)$, where $k$ goes to $+\infty$.
The following Lemma is analogous.
\[milyexoffN\] Let $f$ be a $C^1$-diffeomorphism, $\mu$ be an ergodic invariant probability measure, and $E\subseteq T_{\text{supp}(\mu)}M$ be a $df$-invariant continuous subbundle defined over $\text{supp}(\mu)$. Let $\lambda_E^-$ be the lower Lyapunov exponent in $E$ of the measure $\mu$. Then, for any $\varepsilon>0$, there exists an integer $N_2(\varepsilon)$ such that, for $\mu$ almost every point $x\in M$ and any $N\geq N_2(\varepsilon)$, the Birkhoff averages $$\frac{1}{kN}\sum_{l=0}^{k-1}\log m(d_{f^{lN}(x)}f^N|_{E(f^{lN}(x))})$$ converge towards a number contained in $(\lambda_E^--\varepsilon,\lambda_E^-]$, where $k$ goes to $+\infty$.
It is a slight modification of the proof of Lemma \[malyexoffN\]. We omit it here.
$(\rho,\beta,\gamma)$-rectangle of a compact subset
----------------------------------------------------
Let $(f,M)$ be as above and $\mu$ be a hyperbolic ergodic $f$-invariant Borel probability measure on $M$. For $\mu$ almost every $x\in M$, denote the Lyapunov exponents by $$\lambda_1(\mu)<\cdots<\lambda_s(\mu)<0<\lambda_{s+1}(\mu)<\cdots<\lambda_l(\mu),$$ and the corresponding decomposition of its tangent space by $T_xM=E_1(x)\oplus \cdots \oplus E_l(x)$. Denote $E^{s}=E_1\oplus\cdots\oplus E_s$ and $E^{u}=E_{s+1}\oplus\cdots\oplus E_l$. Let $d_s=\dim E^s$, $d_u=\dim E^u$ with $d_s+d_u=\dim M$. Let $I=[-1,1]$, given $x\in M$, we say $R(x)\subseteq M$ is a *rectangle* in $M$ centered at $x$ if there exists a $C^1$-embedding $\Phi_x:I^{\dim M}\to M$ such that $\Phi_x(I^{\dim M})=R(x)$ and $\Phi_x(0)=x$. A set $\widetilde{H}$ is called an *admissible $u$-rectangle* in $R(x)$, if there exist $0<\widetilde{\lambda}<1$, $C^1$-maps $\phi_1,\phi_2: I^{d_u}\to I^{d_s}$ satisfying $\|\phi_1(u)\|\geq\|\phi_2(u)\|$ for $u\in I^{d_u}$ and $\|d\phi_i\|\leq\widetilde{\lambda}$ for $i=1,2$ such that $\widetilde{H}=\Phi_x(H)$, where $$H=\{(u,v)\in I^{d_u}\times I^{d_s}:v=t\phi_1(u)+(1-t)\phi_2(u),0\leq t\leq1\}.$$ Similarly we can define an *admissible $s$-rectangle* in $R(x)$.
\[smallrectangle\] Let $(f,M)$ be as above and $\Lambda\subseteq M$ be compact. We say $R(x)$ is a $(\rho,\beta,\gamma)$-rectangle of $\Lambda$ for $\rho>\beta>0,\gamma>0$, if there exists $\widetilde{\lambda}=\widetilde{\lambda}(\rho,\beta,\gamma)$ satisfying
- $x\in\Lambda$, $B(x,\beta)\subseteq int\ R(x)$ and ${\operatorname{diam}}R(x)\leq\frac{\rho}{3}$.
- If $z,f^mz\in\Lambda\cap B(x,\beta)$ for some $m>0$, then the connected component $C(z,R(x)\cap f^{-m}R(x))$ of $R(x)\cap f^{-m}R(x)$ containing $z$ is an admissible $s$-rectangle in $R(x)$, and $f^m(C(z,R(x)\cap f^{-m}R(x)))$ is an admissible $u$-rectangle in $R(x)$.
- ${\operatorname{diam}}f^k(C(z,R(x)\cap f^{-m}R(x)))\leq\rho e^{-\gamma\min\{k,m-k\}}$ for $0\leq k\leq m$.
Gelfert [@Gelfert16] proved that there is a finite collection of $(\rho,\beta,\gamma)$-rectangles for $(f,M,\mu)$. We only state the lemma here. See Lemma $2$ in [@Gelfert16] for a proof. We also refer to [@KH95] for more information about $(\rho,\beta,\gamma)$-rectangles.
\[existofrectangle\] Let $f$ be a $C^1$-diffeomorphsim of a compact Riemannian manifold $M$ and $\mu$ be a hyperbolic ergodic $f$-invariant Borel probability measure on $M$. Assume that $T_{{\operatorname{supp}}\mu}M=E^{s}\oplus_< E^{u}$ is dominated. Let $\chi:=\lambda(\mu)=\min_{1\leq i\leq l}|\lambda_i(\mu)|$. Given $\rho>0$ and $\delta>0$, then there exists a compact set $\Lambda_H=\Lambda_H(\rho,\delta,\frac{\chi}{2})$ with $\mu(\Lambda_H)>1-\frac{\delta}{3}$, a constant $\beta=\beta(\rho,\delta)>0$ and a finite collection of $(\rho,\beta,\frac{\chi}{2})$-rectangles $R(q_1),R(q_2),\cdots,R(q_t)$ with $q_j\in\Lambda_H$ so that $\Lambda_H\subseteq\bigcup_{j=1}^tB(q_j,\beta)$.
A necessary and sufficient condition for hyperbolic sets
--------------------------------------------------------
Let $T_xM=F(x)\oplus G(x)$ be a splitting of the tangent space at $x\in M$ and let $\theta\in(0,1)$ be small, we define the cones at $x$ with respect to this decomposition of $TM$ by $$C^F_\theta(x)=\big{\{}v+w \in F(x)\oplus G(x): v\in F(x), w\in G(x)\ \text{and}\ \|w\|\leq\theta\|v\|\big{\}},$$ and $$C^G_\theta(x)=\big{\{}v+w\in F(x)\oplus G(x): v\in F(x), w\in G(x)\ \text{and}\ \|v\|\leq\theta\|w\|\big{\}}.$$ We define a projection $\pi_F(x)$ from $T_xM$ to $F(x)$ with the kernel $G(x)$ by $\pi_F(x)(u)=v$, for every $u=v+w\in T_xM$ with $v\in F(x)$ and $w\in G(x)$. Similarly we can define $\pi_G(x)$. The following Theorem is Theorem $6.1.2$ in [@BP07], which gives a characterization of hyperbolic sets in terms of cones.
\[hyperbolicity\] A compact invariant set $\Lambda\subseteq M$ of a $C^1$ diffeomorphism $f$ of a compact manifold is hyperbolic if and only if there exists a Riemannian metric on $M$, a continuous splitting $$T_xM=F(x)\oplus G(x)\ \text{for each } x\in\Lambda,$$ a constant $\lambda\in(0,1)$, and a continuous function $\theta: \Lambda\to\mathbb{R}^+$ such that for $x\in\Lambda$,
- $d_xf \Big{(} C^G_{\theta(x)}(x)\Big{)}\subseteq C^G_{\theta(fx)}(fx)$ and $d_xf^{-1} \Big{(} C^F_{\theta(x)}(x)\Big)\subseteq C^F_{\theta(f^{-1}x)}(f^{-1}x)$,
- If $v\in C^G_{\theta(x)}(x)$, then $\|d_xf(v)\|\geq\lambda^{-1}\|v\|$. If $v\in C^F_{\theta(x)}(x)$, then $\|d_xf^{-1}(v)\|\geq\lambda^{-1}\|v\|$.
The proof of the main theorem
=============================
This section provides the proof of the main result stated in Section $1$. First of all, we construct the subset $\Lambda^*$ of $M$ by using Katok’s technique [@Ka80] and $\Lambda^*$ is $f^m$-invariant for some sufficiently large positive integer $m$. Denote $\Lambda=\Lambda^*\cup f(\Lambda^*)\cup\cdots\cup f^{m-1}(\Lambda^*)$. We give that $\Lambda$ is in a neighborhood of ${\operatorname{supp}}\mu$. Then we use some properties of the dominated splitting and $C^1$ nonuniform hyperbolic dynamical systems to prove Lemma \[invariantofcone\] and Corollary \[corofle\]. We also obtain a dominated splitting $T_\Lambda M=\widetilde{F}_j\oplus\widetilde{G}_j$ over $\Lambda$ for each $j=1,2,\cdots,l-1$ in the proof of Lemma \[invariantofcone\]. Therefore Lemma \[invariantofcone\] and Theorem \[hyperbolicity\] tell us that $\Lambda^*$ is a hyperbolic set with respect to $f^m$. Finally we prove there is a dominated splitting $T_\Lambda M=\widetilde{E}_1\oplus\widetilde{E}_2\oplus\cdots\oplus\widetilde{E}_{l}$ corresponding to Oseledec subspace over $\Lambda$, where $\widetilde{E}_1=\widetilde{F}_1$, $\widetilde{E}_j=\widetilde{F}_j\cap\widetilde{G}_{j-1}$ for $j=2,3,\cdots,l-1$ and $\widetilde{E}_l=\widetilde{G}_{l-1}$.
Since the splitting $$T_{\Gamma}M=E_1\oplus\cdots\oplus E_l$$ is dominated splitting and ${\operatorname{supp}}(\mu)\subseteq\overline{\Gamma}$ (see P$693$ in [@KH95]), by Remark \[conds\], we have $T_{{\operatorname{supp}}(\mu)}M=E_1\oplus \cdots \oplus E_s\oplus E_{s+1}\oplus \cdots \oplus E_l$ is a dominated splitting. Therefore the angles between $E_i(x)$ and $E_j(x)$ are uniformly bounded from zero for every $i,j \in\{1,2,\cdots, l\}$ with $i\neq j$ and $x \in {\operatorname{supp}}(\mu)$. Combing with Remark \[conds\], there is a small $0<\varepsilon_0<1$ satisfying the two properties:
- we can extend the dominated splitting $$T_{{\operatorname{supp}}(\mu)}M=E_1\oplus_<\cdots\oplus_<E_s\oplus_<E_{s+1}\oplus_<\cdots\oplus_<E_l$$ to a continuous splitting in the $\varepsilon_0$-neighborhood $\mathcal{U}(\varepsilon_0, {\operatorname{supp}}(\mu))$ of ${\operatorname{supp}}(\mu)$.
- for any $x,y\in M$ with $d(x,y)<\varepsilon_0$, there exists a unique geodesic connecting $x$ and $y$.
For every $j\in\{1, \cdots, l-1\}$ we denote $$F_j=E_1\oplus\cdots\oplus E_j \text{ and } G_j=E_{j+1}\oplus\cdots\oplus E_l.$$ It is easy to see $T_xM=F_j(x)\oplus G_j(x)$ for any $x\in\mathcal{U}(\varepsilon_0,{\operatorname{supp}}(\mu))$. For any small $\theta\in(0,1)$, $x,y\in\mathcal{U}(\varepsilon_0,{\operatorname{supp}}(\mu))$ with $d(x,y)<\varepsilon_0$ and every $u\in C^{G_j}_\theta(y)$, let $\widetilde{u}$ be the parallel transport of $u$ from $T_yM$ to $T_xM$ along the geodesic connecting $y$ and $x$. By the Whitney Embedding Theorem, we assume, without loss of generality, that the manifold $M$ is embedded in $\mathbb{R}^{N_3}$ with a sufficiently large $N_3$. Since $f$ is a $C^1$ diffeomorphism and the splitting $T_{\mathcal{U}(\varepsilon_0, {\operatorname{supp}}(\mu))}M=F_j\oplus G_j$ is continuous, for the above $\varepsilon_0>0$, there is a $\rho_0\in(0,\frac{1}{2}\varepsilon_0)$ such that
- for any $x,y\in\mathcal{U}(\varepsilon_0,{\operatorname{supp}}(\mu))$ with $d(x,y)<\rho_0$, if $u\in C^{G_j}_\theta(y)$, then $\widetilde{u}\in C^{G_j}_{\frac32\theta}(x)$.
- let $V=\mathcal{U}(\frac{1}{2}\varepsilon_0,{\operatorname{supp}}(\mu))$, for any $x,y\in V$ with $d(x,y)<\rho_0$, for every $u\in C^{G_j}_\theta(y)$, $\|u\|=1$, $i\in\{F_j,G_j\}$, then $$\|\pi_i(fy)(d_yf(u)) - \pi_i(fx)(d_xf(\widetilde{u}))\|\leq\frac{1}{10}\theta\varepsilon_0 a,$$ where $a=\displaystyle{\min_{\substack{x\in\overline{V}\\ u\in C^{G_j}_{\theta}(x)\\ \|u\|=1}}}\|\pi_{G_j}(fx)\big(d_xf(u)\big)\|$.
By Katok’s entropy formula, for each $\delta\in(0,1)$ we have $$\begin{aligned}
\begin{aligned}
h_\mu(f)&=\lim_{\widetilde{\rho}\to 0}\liminf_{n\to\infty}\frac{1}{n}\log N(\mu,n,\widetilde{\rho},\delta)\\
&=\lim_{\widetilde{\rho}\to 0}\limsup_{n\to\infty}\frac{1}{n}\log N(\mu,n,\widetilde{\rho},\delta),
\end{aligned}\end{aligned}$$ where $N(\mu,n,\widetilde{\rho},\delta)$ denotes the minimal number of $(n,\widetilde{\rho})$-Bowen balls that are needed to cover a set of measure $\mu$ at least $1-\delta$. Denote $\vartheta=\min\{\lambda_2(\mu)-\lambda_1(\mu),\lambda_3(\mu)-\lambda_2(\mu),\cdots,\lambda_l(\mu)-\lambda_{l-1}(\mu)\}$. Fixing $\delta\in(0,1)$, for any $\varepsilon\in(0,\varepsilon_0)$ with $\lambda_l(\mu)-3\varepsilon>0$ and $\vartheta-8\varepsilon>0$, there exists $0<\rho_1<\min\{\rho_0,\frac{\varepsilon}{2}\}$ and a positive integer $N_4$ such that any $\widetilde{\rho}\in(0,\rho_1)$ and $n\geq N_4$ we have $$\label{minimalnumber}
N(\mu,n,\widetilde{\rho},\delta)\geq e^{[h_\mu(f)-\frac12\varepsilon]n}.$$ By Lemma \[malyexoffN\] and Lemma \[milyexoffN\], there exists a subset $\Omega\subseteq M$ with $\mu(\Omega)=1$, and for the previous $\varepsilon>0$, $\exists N_1(\varepsilon)>0$ and $N_2(\varepsilon)>0$ such that any $N\geq\max\{N_1(\varepsilon), N_2(\varepsilon),N_4\}$, any $x\in\Omega$, $j=1,2,\cdots,l-1$, we have $$\begin{aligned}
\lim_{k\to+\infty}\frac {1}{kN}\sum_{\iota=0}^{k-1}\log\|d_{f^{\iota N}x}f^N|_{F_j(f^{\iota N}x)}\|&<\lambda_j(\mu)+\varepsilon, \\
\lim_{k\to+\infty}\frac {1}{kN}\sum_{\iota=0}^{k-1}\log m(d_{f^{\iota N}x}f^N|_{G_j(f^{\iota N}x)})&>\lambda_{j+1}(\mu)-\varepsilon,\\
\lim_{k\to+\infty}\frac {1}{kN}\sum_{\iota=0}^{k-1}\log m(d_{f^{\iota N}x}f^{N}|_{F_j(f^{\iota N}x)})&>\lambda_1(\mu)-\varepsilon, \\
\lim_{k\to+\infty}\frac {1}{kN}\sum_{\iota=0}^{k-1}\log \|d_{f^{\iota N}x}f^{N}|_{G_j(f^{\iota N}x)}\|&<\lambda_l(\mu)+\varepsilon.\\
\end{aligned}$$ Fixing any $L_0>\max\{N_1({\varepsilon}),N_2(\varepsilon),N_4\}$, by the Egornov theorem, for $\delta$ as above, there exists a compact subset $\Omega_\delta\subset\Omega$ with $\mu(\Omega_\delta)>1-\frac{\delta}{3}$ and a positive integer $K_\varepsilon>1$ such that for any $x\in \Omega_\delta$ and $k\geq K_\varepsilon$, $j=1,2,\cdots,l-1$, we have $$\label{lyexoff}
\begin{aligned}
\prod_{\iota =0}^{k-1}\|d_{f^{\iota L_0}x}f^{L_0}|_{F_j(f^{\iota L_0}x)}\|&\leq e^{kL_0[\lambda_j(\mu)+2\varepsilon]},\\
\prod_{\iota =0}^{k-1} m(d_{f^{\iota L_0}x}f^{L_0}|_{G_j(f^{\iota L_0}x)})&\geq e^{kL_0[\lambda_{j+1}(\mu)-2\varepsilon]},
\end{aligned}$$ $$\label{lyexoff-1}
\begin{aligned}
\prod_{\iota=0}^{k-1} m(d_{f^{\iota L_0}x}f^{L_0}|_{F_j(f^{\iota L_0}x)})&\geq e^{kL_0[\lambda_1(\mu)-2\varepsilon]},\\
\prod_{\iota=0}^{k-1} \|d_{f^{\iota L_0}x}f^{L_0}|_{G_j(f^{\iota L_0}x)}\|&\leq e^{kL_0[\lambda_l(\mu)+2\varepsilon]}.
\end{aligned}$$ For the above $\varepsilon>0$, there exist small $0<\rho_2<\rho_1$ and $\zeta_0>0$ so that for every $\iota=1,2,\cdots,L_0$, $$\label{contoff}
\begin{aligned}
e^{-\varepsilon}\leq\frac{\|d_xf^\iota(u)\|}{\|d_yf^\iota(v)\|}\leq e^\varepsilon, e^{-\varepsilon}\leq\frac{m (d_xf^\iota(u) )}{m(d_yf^\iota(v))}\leq e^\varepsilon,
\end{aligned}$$ $$\label{contoff-1}
\begin{aligned}
e^{-\varepsilon}\leq\frac{\|d_xf^{-\iota}(u)\|}{\|d_yf^{-\iota}(v)\|}\leq e^\varepsilon, e^{-\varepsilon}\leq\frac{m (d_xf^{-\iota}(u) )}{m(d_yf^{-\iota}(v))}\leq e^\varepsilon,
\end{aligned}$$ whenever $d(x,y)<\rho_2$ and $\angle(u,v)<\zeta_0$. For any $0<\zeta<\zeta_0$, there exist small $\rho_3\in(0,\rho_2)$ and small $\theta_0\in(0,1)$ such that if $x,y\in\overline{V}$ with $d(x,y)<\rho_3$, then $\angle(w_1,w_2)<\zeta$ for any $0\neq w_1\in C_{\theta_0}^i(x), 0\neq w_2\in C_{\theta_0}^i(y)$ and $i\in\{F_j, G_j\}$.
Pick a countable basis $\{\varphi_i\}_{i\geq 1}$ (nonzero) of the space $C^0(M)$ of all continuous functions on $M$. Recall that the space of $f$-invariant probabilities $\mathcal{M}_f(M)$ can be endowed with the metric $d: \mathcal{M}_f(M)\times\mathcal{M}_f(M)\to [0,1]$, $$d(\mu,\nu):=\sum_{j=1}^\infty 2^{-j}\frac{1}{2\|\varphi_j\|_\infty}\Big|\int\varphi_jd\mu-\int\varphi_jd\nu\Big|$$ where $\|\varphi\|_\infty:=\sup_{x\in M}|\varphi(x)|$. Let $J$ be a positive integer satisfying $\frac{1}{2^J}<\frac{\varepsilon}{8}$ and $\rho\in(0,\frac{\rho_3}{2})$ such that $$\label{continuous}
\begin{aligned}
|\varphi_j(x)-\varphi_j(y)|\leq\frac{\varepsilon}{4}\|\varphi_j\|_\infty
\end{aligned}$$ for any $x,y$ with $d(x,y)\leq\rho$, $j=1,2,\cdots,J$.
Let $\chi:=\lambda(\mu)=\min_{1\leq i\leq l}|\lambda_i(\mu)|$. For $\rho>0$ and $\delta>0$ as above, by Lemma \[existofrectangle\], there exists a compact set $\Lambda_H=\Lambda_H(\rho,\delta,\frac{\chi}{2})$ with $\mu(\Lambda_H)>1-\frac{\delta}{3}$, a constant $\beta=\beta(\rho,\delta)>0$ and a finite collection of $(\rho,\beta,\frac{\chi}{2})$-rectangles $R(q_1),R(q_2),\cdots,R(q_t)$ with $q_j\in\Lambda_H$ so that $\Lambda_H\subseteq\cup_{j=1}^tB(q_j,\beta)$.
Let $\mathcal{P}$ be a finite measurable partition of $\Lambda_H\cap\Omega_\delta\cap{\operatorname{supp}}(\mu)$ so that $P(q_j)\subseteq B(q_j,\beta)$ for $j=1,2,\cdots,t$. We can take $\mathcal{P}$ as follows: $$\begin{aligned}
\begin{aligned}
P_1&=B(q_1,\beta)\cap\Lambda_H\cap\Omega_\delta\cap{\operatorname{supp}}(\mu),\\
P_k&=\big{(}B(q_k,\beta)\cap\Lambda_H\cap\Omega_\delta\cap{\operatorname{supp}}(\mu)\big{)}\setminus\big{(}\cup_{j=1}^{k-1}P_j\big{)}\ \mbox{for}\ k=2,3,\cdots,t.
\end{aligned}\end{aligned}$$ Denote $$\begin{aligned}
\begin{aligned}
\Lambda_{H, \delta,n}=&\Big\{x\in\Lambda_H\cap\Omega_\delta\cap{\operatorname{supp}}(\mu): f^kx\in\mathcal{P}(x)\ \mbox{for some integer}\ k\in[n,(1+\varepsilon)n)\ \mbox{and} \\ &\Big|\frac{1}{m}\sum_{i=0}^{m-1}\varphi_j(f^ix)-\int\varphi_jd\mu\Big|\leq\frac{\varepsilon}{4}\|\varphi_j\|_\infty\ \mbox{for any}\ m\geq n\ \mbox{and}\ j=1,2,\cdots,J\Big\}.
\end{aligned}\end{aligned}$$
$\lim_{n\to\infty}\mu(\Lambda_{H,\delta,n})=\mu(\Lambda_H\cap\Omega_\delta\cap{\operatorname{supp}}(\mu)).$
Let $$\begin{aligned}
\begin{aligned}
A_n&=\Big\{x\in\Lambda_H\cap\Omega_\delta\cap{\operatorname{supp}}(\mu): f^kx\in P(x)\ \mbox{for some integer}\ k\in[n,(1+\varepsilon)n)\Big\},\\
A_{n,j}&=\Big\{x\in P_j: f^kx\in P_j\ \mbox{for some integer}\ k\in[n,(1+\varepsilon)n)\Big\}.
\end{aligned}\end{aligned}$$ It is easy to see $A_n=\cup_{j=1}^tA_{n,j}$. For any $P_j$ with $\mu(P_j)>0$ and small $\tau>0$, denote $$A_{n,j}^\tau=\Big\{x\in P_j: \mu(P_j)-\tau\leq\frac{1}{m}\sum_{j=0}^{m-1}\chi_{P_j}(f^jx)\leq\mu(P_j)+\tau\ \mbox{for any}\ m\geq n\Big\}.$$ By Birkhoff Ergodic Theorem, $\mu(\cup_{n\geq 1}A_{n,j}^\tau)=\mu(P_j)$. Since $A_{1,j}^\tau\subseteq A_{2,j}^\tau\subseteq\cdots\subseteq A_{n,j}^\tau\subseteq \cdots$, we conclude $\lim_{n\to\infty}\mu(A_{n,j}^\tau)=\mu(P_j)$. For any $x\in A_{n,j}^\tau$, $$\begin{aligned}
\begin{aligned}
&\ \ \ \ Card\Big\{k\in[n,(1+\varepsilon)n): f^kx\in P_j\Big\}\\
&\geq Card\Big\{k\in[0,(1+\varepsilon)n): f^kx\in P_j\Big\} - Card\Big\{k\in[0,n): f^kx\in P_j\Big\}\\
&\geq \big[\mu(P_j)-\tau\big]\cdot(1+\varepsilon)n - \big[\mu(P_j)+\tau\big]n\\
&= n\cdot\big[\mu(P_j)\varepsilon-2\tau-\tau\varepsilon\big].
\end{aligned}\end{aligned}$$ Taking $0<\tau<\frac{\varepsilon}{2+\varepsilon}\min\big\{\mu(P_j): \mu(P_j)>0, j=1,2,\cdots,t\big\}$, therefore $$Card\big\{k\in[n,(1+\varepsilon)n): f^kx\in P_j\big\}>1$$ for $n$ large enough. This yields that $x\in A_{n,j}$. Thus $A_{n,j}^\tau\subseteq A_{n,j}$. Then we have $$\label{limset}
\begin{aligned}
\lim_{n\to\infty}\mu(A_n)=\mu(\Lambda_H\cap\Omega_\delta\cap{\operatorname{supp}}(\mu)).
\end{aligned}$$
Let $$\begin{aligned}
B_n=&\Big\{x\in\Lambda_H\cap\Omega_\delta\cap{\operatorname{supp}}(\mu): \Big|\frac{1}{m}\sum_{k=0}^{m-1}\varphi_j(f^kx)-\int\varphi_jd\mu\Big|\leq\frac{\varepsilon}{4}\|\varphi_j\|_\infty\\
&\mbox{for any}\ m\geq n\ \mbox{and}\ 1\leq j\leq J\Big\}.
\end{aligned}$$ Birkhoff Ergodic Theorem tells us that $\mu(\cup_{n\geq 1}B_n)=\mu(\Lambda_H\cap\Omega_\delta\cap{\operatorname{supp}}(\mu))$. Since $B_1\subseteq B_2\subseteq\cdots\subseteq B_n\subseteq\cdots$, we have $\lim_{n\to\infty}\mu(B_n)=\mu(\Lambda_H\cap\Omega_\delta\cap{\operatorname{supp}}(\mu))$. Combining with (\[limset\]), $$\lim_{n\to\infty}\mu(\Lambda_{H,\delta,n})=\lim_{n\to\infty}\mu(A_n\cap B_n)=\mu(\Lambda_H\cap\Omega_\delta\cap{\operatorname{supp}}(\mu)).$$ This shows the lemma.
We proceed to prove the main theorem. Taking $$\label{star}
\begin{aligned}
N_5>\max\Big{\{}&\frac{1}{\varepsilon}, 2(K_\varepsilon+1) L_0, \frac{4\log Q}{\varepsilon}, \frac{4\log Q_1}{\varepsilon}, \frac{4\log Q_2}{\varepsilon}, \frac{4}{\varepsilon}\log t,\\
&\frac{4(K_\varepsilon+1)L_0[\lambda_l(\mu)-3\varepsilon]}{\varepsilon}, \frac{2(K_\varepsilon+1)L_0 (\overline{\vartheta}-6\varepsilon)}{\varepsilon}, N_4\Big{\}}
\end{aligned}$$ large enough with $\mu(\Lambda_{H,\delta,N_5})>\mu(\Lambda_H\cap\Omega_\delta\cap{\operatorname{supp}}(\mu))-\frac{\delta}{3}>1-\delta$ and $N_5\varepsilon<e^{\frac12N_5\varepsilon}$, where $$\begin{aligned}
Q=\sup\Big\{&\prod_{\iota=0}^{k-1}\|d_{f^{\iota L_0}x}f^{L_0}|_{F_j(f^{\iota L_0}x)}\|, \prod_{\iota=0}^{k-1}m(d_{f^{\iota L_0}x}f^{L_0}|_{G_j(f^{\iota L_0}x)})^{-1},\\
&\prod_{\iota=0}^{k-1} m(d_{f^{\iota L_0}x}f^{L_0}|_{F_1(f^{\iota L_0}x)})^{-1}, \prod_{\iota=0}^{k-1}\|d_{f^{\iota L_0}x}f^{L_0}|_{G_{l-1}(f^{\iota L_0}x)}\|:\\
& k=1,2,\cdots, K_\varepsilon-1, \text{ and }j=1,2,\cdots,l-1, x\in\Omega_\delta\Big\},\\
Q_1=\sup\Big\{&1,m(d_xf|_{G_j(x)})^{-1},m(d_xf^2|_{G_j(x)})^{-1},\cdots,m(d_xf^{L_0-1}|_{G_j(x)})^{-1}, \\
&m(d_xf^{-1}|_{G_j(x)})^{-1},m(d_xf^{-2}|_{G_j(x)})^{-1},\cdots,m(d_xf^{-(L_0-1)}|_{G_j(x)})^{-1}:\\
&j=1,2, \cdots,l-1, \text{ and } x \text{ belongs to the }\varepsilon\text{-neighborhood of } {\operatorname{supp}}(\mu)\Big\},\\
Q_2=\sup\Big\{&1,\|d_xf|_{F_j(x)}\|,\cdots,\|d_xf^{L_0-1}|_{F_j(x)}\|,\|d_xf^{-1}|_{F_j(x)}\|,\cdots,\|d_xf^{-L_0+1}|_{F_j(x)}\|:\\
&j=1,2,\cdots,l-1, \text{ and } x \text{ belongs to the }\varepsilon\text{-neighborhood of } {\operatorname{supp}}(\mu)\Big\},\\
\overline{\vartheta}=\max\Big\{&\lambda_2(\mu)-\lambda_1(\mu), \lambda_3(\mu)-\lambda_2(\mu), \cdots, \lambda_l(\mu)-\lambda_{l-1}(\mu)\Big\}.
\end{aligned}$$ Choose a maximal $(N_5,2\rho)$-separated subset $E\subseteq\Lambda_{H,\delta,N_5}$. Hence $\cup_{x\in E}B_{N_5}(x,2\rho)\supseteq\Lambda_{H,\delta,N_5}$. By (\[minimalnumber\]), $$\begin{aligned}
\begin{aligned}
Card(E)&\geq N\Big(\mu,N_5,2\rho,1-\mu(\Lambda_{H,\delta,N_5})\Big)\\
&\geq N(\mu,N_5,2\rho,\delta)\\
&\geq e^{[h_\mu(f)-\frac12\varepsilon]N_5}.
\end{aligned}\end{aligned}$$ For $N_5\leq k< (1+\varepsilon)N_5$, let $\Delta_k=\big\{x\in E: f^kx\in P(x)\big\}$. Take $m\in[N_5,(1+\varepsilon)N_5)$ with $Card( \Delta_m)=\max\big\{Card (\Delta_k): N_5\leq k<(1+\varepsilon)N_5\big\}$. Then $$\begin{aligned}
\begin{aligned}
Card (\Delta_m)&\geq \frac{1}{N_5\varepsilon}Card (E)\\
&\geq \frac{1}{N_5\varepsilon}e^{[h_\mu(f)-\frac12\varepsilon]\cdot N_5}\\
&\geq e^{[h_\mu(f)-\varepsilon]\cdot N_5}.
\end{aligned}\end{aligned}$$ Choose $P\in\mathcal{P}$ with $$Card(\Delta_m\cap P)=\max\Big\{Card(\Delta_m\cap P_k): 1\leq k\leq t\Big\}.$$ Thus $Card(\Delta_m\cap P)\geq\frac{1}{t}Card(\Delta_m)\geq\frac{1}{t}e^{[h_\mu(f)-\varepsilon]\cdot N_5}$. Possibly neglecting some points in $E$ then we can guarantee that $$\label{uplowbounded}
\frac{1}{t}e^{[h_\mu(f)-\varepsilon]N_5}\leq Card(\Delta_m\cap P)\leq Card(\Delta_m)\leq Card(E) \leq e^{[h_\mu(f)+\varepsilon]N_5}.$$ By the definition of the partition $\mathcal{P}$, there exists $q\in\{q_1,q_2,\cdots,q_t\}$ such that $P\subseteq B(q,\beta)\cap\Lambda_H\cap\Omega_\delta\cap{\operatorname{supp}}(\mu)$. For any $x\in \Delta_m\cap P$, since $x,f^mx\in B(q,\beta)\cap\Lambda_H$, by Definition \[smallrectangle\], $$C\Big(x,R(q)\cap f^{-m}R(q)\Big)\ \mbox{and}\ f^m\Big(C\big(x,R(q)\cap f^{-m}R(q)\big)\Big)$$ are admissible $s$-rectangle and $u$-rectangle in $R(q)$ respectively for some number $\widetilde{\lambda}>0$.
Notice that for any $x_1,x_2\in \Delta_m\cap P$ with $x_1\neq x_2$, $$C\Big(x_1,R(q)\cap f^{-m}R(q)\Big)\cap C\Big(x_2,R(q)\cap f^{-m}R(q)\Big)=\emptyset.$$ In fact, suppose $y\in C\Big(x_1,R(q)\cap f^{-m}R(q)\Big)\cap C\Big(x_2,R(q)\cap f^{-m}R(q)\Big)$, by Definition \[smallrectangle\], we obtain $d(f^kx_j,f^ky)\leq\rho\ \mbox{for}\ k=0,1,2,\cdots,m, \text{ and } j=1,2.$ Then we have $$d_m(x_1,x_2)\leq d_m(x_1,y)+d_m(y,x_2)\leq 2\rho.$$ Since $\Delta_m\cap P$ is a $(N_5,2\rho)$-separate set, $$d_m(x_1,x_2)\geq d_n(x_1,x_2)>2\rho$$ which contradicts $d_m(x_1,x_2)\leq 2\rho$. This implies that there are $Card(\Delta_m\cap P)$ disjoint admissible $s$-rectangles which mapped under $f^m$ onto $Card(\Delta_m\cap P)$ disjoint admissible $u$-rectangles.
Let $$\Lambda^*=\cap_{n\in\mathbb{Z}}f^{-nm}\Big(\cup_{x\in \Delta_m\cap P}C\big(x,R(q)\cap f^{-m}R(q)\big)\Big).$$ By the construction, $\Lambda^*$ is locally maximal with respect to $f^m$ and to the closed neighborhood $\cup_{i=1}^tR(q_i)$. It also implies that $f^m|_{\Lambda^*}$ is topologically conjugate to a full two-side shift in the symbolic space with $Card(\Delta_m\cap P)$ symbols. Let $\Lambda=\Lambda^*\cup f(\Lambda^*)\cup\cdots\cup f^{m-1}(\Lambda^*)$. We claim that $\Lambda$ is in a neighborhood of ${\operatorname{supp}}(\mu)$.
\[hyperinne\] $\Lambda \subseteq \mathcal{U}(\rho,{\operatorname{supp}}(\mu))$.
By the construction of $\Lambda$ and the definition of $R(q)$, for any $y\in\Lambda$, $\exists x\in \Delta_m\cap P$, $$d(y,f^kx)\leq\rho \text{ for some integer } k\in[0,m-1].$$ Since $x\in \Delta_m\cap P\subseteq\Lambda_H\cap\Omega_\delta\cap{\operatorname{supp}}(\mu)\subseteq{\operatorname{supp}}(\mu)$ and ${\operatorname{supp}}(\mu)$ is $f$-invariant, $\Lambda$ is contained in the $\rho$-neighborhood of ${\operatorname{supp}}(\mu)$.
It follows from Lemma \[invariantofcone\] and Corollary \[corofle\] below that there is a dominated splitting, corresponding to Oseledec subspace on horseshoes, and the approximation of Lyapunov exponents by horseshoes.
\[invariantofcone\] Let $\varepsilon$, $\theta_0$, $\rho$, $m$ and $\Lambda^*$ be as above. For any small $\theta\in(0,\theta_0)$, there exists $0<\eta<1$ such that for every $y\in\Lambda^*$, $j\in\{1,2,\cdots,l-1\}$ and $n\in\mathbb{Z}$, $$\begin{aligned}
d_{f^{nm}y}f^m C^{G_j}_{\theta}(f^{nm}y)\subseteq C^{G_j}_{\eta^m\theta}(f^{(n+1)m}y),\\
d_{f^{nm}y}f^{-m} C^{F_j}_{\theta}(f^{nm}y)\subseteq C^{F_j}_{\eta^m\theta}(f^{(n-1)m}y).
\end{aligned}$$ Moreover for any nonzero vectors $v\in C^{G_j}_{\theta}(f^{nm}y)$, $w\in C^{F_j}_{\theta}(f^{nm}y)$, $$\begin{aligned}
\|d_{f^{nm}y}f^m(v)\|&\geq e^{[\lambda_{j+1}(\mu)-6\varepsilon]m}\|v\|, \\
\|d_{f^{nm}y}f^{-m}(w)\|&\geq e^{[-\lambda_{j}(\mu)-6\varepsilon]m}\|w\|.
\end{aligned}$$ Furthermore, for any nonzero vectors $v\in C^{G_{l-1}}_{\theta}(f^{nm}y)$, $w\in C^{F_1}_{\theta}(f^{nm}y)$, $$\begin{aligned}
\|d_{f^{nm}y}f^m(v)\|&\leq e^{[\lambda_l(\mu)+6\varepsilon]m}\|v\|, \\
\|d_{f^{nm}y}f^{-m}(w)\|&\leq e^{[-\lambda_{1}(\mu)+6\varepsilon]m}\|w\|.
\end{aligned}$$
We only prove the statements for $G_j$ with respective to $f^m$, since the other statements for $F_j$ with respective to $f^{-m}$ can be proven in a similar fashion.
First of all, we prove the $df^m$-invariance of the cones of $G_j$. For any $y\in\Lambda$, by Claim \[hyperinne\], there is $x\in{\operatorname{supp}}(\mu)$ such that $d(x,y)<\rho$. Since $x\in{\operatorname{supp}}(\mu)$, $T_xM=F_j(x)\oplus_<G_j(x)$ is dominated. By Remark \[conds\], choosing a approximate norm with $N=1$, for each pair of unit vectors $u_F\in F_j(x)$ and $v_G\in G_j(x)$ $$\label{1}
\|d_xf(u_F)\|\leq\frac12\|d_xf(v_G)\|.$$ As $d(x,y)<\rho<\rho_0$, applying $(3)$ and $(4)$ we have for every unit vector $v\in C^{G_j}_{\theta}(y)$, $$\widetilde{v}\in C_{\frac32\theta}^{G_j}(x) \text{ and } \|\pi_i(fy)\big(d_yf(v)\big)- \pi_i(fx)\big(d_xf(\widetilde{v})\big)\|\leq\frac1{10}\theta\varepsilon_0 a$$ where $i\in\{F_j,G_j\}$. Therefore $$\begin{aligned}
&\|\pi_{F_j}(fy)\big( d_yf(v) \big)\|\\
\leq &\| \pi_{F_j}(fy)\big( d_yf(v) \big) - \pi_{F_j}(fx)\big( d_xf(\widetilde{v}) \big)\| + \|\pi_{F_j}(fx)\big( d_xf(\widetilde{v}) \big)\|\\
\leq & \frac1{10}\theta\varepsilon_0a + \|d_xf(\widetilde{v}_F)\|\\
\leq & \frac1{10}\theta\varepsilon_0a + \frac12 \|d_xf(\widetilde{v}_G)\| \cdot \frac{\|\widetilde{v}_F\|}{\|\widetilde{v}_G\|}\\
\leq & \frac1{10}\theta\varepsilon_0 a + \frac34\theta \|d_xf(\widetilde{v}_G)\|\\
= & \frac1{10}\theta\varepsilon_0 a + \frac34\theta \|\pi_{G_j}(fx)\big(d_xf(\widetilde{v})\big)\|\\
\leq & \frac1{10}\theta\varepsilon_0 a + \frac34\theta \Big(\|\pi_{G_j}(fx)\big(d_xf(\widetilde{v})\big) - \pi_{G_j}(fy)\big(d_yf(v)\big)\| + \|\pi_{G_j}(fy)\big(d_yf(v)\big) \| \Big)\\
\leq & \frac1{10}\theta\varepsilon_0a + \frac34\theta \cdot\frac1{10}\theta\varepsilon_0 a + \frac34\theta \|\pi_{G_j}(fy)\big(d_yf(v)\big) \| \\
\leq & \Big( \frac1{10}\varepsilon_0 + \frac34 \cdot\frac1{10}\theta\varepsilon_0 + \frac34 \Big)\theta \|\pi_{G_j}(fy)\big(d_yf(v)\big) \| \\
\leq & \eta\theta\cdot \|\pi_{G_j}(fy)\big(d_yf(v)\big) \|,
\end{aligned}$$ where $\widetilde{v}=\widetilde{v}_F+\widetilde{v}_G$, $\widetilde{v}_F\in F_j(x)$, $\widetilde{v}_G\in G_j(x)$ and $\eta=\frac{37}{40}\in(0,1)$. This yields that for any $y\in\Lambda$, $$\label{2}
d_yfC^{G_j}_{\theta}(y)\subseteq C^{G_j}_{\eta\theta}(fy).$$ Similarly we can prove $d_yf^{-1}C^{F_j}_{\theta}(y)\subseteq C^{F_j}_{\eta\theta}(f^{-1}y)$ for any $y\in\Lambda$. Then it is obvious that $$\begin{aligned}
d_{f^{nm}y}f^m C^{G_j}_{\theta}(f^{nm}y)\subseteq C^{G_j}_{\eta^m\theta}(f^{(n+1)m}y),\\
d_{f^{nm}y}f^{-m} C^{F_j}_{\theta}(f^{nm}y)\subseteq C^{F_j}_{\eta^m\theta}(f^{(n-1)m}y)
\end{aligned}$$ for any $y\in\Lambda$ and $n\in\mathbb{Z}$. Since $\Lambda^*\subseteq\Lambda$, we complete the proof of the first statement of this lemma.
To prove the second statement of the lemma, we first define the families of sets $\widetilde{F}_j$ and $\widetilde{G}_j$ which are in the cone of $F_j$ and $G_j$ respectively. For any $y\in\Lambda^*$, let $$\widetilde{F}_j(y) = \cap_{n=0}^{\infty} d_{f^{nm}y}f^{-nm} C_\theta^{F_{j}}(f^{nm}y) \text{ and } \widetilde{G}_j(y) = \cap_{n=0}^{\infty} d_{f^{-nm}y}f^{nm} C_\theta^{G_{j}}(f^{-nm}y)$$ for $j=1,2,\cdots,l-1$. For any $z\in (f\Lambda^* \cup f^2\Lambda^*\cup\cdots\cup f^{m-1}\Lambda^*)\setminus\Lambda^*$, then there exists $k\in\{1,2,\cdots,m-1\}$ such that $z\in f^k\Lambda^*$ but $z\notin \Lambda^*\cup\cdots\cup f^{k-1}\Lambda^*$. Define $$\widetilde{F}_j(z) = d_{f^{-k}z}f^k \widetilde{F}_j(f^{-k}z) \text{ and } \widetilde{G}_j(z) = d_{f^{-k}z}f^k \widetilde{G}_j(f^{-k}z).$$ Since $\Lambda=\Lambda^*\cup f\Lambda^*\cup\cdots\cup f^{m-1}\Lambda^*$ and $\Lambda^*$ is $f^m$-invariant, it yields that the families of sets $\widetilde{F}_j$ and $ \widetilde{G}_j$ are $df$-invariant. By Lemma \[ds\] below, the splitting $T_\Lambda M=\widetilde{F}_j\oplus\widetilde{G}_j$ is dominated on $\Lambda$. Therefore the splitting $T_zM=\widetilde{F}_j(z) \oplus \widetilde{G}_j(z)$ is continuous for any $z\in\Lambda$. From the construction of $\widetilde{F}_j$, $\widetilde{G}_j$ and (\[2\]) we obtain $$\label{biaohao}
\widetilde{F}_j(z)\subseteq C_\theta^{F_j}(z) \text{ and } \widetilde{G}_j(z)\subseteq C_\theta^{G_j}(z) \text{ for any } z\in\Lambda.$$ Since $\theta_0$ is small, $\theta\in(0,\theta_0)$ and the splitting $T_zM=F_j(z) \oplus G_j(z)$ is continuous for any $z\in\Lambda$, then there exists a constant $\kappa>0$ (independent of $z\in\Lambda$) satisfying $\kappa\theta<\frac78$ such that for any nonzero vector $v\in C^{G_j}_{\theta}(z)$, we have $\|v^s\|\leq \kappa\theta\|v^u\|$ where $v=v^s+v^u$, $v^s\in \widetilde{F}_j(z)$, $v^u\in \widetilde{G}_j(z)$. For any $y\in\Lambda^*$, any $n\in\mathbb{Z}$, there exists $x_n\in \Delta_m\cap P$ such that $f^{nm}y\in C(x_n, R(q)\cap f^{-m}R(q))$. We may as well assume $n=0$ and denote $x_0=x$. The proof of $n\neq0$ is parallel to that of $n=0$. By definition \[smallrectangle\], $d(f^kx,f^ky)\leq\rho$ for $k=0,1,\cdots,m$. For any $0\neq v\in C_\theta^{G_j}(y)$, then $v=v^s+v^u$, $v^s\in \widetilde{F}_j(y)$, $v^u\in \widetilde{G}_j(y)$ with $\|v^s\|\leq \kappa\theta\|v^u\|$. Therefore $$\begin{aligned}
\|d_yf^m(v^s)\|&\leq\|d_yf^m|_{\widetilde{F}_j(y)}\|\cdot\|v^s\|\\
&\leq\Big(\prod_{\iota=0}^{p-1}\|d_{f^{\iota L_0}y}f^{L_0}|_{\widetilde{F}_j(f^{\iota L_0}y)}\|\Big)\cdot Q_2 e^\varepsilon\cdot\|v^s\|
\end{aligned}$$ where $m=pL_0+q$, $p,q\in\mathbb{N}$ and $0\leq q<L_0$. Lemma \[ds\] and Remark \[conds\] tell us that $\widetilde{F}_j(z)=F_j(z)$ and $\widetilde{G}_j(z)=G_j(z)$ if $z\in\Lambda\cap{\operatorname{supp}}(\mu)$. For $\iota=0,1,2,\cdots,p-1$, we have $ \widetilde{F}_j(f^{\iota L_0}y)\subseteq C_\theta^{F_j}(f^{\iota L_0}y)$. Since $F_j(f^{\iota L_0}x)\subseteq C_\theta^{F_j}(f^{\iota L_0}x)$ and $m> K_\varepsilon L_0$, combining (\[contoff\]) and (\[lyexoff\]) we obtain $$\label{4}
\begin{aligned}
\|d_yf^m(v^s)\|&\leq\Big(\prod_{\iota=0}^{p-1}\|d_{f^{\iota L_0}x}f^{L_0}|_{F_j(f^{\iota L_0}x)}\|\Big)\cdot e^{p\varepsilon}\cdot Q_2 e^\varepsilon\cdot \|v^s\|\\
&\leq e^{pL_0[\lambda_j(\mu)+2\varepsilon]}\cdot e^{pL_0\varepsilon}\cdot Q_2\cdot\|v^s\|\\
&\leq \kappa\theta \cdot Q_2\cdot e^{pL_0[\lambda_j(\mu)+3\varepsilon]}\cdot\|v^u\|
\end{aligned}$$ and $$\begin{aligned}
\|v^u\|&=\|d_{f^{m}y}f^{-m}\big(d_yf^m(v^u)\big)\|\\
&\leq\|d_{f^my}f^{-m}|_{\widetilde{G}_j(f^my)}\|\cdot\|d_yf^m(v^u)\|\\
&=m(d_yf^m|_{\widetilde{G}_j(y)})^{-1}\cdot\|d_yf^m(v^u)\|\\
&\leq \Big(\prod_{\iota=0}^{p-1}m (d_{f^{\iota L_0}y}f^{L_0}|_{\widetilde{G}_j(f^{\iota L_0}y)})^{-1}\Big)\cdot Q_1 e^\varepsilon\cdot\|d_yf^m(v^u)\|
\end{aligned}$$ $$\begin{aligned}
&\leq \Big(\prod_{\iota=0}^{p-1}m (d_{f^{\iota L_0}x}f^{L_0}|_{G_j(f^{\iota L_0}x)})^{-1}\Big)\cdot e^{p\varepsilon}\cdot Q_1 e^\varepsilon\cdot\|d_yf^m(v^u)\|\\
&\leq e^{-pL_0[\lambda_{j+1}(\mu)-2\varepsilon]}\cdot e^{pL_0\varepsilon}\cdot Q_1\cdot \|d_yf^m(v^u)\|\\
&= e^{pL_0[-\lambda_{j+1}(\mu)+3\varepsilon]}\cdot Q_1\cdot \|d_yf^m(v^u)\|.
\end{aligned}$$ This implies $$\label{unequalityofsu}
\begin{aligned}
\|d_yf^m(v^s)\|\leq \kappa\theta Q_1\cdot Q_2\cdot e^{pL_0[\lambda_j(\mu)-\lambda_{j+1}(\mu)+6\varepsilon]}\cdot\|d_yf^m(v^u)\|.
\end{aligned}$$ (\[star\]) and $m\geq N_5$ tell us that $$m > \max\Big\{\frac{4\log Q_1}{\varepsilon}, \frac{4\log Q_2}{\varepsilon}, \frac{L_0[\lambda_l(\mu)-3\varepsilon]}{\varepsilon}, \frac{L_0 (\overline{\vartheta}-6\varepsilon)}{\varepsilon}\Big\}.$$ It follows that $$\begin{aligned}
\frac{\|d_yf^m(v)\|}{\|v\|}&=\frac{\|v^u\|}{\|v\|}\cdot\frac{\|d_yf^m(v)\|}{\|v^u\|}\\
&\geq\frac{\|v^u\|}{\|v\|}\cdot\frac{\|d_yf^m(v^u)\|-\|d_yf^m(v^s)\|}{\|v^u\|}\\
&\geq\frac{1}{1+(\kappa\theta)^2}\cdot\Big[ 1-\kappa\theta Q_1 Q_2\cdot e^{pL_0[\lambda_j(\mu) - \lambda_{j+1}(\mu)+6\varepsilon]}\Big]\cdot\frac{\|d_yf^m(v^u)\|}{\|v^u\|}\\
&\geq\frac{1}{2}\cdot\Big\{1- e^{m[\lambda_j(\mu)-\lambda_{j+1}(\mu)+8\varepsilon]}\Big\}\cdot e^{m[\lambda_{j+1}(\mu)-5\varepsilon]}.
\end{aligned}$$ Since $\lambda_j(\mu)<\lambda_{j+1}(\mu)$ and $\varepsilon>0$ small enough with $-\vartheta+8\varepsilon<0$, then $$e^{m[\lambda_j(\mu)-\lambda_{j+1}(\mu)+8\varepsilon]}$$ can be small enough for $m$ large enough. Thus $$\begin{aligned}
\frac{\|d_yf^m(v)\|}{\|v\|}\geq e^{m[\lambda_{j+1}(\mu)-6\varepsilon]}.
\end{aligned}$$ We can prove for any nonzero vector $w\in C^{F_j}_{\theta}(y)$, $$\begin{aligned}
\|d_{y}f^{-m}(w)\|&\geq e^{m[-\lambda_{j}(\mu)-6\varepsilon]}\|w\|
\end{aligned}$$ by the same way. Therefore we complete the second statement of the lemma.
Last but not the least, we prove the third statement of the lemma. For every $y\in\Lambda^*$, $0\neq v\in C_\theta^{G_{l-1}}(y)$ with $v=v^s+v^u$, $v^s\in \widetilde{F}_{l-1}(y)$, $v^u\in \widetilde{G}_{l-1}(y)$ with $\|v^s\|\leq \kappa\theta\|v^u\|$. Since $m>\frac{4\log Q_2}{\varepsilon}$, by (\[4\]), we conclude $$\begin{aligned}
\|d_yf^m(v^s)\|&\leq e^{pL_0[\lambda_{l-1}(\mu)+3\varepsilon]}\cdot Q_2 \cdot \|v^s\|\\
&\leq e^{m[\lambda_{l}(\mu)+4\varepsilon]} \|v^s\|.
\end{aligned}$$ By (\[contoff\]), (\[lyexoff-1\]) and $m>\frac{4\log Q_1}{\varepsilon}$, we have $$\begin{aligned}
\|d_yf^m(v^u)\|&\leq \|d_yf^m|_{\widetilde{G}_{l-1}(y)}\|\cdot\|v^u\|\\
&\leq\prod_{\iota=0}^{p-1} \|d_{f^{\iota L_0}y}f^m|_{\widetilde{G}_{l-1}(f^{\iota L_0}y)}\|\cdot Q_1e^\varepsilon\cdot\|v^u\|
\end{aligned}$$ $$\begin{aligned}
&\leq\prod_{\iota=0}^{p-1} \|d_{f^{\iota L_0}x}f^m|_{G_{l-1}(f^{\iota L_0}x)}\|\cdot e^{p\varepsilon} Q_1e^\varepsilon\cdot\|v^u\|\\
&\leq e^{pL_0[\lambda_l(\mu)+3\varepsilon]}Q_1\cdot\|v^u\|\\
&\leq e^{m[\lambda_l(\mu)+4\varepsilon]}\cdot\|v^u\|.
\end{aligned}$$ It follows that $$\begin{aligned}
\frac{\|d_yf^m(v)\|}{\|v\|} &\leq \frac{\|d_yf^m(v^u)\|+\|d_yf^m(v^s)\|}{\|v\|}\\
&\leq\frac{\|v^u\|}{\|v\|}\cdot \frac{\|d_yf^m(v^u)\|+\|d_yf^m(v^s)\|}{\|v^u\|}\\
&\leq \frac{1}{1-\kappa\theta} \frac{\|d_yf^m(v^u)\|+\|d_yf^m(v^s)\|}{\|v^u\|}\\
&\leq \frac{1}{1-\kappa\theta} (1+\kappa\theta)e^{m[\lambda_l(\mu)+4\varepsilon]}\\
&\leq e^{m[\lambda_l(\mu)+6\varepsilon]},
\end{aligned}$$ the last inequality is because that $m$ is large enough. Similarly we can also prove $ \|d_{f^{nm}y}f^{-m}(w)\|\leq e^{[-\lambda_{1}(\mu)+6\varepsilon]m}\|w\|$ for any nonzero vector $w\in C^{F_1}_{\theta}(f^{nm}y)$. Therefore the proof of Lemma \[invariantofcone\] is completed.
\[ds\] The splitting $T_\Lambda M=\widetilde{F}_j\oplus\widetilde{G}_j$ is dominated on $\Lambda$ for $j=1,2,\cdots,l-1$.
For any $z\in\Lambda$, there is $x\in{\operatorname{supp}}(\mu)$ such that $d(x,z)<\rho$. Since $x\in{\operatorname{supp}}(\mu)$, $T_xM=F_j(x)\oplus_<G_j(x)$ is dominated. By (\[1\]) for small $\theta\in(0,\theta_0)$, each pair of unit vectors $\overline{u}\in C_{\frac32\theta}^{F_j}(x)$ and $\overline{v}\in C_{\frac32\theta}^{G_j}(x)$, $$\|d_x f(\overline{u})\|\leq\frac58\|d_x f(\overline{v})\|.$$ For any unit vectors $u\in\widetilde{F}_j(z)$ and $v\in\widetilde{G}_j(z)$, from (\[biaohao\]) we obtain $$u\in\widetilde{F}_j(z)\subseteq C_\theta^{F_j}(z) \text{ and } v\in\widetilde{G}_j(z)\subseteq C_\theta^{G_j}(z).$$ Since $d(z,x)<\rho$, by $(3)$, we have $\widetilde{u}\in C_{\frac32\theta}^{F_j}(x)$ and $\widetilde{v}\in C_{\frac32\theta}^{G_j}(x)$. Combining $(4)$, we conclude $$\frac{\|d_zf(u)\|}{\|d_zf(v)\|}\leq \frac{\|d_xf(\widetilde{u})\|+\frac15\theta\varepsilon_0a}{\|d_xf(\widetilde{v})\|-\frac15\theta\varepsilon_0a}.$$ Since $\varepsilon_0$ is small, it yields that $$\label{star2}
\frac{\|d_zf(u)\|}{\|d_zf(v)\|}\leq\frac34.$$
By construction of $\widetilde{F}_j(z), \widetilde{G}_j(z)$ and the continuity of the decomposition $T_zM=F_j(z)\oplus G_j(z)$, it follows that $\widetilde{F}_j(z)$ and $\widetilde{G}_j(z)$ contain two subspaces $\overline{F}_j(z)$ and $\overline{G}_j(z)$, respectively of the same dimension as that of $F_j(z)$ and $G_j(z)$. Since $\widetilde{F}_j(z)\cap\widetilde{G}_j(z)=\{0\}$, $\overline{F}_j(z)\cap\overline{G}_j(z)=\{0\}$. Therefore $T_zM=\overline{F}_j(z)\oplus\overline{G}_j(z)$. We claim that $\overline{F}_j(z)=\widetilde{F}_j(z)$. In fact, let $w\in\widetilde{F}_j(z)$ with $w=w_{\overline{F}_j}+w_{\overline{G}_j}$, $w_{\overline{F}_j}\in\overline{F}_j(z)$, $w_{\overline{G}_j}\in\overline{G}_j(z)$ and $\|w\|=1$. If $w_{\overline{G}_j}\neq0$, by (\[star2\]), we have for any $n\geq 1$ $$\frac{\|d_zf^{n}(w_{\overline{F}_j})\|}{\|w_{\overline{F}_j}\|}\leq(\frac34)^{n}\frac{\|d_zf^{n}(w_{\overline{G}_j})\|}{\|w_{\overline{G}_j}\|} \text{ and }
\|d_zf^{n}(w)\|\leq(\frac34)^{n}\frac{\|d_zf^{n}(w_{\overline{G}_j})\|}{\|w_{\overline{G}_j}\|},$$ because $w_{\overline{F}_j}\in\overline{F}_j(z)\subseteq\widetilde{F}_j(z)$, $w\in\widetilde{F}_j(z)$ and $w_{\overline{G}_j}\in\overline{G}_j(z)\subseteq\widetilde{G}_j(z)$. Therefore $$\begin{aligned}
\begin{aligned}
\|w_{\overline{G}_j}\|&\leq(\frac34)^n\frac{\|d_zf^n(w_{\overline{G}_j})\|}{\|d_zf^n(w)\|}\\
&\leq(\frac34)^n\frac{\|d_zf^n(w_{\overline{G}_j})\|}{\|d_zf^n(w_{\overline{G}_j})\|-\|d_zf^n(w_{\overline{F}_j})\|}\\
&\leq(\frac34)^n\frac{1}{1 - \frac{\|d_zf^n(w_{\overline{F}_j})\|}{\|d_zf^n(w_{\overline{G}_j})\|}}\\
&\leq(\frac34)^n\frac{1}{1 - (\frac{3}{4})^n\frac{\|w_{\overline{F}_j}\|}{\|w_{\overline{G}_j}\|}}.
\end{aligned}\end{aligned}$$ Let $n\to\infty$, it follows that $w_{\overline{G}_j}=0$. Thus $\overline{F}_j(z)=\widetilde{F}_j(z)$. Similarly we have $\overline{G}_j(z)=\widetilde{G}_j(z)$. Therefore the splitting $T_{\Lambda}M=\widetilde{F}_j\oplus\widetilde{G}_j$ is dominated.
The following result is extended Lemma \[invariantofcone\] to $\Lambda$.
\[corofle\] Let $\varepsilon$, $\theta_0$, $\rho$, $m$ and $\Lambda$ be as above. For any small $\theta\in(0,\theta_0)$, there exists $0<\eta<1$ such that for every $z\in\Lambda$, $j\in\{1,2,\cdots,l-1\}$ and $n\in\mathbb{Z}$, $$\begin{aligned}
d_{f^{nm}z}f^m C^{G_j}_{\theta}(f^{nm}z)\subseteq C^{G_j}_{\eta^m\theta}(f^{(n+1)m}z),\\
d_{f^{nm}z}f^{-m} C^{F_j}_{\theta}(f^{nm}z)\subseteq C^{F_j}_{\eta^m\theta}(f^{(n-1)m}z).
\end{aligned}$$ Moreover for any nonzero vectors $v\in C^{G_j}_{\theta}(f^{nm}z)$, $w\in C^{F_j}_{\theta}(f^{nm}z)$, $$\begin{aligned}
\|d_{f^{nm}z}f^m(v)\|&\geq e^{[\lambda_{j+1}(\mu)-6\varepsilon]m}\|v\|, \\
\|d_{f^{nm}z}f^{-m}(w)\|&\geq e^{[-\lambda_{j}(\mu)-6\varepsilon]m}\|w\|.
\end{aligned}$$ Furthermore, for any nonzero vectors $v\in C^{G_{l-1}}_{\theta}(f^{nm}z)$, $w\in C^{F_1}_{\theta}(f^{nm}z)$, $$\begin{aligned}
\|d_{f^{nm}z}f^m(v)\|&\leq e^{[\lambda_l(\mu)+6\varepsilon]m}\|v\|, \\
\|d_{f^{nm}z}f^{-m}(w)\|&\leq e^{[-\lambda_{1}(\mu)+6\varepsilon]m}\|w\|.
\end{aligned}$$
We only prove the statements for $G_j$ with respective to $f^m$, since the other statements for $F_j$ with respective to $f^{-m}$ can be proven in a similar fashion. The proof of the first statement is identical to that of Lemma \[invariantofcone\]. It suffices to prove for any $0\neq v\in C^{G_j}_{\theta}(f^{nm}z)$, $j=1,2,\cdots,l-1$, $$\|d_{f^{nm}z}f^m(v)\|\geq e^{[\lambda_{j+1}(\mu)-6\varepsilon]m}\|v\|$$ and for any $0\neq v\in C^{G_{l-1}}_{\theta}(f^{nm}z)$, $$\|d_{f^{nm}z}f^m(v)\|\leq e^{[\lambda_l(\mu)+6\varepsilon]m}\|v\|.$$
For every $z\in\Lambda$, there exist $y\in\Lambda^*$ and $k\in\{0,1,2,\cdots,m-1\}$ such that $z=f^ky$. As $y\in\Lambda^*$, for any $n\geq 0$ there is $x_n\in\Delta_m\cap P$ such that $$d(f^Kx_n,f^K(f^{nm}y))\leq\rho, \text{ for }K=0,1,2,\cdots,m.$$ We may as well assume $n=0$. The proof of $n\neq0$ is parallel to that of $n=0$. For any $0\neq v\in C_\theta^{G_j}(z)$, then $v=v^s+v^u$ with $v^s\in\widetilde{F}_j(z)$, $v^u\in\widetilde{G}_j(z)$ and $\|v^s\|\leq\kappa\theta\|v^u\|$. Thus $$\label{q}
\begin{aligned}
\|d_zf^m(v^s)\|&=\|d_{f^{m-k}z}f^k\Big( d_zf^{m-k}(v^s)\Big)\|\\
&\leq \|d_{f^{m-k}z}f^k|_{\widetilde{F}_j(f^{m-k}z)}\|\cdot \|d_zf^{m-k}|_{\widetilde{F}_j(z)}\| \cdot \|v^s\|
\end{aligned}$$ and $$\label{r}
\begin{aligned}
\|v^u\|&\leq m(d_zf^m|_{\widetilde{G}_j(z)})^{-1}\cdot \|d_zf^m(v^u)\|\\
&\leq m(d_zf^{m-k}|_{\widetilde{G}_j(z)})^{-1}\cdot m(d_{f^{m-k}z}f^k|_{\widetilde{G}_j(f^{m-k}z)})^{-1}\cdot \|d_zf^m(v^u)\|.
\end{aligned}$$ It follows from (\[star\]) and $m\geq N_5$ that $\max\{m-k,k\}\geq K_\varepsilon L_0$. Let $m-k=p_1L_0+q_1$ and $k=p_2L_0+q_2$ where $p_i, q_i\in\mathbb{N}$, $0\leq q_i<L_0$ and $i=1,2$. If $m-k\geq K_\varepsilon L_0$, applying (\[contoff\]) and (\[lyexoff\]), then we have $p_1\geq K_\varepsilon$, $$\label{a}
\begin{aligned}
\|d_zf^{m-k}|_{\widetilde{F}_j(z)}\|&\leq \prod_{\iota=0}^{p_1-1}\|d_{f^{\iota L_0}z}f^{L_0}|_{\widetilde{F}_j(f^{\iota L_0}z)}\|\cdot Q_2 e^\varepsilon\\
&\leq \Big(\prod_{\iota=0}^{p_1-1}\|d_{f^{\iota L_0}(f^kx_0)}f^{L_0}|_{F_j(f^{\iota L_0}(f^kx_0))}\|\Big) e^{p_1\varepsilon}\cdot Q_2 e^\varepsilon \\
&\leq e^{p_1L_0[\lambda_j(\mu)+2\varepsilon]}\cdot e^{p_1L_0\varepsilon}\cdot Q_2 \\
&\leq e^{p_1L_0[\lambda_j(\mu)+3\varepsilon]}\cdot Q_2
\end{aligned}$$ and $$\label{b}
\begin{aligned}
m(d_zf^{m-k}|_{\widetilde{G}_j(z)})^{-1}&\leq \prod_{\iota=0}^{p_1-1} m(d_{f^{\iota L_0}z}f^{L_0}|_{\widetilde{G}_j(f^{\iota L_0}z)})^{-1}\cdot Q_1 e^\varepsilon\\
&\leq \prod_{\iota=0}^{p_1-1} m(d_{f^{\iota L_0}x_0}f^{L_0}|_{G_j(f^{\iota L_0}x_0)})^{-1}\cdot e^{p_1\varepsilon}\cdot Q_1 e^\varepsilon \\
&\leq e^{p_1L_0[-\lambda_{j+1}(\mu)+3\varepsilon]}\cdot Q_1.
\end{aligned}$$ If $m-k\geq K_\varepsilon L_0$, by (\[contoff\]) and (\[lyexoff-1\]), we obtain $$\label{c}
\begin{aligned}
m(d_zf^{m-k}|_{\widetilde{F}_1(z)}) &\geq \prod_{\iota=0}^{p_1-1} m(d_{f^{\iota L_0}z}f^{L_0}|_{\widetilde{F}_1(f^{\iota L_0}z)}) \cdot Q_2^{-1} e^{-\varepsilon}\\
&\geq \Big(\prod_{\iota=0}^{p_1-1} m( d_{f^{\iota L_0}(f^kx_0)}f^{L_0}|_{F_1(f^{\iota L_0}(f^kx_0))})\Big) e^{-p_1\varepsilon}\cdot Q_2^{-1} e^{-\varepsilon} \\
&\geq e^{p_1L_0[\lambda_1(\mu)-2\varepsilon]}\cdot e^{-p_1L_0\varepsilon}\cdot Q_2^{-1} \\
&\geq e^{p_1L_0[\lambda_1(\mu)-3\varepsilon]}\cdot Q_2^{-1}
\end{aligned}$$ and $$\label{d}
\begin{aligned}
\|d_zf^{m-k}|_{\widetilde{G}_{l-1}(z)}\| &\leq \prod_{\iota=0}^{p_1-1} \|d_{f^{\iota L_0}z}f^{L_0}|_{\widetilde{G}_{l-1}(f^{\iota L_0}z)}\| \cdot Q_1 e^\varepsilon\\
&\leq \prod_{\iota=0}^{p_1-1} \|d_{f^{\iota L_0}x_0}f^{L_0}|_{G_{l-1}(f^{\iota L_0}x_0)}\| \cdot e^{p_1\varepsilon}\cdot Q_1 e^\varepsilon \\
&\leq e^{p_1L_0[\lambda_{l}(\mu)+3\varepsilon]}\cdot Q_1.
\end{aligned}$$ If $m-k<K_\varepsilon L_0$, combining (\[contoff\]) and the definition of $Q$, then we have $p_1<K_\varepsilon$, $$\label{e}
\begin{aligned}
\|d_zf^{m-k}|_{\widetilde{F}_j(z)}\| &\leq Q\cdot Q_2\cdot e^{p_1L_0\varepsilon},
\end{aligned}$$ $$\label{f}
\begin{aligned}
m(d_zf^{m-k}|_{\widetilde{G}_j(z)})^{-1}& \leq Q\cdot Q_1\cdot e^{p_1L_0\varepsilon},
\end{aligned}$$ $$\label{g}
\begin{aligned}
m(d_zf^{m-k}|_{\widetilde{F}_1(z)}) &\geq Q^{-1}\cdot Q_2^{-1}\cdot e^{-p_1L_0\varepsilon}
\end{aligned}$$ and $$\label{h}
\begin{aligned}
\|d_zf^{m-k}|_{\widetilde{G}_{l-1}(z)}\|& \leq Q\cdot Q_1\cdot e^{p_1L_0\varepsilon}.
\end{aligned}$$ Similarly we obtain that if $k\geq K_\varepsilon L_0$, then $p_2\geq K_\varepsilon$ and $$\label{i}
\begin{aligned}
\|d_{f^{m-k}z}f^k|_{\widetilde{F}_j(f^{m-k}z)}\|&\leq e^{p_2L_0[\lambda_j(\mu)+3\varepsilon]}\cdot Q_2,
\end{aligned}$$ $$\label{j}
\begin{aligned}
m(d_{f^{m-k}z}f^k|_{\widetilde{G}_j(f^{m-k}z)})^{-1}&\leq e^{p_2L_0[-\lambda_{j+1}(\mu)+3\varepsilon]}\cdot Q_1,
\end{aligned}$$ $$\label{k}
\begin{aligned}
m(d_{f^{m-k}z}f^k|_{\widetilde{F}_1(f^{m-k}z)}) &\geq e^{p_2L_0[\lambda_1(\mu)-3\varepsilon]}\cdot Q_2^{-1}
\end{aligned}$$ and $$\label{l}
\begin{aligned}
\|d_{f^{m-k}z}f^k|_{\widetilde{G}_{l-1}(f^{m-k}z)}\| &\leq e^{p_2L_0[\lambda_{l}(\mu)+3\varepsilon]}\cdot Q_1;
\end{aligned}$$ if $k<K_\varepsilon L_0$, then $p_2< K_\varepsilon$, $$\label{m}
\begin{aligned}
\|d_{f^{m-k}z}f^k|_{\widetilde{F}_j(f^{m-k}z)}\|&\leq Q\cdot Q_2\cdot e^{p_2L_0\varepsilon},
\end{aligned}$$ $$\label{n}
\begin{aligned}
m(d_{f^{m-k}z}f^k|_{\widetilde{G}_j(f^{m-k}z)})^{-1}&\leq Q\cdot Q_1\cdot e^{p_2L_0\varepsilon},
\end{aligned}$$ $$\label{o}
\begin{aligned}
m(d_{f^{m-k}z}f^k|_{\widetilde{F}_1(f^{m-k}z)})&\geq Q^{-1}\cdot Q_2^{-1}\cdot e^{-p_2L_0\varepsilon}
\end{aligned}$$ and $$\label{p}
\begin{aligned}
\|d_{f^{m-k}z}f^k|_{\widetilde{G}_{l-1}(f^{m-k}z)}\| &\leq Q\cdot Q_1\cdot e^{p_2L_0\varepsilon}.
\end{aligned}$$ Then the proof of the second statement can be divided into three cases:
[**Case I:**]{} If $m-k\geq K_\varepsilon L_0$ and $k\geq K_\varepsilon L_0$, then from (\[q\]), (\[a\]) and (\[i\]) we have $$\|d_zf^m(v^s)\|\leq \kappa\theta Q_2^2\cdot e^{(p_1+p_2)L_0[\lambda_j(\mu)+3\varepsilon]} \cdot\|v^u\|.$$ By (\[r\]), (\[b\]) and (\[j\]) we obtain $$\|v^u\|\leq Q_1^2\cdot e^{(p_1+p_2)L_0[-\lambda_{j+1}(\mu)+3\varepsilon]}\cdot \|d_zf^m(v^u)\|.$$ Thus $$\|d_zf^m(v^s)\|\leq \kappa\theta Q_1^2 Q_2^2 \cdot e^{(p_1+p_2)L_0[\lambda_j(\mu)-\lambda_{j+1}(\mu)+6\varepsilon]} \cdot \|d_zf^m(v^u)\|.$$ Therefore $$\frac{\|d_zf^m(v)\|}{\|v\|}\geq \frac{1}{1+\kappa^2\theta^2} \Big\{1-\kappa\theta Q_1^2 Q_2^2 e^{(p_1+p_2)L_0[\lambda_j(\mu)-\lambda_{j+1}(\mu)+6\varepsilon]}\Big\} \cdot \frac{\|d_zf^m(v^u)\|}{\|v^u\|}.$$ It follows from $(\ref{star})$ and $m\geq N_5$ that $$m>\max\{\frac{4\log Q_1}{\varepsilon}, \frac{4\log Q_2}{\varepsilon}, \frac{2L_0[\lambda_l(\mu)-3\varepsilon]}{\varepsilon}, \frac{2L_0(\overline{\vartheta}-6\varepsilon)}{\varepsilon}\}.$$ Then we conclude $$\frac{\|d_zf^m(v)\|}{\|v\|}\geq \frac{1}{1+\kappa^2\theta^2} \Big\{1-\kappa\theta e^{m[\lambda_j(\mu)-\lambda_{j+1}(\mu)+8\varepsilon]}\Big\} \cdot e^{m[\lambda_{j+1}(\mu)-5\varepsilon]}.$$ As $m$ is large enough, it yields that $$\frac{\|d_zf^m(v)\|}{\|v\|}\geq e^{m[\lambda_{j+1}(\mu)-6\varepsilon]}.$$
[**Case II:**]{} If $m-k\geq K_\varepsilon L_0$ and $k<K_\varepsilon L_0$, by $(\ref{q})$, $(\ref{a})$ and $(\ref{m})$, we have $$\|d_zf^m(v^s)\|\leq \kappa\theta Q Q_2^2\cdot e^{p_1L_0[\lambda_j(\mu)+3\varepsilon]+p_2L_0\varepsilon} \cdot\|v^u\|.$$ By $(\ref{r})$, $(\ref{b})$ and $(\ref{n})$, we conclude that $$\|v^u\|\leq Q Q_1^2\cdot e^{p_1L_0[-\lambda_{j+1}(\mu)+3\varepsilon]+p_2L_0\varepsilon}\cdot \|d_zf^m(v^u)\|.$$ Thus $$\|d_zf^m(v^s)\|\leq \kappa\theta Q^2 Q_1^2 Q_2^2 \cdot e^{p_1L_0[\lambda_j(\mu)-\lambda_{j+1}(\mu)+6\varepsilon]+m\varepsilon} \cdot \|d_zf^m(v^u)\|.$$ Therefore $$\frac{\|d_zf^m(v)\|}{\|v\|}\geq \frac{1}{1+\kappa^2\theta^2} \Big\{1-\kappa\theta Q^2 Q_1^2 Q_2^2 e^{p_1L_0[\lambda_j(\mu)-\lambda_{j+1}(\mu)+6\varepsilon]+m\varepsilon}\Big\} \cdot \frac{\|d_zf^m(v^u)\|}{\|v^u\|}.$$ Since $m\geq N_5>\max\{\frac{4\log Q}{\varepsilon}, \frac{4\log Q_1}{\varepsilon}, \frac{4\log Q_2}{\varepsilon}, \frac{2(K_\varepsilon+1)L_0(\overline{\vartheta}-6\varepsilon)}{\varepsilon}, \frac{4(K_\varepsilon+1)L_0[\lambda_l(\mu)-3\varepsilon]}{\varepsilon}\}$, we obtain $$\frac{\|d_zf^m(v)\|}{\|v\|}\geq \frac{1}{1+\kappa^2\theta^2} \Big\{1-\kappa\theta e^{m[\lambda_j(\mu)-\lambda_{j+1}(\mu)+9\varepsilon]}\Big\} \cdot e^{m[\lambda_{j+1}(\mu)-5\varepsilon]}.$$ It yields that $$\frac{\|d_zf^m(v)\|}{\|v\|}\geq e^{m[\lambda_{j+1}(\mu)-6\varepsilon]}.$$
[**Case III:**]{} If $m-k<K_\varepsilon L_0$ and $k\geq K_\varepsilon L_0$, it is parallel to Case II. It follows from $(\ref{q})$, $(\ref{r})$, $(\ref{e})$ $(\ref{f})$, $(\ref{i})$ and $(\ref{j})$ that $$\frac{\|d_zf^m(v)\|}{\|v\|}\geq e^{m[\lambda_{j+1}(\mu)-6\varepsilon]}.$$
Finally, we prove the third statement of the corollary. For any $z\in\Lambda$, $0\neq v\in C_\theta^{G_{l-1}}(z)$ with $v=v^s+v^u$ with $v^s\in\widetilde{F}_{l-1}(z)$, $v^u\in\widetilde{G}_{l-1}(z)$ and $\|v^s\|\leq\kappa\theta\|v^u\|$. If $m-k\geq K_\varepsilon L_0$ and $k\geq K_\varepsilon L_0$, by $(\ref{q})$, $(\ref{a})$, $(\ref{i})$ and $(\ref{d})$, $(\ref{l})$, we have $$\begin{aligned}
\|d_zf^m(v^s)\|&\leq Q_2^2\cdot e^{(p_1+p_2)L_0[\lambda_{l-1}(\mu)+3\varepsilon]}\cdot \|v^s\|,\\
\|d_zf^m(v^u)\|&=\|d_{f^{m-k}z}f^k\circ d_zf^{m-k}(v^u)\|\\
&\leq \|d_{f^{m-k}z}f^k|_{\widetilde{G}_{l-1}(f^{m-k}z)}\| \cdot \|d_zf^{m-k}|_{\widetilde{G}_{l-1}(z)}\| \cdot \|v^u\|\\
&\leq e^{(p_1+p_2)L_0[\lambda_l(\mu)+3\varepsilon]} \cdot Q^2_1\cdot \|v^u\|.
\end{aligned}$$ Since $m\geq \max\{\frac{4\log Q_1}{\varepsilon}, \frac{4\log Q_2}{\varepsilon}, \frac{4(K_\varepsilon+1)L_0[\lambda_l(\mu)-3\varepsilon]}{\varepsilon}\}$, then we conclude $$\begin{aligned}
\|d_zf^m(v^s)\|&\leq e^{m[\lambda_l(\mu)+4\varepsilon]}\cdot \|v^s\|,\\
\|d_zf^m(v^u)\|&\leq e^{m[\lambda_l(\mu)+4\varepsilon]}\cdot \|v^u\|.
\end{aligned}$$ It yields that $$\begin{aligned}
\frac{\|d_zf^m(v)\|}{\|v\|} &\leq \frac{\|d_zf^m(v^u)\|+\|d_zf^m(v^s)\|}{\|v\|}\\
&\leq\frac{\|v^u\|}{\|v\|}\cdot \frac{\|d_zf^m(v^u)\|+\|d_zf^m(v^s)\|}{\|v^u\|}\\
&\leq \frac{1}{1-\kappa\theta} \frac{\|d_zf^m(v^u)\|+\|d_zf^m(v^s)\|}{\|v^u\|}\\
&\leq \frac{1}{1-\kappa\theta} (1+\kappa\theta)e^{m[\lambda_l(\mu)+4\varepsilon]}\\
&\leq e^{m[\lambda_l(\mu)+6\varepsilon]}
\end{aligned}$$ the last inequality is because that $m$ is large enough. If $m-k\geq K_\varepsilon L_0$ and $k<K_\varepsilon L_0$, by $(\ref{q})$, $(\ref{a})$, $(\ref{m})$ and $(\ref{d})$, $(\ref{p})$, $(\ref{star})$, $m\geq N_5$, then we conclude that $$\begin{aligned}
\|d_zf^m(v^s)\|&\leq Q Q_2^2\cdot e^{p_1L_0[\lambda_{l-1}(\mu)+3\varepsilon]+p_2L_0\varepsilon}\cdot \|v^s\|\\
&\leq e^{m[\lambda_l(\mu)+4\varepsilon]}\cdot \|v^s\|,\\
\|d_zf^m(v^u)\|&\leq Q_1\cdot e^{p_1L_0[\lambda_l(\mu)+3\varepsilon]} \cdot QQ_1 e^{p_2L_0\varepsilon} \cdot \|v^u\|\\
&\leq e^{m[\lambda_l(\mu)+4\varepsilon]}\cdot \|v^u\|.
\end{aligned}$$ Therefore $$\frac{\|d_zf^m(v)\|}{\|v\|}\leq e^{m[\lambda_l(\mu)+6\varepsilon]}.$$ If $m-k<K_\varepsilon L_0$ and $k\geq K_\varepsilon L_0$, then we can also prove $$\frac{\|d_zf^m(v)\|}{\|v\|}\leq e^{m[\lambda_l(\mu)+6\varepsilon]}$$ by the same way as in the case of $m-k\geq K_\varepsilon L_0$ and $k<K_\varepsilon L_0$. Thus we proved for any $z\in\Lambda$, $0\neq v\in C_\theta^{G_{l-1}}(z)$ $$\|d_zf^m(v)\| \leq e^{m[\lambda_l(\mu)+6\varepsilon]} \|v\|.$$
By considering $f^{-1}$, we can similarly show that for any $n\in\mathbb{Z}$, $0\neq w\in C^{F_j}_{\theta}(f^{nm}z)$, $j=1,2,\cdots,l-1$, $$\|d_{f^{nm}z}f^{-m}(w)\|\geq e^{[-\lambda_{j}(\mu)-6\varepsilon]m}\|w\|,$$ and for any nonzero vector $w\in C^{F_1}_{\theta}(f^{nm}z)$, $$\|d_{f^{nm}z}f^{-m}(w)\|\leq e^{[-\lambda_{1}(\mu)+6\varepsilon]m}\|w\|.$$ Therefore we complete the proof of the corollary.
In the paragraphs to follow, we will construct a dominated splitting corresponding to Oseledec subspace on $\Lambda$. For any $z\in\Lambda$, let $$\begin{aligned}
\widetilde{E}_1(z) &= \widetilde{F}_1(z),\\
\widetilde{E}_j(z) &= \widetilde{F}_j(z) \cap \widetilde{G}_{j-1}(z),\ j=2,3,\cdots,l-1,\\
\widetilde{E}_{l}(z) &= \widetilde{G}_{l-1}(z).
\end{aligned}$$ By Lemma \[ds\], we conclude that the splitting $$T_\Lambda M=\widetilde{E}_1\oplus \widetilde{E}_2\oplus\cdots\oplus \widetilde{E}_l$$ is dominated. Therefore the splitting $$T_z M=\widetilde{E}_1(z)\oplus \widetilde{E}_2(z)\oplus\cdots\oplus \widetilde{E}_l(z)$$ varies continuously with the point $z\in\Lambda$. By Corollary \[corofle\] and (\[biaohao\]), we have $$\label{3}
e^{m[\lambda_j(\mu)-6\varepsilon]}\|w\| \leq \|d_zf^m(w)\| \leq e^{m[\lambda_j(\mu)+6\varepsilon]}\|w\|$$ for any nonzero vector $w\in \widetilde{E}_j(z)$, $j=1,2,\cdots,l$ and $z\in\Lambda$.
(i)By Lemma \[invariantofcone\] and Theorem \[hyperbolicity\], $\Lambda^*$ is a hyperbolic set with respect to $f^m$. Since $f^m|_{\Lambda^*}$ is topologically conjugate to a full two-side shift in the symbolic space with $Card(\Delta_m\cap P)$ symbols, it is topologically mixing with respect to $f^m$. We proved (i) of the main Theorem.
(ii)Since $ h_{top}(f|_\Lambda)= \frac{1}{m}h_{top}(f^m|\Lambda^*)= \frac{1}{m}\log Card(\Delta_m\cap P)$, by (\[uplowbounded\]), it yields that $$h_{top}(f|_\Lambda)\geq-\frac{1}{m}\log t+\frac{N_5}{m}\cdot[h_\mu(f)-\varepsilon].$$ Combining $N_5>\frac{4}{\varepsilon}\log t$ and $N_5\leq m< N_5(1+\varepsilon)$, we have $$\begin{aligned}
\begin{aligned}
h_{top}(f|_\Lambda)&\geq -\frac14\varepsilon+\frac{1}{1+\varepsilon}\cdot[h_\mu(f)-\varepsilon]\geq h_\mu(f)-[h_\mu(f)+2]\varepsilon.
\end{aligned}\end{aligned}$$ Applying (\[uplowbounded\]), we conclude that $$h_{top}(f|_\Lambda)\leq \frac{N_5}{m}\cdot[h_\mu(f)+\varepsilon]\leq h_\mu(f)+\varepsilon.$$ This shows (ii) of the theorem.
\(iii) Since $\rho<\frac{\varepsilon}{2}$, combining Claim \[hyperinne\], we prove (iii) of the main theorem.
(iv)For any $f$-invariant probability measure $\nu$ supported on $\Lambda$, suppose $\nu$ is ergodic, $$\begin{aligned}
\begin{aligned}
d(\nu,\mu)&= \sum_{j=1}^\infty 2^{-j}\frac{1}{2\|\varphi_j\|_\infty}\cdot\Big|\int\varphi_jd\mu-\int\varphi_j d\nu\Big|\\
&= \sum_{j=1}^J 2^{-j}\frac{1}{2\|\varphi_j\|_\infty}\cdot\Big|\int\varphi_jd\mu-\int\varphi_j d\nu\Big|\\
&\ \ \ \ +\sum_{j=J+1}^\infty 2^{-j}\frac{1}{2\|\varphi_j\|_\infty}\cdot\Big|\int\varphi_jd\mu-\int\varphi_j d\nu\Big|\\
&\leq \sum_{j=1}^J 2^{-j}\frac{1}{2\|\varphi_j\|_\infty}\cdot\Big|\int\varphi_jd\mu-\int\varphi_j d\nu\Big|+\frac{1}{2^{J-1}}\\
&\leq \sum_{j=1}^J 2^{-j}\frac{1}{2\|\varphi_j\|_\infty}\cdot\Big|\int\varphi_jd\mu-\int\varphi_j d\nu\Big|+\frac{\varepsilon}{4}.
\end{aligned}\end{aligned}$$ We claim that $|\int\varphi_jd\mu-\int\varphi_jd\nu|\leq\frac{3\|\varphi_j\|_\infty}{4}\varepsilon \text{ for } j=1,2,\cdots,J.$ In fact, choosing $y\in\Lambda^*$ and $s\in\mathbb{N}$ large enough such that $$\Big|\frac{1}{ms}\sum_{k=0}^{ms-1}\varphi_j(f^ky)-\int\varphi_jd\nu\Big|\leq\frac{\|\varphi_j\|_\infty}{4}\varepsilon$$ for $j=1,2,\cdots,J$. Then there exist $x_0,x_1,\cdots,x_{s-1}\in \Delta_m\cap P$ such that $d(f^{km+t}y,f^tx_k)\leq\rho$ for $0\leq k\leq s-1$ and $0\leq t\leq m-1$. By (\[continuous\]) and the construction of $\Lambda_{H,\delta,m}$, $$\begin{aligned}
\begin{aligned}
&\ \ \ \ \Big|\int\varphi_jd\mu-\int\varphi_jd\nu\Big|\\
&\leq \Big|\frac{1}{s}s\int\varphi_jd\mu-\frac{1}{s}\cdot\Big(\frac{1}{m}\sum_{k=0}^{m-1}\varphi_j(f^kx_0)+\frac{1}{m}\sum_{k=0}^{m-1}\varphi_j(f^kx_1)+\cdots+\frac{1}{m}\sum_{k=0}^{m-1}\varphi_j(f^kx_{s-1})\Big)\Big|\\
& +\Big|\frac{1}{ms}\Big(\sum_{k=0}^{m-1}\varphi_j(f^kx_0)+\cdots+\sum_{k=0}^{m-1}\varphi_j(f^kx_{s-1})\Big)-\frac{1}{ms}\sum_{k=0}^{ms-1}\varphi_j(f^ky)\Big|\\
& +\Big|\frac{1}{ms}\sum_{k=0}^{ms-1}\varphi_j(f^ky)-\int\varphi_jd\nu\Big|\\
&\leq \frac{\|\varphi_j\|_\infty}{4}\varepsilon+\frac{\|\varphi_j\|_\infty}{4}\varepsilon+\frac{\|\varphi_j\|_\infty}{4}\varepsilon=\frac{3\|\varphi_j\|_\infty}{4}\varepsilon.
\end{aligned}\end{aligned}$$ Therefore the proof of the claim is complete. It implies that $d(\mu,\nu)\leq\frac{3}{4}\varepsilon+\frac{\varepsilon}{4}=\varepsilon$ for any ergodic measure $\nu$. If $\nu$ is not ergodic, the ergodic decompositional theorem tells us that $\nu$-almost every ergodic component is supported on $\Lambda$, thus $$\begin{aligned}
\begin{aligned}
d(\mu,\nu)&= \sum_{j=1}^\infty\frac{\Big|\int\varphi_jd\mu-\int\varphi_jd\nu\Big|}{2^j\cdot2\cdot\|\varphi_j\|_\infty}= \sum_{j=1}^\infty\frac{\Big|\int\big(\int\varphi_jd\mu-\int\varphi_jd\nu_x\big)d\nu(x)\Big|}{2^j\cdot2\cdot\|\varphi_j\|_\infty}\\
&\leq \int d(\mu,\nu_x)d\nu(x)\leq \varepsilon.
\end{aligned}\end{aligned}$$ This proves (iv) of the theorem.
(v)Since $\Lambda^*$ is a hyperbolic set with respective to $f^m$ and $\Lambda=\Lambda^*\cup f(\Lambda^*)\cup\cdots\cup f^{m-1}(\Lambda^*)$, combining with (\[3\]), we obtain conclusion (v) of Theorem \[maintheorem\].
[2]{}
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[^1]: The first author is partially supported by NSFC (11501400, 11871361) and the Talent Program of Shanghai University of Engineering Science. The third author is partially supported by NSFC (11771317, 11790274).
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abstract: 'A general method of the Foldy-Wouthyusen (FW) transformation for relativistic particles of arbitrary spin in strong external fields has been developed. The use of the found transformation operator is not restricted by any definite commutation relations between even and odd operators. The final FW Hamiltonian can be expanded into a power series in the Planck constant which characterizes the order of magnitude of quantum corrections. Exact expressions for low-order terms in the Planck constant can be derived. Finding these expressions allows to perform a simple transition to the semiclassical approximation which defines a classical limit of the relativistic quantum mechanics. As an example, interactions of spin-1/2 and scalar particles with a strong electromagnetic field have been considered. Quantum and semiclassical equations of motion of particles and their spins have been deduced. Full agreement between quantum and classical theories has been established.'
author:
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title: 'Foldy-Wouthyusen Transformation and Semiclassical Limit for Relativistic Particles in Strong External Fields'
---
Introduction
============
The Foldy-Wouthuysen (FW) representation [@FW] occupies a special place in the quantum theory. Properties of this representation are unique. The Hamiltonian and all operators are block-diagonal (diagonal in two spinors). Relations between the operators in the FW representation are similar to those between the respective classical quantities. For relativistic particles in external fields, operators have the same form as in the nonrelativistic quantum theory. For example, the position operator is $\bm r$ and the momentum one is $\bm p=-i\hbar\nabla$. These properties considerably simplify the transition to the semiclassical description. As a result, the FW representation provides the best possibility of obtaining a meaningful classical limit of the relativistic quantum mechanics. The basic advantages of the FW representation are described in Refs. [@FW; @CMcK; @JMP].
Interactions of relativistic particles with strong external fields can be considered on three levels: (i) classical physics, (ii) relativistic quantum mechanics, and (iii) quantum field theory. The investigation of such interactions on every level is necessary. The use of the FW representation allows to describe strong-field effects on level (ii) and to find a unambiguous connection between classical physics and relativistic quantum mechanics. To solve the problem, one should carry out an appropriate FW transformation (transformation to the FW representation). The deduced Hamiltonian should be exact up to first-order terms in the Planck constant $\hbar$. This precision is necessary for establishment of exact connection between the classical physics and the relativistic quantum mechanics. However, known methods of exact FW transformation either can be used only for some definite classes of initial Hamiltonians in the Dirac representation [@Nikitin; @JMP] or need too cumbersome derivations [@E].
In the present work, new general method of the FW transformation for relativistic particles in strong external fields is proposed. This method gives exact expressions for low-order terms in $\hbar$. The proposed method is based on the developments performed in Ref. [@JMP] and can be utilized for particles of arbitrary spin. Any definite commutation relations between even and odd operators in the initial Hamiltonian are not needed. An expansion of the FW Hamiltonian into a power series in the Planck constant is used. Since just this constant defines the order of magnitude of quantum corrections, the transition to the semiclassical approximation becomes trivial. As an example, interaction of scalar and spin-1/2 particles with a strong electromagnetic field is considered. We use the designations $[\dots,\dots]$ and $\{\dots,\dots\}$ for commutators and anticommutators, respectively.
Foldy-Wouthyusen transformation for particles in external fields
================================================================
In this section, we review previously developed methods of the FW transformation for particles in external fields.
The relativistic quantum mechanics is based on the Klein-Gordon equation for scalar particles, the Dirac equation for spin-1/2 particles, and corresponding relativistic wave equations for particles with higher spins (see, e.g., Ref. [@FN]). The quantum field theory is not based on these fundamental equations of the relativistic quantum mechanics (see Ref. [@W]). Relativistic wave equations for particles with any spin can be presented in the Hamilton form. In this case, the Hamilton operator acts on the bispinor wave function $\Psi=\left(\begin{array}{c} \phi \\ \chi \end{array}\right)$: $$i\hbar\frac{\partial\Psi}{\partial t}={\cal
H}\Psi. \label{eqi}$$ A particular case of Eq. (\[eqi\]) is the Dirac equation. We can introduce the unit matrix $I$ and the Pauli matrices those components act on the spinors: $$I=\left(\begin{array}{cc}1&0\\0&1\end{array}\right),
~~~\rho_1=\left(\begin{array}{cc}0&1\\1&0\end{array}\right),
~~~ \rho_2=\left(\begin{array}{cc}0&-i\\i&0\end{array}\right),
~~~ \rho_3\equiv\beta=\left(\begin{array}{cc}1&0\\0&-1\end{array}\right).$$ The Hamiltonian can be split into operators commuting and noncommuting with the operator $\beta$: $${\cal H}=\beta {\cal M}+{\cal E}+{\cal O},~~~\beta{\cal M}={\cal M}\beta,
~~~\beta{\cal E}={\cal E}\beta, ~~~\beta{\cal O}=-{\cal O}\beta,
\label{eq3}$$ where the operators ${\cal M}$ and ${\cal E}$ are even and the operator ${\cal O}$ is odd. We suppose that the operator ${\cal E}$ is multiplied by the unit matrix $I$ which is everywhere omitted.
Explicit form of the Hamilton operators for particles with arbitrary half-integer spin has been obtained in Ref. [@NGNN]. Similar equations have been derived for spin-0 [@FV] and spin-1 [@CS; @YB] particles. To study semiclassical limits of these equations, one should perform appropriate FW transformations.
The wave function of a spin-1/2 particle can be transformed to a new representation with the unitary operator $U$: $$\Psi'=U\Psi.$$ The Hamilton operator in the new representation takes the form [@FW; @JMP; @Gol] $${\cal H}'=U{\cal H}U^{-1}-i\hbar U\frac{ \partial U^{-1}}{\partial
t}, \label{eq2}$$ or $${\cal H}'=U\left({\cal
H}-i\hbar\frac{\partial}{\partial t}\right)U^{-1}+
i\hbar\frac{\partial}{\partial t}.$$
The FW transformation has been justified in the best way. In the classical work by Foldy and Wouthuysen [@FW], the exact transformation for free relativistic particles and the approximate transformation for nonrelativistic particles in electromagnetic fields have been carried out. There exist several other nonrelativistic transformation methods which give the same results (see Ref. [@JMP] and references therein). A few methods can be applied for relativistic particles in external fields. However, the transformation methods explained in Refs. [@B; @GS] require cumbersome calculations. The block-diagonalization of two-body Hamiltonians for a system of two spin-1/2 particles and a system of spin-0 and spin-1/2 particles can be performed by the methods found by Chraplyvy [@Chraplyvy] and Tanaka *et al* [@Tanaka], respectively.
Some methods allow to reach the FW representation without the use of unitary transformations. The so-called elimination method [@Pa] makes it possible to exclude the lower spinor from relativistic wave equations. Variants of this method useful for relativistic particles have been elaborated in Refs. [@Neznamov; @STMP]. Another method which essentially differs from the FW and elimination methods has been presented in Ref. [@Gos]. This method defines a diagonalization procedure based on a formal expansion in powers of the Planck constant $\hbar$ and can be used for a large class of Hamiltonians directly inducing Berry phase corrections [@Gos]. An important feature of this method is a possibility to take into account strong-field effects.
Any method different from the FW one should also be justified. The validity of the elimination method is proved only by the coincidence of results obtained by this method and the FW one [@VJ]. Impressive agreement between results presented in Ref. [@Gos] and corresponding results obtained by the FW method is reached for first-order terms in $\hbar$. These terms define momentum and spin dynamics that can be well described in the framework of classical physics. To prove the validity of the method, one should show such an agreement for terms derived from second-order commutators (e.g., for the Darwin term [@FW]). Therefore, we desist from a definitive estimate of the method developed in Ref. [@Gos].
We suppose the consistence with the genuine FW transformation to be necessary for any diagonalization method. For example, the Eriksen-Korlsrud method [@EK] does not transform the wave function to the FW representation even for free particles [@PRD]. The use of this method in Refs. [@Obukhov; @Heidenreich] instead of the FW one could cause a misunderstanding of the nature of spin-gravity coupling (see the discussion in Refs. [@PRD; @Mashhoon2]).
In the general case, the exact FW transformation has been found by Eriksen [@E]. The validity of the Eriksen transformation has also been argued by de Vries and Jonker [@VJ]. The Eriksen transformation operator has the form [@E] $$U=\frac12(1+\beta\lambda)\left[1+\frac14(\beta\lambda+\lambda\beta-2)\right]^{-1/2},
~~~ \lambda=\frac{{\cal H}}{({\cal H}^2)^{1/2}}, \label{E}$$ where ${\cal H}$ is the Hamiltonian in the Dirac representation. This operator brings the Dirac wave function and the Dirac Hamiltonian to the FW representation in one step. However, it is difficult to use the Eriksen method for obtaining an explicit form of the relativistic FW Hamiltonian because the general final formula is very cumbersome and contains roots of Dirac matrix operators. Therefore, the Eriksen method was not used for relativistic particles in external fields.
To perform the FW transformation in the strong external fields, we develop the much simpler method elaborated in Ref. [@JMP] for relativistic spin-1/2 particles. In this work, the initial Dirac Hamiltonian is given by $${\cal H}=\beta m+{\cal E}+{\cal O}, \label{eq3N}$$ where $m$ is the particle mass. In Eqs. (\[eq3N\])–(\[eq31\]), the system of units $\hbar=c=1$ is used.
When $[{\cal E},{\cal O}]=0$, the FW transformation is exact [@JMP]. This transformation is fulfilled with the operator $$U=\frac{\epsilon+m+\beta{\cal
O}}{\sqrt{2\epsilon(\epsilon+m)}}, ~~~\epsilon=\sqrt{m^2+{\cal
O}^2} \label{eq18}$$ and the transformed Hamiltonian takes the form $${\cal H}_{FW}=\beta \epsilon+{\cal
E}. \label{eq17}$$
The same transformation is valid for Hamiltonian (\[eq3\]) when not only does the operator ${\cal E}$ commutates with ${\cal O}$ but also the operator ${\cal M}$: $$[{\cal M},{\cal O}]=0. \label{comm}$$ In this case, Eq. (\[eq17\]) remains valid but the operator $\epsilon$ takes the form $$\epsilon=\sqrt{{\cal M}^2+{\cal O}^2}.$$
In the general case, the FW Hamiltonian has been obtained as a power series in external field potentials and their derivatives [@JMP]. As a result of the first stage of transformation performed with operator (\[eq18\]), the following Hamiltonian can be found: $${\cal H}'=\beta\epsilon+{\cal
E}'+{\cal O}',~~~\beta{\cal E}'={\cal E}'\beta, ~~~\beta{\cal
O}'=-{\cal O}'\beta. \label{eq7}$$ The odd operator ${\cal O}'$ is now comparatively small: $$\begin{array}{c} \epsilon=\sqrt{m^2+{\cal O}^2}, \\ {\cal
E}'=i\frac{\partial}{\partial t}+\frac{\epsilon+m}
{\sqrt{2\epsilon(\epsilon+m)}}\left({\cal
E}-i\frac{\partial}{\partial t}
\right)\frac{\epsilon+m}{\sqrt{2\epsilon(\epsilon+m)}}\\-\frac{\beta{\cal
O}} {\sqrt{2\epsilon(\epsilon+m)}}\left({\cal
E}-i\frac{\partial}{\partial t} \right)\frac{\beta{\cal
O}}{\sqrt{2\epsilon(\epsilon+m)}}, \\ {\cal O}'=\frac{\beta{\cal
O}}{\sqrt{2\epsilon(\epsilon+m)}} \left({\cal
E}-i\frac{\partial}{\partial t}
\right)\frac{\epsilon+m}{\sqrt{2\epsilon(\epsilon+m)}}\\-
\frac{\epsilon+m}{\sqrt{2\epsilon(\epsilon+m)}}\left({\cal
E}-i\frac{\partial} {\partial t}\right)\frac{\beta{\cal
O}}{\sqrt{2\epsilon(\epsilon+m)}}. \end{array} \label{eq28}$$
The second stage of transformation leads to the approximate equation for the FW Hamiltonian: $${\cal
H}_{FW}=\beta\epsilon+{\cal E}'+\frac14\beta\left\{{\cal
O}'^2,\frac{1}{\epsilon}\right\}. \label{eq31}$$
To reach a better precision, additional transformations can be used [@JMP].
This method has been applied for deriving the Hamiltonian and the quantum mechanical equations of momentum and spin motion for Dirac particles interacting with electroweak [@JMP] and gravitational [@PRD; @PRD2] fields. The semiclassical limit of these equations has been obtained [@JMP; @PRD; @PRD2]. To determine the exact classical limit of the relativistic quantum mechanics of arbitrary-spin particles in *strong* external fields, we need to generalize the method.
General properties of the Hamiltonian depend on the particle spin. The Hamiltonian is hermitian (${\cal H}={\cal H}^\dagger$) for spin-1/2 particles and pseudo-hermitian for spin-0 and spin-1 ones (more precisely, $\beta$-pseudo-hermitian, see Ref. [@Mostafazadeh] and references therein). In the latter case, it possesses the property ($\beta^{-1}=\beta$) $${\cal
H}^\dagger=\beta{\cal H}\beta$$ that is equivalent to $${\cal
H}^\ddagger\equiv\beta{\cal H}^\dagger\beta={\cal H}.$$
The normalization of wave functions is given by $$\int{\Psi^\dagger\Psi dV}=\int{(\phi\phi^\ast+\chi\chi^\ast)
dV}=1$$ for spin-1/2 particles and $$\int{\Psi^\ddagger\Psi
dV}\equiv\int{\Psi^\dagger\beta\Psi
dV}=\int{(\phi\phi^\ast-\chi\chi^\ast) dV}=1$$ for spin-0 and spin-1 particles.
We suppose ${\cal M}={\cal M}^\dagger,~{\cal E}={\cal
E}^\dagger,~{\cal O}={\cal O}^\dagger$ when ${\cal H}={\cal
H}^\dagger$ and ${\cal M}={\cal M}^\ddagger,~{\cal E}={\cal
E}^\ddagger,~{\cal O}={\cal O}^\ddagger$ when ${\cal H}={\cal
H}^\ddagger$. These conditions can be satisfied in any case.
Since the FW Hamiltonian is block-diagonal and a lower spinor describes negative-energy states, this spinor should be equal to zero. The FW transformation should be performed with the unitary operator $U^\dagger =U^{-1}$ for spin-1/2 particles and with the pseudo-unitary operator $U^\ddagger\equiv\beta U^\dagger\beta
=U^{-1}$ for spin-0 and spin-1 particles.
FOLDY-WOUTHYUSEN TRANSFORMATION IN STRONG EXTERNAL FIELDS
=========================================================
We propose the method of the FW transformation for relativistic particles in strong external fields which can be used for particles of arbitrary spin. The FW Hamiltonian can be expanded into a power series in the Planck constant which defines the order of magnitude of quantum corrections. The obtained expressions for low-order terms in $\hbar$ are exact. The proposed FW transformation makes the transition to the semiclassical approximation to be trivial. The power expansion can be available only if $$pl\gg\hbar, \label{rel1}$$ where $p$ is the momentum of the particle and $l$ is the characteristic size of the nonuniformity region of the external field. This relation is equivalent to $$\lambda\ll l, \label{rel2}$$ where $\lambda$ is the de Broglie wavelength. Eqs. (\[rel1\]),(\[rel2\]) result from the fact that the Planck constant appears in the final Hamiltonian due to commutators between the operators ${\cal M},{\cal E}$, and ${\cal O}$.
The expansion of the FW Hamiltonian into the power series in the Planck constant is formally similar to the previously obtained expansion [@JMP] into a power series in the external field potentials and their derivatives. However, the equations derived in Ref. [@JMP] do not define the semiclassical limit of the Dirac equation for particles in strong external fields, while these equations exhaustively describe the weak-field expansion. The proposed method can also be used in the weak-field expansion even when relations (\[rel1\]),(\[rel2\]) are not valid.
When the power series in the Planck constant is deduced, zero power terms define the quantum analogue of the classical Hamiltonian. On this level, classical and quantum expressions should be very similar because the classical theory gives the right limit of the quantum theory. Terms proportional to powers of $\hbar$ may describe quantum corrections. As a rule, interactions described by these terms also exist in the classical theory. However, classical expressions may differ from the corresponding quantum ones because the quantum corrections to the classical theory may appear. We generalize the method developed in Ref. [@JMP] in order to take into account a possible non-commutativity of the operators ${\cal M}$ and ${\cal O}$. The natural generalization of transformation operator (\[eq18\]) used in Ref. [@JMP] is $$U=\frac{\beta\epsilon+\beta {\cal M}-{\cal
O}}{\sqrt{(\beta\epsilon+\beta {\cal M}-{\cal O})^2}}\,\beta,~~~
U^{-1}=\beta\,\frac{\beta\epsilon+\beta{\cal M}-{\cal
O}}{\sqrt{(\beta\epsilon+\beta{\cal M}-{\cal O})^2}},
\label{eq18N}$$ where $U^\dagger=U^{-1}$ when ${\cal
H}={\cal H}^\dagger$ and $U^\ddagger=U^{-1}$ when ${\cal H}={\cal
H}^\ddagger$. This form of the transformation operator allows to perform the FW transformation in the general case. The special case ${\cal M}=mc^2$ has been considered in Ref. [@JMP] and commutation relation (\[comm\]) has been used in Refs. [@PRD; @JETP].
We consider the general case when external fields are nonstationary. The exact formula for the transformed Hamiltonian has the form $$\begin{array}{c} {\cal H}'=\beta\epsilon+{\cal
E}+ \frac{1}{2T}\Biggl(\left[T,\left[T,(\beta\epsilon+{\cal
F})\right]\right] +\beta\left[{\cal O},[{\cal O},{\cal
M}]\right]\\- \left[{\cal O},\left[{\cal O},{\cal F}\right]\right]
- \left[(\epsilon+{\cal M}),\left[(\epsilon+{\cal M}),{\cal
F}\right]\right] - \left[(\epsilon+{\cal M}),\left[{\cal M},{\cal
O}\right]\right]\\-\beta \left\{{\cal O},\left[(\epsilon+{\cal
M}),{\cal F}\right]\right\}+\beta \left\{(\epsilon+{\cal
M}),\left[{\cal O},{\cal F}\right]\right\} \Biggr)\frac{1}{T},
\end{array} \label{eq28N}$$ where ${\cal F}={\cal E}-i\hbar\frac{\partial}{\partial t}$ and $T=\sqrt{(\beta\epsilon+\beta{\cal M}-{\cal O})^2}$.
Hamiltonian (\[eq28N\]) still contains odd terms proportional to the first and higher powers of the Planck constant. This Hamiltonian can be presented in the form $${\cal
H}'=\beta\epsilon+{\cal E}'+{\cal O}',~~~\beta{\cal E}'={\cal
E}'\beta, ~~~\beta{\cal O}'=-{\cal O}'\beta,
\label{eq27}$$ where $\epsilon=\sqrt{{\cal M}^2+{\cal
O}^2}.$ The even and odd parts of Hamiltonian (\[eq27\]) are defined by the well-known relations: $${\cal
E}'=\frac12\left({\cal H}'+\beta{\cal
H}'\beta\right)-\beta\epsilon,~~~ {\cal O}'=\frac12\left({\cal
H}'-\beta{\cal H}'\beta\right).$$
Additional transformations performed according to Ref. [@JMP] bring ${\cal H}'$ to the block-diagonal form. The approximate formula for the final FW Hamiltonian is $${\cal
H}_{FW}=\beta\epsilon+{\cal E}'+\frac14\beta\left\{{\cal
O}'^2,\frac{1}{\epsilon}\right\}. \label{eqf}$$ This formula is similar to the corresponding one obtained in Ref. [@JMP]. The additional transformations allow to obtain more precise expression for the FW Hamiltonian.
Eqs. (\[eq28N\])–(\[eqf\]) solve the problem of the FW transformation for relativistic particles of arbitrary spin in strong external fields.
Eq. (\[eq28N\]) can be significantly simplified in some special cases. When $[{\cal M},{\cal O}]=0$ and the external fields are stationary, it is reduced to $$\begin{array}{c} {\cal H}'=\beta\epsilon+{\cal
E}+ \frac{1}{2T}\Biggl(\left[T,\left[T,{\cal E}\right]\right] \\-
\left[{\cal O},\left[{\cal O},{\cal E}\right]\right] -
\left[(\epsilon+{\cal M}),\left[(\epsilon+{\cal M}),{\cal
E}\right]\right] \\-\beta \left\{{\cal O},\left[(\epsilon+{\cal
M}),{\cal E}\right]\right\}+\beta \left\{(\epsilon+{\cal
M}),\left[{\cal O},{\cal E}\right]\right\} \Biggr)\frac{1}{T}.
\end{array} \label{eqrd}$$ In this case, $[\epsilon,{\cal M}]=[\epsilon,{\cal O}]=0$ and the operator $T=\sqrt{2\epsilon(\epsilon+{\cal M})}$ is even.
SPIN-1/2 AND SCALAR PARTICLES IN STRONG ELECTROMAGNETIC FIELD
=============================================================
As an example, the FW transformation for spin-1/2 and scalar particles interacting with a strong electromagnetic field can be considered. The initial Dirac-Pauli Hamiltonian for a particle possessing an anomalous magnetic moment (AMM) has the form [@P] $$\begin{array}{c} {\cal
H}_{DP}=c\bm{\alpha}\cdot\bm{\pi}+\beta mc^2+e\Phi+\mu'(-\bm
{\Pi}\cdot \bm{H}+i\bm{\gamma}\cdot\bm{ E}),\\
\bm{\pi}=\bm{p}-\frac{e}{c}\bm{ A}, ~~~
\mu'=\frac{g-2}{2}\cdot\frac{e\hbar}{2mc}, \end{array}
\label{eqDP}$$ where $\mu'$ is the AMM, $\Phi,\bm{
A}$ and $\bm{ E},\bm{H}$ are the potentials and strengths of the electromagnetic field.
Here and below the following designations for the matrices are used: $$\begin{array}{c}\bm{\gamma}=\left(\begin{array}{cc} 0 & \bm{\sigma} \\ -\bm{\sigma} & 0
\end{array}\right), ~~~ {\beta}\equiv\gamma^0=\left(\begin{array}{cc} 1 & 0
\\ 0 & -1 \end{array}\right), ~~~\bm{\alpha}=\beta\bm\gamma=
\left(\begin{array}{cc} 0 & \bm{\sigma} \\ \bm{\sigma} & 0
\end{array}\right), \\ \bm{\Sigma}
=\left(\begin{array}{cc} \bm{\sigma} & 0 \\ 0 &
\bm{\sigma}\end{array}\right), ~~~\bm{\Pi}=\beta\bm\Sigma
=\left(\begin{array}{cc} \bm{\sigma} & 0 \\ 0 &
-\bm{\sigma}\end{array}\right), \end{array}$$ where $0,1,-1$ mean the corresponding 2$\times$2 matrices and $\bm{\sigma}$ is the Pauli matrix.
Terms describing the electric dipole moment (EDM) $d$ have been added in Ref. [@RPJ]. The resulting Hamiltonian is given by $$\begin{array}{c} {\cal
H}=c\bm{\alpha}\cdot\bm{\pi}+\beta mc^2+e\Phi+\mu'(-\bm {\Pi}\cdot
\bm{H}+i\bm{\gamma}\cdot\bm{ E}) %\\
-d(\bm {\Pi}\cdot
\bm{E}+i\bm{\gamma}\cdot\bm{ H}), ~~~
d=\frac{\eta}{2}\cdot\frac{e\hbar}{2mc}, \end{array} \label{eqEDM}$$ where $\eta$ factor for the EDM is an analogue of $g$ factor for the magnetic moment. It is important that $\mu'$ and $d$ are proportional to $\hbar$.
In the considered case $$\begin{array}{c} {\cal M}=mc^2,~~~ {\cal
E}=e\Phi-\mu'\bm {\Pi}\cdot \bm{H}-d\bm {\Pi}\cdot \bm{E}, ~~~%\\
{\cal O}=c\bm{\alpha}\cdot\bm{\pi}+i\mu'\bm{\gamma}\cdot\bm{
E}-id\bm{\gamma}\cdot\bm{ H}. \end{array}$$
Since only terms of zero and first powers in the Planck constant define the semiclassical equations of motion of particles and their spins, we retain only such terms in the FW Hamiltonian. The terms of order of $\hbar$ are proportional either to field gradients or to products of field strengths ($H^2,~E^2$ and $EH$). We do not calculate the terms proportional to products of field strengths because they are usually small in comparison with the terms proportional to field gradients.
The calculated Hamiltonian is given by $$\begin{array}{c} {\cal
H}_{FW}=\beta\epsilon'+e\Phi- \mu'\bm\Pi\cdot\bm
H-\frac{\mu_0}{2}\left\{\frac{mc^2}{\epsilon'}, \bm\Pi\cdot\bm
H\right\}\\+\frac{\mu'c}{4}\left\{\frac{1}{\epsilon'},
\left[\bm\Sigma\cdot(\bm \pi\times \bm E)-\bm\Sigma\cdot(\bm
E\times \bm\pi) \right]\right\}\\+
\frac{\mu_0mc^3}{\sqrt{2\epsilon'(\epsilon'+mc^2)}}\left[\bm\Sigma\cdot(\bm
\pi\times \bm E)-\bm\Sigma\cdot(\bm E\times \bm\pi)
\right]\frac{1}{\sqrt{2\epsilon'(\epsilon'+mc^2)}}\\+
\frac{\mu'c^2}{2\sqrt{2\epsilon'(\epsilon'+mc^2)}}\left\{(\bm{\Pi}\cdot\bm\pi),
(\bm{H}\cdot\bm\pi+\bm{\pi}\cdot\bm
H)\right\}\frac{1}{\sqrt{2\epsilon'(\epsilon'+mc^2)}}\\
-d\bm\Pi\cdot\bm E %\\
+\frac{dc^2}{2\sqrt{2\epsilon'(\epsilon'+mc^2)}}\left\{(\bm{\Pi}\cdot\bm\pi),
(\bm{E}\cdot\bm\pi+\bm{\pi}\cdot\bm
E)\right\}\frac{1}{\sqrt{2\epsilon'(\epsilon'+mc^2)}}\\
-\frac{dc}{4}\left\{\frac{1}{\epsilon'}, \left[\bm\Sigma\cdot(\bm
\pi\times \bm H)-\bm\Sigma\cdot(\bm H\times \bm\pi)
\right]\right\}, \end{array} \label{eq33}$$ where $$\epsilon'=\sqrt{m^2c^4+c^2\bm{\pi}^2} \label{eq34}$$ and $\mu_0=\frac{e\hbar}{2mc}$ is the Dirac magnetic moment.
The quantum evolution of the kinetic momentum operator, $\bm\pi$, is defined by the operator equation of particle motion: $$\frac{d\bm\pi}{dt}=\frac{i}{\hbar}[{\cal
H}_{FW},\bm\pi] -\frac{e}{c}\cdot\frac{\partial\bm A}{\partial t}.
\label{eqme}$$
The equation of spin motion describes the evolution of the polarization operator $\bm\Pi$: $$\frac{d\bm\Pi}{dt}=\frac{i}{\hbar}[{\cal
H}_{FW},\bm\Pi]. \label{eqpoe}$$
Because the operator $\bm\pi$ does not contain the Dirac spin matrices, the commutator of this operator with the Hamiltonian is proportional to $\hbar$. The equation of spin-1/2 particle motion in the strong electromagnetic field to within first-order terms in the Planck constant has the form $$\begin{array}{c} \frac{d\bm
\pi}{dt}=e\bm E+\beta\frac{ec}{4}\left\{\frac{1}{\epsilon'},
\left([\bm\pi\times\bm H]-[\bm H\times\bm\pi]\right)\right\}
%\\
+\mu' \nabla(\bm\Pi\cdot\bm H)+
\frac{\mu_0}{2}\left\{\frac{mc^2}{\epsilon'},
\nabla(\bm\Pi\cdot\bm
H)\right\}\\-\frac{\mu'c}{4}\left\{\frac{1}{\epsilon'},
\left[\nabla(\bm\Sigma\cdot[\bm\pi\times\bm E])-
\nabla(\bm\Sigma\cdot[\bm E\times\bm\pi]) \right]\right\}\\-
\frac{\mu_0mc^3}{\sqrt{2\epsilon'(\epsilon'+mc^2)}}\left[\nabla(\bm\Sigma\cdot[\bm\pi\times\bm
E])- \nabla(\bm\Sigma\cdot[\bm
E\times\bm\pi])\right]\frac{1}{\sqrt{2\epsilon'(\epsilon'+mc^2)}}\\
-\frac{\mu'c^2}{2\sqrt{2\epsilon'(\epsilon'+mc^2)}}\left\{(\bm{\Pi}\cdot\bm\pi),\left[\nabla
(\bm{H}\cdot\bm\pi)+\nabla(\bm{\pi}\cdot\bm H)\right]\right\}
\frac{1}{\sqrt{2\epsilon'(\epsilon'+mc^2)}}. \end{array}
\label{eq35}$$
This equation can be divided into two parts. The first part does not contain the Planck constant and describes the quantum equivalent of the Lorentz force. The second part is of order of $\hbar$. This part defines the relativistic expression for the Stern-Gerlach force. Since terms proportional to $d$ are small, they are omitted.
The equation of spin motion is given by $$\begin{array}{c} \frac{d\bm{\Pi}}{dt}=2\mu'\bm\Sigma\times\bm H+\mu_0 \left\{\frac{mc^2}{\epsilon'},
\bm\Sigma\times\bm H\right\} %\\
-\frac{\mu'c}{2}\left\{\frac{1}{\epsilon'},
\left[\bm\Pi\times(\bm \pi\times \bm E)-\bm\Pi\times(\bm E\times \bm\pi) \right]\right\}\\
- \frac{\mu_0mc^3}{\sqrt{\epsilon'(\epsilon'+mc^2)}}\left[\bm\Pi\times(\bm \pi\times \bm E)
-\bm\Pi\times(\bm E\times \bm\pi) \right]\frac{1}{\sqrt{\epsilon'(\epsilon'+mc^2)}}\\
- \frac{\mu'c^2}{\sqrt{2\epsilon'(\epsilon'+mc^2)}} \left\{(\bm\Sigma\times\bm \pi), (\bm{H}\cdot\bm\pi+\bm{\pi}\cdot\bm H)\right\}
\frac{1}{\sqrt{2\epsilon'(\epsilon'+mc^2)}} \\
+2d\bm\Sigma\times\bm E %\\
-\frac{dc^2}{\sqrt{2\epsilon'(\epsilon'+mc^2)}} \left\{(\bm\Sigma\times\bm \pi),
(\bm{E}\cdot\bm\pi+\bm{\pi}\cdot\bm E)\right\} \frac{1}{\sqrt{2\epsilon'(\epsilon'+mc^2)}} \\+\frac{dc}{2}\left\{\frac{1}{\epsilon'},
\left[\bm\Pi\times(\bm \pi\times \bm H)-\bm\Pi\times(\bm H\times \bm\pi) \right]\right\}.
\end{array} \label{eq36}$$
Eqs. (\[eq33\]),(\[eq35\]),(\[eq36\]) agree with the corresponding equations derived in Refs. [@JMP; @RPJ]. However, unlike the latter equations, Eqs. (\[eq33\]),(\[eq35\]),(\[eq36\]) describe strong-field effects.
We can also consider the interaction of spinless particles with the strong electromagnetic field. The initial Klein-Gordon equation describing this interaction has been transformed to the Hamilton form in Ref. [@FV].
In this case, the Hamiltonian acts on the two-component wave function which is the analogue of the spinor. The explicit form of this Hamiltonian is [@FV] $${\cal
H}=\rho_3mc^2+(\rho_3+i\rho_2)\frac{\bm\pi^2}{2m}+e\Phi.
\label{FV}$$ Therefore, $${\cal
M}=mc^2+\frac{\bm\pi^2}{2m}, ~~~{\cal E}=e\Phi, ~~~{\cal
O}=i\rho_2\frac{\bm\pi^2}{2m}, ~~~[{\cal M},{\cal O}]=0.
\label{MEO}$$ For spinless particles, $$\epsilon=\sqrt{m^2c^4+c^2\bm\pi^2}, ~~~
T=\sqrt{\frac{\epsilon}{mc^2}}\left(\epsilon+mc^2\right).
\label{eqep}$$
The Hamiltonian transformed to the FW representation is given by $${\cal H}_{FW}=\beta\epsilon+{\cal
E}=\beta\sqrt{m^2c^4+c^2\bm\pi^2}+e\Phi. \label{eqfz}$$ There are not any terms of order of $\hbar$ in this Hamiltonian, while it contains terms of second and higher orders in the Planck constant. We do not calculate the latter terms because their contribution into equations of particle motion is usually negligible.
The operator equation of particle motion takes the form $$\begin{array}{c} \frac{d\bm \pi}{dt}=e\bm
E+\beta\frac{ec}{4}\left\{\frac{1}{\epsilon},
\left([\bm\pi\times\bm H]-[\bm H\times\bm\pi]\right)\right\}.
\end{array} \label{eqpz}$$
The right hand side of this equation coincides with the spin-independent part of the corresponding equation for spin-1/2 particles. Eq. (\[eqfz\]) for the FW Hamiltonian agrees with Eq. (12) in Ref. [@PAN]. In this reference, the weak-field approximation has been used and the operator equation of particle motion in the strong electromagnetic field has not been obtained.
SEMICLASSICAL LIMIT OF RELATIVISTIC QUANTUM MECHANICS FOR PARTICLES IN STRONG EXTERNAL FIELDS
==============================================================================================
To obtain the semiclassical limit of the relativistic quantum mechanics, one needs to average the operators in the quantum mechanical equations. When the FW representation is used and relations (\[rel1\]),(\[rel2\]) are valid, the semiclassical transition consists in trivial replacing operators by corresponding classical quantities. In this representation, the problem of extracting even parts of the operators does not appear. Therefore, the derivation of equations for particles of arbitrary spin in strong external fields made in the precedent section solves the problem of obtaining the semiclassical limit of the relativistic quantum mechanics. If the momentum and position operators are chosen to be the dynamical variables, relations (\[rel1\]),(\[rel2\]) are equivalent to the condition $$|<p_i>|\cdot|<x_i>|\gg|<[p_i,x_i]>|=\hbar,~~~
i=1,2,3. \label{rel3}$$ The angular brackets which designate averaging in time will be hereinafter omitted.
Obtained semiclassical equations may differ from corresponding classical ones.
As a result of replacing operators by corresponding classical quantities, the semiclassical equations of motion of particles and their spins take the form $$\begin{array}{c}
\frac{d\bm \pi}{dt}=e\bm E+\frac{ec}{\epsilon'}
\left(\bm\pi\times\bm H\right)
%\\
+\mu'\nabla(\bm P\cdot\bm H)+ \frac{\mu_0}{mc^2\epsilon'} \nabla(\bm P\cdot\bm H)\\
-\frac{\mu'c}{\epsilon'} \nabla(\bm P\cdot[\bm\pi\times\bm E])%\\
-\frac{\mu_0mc^3}{\epsilon'(\epsilon'+mc^2)}\nabla(\bm
P\cdot[\bm\pi\times\bm E])\\
-\frac{\mu'c^2}{\epsilon'(\epsilon'+mc^2)}(\bm{P}\cdot\bm\pi)\nabla
(\bm{H}\cdot\bm\pi), ~~~~~~~ \bm P=\frac{\bm S}{S}, \end{array}
\label{eqw}$$ $$\begin{array}{c} \frac{d\bm P}{dt}=2\mu'\bm P\times\bm H+ \frac{2\mu_0mc^2}{\epsilon'}( \bm P\times\bm H)%\\
-\frac{2\mu'c}{\epsilon'} \left(\bm P\times[\bm \pi\times \bm
E]\right)\\ -\frac{2\mu_0mc^3}{\epsilon'(\epsilon'+mc^2)}\left(\bm
P\times[\bm \pi\times \bm E] \right)
%\\
-\frac{2\mu'c^2}{\epsilon'(\epsilon'+mc^2)} (\bm P\times\bm \pi)(\bm{\pi}\cdot\bm H)\\ %
+2d\bm P\times\bm E-\frac{2dc^2}{\epsilon'(\epsilon'+mc^2)} (\bm
P\times\bm \pi)(\bm{\pi}\cdot\bm E) %\\
+\frac{2dc}{\epsilon'} \left(\bm P\times[\bm \pi\times \bm
H]\right). \end{array} \label{eqt}$$ In Eqs. (\[eqw\]),(\[eqt\]), $\epsilon'$ is defined by Eq. (\[eq34\]), $\bm P$ is the polarization vector, and $\bm S$ is the spin vector (i.e., the average spin).
For scalar particles $$\begin{array}{c}
\frac{d\bm \pi}{dt}=e\bm E+\frac{ec}{\sqrt{m^2c^4+c^2\bm\pi^2}}
\left(\bm\pi\times\bm H\right). \end{array} \label{eqwl}$$
Two first terms in right hand sides of Eqs. (\[eqw\]),(\[eqwl\]) are the same as in the classical expression for the Lorentz force. This is a manifestation of the correspondence principle. The part of Eq. (\[eqt\]) dependent on the magnetic moment coincides with the well-known Thomas-Bargmann-Michel-Telegdi (T-BMT) equation. It is natural because the T-BMT equation has been derived without the assumption that the external fields are weak. The relativistic formula for the Stern-Gerlach force can be obtained from the Lagrangian consistent with the T-BMT equation (see Ref. [@PK]). The semiclassical and classical formulae describing this force also coincide. High-order corrections to the quantum equations of motion of particles and their spins should bring a difference between quantum and classical approaches.
DISCUSSION AND SUMMARY
======================
The new method of the FW transformation for relativistic particles of arbitrary spin in strong external fields described in the present work is based on the previous developments [@JMP]. However, the use of transformation operator (\[eq18N\]) is not restricted by any definite commutation relations \[see Eq. (\[comm\])\] between even and odd operators. The proposed method utilizes the expansion of the FW Hamiltonian into a power series in the Planck constant which defines the order of magnitude of quantum corrections. In the FW Hamiltonian, exact expressions for low-order terms in $\hbar$ can be obtained. If the de Broglie wavelength is much less than the characteristic size of the nonuniformity region of the external field \[see Eqs. (\[rel1\]),(\[rel2\])\], the transition to the semiclassical approximation becomes trivial. In this case, it consists in replacing operators by corresponding classical quantities. The simplest semiclassical transition is one of main preferences of the FW representation.
If Eqs. (\[rel1\]),(\[rel2\]) are not valid, the proposed method can be used in the weak-field expansion. This expansion previously used in Ref. [@JMP] presents the FW Hamiltonian as a power series in the external field potentials and their derivatives. In this case, the operator equations characterizing dynamics of the particle momentum and spin can also be derived. Solutions of these equations define the quantum evolution of main operators. Semiclassical evolution of classical quantities corresponding to these operators can be obtained by averaging the operators in the solutions. An example of such an evolution is time dependence of average energy and momentum in a two-level system.
When the FW Hamiltonian can be expanded into a power series in the Planck constant, we obtain the semiclassical limit of the relativistic quantum mechanics. Since the correspondence principle must be satisfied, classical and semiclassical Hamiltonians and equations of motion must agree. As an example, we consider the interaction of scalar and spin-1/2 particles with the strong electromagnetic field. We have carried out the FW transformation and have derived the quantum equations of particle motion. We have also deduced the quantum equations of spin motion for spin-1/2 particles. Averaging operators in the quantum equations consists in substitution of classical quantities for these operators and allows to obtain the semiclassical equations which are in full agreement with the corresponding classical equations. The proved agreement confirms the validity of both the correspondence principle and the aforesaid method. All calculations have been carried out for relativistic particles in strong external fields.
Acknowledgements {#acknowledgements .unnumbered}
================
The author is grateful to O.V. Teryaev for interest in the work and helpful discussions and to T. Tanaka for bringing to his attention Ref. [@Tanaka]. This work was supported by the Belarusian Republican Foundation for Fundamental Research.
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[**LMU 08/97**]{}\
[**CERN-TH/97-200**]{}
**The Gluonic Decay of the $b$–Quark and the $\eta '$–Meson**
[**Harald Fritzsch**]{}\
\
\
\
[*Theresienstrasse 37, D–80333 München*]{}
The observed inclusive decay of $B$–mesons into $\eta ' + X$ is interpreted as the consequence of the gluonic decay of the $b$–quark into an $s$–quark. As a result of the QCD anomaly this decay proceeds partly as the decay $b \rightarrow s + \eta '$, similar to $b \rightarrow s + J / \psi $. The hadronic recoiling system is found to have a relatively large mass. Analogously one expects a decay of the type $b \rightarrow s + \sigma $. The branching ratios for these decays are large (of the order of 10 %). The results indicate that there is no room for an anomalously large chromomagnetic decay mode of the $b$–quark. Gluon jets are expected to exhibit an anomalously large tendency to fragment into $\eta '$- and $\sigma$- mesons.\
[**LMU 08/97**]{}\
[**CERN-TH/97-200**]{}
Recently one has observed a relatively large rate for the decay of $B$–mesons into an $\eta '$–meson and additional hadrons. One finds$^{1)}$:\
$$Br \left( B \rightarrow \eta ' + X \right)
= (7.5 \pm 1.5 \pm 1.1) \times 10^{-4}$$ under the constraint:\
$$2.2 \, GeV \le E (\eta ') \le 2.7 \, GeV.$$ Furthermore the exclusive decay $B^{\pm } \rightarrow \eta ' K^{\pm}$ has been observed$^{2)}$: $$Br (B \rightarrow \eta ' K)
= \left( 7.8^{+2.7}_{-2.2} \pm 1.0 \right) \times 10^{-5}$$ The latter is of the same order (albeit somewhat larger) as the decay $B^{\pm} \rightarrow \pi ^{\pm}
K^0$, which is observed with a branching ratio of $2.3^{+1.1}_{-1.0} \cdot 10^{-5} $.\
\
The observation of an $\eta '$–meson in the final state of a $B$–decay is of interest both with respect to the internal dynamics of such a decay involving nonperturbative aspects of QCD and with respect to the validity of the standard electroweak theory describing the decay.\
\
It is well known that the $\eta '$–meson plays a special role in the dynamics of the strong interactions. Due to its QCD anomaly it evades to be a massless Goldstone boson in the limit of the chiral $SU(3) \times SU(3)$ symmetry$^{3)}$. Due to its special character it is expected that the $\eta '$–meson has an anomalously large coupling to gluonic field configurations.$^{4)}$ Thus a possible connection between this feature, the QCD anomaly and the $B$–decays involving the $\eta '$–meson, might exist. Such possibilities have been discussed recently in a number of papers$^{5, 6, 7, 8, 9)}$. In this note we shall discuss the decay $B \rightarrow \eta ' + X$ from a phenomenological point of view. It is shown that definite conclusions can be drawn using simple and model-independent arguments.\
\
The decay $B \rightarrow \eta ' + X $ (either inclusive or exclusive) is a decay, in which no charmed particles are present in the final state. Thus the decay proceeds either via a charmless decay mode of the $b$–quark (see below) or via the intermediate formation of a $\bar c c$–pair (via the decay mode $b \rightarrow c (\bar c s)$), which annihilates to form the final state containing an $\eta '$–meson. The most important decay channels would be\
a) $B \rightarrow J / \psi + X, \, J / \psi \rightarrow \eta ' + X $ b) $B \rightarrow \eta _c + X, \, \eta _c \rightarrow \eta ' + X$.\
\
The possibility a) can easily be dismissed, since decays of the $J / \psi $ involving $\eta ' $ are seen of the level of $10^{-4}$ in branching ratio, thus the corresponding $B$–decays are of the order of $10^{-6}$. The second possibility was recently studied$^{5)}$, with the conclusion, that the expected branching ratio for $B \rightarrow \eta ' + X $ could be at most $10^{-4}$, i. e. small in comparison to the observed rate.\
\
It is well known that there is a relatively large mixing between the $\eta '$–meson and the $\eta _c$–state$^{10)}$, which accounts for the decay $J / \psi \rightarrow \eta ' \gamma $ (for an updated discussion, including the $\eta - \eta '$–mixing, see ref. (11)). Following ref. (10), we denote the $\eta (\eta ')$–wave function as follows (in the absence of a $\bar c c$–contribution): $$\begin{array}{lll}
\eta & = & \frac{1}{\sqrt{2}} (\bar u u + \bar d d) \, {\rm cos} \,
\alpha - \bar s s \, {\rm sin } \, \alpha\\
\eta ' & = & \frac{1}{\sqrt{2}} (\bar u u + \bar d d) \, {\rm sin} \alpha \,
+ \bar s s \, {\rm cos} \, \alpha
\end{array}$$ The mass spectrum of the pseudoscalar mesons as well as their radiative decays indicate $\alpha \simeq 44^{\circ }$ (possible uncertainties in this value will not be discussed here). The large mixing between the various $(\bar q q)$–contributions is due to the QCD anomaly. The latter is also responsible or the $(\bar c c)$–admixture of the $\eta $ and $\eta '$–meson. As discussed in ref. (10), the $\eta_c$–state can be written as: $$\begin{array}{lll}
\eta_c & = & \bar c c + \varepsilon \cdot \eta + \varepsilon ' \cdot \eta '\\
\varepsilon & \cong & 0.010 \qquad \varepsilon ' \cong 0.024
\end{array}$$ This gives a satisfactory description of the decay: $J / \psi \rightarrow
\eta (\eta ') + \gamma $. Thus the QCD anomaly which is responsible for the large mixing in the pseudoscalar channel, accounts well for these radiative decays. Likewise the decay $B \rightarrow \eta ' + X$ can proceed via the $(\bar c c)$–admixture of the $\eta '$–meson, given by the mixing parameter $\varepsilon '$, and one finds: $$Br (B \rightarrow \eta ' X) = Br (B \rightarrow \eta _c X) \cdot
(\varepsilon ')^2 \preceq 10^{-2} \cdot (\varepsilon')^2
\approx 5 \cdot10^{-6}$$ (Here we used $Br (B \rightarrow \eta _c X) \preceq 10^{-2} $ (see ref.(12)). Thus the $\bar c c$–admixture of the $\eta '$–meson although anomalously large, does not lead to a decay rate for $B \rightarrow \eta ' X$ at the observed rate. Likewise one can see that the $\bar cc$–admixture does not give a sizeable contribution to the exclusive decay $B \rightarrow \eta ' K$. The decay $B \rightarrow \eta _c K$ has not been observed, but its branching ratio is expected to be smaller than for the decay $B \rightarrow J / \psi K$ (branching ratio of order $10^{-3}$). Thus one finds for the decay induced by mixing: $$Br (B \rightarrow \eta ' K) \preceq 10^{-3} \cdot (\varepsilon ')^2 \simeq
5 \cdot 10^{-7}$$ an effect far below the observed signal. In general we conclude that the $\bar c c$-admixture of the $\eta
'$–meson can be neglected for the discussion of the decays $B \rightarrow \eta ' X$ and $B \rightarrow \eta ' K$. Below we shall argue that the observed inclusive decay $B \rightarrow \eta ' X$ is indeed the first observation of the gluonic decay of the $b$–quark into an $s$–quark.\
\
It is known that within the standard electroweak model the $b$–quark can decay into an $s$-quark by emitting an on–shell or off–shell gluon$^{13, 14)}$. The process proceeds via the transition of the $b$–quark into a virtual $t$–quark and a $W$–boson, which combine to give an $s$–quark under gluon–emission. If the gluon is on–shell, this process corresponds to a chromomagnetic $b - s$ transition, described by the effective Lagrangian\
$${\cal L}^{eff}_{b \rightarrow s} = - const. \frac{G_F}{\sqrt{2}} \cdot
\frac{g_s}{4 \pi ^2} \cdot m_b \, \bar b_R \sigma_{\mu \nu }
G^{\mu \nu} s_L \cdot V_{ts}$$\
($G_{\mu \nu }$: gluonic field–strength, $V_{ts}: t-s$–transition element of the weak current, the const. depends on details of perturbate QCD and is estimated to be about 0.15). If the gluon is off–shell, decaying afterwards into a $ \bar q q$–pair or gluon–pair, the decay proceeds by a chromoelectric transition. Taking both effects into account, one finds that about 1% of all $B$–decays should be of the type $b \rightarrow s +$ glue. The rate for this process shall be denoted by $\Gamma ^g$. The low–order QCD–calculation indicates that the chromomagnetic process is smaller than the chromoelectric one.\
\
It is interesting to note that the gluonic decay of the $b$–quark would be the dominant decay mode for the $b$–quark if the $c$–quark would be heavier than the $b$–quark. In such a fictitious world the life–time of the $B$–mesons would be about two orders of magnitude larger than in reality. The question arises how the hadronic final state looks in the gluonic $b$–decay. In the chromomagnetic decay the $b$–quark, essentially at rest inside the $B$–meson, disintegrates into an almost massless $s$–quark and a gluon. Both quanta are emitted back to back, each carrying a momentum of about 2.3 GeV / c. (We shall use for our subsequent consideration an effective $b$–mass of 4.6 GeV). This process is reminiscent of the decay $b \rightarrow s + J / \psi $, discussed a long time ago$^{15)}$, except in the latter process the energy of the emitted $s$–quark is about 1 GeV smaller than in the process discussed here. The hadronic final state is formed by the fragmentation of the $(s \bar q - g)$–system into hadrons.\
\
The chromoelectric decay proceeds in a similar way, except the emitted gluon is not on–shell, i. e. the final system is a $(s \bar q - g^*)$–System. Since the process is dominated by small $g^*$- masses, the fragmentation process should be similar, and we shall not discriminate between the two any longer. In fact, in term of observable quantities a clear–cut distinction between the chromomagnetic and chromoeletric reactions cannot be made.\
The gluonic $b$–decay is a reaction in which for the first time the conversion of a single gluon of relatively low energy into hadrons can be studied experimentally. A single color–octet gluon emitted by the decaying $b$–quark and leaving the hadronic debris of the previous $B$–meson with relativistic speed can, of course, not be converted into a color–singlet system of hadrons. However the color–octet nature of the gluon can easily be “bleached” by the interaction with one or several soft gluons participating in the process. If the mass of the decaying $b$–quark were very high, say 20 GeV or more, one would observe a gluon jet, identical to the gluon jets observed in other hadronic processes. In our case, however, the “gluon jet” has only an energy of about 2.3 GeV. A certain fraction of the decays will lead to glue–mesons, expected to have a mass in the region above 1.6 GeV. These final states will be complicated, involving a fairly large number of hadrons. We shall not consider them any longer. An important fraction of the decays will involve those hadrons which are known to have a strong affinity to the gluonic degrees of freedom. In low energy hadron physics there are only two particles known to have this property:
1. The $\eta '$–meson\
The large $\eta '$–mass reflects the strong coupling of this meson to gluons (see ref. 3, 4). The matrix element $< 0 \mid A \mid \eta ' >$ of the gluonic operator $A = \alpha _s \, G_{\mu, \nu } \,
\stackrel{\sim }{G^{\mu \nu }}$, which gives rise to the QCD–anomaly in the divergence of the axial current bilinear in a particular quark field, e.g.$\bar u \gamma _{\mu }
\gamma _5 u$, is known to be large. The special properties of the $\eta '$–meson due to its strong gluonic coupling as a consequence of the QCD-anomaly have been discussed a long time ago$^{4)}$.
2. The $\sigma (f^0)$–meson\
This $0^{++}$–meson is observed as a broad resonance: $M = (400 - 1200)$ MeV, decaying predominantly into $ \pi \pi $. It is usually associated with the $\sigma $–meson in chiral models of the $\pi $–nucleon–system. In QCD the $\sigma $–state dominates the matrix elements of the trace of the energy–momentum–tensor which is proportional to the scalar gluonic density $G_{\mu \nu } \, G^{\mu \nu }$. Thus the transition element $< 0 \mid \alpha_s \cdot G_{\mu } G _{\mu \nu} \, G ^{\mu \nu } \mid \sigma >$ is also anomalously large, and the $\sigma $–meson shows like the $\eta '$ a strong affinity to gluons.\
The bleaching of the color-octet gluon emitted in the b-s-transition is achieved by its interaction with one or several soft gluons. Finally a color singlet gluonic system is emitted consisting of the original gluon of relatively high energy, accompanied by at least one soft gluon. This system can be described by a complicated sum of products of at least two gluonic field strengths operators and their derivatives, corresponding to the various spin and parity assignments of the multigluon system in question. The hadronic matrix elements of these operators will finally determine which types of hadrons are emitted in the decay. These states will be primarily gluonic mesons, except for the special cases, in which one deals with the pseudoscalar and scalar gluonic densities, which at low frequencies are strongly dominated by the $\eta '$ and $\sigma$–mesons. If the mass of the b-quark would be less than in reality, say only 2.5 GeV, there would be no phase space for producing gluonic mesons. Effectively all gluonic densities would be projected out, except for the scalar and pseudoscalar densities, due to the large coupling to the $\sigma$ and $\eta '$–mesons. In this case the gluonic decay of the b-quark would lead exclusively to $\eta '$ or $\sigma$–mesons!\
\
If we restrict ourselves to the pseudoscalar and scalar densities, we can write down an ansatz for the effective interaction between the quarks and the gluonic densities as follows:\
$${\cal L}^{eff} = -const. \frac{G_F}{\sqrt{2}}V_{ts}
\frac{\alpha_s}{4 \pi ^2} \cdot m_b \, \bar b_R s_L \left( G_{\mu \nu}
G^{\mu \nu} + G_{\mu \nu } \stackrel{\sim} {G^{\mu \nu}}\right)$$ where the const. absorbs the details of the non-perturbative aspects of the strong interactions involved in the decay process. No attempt is made here to calculate this coefficient. Since at low frequecnies the scalar and pseudoscalar densities are dominated by the corresponding mesons, one concludes that a rather sizeable fraction of the gluonic $b$–decay will be given by the decays $b \rightarrow s +
\eta '$ and $b \rightarrow s + \sigma $. The total gluonic decay rate is given by $$\Gamma ^g = \Gamma ^{\eta '} + \Gamma ^{\sigma } + \stackrel{\sim }{\Gamma}$$ where $\stackrel{\sim }{\Gamma} $ denotes in particular the contribution of the large mass states (glue mesons etc.).\
\
It is interesting to study the kinematics of the decay $b \rightarrow s +
\eta '$. As an example we shall use a $b$–quark mass of 4.6 GeV and a mass of 0.2 GeV for the $s$–quark. For a $b$–quark at rest the energy of the emitted $\eta '$–meson is 2.4 GeV, the momenta of both the $\eta '$ and the $s$–quark are 2.2 GeV/c. The invariant mass $M$ of the hadronic system recoiling against the $\eta '$–meson can easily be calculated in the case in which the $b$–quark is taken to be at rest in the rest system of the $B$–meson. One finds M = 1.85 GeV.In reality the momentum of a $b$–quark inside the $B$–Meson varies between zero and about 300 MeV. Thus the momenta of the emitted $\eta '$ should vary around 2.4 GeV $(\pm \sim 300 $ MeV). Likewise the invariant mass of the recoiling system should vary in the range 1.5 $\ldots $ 2.1 GeV. Both the momentum distribution of the $\eta '$–meson and the distribution of the invariant mass should exhibit a Gaussean behaviour, reflecting the momentum distribution of the $b$–quark inside a $B$–meson. In fact, the momentum distribution of the $\eta '$–meson can be interpreted as a measure of the momentum distribution of the $b$–quarks inside the meson.\
\
The decay $b \rightarrow s + \eta '$ differs in this respect from the decay $b \rightarrow s + J / \psi $. There the momentum of the emitted $s$–quark is much less, due to the large $\bar c c$–mass, and the invariant mass of the hadronic system recoiling against the $J / \psi $ is correspondingly much lower such that a sizeable fraction of these decays are of the type $b \rightarrow s K$ or $b \rightarrow s K^*$, in agreement with experiment. In the case of the decays $b \rightarrow s + \eta '$ or $b \rightarrow s +
\sigma $ one has little chance to find solely a $K$ or $K^*$–meson recoiling against the $\eta '$ or $\sigma $, due to the large average mass of the recoiling system. This agrees with the observed fact that the rate for $B \rightarrow K \eta '$ is significantly smaller than the rate for $B \rightarrow \eta ' X$. The decays should be dominated by decays in which the recoiling hadronic system has an invariant mass in the range between 1.5 and 2.1 GeV.\
\
The emitted $\eta '$–meson carries a sizeable energy (between about 2.1 GeV and 2.7 GeV). This energy region essentially coincides with the energy cut made in the experimental analysis. It remains to be seen whether the two main features discussed above (large invariant mass ($\sim $ 1.85 GeV) of the recoiling system, large $\eta '$–energy ($\sim $ 2.4 GeV), narrow momentum and mass distribution) are established in future experiments.\
\
The conversion of the gluon emitted in the $b$–decay into an $\eta '$–meson is intrinsically a truly non–perturbative effect, like the generation of the $\eta '$–mass. We doubt whether it is useful to use perturbative methods, for example by introducing an $\eta '- g - g$ vertex as suggested in refs. (5, 6). In this approach one would not expect that a single $\eta '$–meson is emitted, while in our approach this is the case as a natural consequence of the gluonic anomaly.\
\
The mechanism discussed in refs. (5, 6) leads to the decay $b \rightarrow
s \eta ' g$. Thus the elementary process is not a two–body decay. The momentum distribution for the $\eta '$–meson is expected to be much broader than in our case. While we do not expect that $\eta '$–mesons are found with momenta less than 2.1 GeV and more than 2.7 GeV, in the $b \rightarrow
\eta '
g$–case this is expected. Likewise the invariant masses of the recoiling systems should show a broad distribution. Thus the two possibilities $b \rightarrow s \eta '$ and $b \rightarrow s \eta ' g$ can be distinguished experimentally as soon as more detailed data become available.\
\
What has been said above for the decay $b \rightarrow s + \eta '$ can be repeated for the decay $b \rightarrow s + \sigma $. Since the central value of the $\sigma$–mass (800 MeV) is 160 MeV lower than the $\eta '$–mass, in average the momentum of the emitted $s$–quark is slightly larger, but in view of the large $\sigma $–width and the fluctuation of the $b$–momentum inside the $B$–meson this effect can be neglected. Thus it is expected that the $B$–meson decays into a $\sigma $–meson, such that the total energy is in average about 2.4 GeV, recoiling against a hadronic system involving a $K$–meson with an invariant mass just below 2 GeV. The $\sigma $–meson decays into a $\pi \pi $–system. Due to the large width of the $\sigma $–meson it is not easy to identify. Nevertheless the simple two–body kinematics of the underlying decay $b \rightarrow s + \sigma $ might help in the search for this decay.\
\
Finally we shall discuss the expected rate for the $\eta '$– and $\sigma $–decays of the $b$–quark, by comparing them to the total gluonic decay rate, introducing the ratios $$\begin{aligned}
r ( \eta ') & = & \Gamma (B \rightarrow \eta ' X) / \Gamma ^g\\
r ( \sigma ) & = & \Gamma (B \rightarrow \sigma X) / \Gamma ^g\nonumber\end{aligned}$$ As argued above, these ratios will not be small. To get an order of magnitude estimate for $r(\eta ')$ and $r(\sigma )$, we consider the decay of the $\eta _c$–meson. In lowest order of QCD this meson decays into two gluons, each of them carrying an energy of about 1.5 GeV (about 0.8 GeV less than the gluon emitted in the gluonic $b$–decay). Applying the ideas discussed above to the $\eta_c$–decay, we would expect that it decays relatively often into $\eta ' \eta ', \sigma \eta '$ and $\sigma \, \sigma $. Since the $\eta '$ is detected by its (small) $\gamma \, \gamma $–decay mode, the $\eta ' \eta ' $–mode would be difficult to identify, but the $\sigma \eta '$–mode, giving e. g. a final state $\pi ^+ \pi ^- \eta '$, has indeed been identified as a leading decay mode: $ Br \left( \eta _c \rightarrow \pi \pi \eta ' \right) =
4.1 \pm 1.7 \%$. The kinematics of the process is consistent with the hypothesis that originally an $\eta ' -\sigma $–system is formed. These observations support our idea that gluons at relatively low energy have a sizeable tendency to produce $\eta '$ or $\sigma $–mesons. We see no reason why the fragmentation into an $\eta '$–meson should differ much from the fragmentation into a $\sigma $–meson, and we expect that $r(\eta ')$ and $r (\sigma )$ should be of similar order of magnitude.\
\
If we take $r (\sigma ) = r(\eta ')$, one finds $ r(\eta ') = r (\sigma )
\approx \left( Br (\eta_c \rightarrow \eta ' \sigma ) \right)^{1/2} \approx
0.2$. This estimate cannot be regarded as more than an estimate of order of magnitude for $r (\eta ')$ and $ r(\sigma )$, in view of the large uncertainties involved, but it implies that e. g. $r (\eta ')$ is not small, but surprisingly large, of the order of 20%, perhaps even larger, in complete agreement with our expectation based on the dominance of the gluonic densities by the corresponding isoscalar mesons. The contributions of the exclusive decays $b \rightarrow s + \eta '$ and $b \rightarrow s + \sigma$ to $\Gamma ^g$ could make up a fairly large portion of all gluonic decays, say 20 $\ldots $ 50%.\
\
For the inclusive decay $B \rightarrow \eta ' X$ we find $$Br (B \rightarrow \eta ' X) = r (\eta ') \cdot \Gamma ^g / \Gamma ^{tot}$$ As remarked earlier, the ratio $\Gamma ^g / \Gamma ^{tot}$ is expected in the Standard Model to be about 1 %. For $r (\eta ')$ = 10 % we would obtain $Br (B \rightarrow \eta ' + X) \approx 0.1\%$, not in disagreement with the observed rate, in view of the uncertainty in the calculation of $\Gamma ^g / \Gamma ^{tot}$. The branching ratio for the decay $B \rightarrow
\sigma + X$ should also be about 0.1%.\
\
We add a remark concerning the decay $ B \rightarrow \eta + X$. If SU(3) were an exact symmetry of the strong interactions, the decay $b \rightarrow s +
\eta $ would be forbidden. Due to SU(3) breaking effects it will be induced. These breaking effects will show up in two different ways:
1. The $\eta $–meson is partly a SU(3) singlet and can communicate via its singlet part with the pseudoscalar gluonic operator $G_{\mu \nu} \stackrel{\sim }{G^{\mu \nu}}$. Thus the decay can proceed via the singlet–octet–mixing in the wave function of the $\eta $–meson.
2. The gluonic coupling to $\bar u u / \bar d d$ and to $\bar s s$ in the pseudoscalar channel violates SU(3)$^{4)}$. This effect also influences the decay rate for $b \rightarrow s + \eta$.
We do not attempt a precise calculation of these effects, which in any case would have a sizeable systematic error. However both effects play an analogous role in the decays $J / \psi \rightarrow \eta ' +
\gamma $, which proceed via the gluonic mixing with the $\eta _c$–state. Using the results of ref. (10), we find in terms of the parameters $\varepsilon $ and $\varepsilon '$ (see eq. (4)):\
$$\frac{\Gamma (b \rightarrow s + \eta)}{\Gamma (b \rightarrow s + \eta ')}
\approx \left( \frac{\varepsilon}{\varepsilon '} \right) ^2 \approx
\frac{1}{6}$$\
Phase space effects which are small due to the large $b$–mass have been neglected in eq. (3). Thus we expect the rate for the inclusive decay $B \rightarrow \eta + X$ to be about $10 ^{-4}$. It could hardly be more than an order of magnitude less than the rate for the decay $B \rightarrow \eta '
+ X$ and should be seen eventually in the experiments. The momentum distribution of the $\eta $ as well as the distribution of the invariant masses of the recoiling systems are identical to the $\eta '$–case, if tiny phase space corrections are neglected.\
\
In view of the fact that the semileptonic branching ratio for the $B$–decays seems to be smaller than expected by theory and that there seems to be a charm–deficit in the $B$–decays, it has been suggested that the chromomagnetic decay $b \rightarrow s + g$ is abnormally large (branching ratio 5 $\ldots $ 10%), in disagreement with the Standard Model$^{16, 17, 18)}$. (For recent calculations see refs. (18).) Our results are in good agreement with the expectation within the Standard Model. Therefore we conclude that there is no abnormally large chromomagnetic $b$–decay. A large gluonic decay of the $b$–quark is likely not to be the reason for the apparent charm deficit. Nevertheless it could be that due to the high proportion of exclusive decay modes on the level of quark decays the estimate of the total gluonic decay rate based on perturbative QCD is not correct. The actual rate could be enhanced by the nonperturbative effects discussed here, but due to duality arguments which provide a link between perturbative estimates and resonance effects we doubt that this effect could amount to more than a factor of two.\
\
Our considerations support again the idea that due to the QCD–anomaly the $\eta '$–meson plays an important and interesting dual role in hadronic physics. It is a $\bar q q $–state, but shows like the $\sigma $–meson a strong affinity to the gluonic sector of QCD. Gluons of relatively low energy fragment with a large probability into an isolated $\eta '$–meson. It would be interesting to search for an inclusive $\eta '$–signal in gluon jets at high energies, e.g. at LEP. Our considerations suggest that high energy gluon jets should exhibit a leading $\eta '$–meson (see also refs. (4), (20) and (21)). Whether the strong $\eta$–signal, seen in gluonic jets at LEP by the L3-collaboration$^{22)}$ has anything to do with our expectation remains to be seen. The observation of $\eta $ and $\eta '$–mesons in the three-jet-events by the Aleph-collaboration$^{23}$ does not yet allow a firm conclusion, in view of the large experimental uncertainties.\
\
We conclude: The observed inclusive decay $B \rightarrow \eta ' + X$ could not be due to the primary $b$–quark decay $b \rightarrow $ c + (cs). It is the first signal for the gluonic decay $b \rightarrow s + glue $. About 10 $\ldots $ 20% of these decays lead to the production of a single $\eta '$–meson, which has a strong affinity to gluons due to the QCD–anomaly. The decay can be viewed as the two–body decay $b \rightarrow
s + \eta ' $, analogous to $b \rightarrow s + J / \psi $. The hadronic system recoiling against the $\eta '$ has a relatively large mass just below 2 GeV. Likewise the decay $b \rightarrow s + \sigma $ should occur, which could be identified by looking for decays $B \rightarrow \pi ^+ \pi ^- + X$, with $E (\pi ^+ ) + E(\pi ^- ) \approx 2.3$ GeV and the invariant mass of the $\pi \pi $–system being consistent with the $\sigma $–mass. Our results are in agreement with the prediction of the Standard Model for the total gluonic decay rate of the $b$–quark. There is no or only little room for an abnormally large chromomagnetic decay of the $b$–quark.\
\
[**Acknowledgements**]{}:\
I should like to thank M. Gronau, G. Hou, V. Khoze, P. Minkowski, M. Neubert, U. Nierste, W. Schlatter, J. Steinberger, G. Veneziano and G. Wolf for useful discussions.\
\
1. B. Behrens (CLEO coll.), talk presented at the conference on $B$ Physics and $CP$ violation, Honolulu, Hawaii, March 1997
2. F. Wuerthwein (CLEO coll.), talk presented at Rencontre de Moriond, QCD and High Energy Hadronic Interactions, Les Arcs, March 1997
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---
abstract: 'We have identified 335 galaxy cluster and group candidates, 106 of which are at $z > 1$, using a $4.5\mu$m selected sample of objects from a 7.25 deg$^2$ region in the [*Spitzer*]{} Infrared Array Camera (IRAC) Shallow Survey. Clusters were identified as 3-dimensional overdensities using a wavelet algorithm, based on photometric redshift probability distributions derived from IRAC and NOAO Deep Wide-Field Survey data. We estimate only $\sim10\%$ of the detections are spurious. To date 12 of the $z > 1$ candidates have been confirmed spectroscopically, at redshifts from 1.06 to 1.41. Velocity dispersions of $\sim 750$ km s$^{-1}$ for two of these argue for total cluster masses well above $10^{14} M_\odot$, as does the mass estimated from the rest frame near infrared stellar luminosity. Although not selected to contain a red sequence, some evidence for red sequences is present in the spectroscopically confirmed clusters, and brighter galaxies are systematically redder than the mean galaxy color in clusters at all redshifts. The mean $I - [3.6]$ color for cluster galaxies up to $z \sim 1$ is well matched by a passively evolving model in which stars are formed in a 0.1 Gyr burst starting at redshift $z_f = 3$. At $z > 1$, a wider range of formation histories is needed, but higher formation redshifts (i.e. $z_f > 3$) are favored for most clusters.'
author:
- 'Peter R. M. Eisenhardt, Mark Brodwin, Anthony H. Gonzalez, S. Adam Stanford, Daniel Stern, Pauline Barmby, Michael J. I. Brown, Kyle Dawson, Arjun Dey, Mamoru Doi, Audrey Galametz, B. T. Jannuzi, C. S. Kochanek, Joshua Meyers, Tomoki Morokuma, & Leonidas A. Moustakas'
title: |
Clusters of Galaxies in the First Half of the Universe\
from the IRAC Shallow Survey
---
Introduction\[sec:intro\]
=========================
As the most massive gravitationally bound systems in the Universe, the rate of emergence of galaxy clusters since the Big Bang might be expected to be among the most straightforward predictions of cosmological models. Yet despite the advent of the era of precision cosmology ushered in by observations of SNe and the cosmic microwave background (CMB), significant uncertainty remains in the expected numbers of galaxy clusters at $z > 1$. The CMB temperature anisotropies on scales corresponding to clusters are not accurately known, leading to a range of values for $\sigma_8$, the rms matter fluctuation in a sphere of radius $8h^{-1}$ Mpc at $z=0$. Estimates of $\sigma_8$ vary significantly, including e.g. $0.67^{+0.18}_{-0.13}$ [@Gladders2007], $0.76\pm0.05$ [@Spergel2007], $0.80\pm0.1$ [@Hetterscheidt2007], $0.85\pm0.06$ [@Hoekstra2006], $0.90\pm0.1$ [@Spergel2003], $0.92\pm0.03$ [@Hoekstra2002], and $0.98\pm0.1$ [@BahcallBode2003]. The range $\sigma_8 = 0.7$ to 1 corresponds to a variation of a factor of nearly 20 in the predicted numbers of $z > 1$ clusters with $M_{\rm tot} > 10^{14} M_\odot$ [e.g. @ShethTormen1999]. Removing this uncertainty is a major goal of upcoming Sunyaev-Zeldovich cluster surveys such as the SZA [Sunyaev-Zeldovich Array; @Loh2005], AMI [Arcminute Microkelvin Imager; @Kneissl2001], ACT [Atacama Cosmology Telescope; @Kosowsky2003], and SPT [South Pole Telescope; @Ruhl2004].
Since by definition[^1] galaxy clusters contain an unusually high density of galaxies, they provide an efficient means of observing substantial numbers of galaxies at a common distance, offering the hope of constructing the analog of the Hertzsprung-Russell (i.e. color-magnitude) diagram for galaxy evolution. Indeed, studies of the relationship between color and magnitude indicate that clusters are the habitat of galaxies with the oldest and most massive stellar populations (e.g. @SED98; @Blakeslee2003), objects reasonably free of the complications associated with starbursts and dust. These studies are consistent with an extremely simple formation history for cluster galaxies, in which their stars are formed in a short burst at high redshift, and they evolve quiescently thereafter (we use the term “red spike model” in referring to this scenario - see Figure \[m\_vs\_z\]). Studies of the near-IR luminosity functions of cluster galaxies reinforce this picture [e.g. @DePropris1999; @DePropris2007; @Toft2004; @Strazzullo2006]. With their large lookback times, therefore, high redshift galaxy clusters also provide an observational pillar for our understanding of the formation and evolution of galaxies.
Obtaining substantial samples of galaxy clusters at $z > 1$ has proved challenging, largely because such objects are difficult to detect using only optical data. Due to their greatly enhanced rate of star formation by $z \sim 1$, the UV emission from modest sized field galaxies overwhelms that from the intrinsically red spectra of quiesecent, early type galaxies preferentially found in clusters. The Red Sequence Cluster Survey [@GladdersYee2000; @GladdersYee2005] uses the observed color-magnitude relationship in cluster galaxies to improve the contrast and has proven highly efficient to $z \sim 1$, but the optical colors of the red sequence become increasingly degenerate at higher redshifts, as they no longer span the rest 4000Å break. @Wilson2006 describe a program to extend the red sequence technique to higher redshift using Spitzer data, but it is also important to [*test*]{} for the existence of red sequences in $z > 1$ clusters rather than preselecting for them, if possible.
The contrast of high redshift clusters over the field improves at longer wavelengths (Figure \[m\_vs\_z\]), but the contrast against atmospheric emission declines, and until recently the relatively small formats of infrared detector arrays made surveying sufficient $z \ga 1$ volume a formidable undertaking. @Stanford1997 reported the discovery of a cluster at $z = 1.27$ in a 100 square arcminute survey to $K_s$(Vega) = 20 (10$\sigma$). But this survey required approximately 2 hours of exposure in both $J$ and $K_s$ per position, and 30 allocated nights of KPNO 4m time to complete. With estimates for the surface density of $10^{14} M_\odot$ clusters at $z > 1$ in the range $0.2 - 4$ per square degree [@ShethTormen1999], the discovery was in hindsight fortuitous.
Such considerations motivated a different approach, where extended sources in deep X-ray surveys lacking prominent optical counterparts were targeted for IR followup. This technique yielded confirmed clusters at $z = 1.10,$ 1.23, and 1.26 [@Stanford2002; @Rosati1999; @Rosati2004]. With the arrival of [*XMM*]{}, X-ray surveys offer renewed promise, leading recently to the identification of galaxy clusters at $z = 1.39$ [@Mullis2005] and 1.45 [@Stanford2006]. With exposure times $> 20$ ksec and a 30 arcmin field of view, a discovery rate of approximately 30 hours per candidate $z > 1$ cluster above $10^{14} M_\odot$ is expected (assuming one such cluster per square degree, which corresponds to $\sigma_8 = 0.83$).
Searches for clusters around radio galaxies have yielded protoclusters with redshifts as high as 4.1 [@Pentericci2000; @Venemans2002; @Venemans2005] and possibly even 5.2 [@Overzier2006]. The very large redshifts of these systems enable powerful inferences to be drawn regarding the formation of cluster galaxies, but they are less useful as probes of the cosmological growth of structure. Lyman Break Galaxy surveys with intensive followup spectroscopy on the Keck telescopes have also identified highly overdense structures at $z=2.30$ and 3.09 [@Steidel1998; @Steidel2005], and @Ouchi2007 discuss a $z = 5.7$ structure identified via Ly-$\alpha$ emission in a narrow band imaging survey.
Recent advancements in IR detector array formats have renewed interest in ground-based IR surveys. In one example of the state of the art, Elston, Gonzalez et al. (2006) use the $2048 \times 2048$ pixel FLAMINGOS camera to map 4 deg$^2$ to a 50% completeness limit of $K_s = 19.2$ (Vega). With 2 hour exposures on the KPNO 2.1m each covering 1/10th deg$^2$, this leads to an expected discovery rate for high redshift clusters per useful hour of observing which is similar to [*XMM*]{}. The UKIDSS Ultra Deep Survey provides another recent example, finding 13 cluster candidates with $0.6 < z < 1.4$ in a 0.5 deg$^2$ survey [@vanBreukelen2006], one of which has 4 spectroscopic redshifts at $z = 0.93$ [@Yamada2005], and @Zatloukal2007 find 12 candidates with $1.23 < z < 1.55$ in a 0.66 deg$^2$ $H$-band survey in the COSMOS field. @McCarthy2007 present a system with a high density of galaxies with red optical to near-IR colors surrounding a galaxy at $z=1.51$, identified in the 120 arcmin$^2$ Gemini Deep Deep Survey. Candidates drawn from surveys of less than a square degree are unlikely to include many rich clusters, however.
With the launch of the [*Spitzer Space Telescope*]{} in 2003 [@Werner2004], sensitive infrared arrays free from foreground thermal emission were put into operation [@Fazio2004a]. A major scientific driver for the [*Spitzer*]{} Infrared Array Camera (IRAC) Shallow Survey [@Eisenhardt2004] was the detection of $z > 1$ galaxy clusters. The IRAC Shallow Survey uses 90 second exposures per position and covers 8.5 deg$^2$, leading to an expected discovery rate of $< 8$ hours per $z > 1$ cluster. Here we present results from the IRAC Shallow Survey cluster search, finding 106 cluster and group candidates at $z > 1$, of which we estimate only $\sim10\%$ are spurious.
A surface density of over 10 systems per square degree at $z > 1$ is higher than expected for bound systems with masses above $10^{14} M_\odot$ for the current range of plausible $\sigma_8$ estimates. While we present evidence that at least two of the $z > 1$ clusters have masses well above $10^{14} M_\odot$, it is likely that our sample includes systems with masses below $10^{14} M_\odot$ (i.e. groups), and perhaps some unbound filaments viewed end-on. In the remainder of this paper, for brevity the terms “clusters" and “candidates" are used to refer to all such objects which meet our selection criteria, unless otherwise stated.
This paper describes how the cluster sample was identified, and some of the overall photometric properties of the sample. We also provide new spectroscopic evidence supporting nine of these clusters, from $z = 1.057$ to 1.373. Spectroscopic evidence in support of IRAC Shallow Survey selected clusters at $z = 1.112,$ 1.243, and 1.413 was presented in @Elston2006, @Brodwin2006, and @Stanford2005 respectively, and we provide additional previously unpublished spectroscopy on those clusters here, for completeness. @Brodwin2007 discuss the clustering of the clusters, and Galametz et al. (in preparation) report on AGN incidence vs cluster-centric distance. Followup imaging with [*HST*]{} (GO 10496, Perlmutter; 10836, Stanford; and 11002, Eisenhardt) and [*Spitzer*]{} (GO 30950, Eisenhardt) is underway, and future papers will examine the scatter in the color-magnitude relation as a function of morphological type, the dependence of cluster galaxy size on redshift, starburst activity in clusters vs. redshift, and the dependence of mean galaxy properties on surface density.
A cosmology with $H_0 = 70$ km s$^{-1}$, $\Omega_m = 0.3$, $\Lambda = 0.7$ is assumed, and magnitudes are on the Vega system [defined in @Reach2005 for IRAC]. At $z = 1 - 2$, this means that one arcminute corresponds to a physical scale of $480 - 502$ kpc, peaking at 508 kpc at $z = 1.6$. Unless otherwise specified, physical (rather than co-moving) scales are used throughout.
Data\[sec:data\]
================
IRAC Shallow Survey
-------------------
The IRAC Shallow Survey [@Eisenhardt2004] was designed to maximize the number of reliable sources detected per unit time and to cover sufficient area to detect significant numbers of $z \ga 1$ galaxy clusters. As explained in @Eisenhardt2004, a 30 second exposure time per pointing is close to optimum for maximizing source detections, and reliability was obtained by requiring three independent exposures separated by hours at each position. The survey covers $\approx$ 8.5 deg$^2$ and reaches an aperture-corrected 5$\sigma$ depth of $\approx$ 19.1 and 18.3 mag (Vega) at 3.6 and $4.5 \mu$m in 3 diameter apertures. It is a remarkable fact that 90 seconds of combined exposure with IRAC on the 85 cm [*Spitzer Space Telescope*]{} provides sufficient sensitivity to detect evolving $L^*$ galaxies to $z = 2$ (Figure 1).
NOAO Deep Wide-Field Survey
---------------------------
The survey was carried out in the Boötes region of the NOAO Deep Wide-Field Survey [NDWFS; @JannuziDey1999] to allow photometric redshifts to be derived using the deep optical imaging available for this field. The NDWFS reaches 5$\sigma$ point-source depths in the $B_W, R,$ and $I$ bands of $\approx$ 27.1, 26.1, and 25.4 respectively. Typical exposure times per position with the Mosaic-1 camera on the KPNO Mayall 4-m telescope were 1 – 2 hours in $B_W$, 1 – 2 hours in $R$, and 2 – 4 hours in $I$, and the seeing ranged from 0.7 to 1.5. The data acquisition, reduction, and catalog generation are discussed in detail by B. Jannuzi et al. (in preparation) and A. Dey et al. (in preparation). This paper uses the NDWFS third data release (DR3) images and SExtractor catalogs which can be obtained through the NOAO data archive[^2].
AGN and Galaxy Evolution Survey, FLAMINGOS Extragalactic Survey, and Other Surveys
----------------------------------------------------------------------------------
The AGN and Galaxy Evolution Survey (AGES, C. Kochanek et al. in preparation) provides spectroscopic redshifts for $\approx 17,000$ objects (using the version 2.0 catalog) in the IRAC Shallow Survey. AGES is highly complete for sources brighter than 15.7 mag at $4.5 \mu$m, and also for sources brighter than $I=18.5$, with many redshifts for sources up to $I=20$, enabling excellent assessment of the photometric redshifts to $z \sim 0.5$ (Figure 1).
Deep near infrared imaging from the FLAMINGOS Extragalactic Survey (FLAMEX) is available for half the Boötes region [@Elston2006], and was used for deriving a prior on the redshift likelihood functions (see §\[sec:photz\]).
Imaging of the Boötes NDWFS field has also been obtained in the radio [@deVries2002], at 24, 70, and $160 \mu$m with the MIPS instrument on [*Spitzer*]{} [@Houck2005], in the $z-$band [@Cool2007], in the $UV$ with $GALEX$, and in X-rays to a depth of 5 ksec with the ACIS instrument on the [*Chandra X-ray Observatory*]{} [@Murray2005], but these data are not used in this paper.
Sample\[sec:sample\]
====================
Object selection for the cluster search was carried out in the $4.5\mu$m band, because the negative K-correction as the rest–frame $1.6\mu$m peak shifts into this band leads to a flux which is nearly independent of redshift for $0.7 < z < 2$ (Figure 1). Object detection and photometry was carried out using SExtractor [@sextractor] in double–image mode, allowing matched aperture photometry in the other IRAC bands. While smaller apertures maximize the depth of the survey, Monte Carlo simulations showed that 5 diameter aperture magnitudes are necessary to provide sufficiently reliable color measurements and photometric redshifts [@Brodwin2006]. This flux limit (5$\sigma$ at $4.5 \mu$m in 5) corresponds to $13.3$ [$\mu$Jy]{}, or a Vega–based magnitude of $17.8$.
Matching to the NDWFS Catalog
-----------------------------
The NDWFS catalogs were also generated using SExtractor, but run in single-image mode in each band. Detections in the different optical bands and between the optical and IRAC catalogs were matched if the centroids were within $1\arcsec$ of each other, using the closest optical source if more than one satisfied this criterion. For very extended objects (generally at $z\le 0.2$), detections in the different bands were matched if the centroids were within an ellipse defined using the second order moments of the light distribution of the object [@Brown2005].
$\mathbf B_W$RI\[3.6\] Flux Limits and Photometric Errors\[sec:phot-limits\]
----------------------------------------------------------------------------
The NDWFS data were taken over several years in variable conditions, and therefore the photometric depths vary somewhat from pointing to pointing. Average 50% completeness limits for the $B_W$, $R$, and $I$–bands were 26.7, 25.6, and 25.0 respectively. Average 5$\sigma$ flux (magnitude) limits in a 5aperture (which is significantly larger than the optimum detection aperture for these data) were measured via Monte Carlo simulations to be $0.45$ [$\mu$Jy]{} ($24.8$ mag) in $B_W$, $1.03$ [$\mu$Jy]{} ($23.7$ mag) in $R$, and $1.45$ [$\mu$Jy]{} ($23.1$ mag) in the $I$–band [@Brodwin2006]. In the space–based IRAC Shallow survey the depth is more uniform, and in the $3.6\mu$m band it is $10.0$ [$\mu$Jy]{} ($18.6$ mag), also derived from a Monte Carlo simulation.
Although the majority of objects in the $4.5\mu$m-selected catalog are well detected at shorter wavelengths, this is not generally the case for $z>1$ red ellipticals. Objects at these redshifts are often quite faint in the optical as the 4000Å break is longward of the $I$–band (see Figure 1). Where sources were observed but not detected, the flux was taken to be zero and a Monte Carlo 1$\sigma$ error was adopted. This approach is optimal for photometric redshift fitting, where the non-detection provides important contraints on the galaxy spectral energy distribution (SED).
Sample for Photometric Redshift Estimation and Cluster Search\[sec:cluster\_search\_sample\]
--------------------------------------------------------------------------------------------
The photometry used for photometric redshift estimation (§\[sec:photz\]) consists of $B_WRI[3.6][4.5]$ data with the Monte Carlo photometric errors and limits noted above. The $5.8 \mu$m and $8.0 \mu$m bands were not used because they do not have the sensitivity to detect $z>1$ cluster $L^*$ galaxies in the IRAC Shallow Survey. While the IRAC Shallow Survey covers 8.5 $\deg^2$ in each band, the overlap area observed in both the 3.6 and $4.5 \mu$m IRAC bands is 8.0 $\deg^2$. All of this falls within the optical NDWFS [Boötes]{} region. However, due to haloes around bright objects, shorter observation times for regions at the edges of individual mosaic camera pointings, and residual CCD defects, 0.75 $\deg^2$ may have lower quality optical photometry and hence photometric redshifts. Hence these regions are also excluded from the sample.
Finally, 14,044 stars were removed from the catalog using the SExtractor stellarity index in the best seeing optical data for each location. Comparison to the star count model of @Arendt1998, as tabulated in @Fazio2004b, suggests that $\sim80\%$ of stars were identified via this approach, leaving $\sim 2\%$ of the $4.5\mu$m sample as unrecognized stars.
In summary, photometric redshifts were estimated for a total of 175,431 objects brighter than $13.3$ [$\mu$Jy]{} ($5\sigma$) at $4.5 \mu$m in 5 in a 7.25 $\deg^2$ region.
Photometric Redshifts\[sec:photz\]
==================================
A full description of the photometric redshift methodology is given in @Brodwin2006. A summary of those aspects most relevant to cluster detection is provided here. Photometric redshifts were computed using an empirical template–fitting algorithm which linearly interpolates between the four Coleman, Wu, & Weedman (1980) SEDs (E, Sbc, Scd, and Im), augmented by the @Kinney1996 SB3 and SB2 starburst templates. These SEDs were extended to the far–UV and near–IR using @BC03 models. These stellar photospheric models do not include emission from dust, in particular the PAH features which dominate the $3 < \lambda < 12 \mu$m portion of the spectrum in starforming galaxies, but at $z > 1$ the photometry used does not sample these rest wavelengths.
In addition to the large AGES (C. Kochanek et al. in preparation) survey, there are $\sim 500$ spectroscopic redshifts extending to $z\sim1.5$ gleaned from several ongoing surveys in the Boötes field. These were used as training sets to adjust the templates and photometric zero points to improve overall redshift accuracy and reliability [see @Brodwin2006 for details]. Comparison with these spectroscopic samples shows that an rms dispersion of $\sigma_z \sim 0.06(1+z)$ is achieved for 95% of galaxies to at least $z=1.5$. Subsequent follow–up spectroscopy of high redshift candidate clusters [§\[sec:Keck\]; see also @Stanford2005; @Brodwin2006; @Elston2006] verify this accuracy.
A key output of the @Brodwin2006 technique is the redshift probability function for each object, $P(z)$, derived directly from the redshift-axis projection of the full redshift–SED likelihood surface. To transform these simple redshift likelihood functions into true probability distribution functions, a prior consisting of the observed redshift distribution was applied. This was measured in the FLAMEX region using high–quality $B_WRIJK_s[3.6][4.5]$ photometric redshifts. Comparison to the spectroscopic sample illustrates that the resulting distribution functions are statistically valid in the sense that integrated areas accurately represent redshift probabilities at the 1, 2, and 3$\sigma$ levels. These $P(z)$ functions are the input to the wavelet detection algorithm discussed in § \[sec:detn\] and are also used in §\[sec:discussion\].
Note that the $P(z)$ distributions shows relatively little dependence on galaxy type [@Brodwin2006]. The reason is that while all galaxies are selected to have at least 5 sigma detections in \[4.5\], red galaxies are often only marginally detected or even undetected in our optical images, since they have very little blue light. This is particularly true at $z>1$ where the $k$-correction for early types is large in the optical. Blue galaxies have good $B_wRI$\[3.6\]\[4.5\] photometry, because they have more blue light. Thus even though blue, late-type galaxy SED’s have smaller breaks, they have more extensive (useful) photometry. These compensating effects lead to effectively type-independent photometric redshifts for $z>1$ galaxies.
Cluster Detection\[sec:detn\]
=============================
We employed a wavelet analysis to identify galaxy clusters within the [Boötes]{} region. Wavelet decomposition is a commonly used technique for cluster identification in X-ray images [for example, see @Vikhlinin1998; @Valtchanov2004; @Andreon2005; @Kenter2005], where it provides an effective means of identifying extended sources in the presence of contaminating point sources. In principle, a similar analysis can be used with optical and infrared data sets, using galaxies rather than X-ray photons to identify extended sources. As is well known, galaxy number counts are more susceptible to projection effects than is bremsstrahlung emission from the ICM. This issue is one that must be dealt with for all optical and infrared cluster searches, and consequently most such searches for distant clusters make explicit assumptions about the properties of the distant cluster population, such as an assumed density profile [e.g., @Postman1996; @Olsen1999; @Scodeggio1999] or the presence of a red sequence [@GladdersYee2000; @GladdersYee2005]. The SDSS C4 technique [@Miller2005] does not require a [*red*]{} sequence, but it does demand that the colors of cluster galaxies are similar to one another.
A significant advantage of the [Boötes]{} data set is that the photometric redshifts permit such assumptions to be minimized. The full photometric redshift probability distributions $P(z)$ were used to construct weighted galaxy density maps within overlapping redshift slices of width $\Delta z=0.2$, stepping through redshift space in increments of $\delta z=0.1$. For each galaxy the weight in the map corresponds to the probability that the galaxy lies within the given redshift slice. Weighting in this fashion de-emphasizes sources for which the redshift is poorly constrained. It is worth emphasizing that [*all*]{} galaxies were included in construction of the density map, and consequently cluster detection should be relatively independent of SED type and hence independent of morphology. Finally, cluster detection will be insensitive to the resolution of the density maps provided that the pixel size is small compared to the angular extent of cluster cores at all redshifts. A resolution of 12$\arcsec\ (\sim 100$ kpc) per pixel was used, which satisfies this criterion while being sufficiently large to keep computational overhead manageable.
Galaxy cluster candidates were detected within each redshift slice by convolving the density map with the wavelet kernel. We use a Gaussian difference kernel of the form $$k(r) = \frac{e^{-r^2/(2 \sigma_1^2)} }{ \sigma_1^2} - \frac{e^{-r^2/(2 \sigma_2^2)} }{ \sigma_2^2},$$ where $\sigma_1=400$ kpc and $\sigma_2=1600$ kpc, and which crosses zero near $r = 1$ Mpc. The scale of the kernel is fixed in physical rather than angular units, preserving our ability to uniformly identify comparable systems at different redshifts. The precise physical values of $\sigma$ are subject to refinement, but the selected values effectively isolate overdensities on the scale of clusters or groups.
Galaxy cluster candidates were detected in each redshift slice of these wavelet smoothed galaxy density maps using a simple peak-finding algorithm. To establish a consistent significance level for the candidates, 1000 bootstrap simulations were carried out within each redshift slice. The existing P(z) distributions, right ascensions, and declinations were repeatedly shuffled, convolved with the wavelet kernel, and candidates detected to find the threshold corresponding to one false positive per redshift slice within the [Boötes]{} field. A list of detections above this significance threshold was generated for each redshift slice. Because the right ascensions and declinations were shuffled independently, the true correlated background was not preserved. Consequently, the contamination rate may be somewhat higher than the one false positive per redshift slice expected if we had preserved the correlated background. For the current analysis we accept the somewhat higher contamination rate in exchange for improved completeness, particularly for the highest redshift clusters. The majority of clusters are detected in multiple redshift slices (a natural consequence of sampling in slices separated by step sizes finer than the galaxy redshift uncertainties). Multiple detections with small separations in positions and redshift slices were considered to be a single cluster, which was assigned an estimated redshift and position corresponding to the slice with the highest statistical significance. A total of 335 candidates were found in redshift slices with centers from 0.1 to 1.9, including 98 in slices with $z>1$.
While the detection technique is on three dimensional overdensities, it is not immune to projection effects. The expected number of clusters in each cylindrical bin of length $\Delta z=0.2$ and radius 1 Mpc (the radius of the wavelet detection kernel) was computed to quantify the extent of projection. This calculation included both the random expectation due to the observed number density and the excess clusters expected due to the observed clustering of this sample [@Brodwin2007]. The projection rate is most significant for lower redshifts because of the large angular size corresponding to 1 Mpc, affecting $\sim 10\%$ of systems at $z=0.5$ to $\sim 20\%$ at $z = 0.1$. The projection rate decreases to below $\sim
4$% at $z =1$ and is negligible at $z > 1.5$. Overall we estimate that $\sim10\%$ of the candidates may be spurious, including the contamination noted in the previous paragraph.
Cluster Redshifts and Members\[clusterz\]
=========================================
The peak of the summed $P(z)$ distribution at the cluster detection location was taken as the initial estimate of each cluster’s redshift. To improve this estimate, individual objects within a 1 Mpc radius of the cluster which included the cluster redshift within the $1\sigma$ range of their $P(z)$ functions were considered candidate cluster members. A refined estimate ($z_{\rm est}$) of the mean cluster redshift was obtained from the peak of the summed $P(z)$ distribution for these members. From this analysis, 104 candidates have $z_{\rm est} > 1$.
In Table \[z1table\] and Figures 2 - 13 we present 12 of the cluster candidates that were detected via the above criteria, and subsequently spectroscopically confirmed at the Keck Observatory to lie at $z>1$ (see §\[sec:Keck\]). Two of these, ISCS J1434.1+3328 and ISCS J1429.2+3357, have $z_{\rm est} = 0.98$ but the spectroscopic mean redshifts are $z = 1.057$ and $1.058$ respectively. Hence we report a total of 106 clusters at $z > 1$, of which roughly 10% may be expected to arise by chance or from projection effects, for the reasons noted in §\[sec:detn\]. Column 1 of Table 1 provides the catalog number of each cluster. The catalog numbers increase with decreasing detection significance (§\[sec:detn\])[^3]. Column 2 is the IAU designation for each cluster, based on the (J2000) coordinates of the detection given in columns 3 and 4. Column 5 provides the $z_{\rm est}$ value described above, and column 6 gives the mean redshift of the spectroscopically confirmed members. Column 7 provides the number of photometric redshift members of the cluster, defined as galaxies within 1 Mpc of the cluster center and with integrated $P(z) \ge 0.3$ in the range $z_{\rm est} \pm 0.06(1 + z_{\rm est})$. These galaxies are used to calculate mean cluster colors in §\[sec:Im3p6\_vs\_z\]. Column 8 reduces the number in column 7 by the number of galaxies which satisfy these criteria for each cluster redshift over the entire field, scaled to the 1 Mpc radius area for each cluster redshift. Column 9 gives the number of spectroscopically confirmed member galaxies in each cluster (see below). Column 10 gives the mean $I - [3.6]$ color for the photometric redshift member galaxies, and column 11 provides the sum of their luminosities in \[4.5\] relative to an $L^*$ value which evolves according to the “red spike” model shown in Figure \[m\_vs\_z\], corrected for the average over the field. Column 12 gives the luminosity in \[4.5\] of the brightest photometric redshift member galaxy relative to $L^*$.
We define a $z > 1$ cluster as spectroscopically confirmed if it contains at least 5 galaxies in the range $z_{\rm est} \pm 0.06(1 + z_{\rm est})$ and within a radius of 2 Mpc, whose spectroscopic redshifts match to within $\pm 2000(1 + \left<z_{\rm sp}\right>)$ km/s. The spectroscopic redshifts must also be of class A or B. Class A spectra have unambiguous redshift determinations, typically relying upon multiple well detected emission or absorption features. Class B spectral features are reliable but are less well detected. The radius threshold for spectroscopic membership is larger than the 1 Mpc used for photometric redshift members for practical reasons: most of the $z > 1$ redshifts reported here were obtained with slitmasks extending out to approximately 2 Mpc from the cluster center. Typically $< 10$ photometric redshift members within 1 Mpc could be accommodated in the mask design, and often no redshift could be determined from the resulting spectra.
The least significant cluster in Table 1 is ISCS J1434.5+3427, which is the 327th most significant detection out of the sample of 335, and the 100th most significant detection at $z > 1$. Despite its relatively low detection significance, ISCS J1434.5+3427 has a striking filamentary morphology (Figure \[color10.342\]), and has eleven spectroscopically confirmed members [see also @Brodwin2006].
[lcrccccccccc]{}
152 & ISCS\_J1434.1+3328 & 14:34:10.37 & +33:28:18.3 & 0.98 & 1.057 & 32 & 13 & 6 & 4.83 & 15 & 2.5\
51 & ISCS\_J1429.2+3357 & 14:29:15.16 & +33:57:08.5 & 0.98 & 1.058 & 45 & 26 & 7 & 4.80 & 34 & 4.2\
19 & ISCS\_J1433.1+3334 & 14:33:06.81 & +33:34:14.2 & 1.02 & 1.070 & 57 & 38 & 20 & 4.93 & 45 & 4.3\
123 & ISCS\_J1433.2+3324 & 14:33:16.01 & +33:24:37.4 & 1.01 & 1.096 & 31 & 11 & 6 & 4.94 & 15 & 4.6\
17 & ISCS\_J1432.4+3332 & 14:32:29.18 & +33:32:36.0 & 1.08 & 1.112 & 49 & 31 & 23 & 5.16 & 47 & 5.3\
34 & ISCS\_J1426.1+3403 & 14:26:09.51 & +34:03:41.1 & 1.08 & 1.135 & 31 & 13 & 7 & 5.18 & 26 & 4.0\
14 & ISCS\_J1426.5+3339 & 14:26:30.42 & +33:39:33.2 & 1.11 & 1.161 & 52 & 35 & 5 & 5.15 & 47 & 2.8\
342 & ISCS\_J1434.5+3427 & 14:34:30.44 & +34:27:12.3 & 1.20 & 1.243 & 27 & 12 & 11 & 5.51 & 27 & 6.0\
30 & ISCS\_J1429.3+3437 & 14:29:18.51 & +34:37:25.8 & 1.14 & 1.258 & 23 & 7 & 9 & 5.30 & 11 & 6.5\
29 & ISCS\_J1432.6+3436 & 14:32:38.38 & +34:36:49.0 & 1.24 & 1.347 & 30 & 17 & 8 & 5.65 & 31 & 4.3\
25 & ISCS\_J1434.7+3519 & 14:34:46.33 & +35:19:33.5 & 1.37 & 1.373 & 19 & 9 & 5 & 5.77 & 17 & 3.9\
22 & ISCS\_J1438.1+3414 & 14:38:08.71 & +34:14:19.2 & 1.33 & 1.413 & 25 & 15 & 10 & 5.75 & 24 & 3.5\
AGES Spectroscopy {#sec:AGES}
-----------------
The extensive AGES spectroscopic database (C. Kochanek et al. in preparation) can be used to spectroscopically confirm clusters at $z < 0.5$, a redshift at which $L^*$ corresponds to $I = 20$ (Figure 1), the magnitude limit for AGES spectroscopy. Note that while AGES is 94% complete to $I = 18.5$, sparse sampling is used for fainter galaxies, and currently spectra are available for $\sim40\%$ of galaxies with $18.5 < I < 20$. Because of the high surface density of AGES redshifts and the larger angular scale at $z < 0.5$, and the fact that the \[4.5\] flux limit samples substantially less luminous galaxies at these redshifts, two additions to the criteria for spectroscopic confirmation used at $z > 1$ were imposed. To contribute to spectroscopic confirmation for cluster candidates with $z_{\rm est} < 0.5$, galaxies were required to be more luminous than L\* + 1 in \[4.5\] for a red spike evolving model. Also, the average surface density of AGES galaxies over the [Boötes]{} field which would satisfy our high redshift confirmation criteria was calculated as a function of redshift, and 5 galaxies above this field level were required for a cluster to be confirmed.
Of the 335 cluster candidates, 80 have $z_{\rm est} < 0.5$. However seven lie near the edges of the [Boötes]{} field and hence were not well observed in AGES. Of the remaining 73, 61 candidates are confirmed by AGES spectroscopy using the criteria just described. This criterion is perhaps overly stringent: it rejects two clusters at $z=0.2$ with 17 or more matching AGES redshifts, but with only 4 (rather than 5) for galaxies brighter than L\* + 1 after field correction. All of the other 10 candidates observed by AGES which do not meet the confirmation criteria have $z_{\rm est} \ge 0.37$, and half have $z_{\rm est} \ge 0.45$.
Given the sparse sampling of AGES at $I > 18.5$, which corresponds to $L^*$ at $z > 0.3$ (Figure 1), we believe this confirmation rate validates our estimate that only $\sim 10\%$ of our cluster candidates arise by chance or due to projection effects. Further details on $z < 1$ clusters will be provided in A. Gonzalez et al. (in preparation).
Keck Spectroscopy {#sec:Keck}
-----------------
Most of the high redshift spectroscopic confirmation of ISCS clusters has been obtained at Keck Observatory. Three clusters observed with Keck have been reported previously: ISCS J1432.4+3332 ($z =
1.112$), ISCS J1434.5+3427 ($z = 1.243$), and ISCS J1438.1+3414 ($z =
1.413$) are presented in @Elston2006, @Brodwin2006, and @Stanford2005, respectively. Here we provide spectroscopic confirmation for an additional nine clusters at $z > 1$, as well as some new spectroscopic information for the initial three clusters. Table \[SpecObsTable\] details new observations of ISCS clusters. Table \[SpecTable\] provides properties of previously unreported, spectroscopically confirmed cluster members, in declination order for IRAC sources, followed by serendipitous sources. Figure \[spectra\] shows three example spectra obtained in April 2007 with Keck/DEIMOS.
[lcrcccccccc]{} ISCS\_J1434.1+3328 & 1.057 & DEIMOS & 2007 Apr 18$-$19 & 6$\times$1800 & clear; 08 - 20\
ISCS\_J1429.2+3357 & 1.058 & DEIMOS & 2006 Apr 26 & 4$\times$1500 & not photometric\
ISCS\_J1433.1+3334 & 1.070 & DEIMOS & 2007 Apr 19 & 3$\times$1200 & clear; 08\
ISCS\_J1433.2+3324 & 1.096 & DEIMOS & 2007 Apr 18$-$19 & 6$\times$1800 & clear; 08 - 20\
ISCS\_J1432.4+3332 & 1.112 & LRIS & 2005 Jun 03 & 5$\times$1800 & clear; 09 (Elston et al. 2006)\
& & FOCAS & 2006 Apr 21 & 5$\times$1200 &\
& & DEIMOS & 2007 Apr 19 & 3$\times$1200 & clear; 08\
ISCS\_J1426.1+3403 & 1.135 & LRIS & 2007 May 19 & 3$\times$1800 & clear; 09\
ISCS\_J1426.5+3339 & 1.161 & LRIS & 2006 Apr 04 & 4$\times$1800 & clear\
ISCS\_J1434.5+3427 & 1.243 & FOCAS & 2006 Apr 22 & 5$\times$1200 &\
& & DEIMOS & 2007 Apr 18 & 8$\times$1800 & likely cirrus; 10 - 15\
ISCS\_J1429.3+3437 & 1.258 & LRIS & 2006 Apr 05 & 3$\times$1200 &\
ISCS\_J1432.6+3436 & 1.347 & LRIS & 2007 May 21 & 7$\times$1800 & mostly clear; 08\
ISCS\_J1434.7+3519 & 1.373 & LRIS & 2005 Jun 02 & 7$\times$1800 &\
ISCS\_J1438.1+3414 & 1.413 & FOCAS & 2006 Jun 28 & 9$\times$1200 &\
& & DEIMOS & 2007 Apr 19 & 7$\times$1800 & clear; 09\
### LRIS Observations
We obtained deep optical slitmask spectroscopy for several clusters using the dual-beam Low Resolution Imaging Spectrograph [LRIS; @Oke1995] on the 10 m Keck I telescope during three observing runs between 2005 and 2007. Slitmasks generally included approximately 15 objects with photometric redshifts consistent with cluster membership and within 4 arcmin of the nominal cluster centers. Additional IRAC 4.5$\mu$m selected sources were included to fill out the slitmasks. Slitlets had widths of 1.3 arcsec and minimum lengths of 10 arcsec. We employed the D580 dichroic which splits the light at $\sim 5800$ Å between the two channels of LRIS. On the red side, the 400 line grating, blazed at 8500 Å, was used to cover a nominal wavelength range of 5800 to 9800 Å, varying somewhat depending on the position of a slit in the mask. For objects filling a slitlet, the spectral resolution for this instrument configuration is $\sim 9$ Å ($R \sim 900$), as determined from arc lamp spectra. On the blue side, the 400 line grism, blazed at 3400 Å, provided coverage from the atmospheric cutoff ($\sim 3200$ Å) up to the dichroic cut off. For objects filling a slitlet, the spectral resolution for this instrument configuration is $\sim 8$ Å ($R \sim 450$).
We obtained multiple exposures for each mask, usually with 1800 s per individual exposure. Table \[SpecObsTable\] details the exposure times and observing conditions. The observations were carried out with the slitlets aligned close to the parallactic angle, and objects were shifted along the long axis of the slitlets between exposures to enable better sky subtraction and fringe correction.
The slitmask data were separated into individual spectra and then reduced using standard longslit techniques. The multiple exposures for each slitlet were reduced separately and then coadded. The spectra were reduced both without and with a fringe correction; the former tends to yield higher quality object spectra at the shorter wavelengths, while the latter is necessary at the longer wavelengths. Calibrations were obtained from arc lamp exposures taken immediately after the object exposures for the red side, and from arc lamp exposures taken during the afternoon for the blue side. Corrections for small offsets in the wavelength calibration were obtained by inspection of the positions of sky lines in the object spectra. Using longslit observations of the standard stars from @Massey90 obtained during the same observing runs, we achieved relative flux calibration of the spectroscopy. While slit losses for resolved sources preclude absolute spectrophotometry from the slit mask data, the relative calibration of the spectral shapes should be accurate. One–dimensional spectra were extracted from the sum of all the reduced data for each slitlet for both the red and blue sides. For the targets in the high-redshift clusters, generally only the red side data proved useful.
### DEIMOS Observations
Additional spectroscopy was obtained with the Deep Imaging Multi-Object Spectrograph [DEIMOS; @Faber2003] on the 10 m Keck II telescope, a second generation instrument with significantly more multiplexing capabilities as compared to LRIS, albeit without the blue sensitivity. During an observing run in April 2006, we targeted Boötes active galaxy candidates selected on the basis of mid-infrared colors [e.g., @Stern2005], but included candidate cluster members in one mask. In April 2007, while observing host galaxies of high redshift cluster supernovae (Perlmutter et al., in preparation), we also observed candidate cluster members, typically with more than one high redshift cluster candidate observed in each wide area mask.
For both observing runs, the 600ZD grating ($\lambda_{\rm blaze}
= 7500$ Å; $\Delta \lambda_{\rm FWHM} = 3.7$ Å) and a GG455 order-blocking filter were used. DEIMOS data were processed using a slightly modified version of the pipeline developed by the DEEP2 team at UC-Berkeley[^4]. Although neither run was completely photometric, relative flux calibration was achieved from observations of standard stars from @Massey90.
Subaru Spectroscopy {#sec:Subaru}
-------------------
Followup spectroscopic observations were also obtained with the FOCAS spectrograph [@Kashikawa2002] on the 8.2-m Subaru telescope in April and June 2006. For these observations, we used the instrument in multi-object spectroscopy mode with 0.8 arcsec width slits, 300R grism, and SO58 order-sorting filter, which provided about 15 spectra from 5800Å to 10000Å with a spectral resolution $R\sim500$. Total exposure times were 2-6 hours. The FOCAS data were reduced with IRAF using standard methods including a fringe correction. Wavelength calibration was done using OH airglow emission lines. Absolute flux was calibrated using standard star (Feige 34, Wolf 1346, Hz 44) spectra taken during the same nights.
Notes on Individual Clusters {#sec:Notes}
----------------------------
### ISCS J1434.1+3328 (z = 1.057)\[1434.1+3328\]
ISCS J1434.1+3328 was observed with DEIMOS on a mask optimized for observing a high redshift supernova identified in a different ISCS high redshift cluster (Perlmutter et al., in preparation) which has not yet been spectroscopically confirmed. Four candidate members of ISCS J1434.1+3328 were targeted, and two more spectroscopic members were identified from additional slitlets targeting 4.5 $\mu$m selected sources to fill the mask. All six confirmed members have clearly identified Ca H and K absorption lines and D4000 breaks, and none of the galaxies exhibit emission features in the wavelengths covered by the DEIMOS observations ($\approx 5500 -
9500$ Å).
### ISCS J1429.2+3357 (z = 1.058)\[1429.2+3357\]
ISCS J1429.2+3357 was observed with DEIMOS, on a slitmask optimized for observing faint, mid-infrared selected AGN candidates [e.g., @Stern2005]. Eight candidate cluster members were included in the slitmask, six of which were confirmed spectroscopically to reside at $z \approx 1.06$. Only the brightest galaxy, IRAC J142912.9+335808, shows \[\] emission; the remaining redshifts were derived on the basis of Ca H and K absorption lines and/or D4000 breaks. In addition, one galaxy targeted spectroscopically as an IRAC-selected AGN candidate, IRAC J142916.1+335537, is a cluster member, bringing the tally to seven spectroscopically confirmed cluster members. The spectrum of this source shows strong, narrow (400 km s$^{-1}$) emission lines from \[\] $\lambda 3426$, indicating that it is indeed an active galaxy.
### ISCS J1433.1+3334 (z = 1.070)\[1433.1+3334\]
ISCS J1433.1+3334 was observed with DEIMOS on a mask optimized for observing a high redshift supernova identified in ISCS J1432.4+3332 (Perlmutter et al., in preparation). Eight candidate members of ISCS J1434.1+3328 were targeted, of which six were confirmed as cluster members, one was found to be slightly foreground to the cluster, and one was slightly behind the cluster. All eight candidates show clear D4000 breaks. Many additional cluster members were identified on this mask, either serendipitously or as targeted IRAC-selected, $z > 1$ galaxies, bringing the total number of spectroscopically confirmed cluster members to 20. As seen in the bottom two panels of Figure \[spectra\], the confirmed sources show a range of spectral properties. While most show Ca H and K absorption and D4000 breaks, some also show emission features likely due to either star formation or AGN activity.
### ISCS J1433.2+3324 (z = 1.096)\[1433.2+3324\]
ISCS J1433.2+3324 was observed with DEIMOS on the same mask as ISCS J1434.1+3328. Five candidate members of ISCS J1434.1+3328 were targeted, of which two were confirmed as cluster members, one was found to be foreground, and two yielded inconclusive spectra. Four additional cluster members were identified in the same mask, from IRAC-selected sources. The two confirmed cluster members which were specifically targeted show strong Ca H and K absorption and lack emission lines. The other four confirmed members all show \[\] emission, with three of the four also showing D4000 breaks.
### ISCS J1432.4+3332 (z = 1.112)\[1432.4+3332\]
The spectroscopic observations which confirmed ISCS J1432.4+3332 at $z = 1.11$ are described in @Elston2006, but no data on individual sources was presented there. The @Elston2006 result was based on nine spectroscopically confirmed cluster members, two of which were selected not as cluster members, but rather as mid-IR selected AGN, and that data is now included in Table \[SpecTable\] of this paper. One of the candidate members of this cluster hosted a high-redshift supernova (Perlmutter et al., in preparation) and was thus targeted for additional DEIMOS and FOCAS slitmask spectroscopy. An example DEIMOS spectrum for this cluster is shown in the top panel of Figure \[spectra\]. In total, there are now 23 spectroscopically confirmed members in the cluster.
### ISCS J1426.1+3403 (z = 1.135)\[1426.1+3403\]
Seven candidate members of this cluster were confirmed spectroscopically during our LRIS observations in May 2007. All seven galaxies show red continuum emission and/or a clear D4000 break. Two of the sources also show \[\] emission.
### ISCS J1426.5+3339 (z = 1.161)\[1426.5+3339\]
Four of the five spectroscopically confirmed members of this cluster show \[\] emission, a higher proportion than is typical for this program. This is possibly a selection effect; sources with line emission are the easiest to spectroscopically confirm. All five confirmed members show breaks typical of early-type galaxies (e.g., D2900 and/or D4000).
### ISCS J1434.5+3427 (z = 1.243)\[1434.5+3427\]
This cluster is discussed in @Brodwin2006, where eight spectroscopic members were presented. Table \[SpecTable\] of the current paper does not duplicate those data, and instead lists three additional spectroscopic members that have since been identified.
### ISCS J1429.3+3437 (z = 1.258)\[1429.3+3437\]
This cluster has nine spectroscopically confirmed members. One is an optically-bright AGN from the AGES survey, while the rest were confirmed spectroscopically by LRIS. Four of the Keck/LRIS sample show \[\] emission, while the other four show only spectral breaks and absorption lines.
### ISCS J1432.6+3436 (z = 1.347)\[1432.6+3436\]
This cluster has eight spectroscopically confirmed members, all from our Keck/LRIS observations in May 2007. Only one of the sources shows \[\] emission; the rest of the redshifts are on the basis of continuum breaks at 2640 and 2900 Å, as well as $\lambda$ 2800 absorption.
### ISCS J1434.7+3519 (z = 1.373)\[1434.7+3519\]
This cluster has five confirmed members, one of which was serendipitously identified in the spectroscopy, all from our Keck/LRIS spectroscopy in June 2005. Three of the members show \[\], and the other two show spectral breaks characteristic of early-type galaxy spectra.
### ISCS J1438.1+3414 (z = 1.413)\[1438.1+3414\]
This cluster was first published in @Stanford2005, at which time it was the highest redshift galaxy cluster known. Two candidate cluster members hosted supernovae in the [*HST*]{}/ACS program of Perlmutter et al. (in preparation), and this field thus has been the target of additional spectroscopy from Subaru and Keck. Five new cluster members have been confirmed, listed in Table \[SpecTable\]. Data on the original five members is not duplicated from @Stanford2005. Of the five new members, one is an \[\] emitter, serendipitously identified in a slitlet targeting another source. Three show only Ca H and K absorption lines, with no emission lines identified. IRAC J143816.8+341440 (22.3 in Table 3) is an AGN, showing \[\], \[\], and \[\] emission lines.
Cluster Masses\[sec:masses\]
----------------------------
With 20 or more spectroscopic redshifts, it is possible to estimate cluster masses via scaling relations using the velocity dispersion. The line of sight velocity dispersion for the 20 spectroscopic member galaxies in cluster 19 (ISCS J1433.1+3334) at $z=1.070$ is 760 km s$^{-1}$ in the rest frame, and for the 23 spectroscopic member galaxies in cluster 17 (ISCS J1432.4+3332) at $z=1.112$ it is 734 km s$^{-1}$.
For these two clusters, which are among the richest in the sample, the velocity dispersion of $\sim750$ km s$^{-1}$ corresponds to a virial mass of $\sim 3.8~{\rm r_v} \times 10^{14} M_\odot$ where ${\rm r_v}$ is the virial radius in Mpc [e.g. equation 4 of @Carlberg1996]. The x-ray temperature corresponding to $\sigma=750$ km s$^{-1}$ is 3.9 keV [Table 4 of @XueWu2000], which in turn gives a mass of $\sim 4 \times 10^{14} M_\odot$ [@Shimizu2003]. This is consistent with the average halo mass of $\sim 10^{14}M_\odot$ estimated by @Brodwin2007 for the sample.
Stellar luminosities can also be used to make a rough estimate of cluster masses, scaling to the Coma cluster via the red spike model. Within the $650\times850$ kpc region of Coma sampled by @Eisenhardt2007 the integrated $K$-band luminosity to $L^*+1$ is $54 L^*$, slightly higher than the values shown in column 11 of Table \[z1table\], but in a significantly smaller effective radius of 0.42 Mpc. In their Figure 1, Geller, Diaferio, & Kurtz (1999) show a mass for the Coma cluster of $\sim 3.5 \times 10^{14} M_\odot$ within this radius (for h=0.7). At r=1 Mpc they show about double that mass, and suggest the total mass for Coma is about double this again. Hence if the profile and $M/L_K$ for our clusters is similar to Coma, allowing for red spike model evolution in $L_K$, the $L_{\rm tot}/L^*$ values in 1 Mpc radius provided in Table 1 scale to total cluster masses of $\sim 1 - 6 \times 10^{14} M_\odot$ (i.e. $\sim 0.1 - 0.4 M_{\rm Coma}$). Clusters 17 and 19 are at the top of this range, again providing mass estimates consistent with those found using the velocity dispersion. More detailed exploration of this approach will require photometry which takes account of the extent to which a 5 diameter aperture fails to include all of the light from such galaxies, or includes light from multiple galaxies (as determined from higher spatial resolution imaging).
Discussion\[sec:discussion\]
============================
As noted in the introduction, out to $z \sim 1$ substantial evidence exists which is consistent with an extremely simple formation history for cluster galaxies, in which their stars are formed in a short burst at high redshift, and they evolve quiescently thereafter (ie a “red spike” model). The colors of luminous cluster galaxies out to $z \sim 1$ typically fall on a tight sequence which is red relative to field galaxies at similar redshift (the “red sequence”), with a mean color which evolves as the red spike model predicts.
The most luminous red sequence galaxies are the reddest, a correlation which is attributed to higher metallicity in higher mass galaxies (@KodamaArimoto97; but see also @FerrerasSilk2003). This correlation (i.e the color-magnitude or mass-metallicity relation) is explained by additional cycles of star formation and enrichment in more massive galaxies, as they are able to retain their gas more effectively against supernovae-driven winds [@ArimotoYoshii87]. In the red spike formation paradigm, this is a natural consequence. In the context of the hierarchical merging galaxy formation models, a unique correlation between stellar mass and metallicity is a greater challenge. While the inclusion of feedback (whether by supernovae or by AGN) in hierarchical models stops the buildup of galaxy mass [e.g. @KauffmannCharlot98; @NagashimaYoshii2004], such models have difficulty reproducing the exact color and slope of the color-magnitude relation in clusters over the full redshift range for which it has been measured. Recent work shows promise, however, as feedback behavior and other “gastrophysical” effects are taken more into account [e.g. @deLucia2006].
If the onset of the star formation “spike" is simultaneous for all cluster galaxies, and if the color-magnitude relation is caused by more protracted spikes in more massive galaxies, then as one approaches the star-forming epoch, massive galaxies might be expected to eventually become bluer, reversing the slope of the color-magnitude relation. In fact no measurable change is seen in color-magnitude slope out to $z \sim 1$ [e.g @SED98; @Mei2006], which suggests either formation redshifts well before $z = 1$, or that the spikes begin earlier in more massive galaxies, perhaps ending rather than beginning simultaneously. The small and unchanging scatter of the red sequence in clusters out to $z\sim1$ [e.g. @SED98; @Blakeslee2003; @Tran2007] also argues for synchronized or very early spikes and against a primarily age-based origin of the color-magnitude relation, since the scatter would increase by a much larger amount, and faster, than has been observed.
Testing to what extent red spike models remain consistent with cluster galaxy data at $z > 1$ is one of the primary motivations for the present study. We begin by constructing the color-magnitude relations for the 12 spectroscopically confirmed clusters in Table \[z1table\].
Color-Magnitude Diagrams\[sec:cmds\]
------------------------------------
Figures \[CMDs1\] and \[CMDs2\] present color magnitude diagrams for the clusters listed in Table \[z1table\]. The $I - [3.6]$ color was selected because these filters bracket the 4000Å break most tightly at $z > 1$. The symbol area is proportional to the integral of the object’s redshift probability distribution over the range $z_{\rm est} \pm 0.06(1+z_{\rm est})$, which is the rms dispersion in individual photometric redshifts [@Brodwin2006]. Circled symbols are spectroscopically confirmed members, while crosses indicate objects known [*not*]{} to be members on the basis of spectroscopy. Figures \[CMDs1\] and \[CMDs2\] include objects within 1 Mpc of the cluster center, with $> 5\sigma$ detections in both \[3.6\] and \[4.5\], and with $> 2\sigma$ detections in $I$; or with spectroscopic redshifts. As noted in §\[sec:phot-limits\], the $5\sigma$ limit in \[3.6\] is 18.6 mag, so the limiting factor in object selection for Figures \[CMDs1\] and \[CMDs2\] is from the $5\sigma$ limit of 17.8 mag in \[4.5\]. We have used the 0.1 Gyr burst, $z_f = 3$ red spike model shown in Figure \[m\_vs\_z\] to calculate the corresponding \[3.6\] mag as a function of redshift, and this is shown in Figures \[CMDs1\] and \[CMDs2\] by the vertical dotted lines. The $I$ limit is shown by the diagonal dotted lines in Figures \[CMDs1\] and \[CMDs2\]. The vertical dashed line plots the expected L\* magnitude at \[3.6\] for the red spike model. The sloped dashed line shows the observed $U - H$ color-magnitude slope of 0.22 for the Coma cluster from the data of @Eisenhardt2007, normalized at the red spike model \[3.6\] magnitude and $I - [3.6]$ color for an L\* galaxy. In this redshift range observed $I - [3.6]$ is close to rest $U - H$. Note that while the normalization evolves according to the red spike model, a constant, [*unevolving*]{} slope value is used.
The brightest galaxy likely to be a member is typically 1 to 2 magnitudes brighter than the expected L\* magnitude for the red spike model (see Table \[z1table\]). It is noteworthy that the brightest galaxy in the Coma cluster is $\approx 6$ times brighter than L\* in the $H$ band [@DePropris1998; @Eisenhardt2007], suggesting less than a factor of two growth in stellar mass in such galaxies since $z \sim 1.5$. [*HST*]{} imaging should be used to assess this more carefully.
Because these clusters were [*not*]{} selected on the basis of containing a red sequence (§\[sec:detn\]), the fact that the highest probability cluster members tend to track the passively evolving Coma cluster sequence shown by the dashed line is significant. Evidently the red sequence persists in dense environments to $z = 1.4$, even when not used as a selection criterion. In addition there is no indication that the slope of the color-magnitude relation has changed sign, and in fact a non-evolving Coma cluster slope appears consistent with the data.
Comparing the data for e.g. cluster 14 (ISCS J1426.5+3339 at $z=1.16$) vs. 29 (ISCS J1432.6+3436 at $z=1.34$) suggests real differences in the scatter of the color-magnitude relation do exist. The presence of luminous galaxies substantially bluer than the red sequence in several clusters, a number of them spectroscopically confirmed, is also noteworthy. These may represent a population of massive star-forming galaxies which fade onto the red sequence in rich clusters, but is not found in field surveys [e.g. @Bell2004]. It should be noted that the very low scatters reported for $ z > 1$ clusters in e.g. @Blakeslee2003 and @Mei2006 are calculated for galaxies known to have early-type morphologies, using much deeper photometry than the survey/discovery data used here. Further investigation of the color-magnitude relations is deferred to a future paper which will use deeper [*Spitzer*]{} and [*HST*]{} imaging presently being obtained.
Although the survey data used to identify the clusters does not enable accurate photometry of individual galaxies in clusters, we can calculate mean properties for galaxies in each cluster with some confidence. In the next section we briefly explore the color evolution and color-magnitude relation of the entire cluster sample.
Color vs. Redshift\[sec:Im3p6\_vs\_z\] {#sec:Im3p6vsz}
--------------------------------------
Figure \[Im3p6vsz\] plots the average $I - [3.6]$ color for galaxies within 1 Mpc of the cluster centers, and whose integrated redshift probability distribution in the range $z_{\rm est} \pm 0.06(1+z_{\rm est})$ exceeds 0.3. The mean values are calculated after iteratively clipping $3\sigma$ outliers. As implied by the color-magnitude diagrams for the spectroscopically confirmed clusters (§\[sec:cmds\]), the red spike model (0.1 Gyr burst at $z_f = 3$) provides a remarkably good fit to the empirical data, validating the survey design assumptions illustrated in Figure \[m\_vs\_z\].
Much of the increase in $I - [3.6]$ with redshift is due to the change in the rest frame wavelengths observed, and in Figure \[Im3p6relNEvsz\] we plot the average color relative to a no-evolution model (i.e. the k-corrected color). The no-evolution model is simply the red spike model at z=0, i.e. at an age of 11.3 Gyr for the stellar population. Clearly the evolving red spike model is a better fit to the data than is the k-correction alone. Note that the photometric redshifts (§\[sec:photz\]) are calculated using [*non*]{}-evolving templates, so this result is not a foregone conclusion, particularly at $z > 1$.
The mean $I - [3.6]$ colors in Figure \[Im3p6vsz\] include galaxies down to the $13.3\mu$Jy survey limit at $4.5\mu$m, and therefore less luminous galaxies contribute to the mean at lower redshifts. The color-magnitude relation shows that less luminous galaxies are bluer, and massive galaxies are quite quiescent today, so a constant flux limit will lead to a systematic bias towards more luminous, massive galaxies and redder mean color with increasing redshift. On the other hand the excellent fit of the red spike model implies that a constant [*luminosity*]{} limit for the sample would select galaxies with smaller stellar masses at high redshift, since a constant stellar mass, passively evolving, will be more luminous as one approaches the formation redshift. To avoid this bias, in Figure \[Im3p6relNEvsz\] we also show (with red squares) the mean colors for galaxies brighter than the red spike passively-evolving L\* in \[3.6\].
It is evident that the mean color of the more luminous galaxies is systematically redder, and hence that the color-magnitude relation continues to hold out to $z \sim 1.5$. The color offset is 0.1 magnitudes, independent of redshift. The persistence of the essentially unchanged color-magnitude relation slope, with a passively evolving intercept, out to lookback times within 4 Gyr of the Big Bang is a phenomenon that models of cluster galaxy formation must account for. The implication is that the correlation between high stellar population metallicity and stellar mass is already in place at $z \sim 1.5$, and that the star formation era remains well in the past for these galaxies.
In Figure \[Im3p6relPEvsz\] the mean $I - [3.6]$ colors are plotted with the predicted $I - [3.6]$ color for the red spike model subtracted. The mean colors are for galaxies more luminous in \[3.6\] than the red spike L\*. Note the red spike model is a good match to $L^*$ galaxy colors in figures \[Im3p6vsz\] and \[Im3p6relNEvsz\], so the offset in color out to $z \sim 1$ is attributable to the color-magnitude relation.
Other effects may be in play. In Table 1 the $z_{\rm est}$ values tend to be lower than the $\left<z_{\rm sp}\right>$ values, which may lead to colors appearing $\sim 0.1$ mag redder relative to the red spike model than if $\left<z_{\rm sp}\right>$ were available for all $z > 1$ clusters. The average colors of cluster members may still be redder than shown, however, as no attempt has been made to correct the average cluster colors for field contamination, and field galaxies tend to be bluer than cluster galaxies. Blending issues could lead to colors which are systematically biased. Given these uncertainties, the excellent agreement with the red spike model is the more remarkable. Nevertheless, at $z > 1$ the trend is towards colors which are increasingly redder than the red spike model, for both the full sample and the spectroscopically confirmed subset, though there is clearly a range. Hence even higher formation redshifts (as high as $z_f = 30$) are favored for most $z > 1$ clusters, within the context of red spike models.
Clusters at $1.5 < z < 2$\[sec:z2clusters\]
-------------------------------------------
Figure \[Im3p6vsz\] shows that that there are relatively few cluster candidates in the IRAC sample with $z_{\rm est} > 1.5$. Why is this? The red spike model fits, circumstantial evidence from BCG luminosity, and the persistence of the color-magnitude relation suggest that the stellar populations were formed – and assembled – well before $z \sim 1.5$. From Figure \[m\_vs\_z\], such clusters should be detectable by the IRAC Shallow Survey.
A potential selection effect against $z > 1.5$ galaxies is that photometric redshifts in this range have increasingly broad redshift probability distributions, due in part to insufficient depth in photometry at shorter wavelengths. Simulations indicate a factor of two reduction in IRAC photometric error can help compensate, leading to a similar reduction in photometric redshift error in this redshift range. With the [*Spitzer*]{} Deep Wide-Field Survey legacy program now underway (PI D. Stern), such data will be available in the near future. The tighter redshift probability distributions would improve the contrast of high redshift clusters over the field, allowing them to meet our detection threshold.
But the lack of massive $z > 1.5$ cluster detections may simply reflect a real decline in the space density of such objects. Models for the hierarchical growth of structure predict only about 1 cluster with $z> 1.4$ above $10^{14} M_{\odot}$ for every 5 clusters with $z > 1$. Full exploration of this possibility will require more careful simulation and assessment of our observational selection effects.
Summary\[sec:summary\]
======================
We have identified 335 galaxy cluster and group candidates from a $4.5 \mu$m selected sample of galaxies in the IRAC Shallow Survey. Candidates were identified by searching for overdensities in photometric redshift slices, and 106 clusters are at $z > 1$. Roughly 10% of these candidates may be expected to arise by chance or from projection effects. To date, 12 clusters have been spectroscopically confirmed at $z > 1$, as have 61 of the 73 clusters observed with AGES at $z < 0.5$. For the two $z > 1$ clusters with 20 or more spectroscopic members, total cluster masses of several $10^{14} {\rm M_\odot}$ are indicated, and the total mass estimated from the stellar luminosity yields comparable values. Color-magnitude diagrams in $I - [3.6]$ vs. \[3.6\] for the $z > 1$ spectroscopically confirmed clusters reveal that a red sequence is generally present, even though clusters were not selected for this. The brightest probable member galaxy (at the spatial resolution of IRAC) in the spectroscopically confirmed $ z > 1$ clusters remains 1 – 2 mag brighter than the passively-evolving $L^*$ luminosity. For the full cluster sample, the mean color of brighter galaxies within each cluster is systematically redder than the mean color of all probable cluster member galaxies, implying that the mass-metallicity relation is already in place at $z \sim 1.5$. The mean $I - [3.6]$ color of probable cluster members is well fit by a simple model in which stars form in a 0.1 Gyr burst beginning at $z_f = 3$. At $z > 1$ there is a tendency for mean cluster colors to favor formation redshifts $z_f > 3$, although a few are consistent with $z_f \sim 2$. This adds to the large body of evidence that galaxies in clusters were established at extremely early times.
We thank Mark Dickinson, Emily MacDonald, and Hyron Spinrad for generously making time available on their scheduled nights for the DEIMOS observations reported here. Naoki Yasuda, Naohiro Takanashi, Yutaka Ihara, Kohki Konishi, and Hiroyuki Utsunomiya assisted with observations at the Subaru Telescope. Roberto De Propris provided the integrated luminosity for Coma cluster galaxies. Thoughtful comments from the anonymous referee improved the presentation of this work. The IRAC Shallow Survey was executed using guaranteed observing time contributed by G. Fazio, G. and M. Rieke, M. Werner, and E. Wright. This work is based in part on observations made with the [*Spitzer Space Telescope*]{}, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA. This work made use of images and data products provided by the NOAO Deep Wide-Field Survey, which is supported by the National Optical Astronomy Observatory (NOAO). NOAO is operated by AURA, Inc., under a cooperative agreement with the National Science Foundation. Some of the data presented herein were obtained at the W.M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W.M. Keck Foundation. Some of the data presented were collected at the Subaru Telescope, which is operated by the National Astronomical Observatory of Japan. The work of SAS was performed under the auspices of the U.S. Department of Energy, National Nuclear Security Administration, by the University of California, Lawrence Livermore National Laboratory, under contract No. W-7405-Eng-48. The work of KD and JM was partially supported by the Director, Office of Science, Department of Energy, under grant DE-AC02-05CH11231.
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[lccc]{}\
152.1 & 15.45 & 1.01 & 1.057\
152.2 & 15.99 & 0.91 & 1.055\
152.3 & 17.00 & 0.96 & 1.054\
152.4 & 17.47 & 1.04 & 1.065\
152.5 & 16.40 & 0.99 & 1.055\
152.6 & 16.76 & 0.96 & 1.055\
\
51.1 & 15.10 & 0.98 & 1.059\
51.2 & 17.06 & 0.95 & 1.056\
51.3 & 17.83 & 1.04 & 1.055\
51.4 & 17.16 & 1.00 & 1.060\
51.5 & 17.01 & 1.03 & 1.059\
51.6 & 17.19 & 1.03 & 1.054\
51.7 & 16.08 & 1.01 & 1.060\
\
19.1 & 17.49 & 0.99 & 1.075\
19.2 & 16.67 & 0.99 & 1.075\
19.3 & 17.04 & 1.06 & 1.076\
19.4 & 16.00 & 1.06 & 1.066\
19.5 & 16.89 & 1.09 & 1.067\
19.6 & 16.47 & 0.94 & 1.065\
19.7 & 17.70 & 1.00 & 1.063\
19.8 & 17.15 & 1.10 & 1.079\
19.9 & 16.75 & 0.98 & 1.074\
19.10 & 16.30 & 1.07 & 1.066\
19.11 & 16.21 & 1.03 & 1.066\
19.12 & 16.66 & 1.03 & 1.075\
19.13 & 15.44 & 1.06 & 1.064\
19.14 & 17.03 & 1.28 & 1.076\
19.15 & 17.51 & 1.00 & 1.065\
19.16 & 17.01 & 1.27 & 1.069\
19.17 & 17.36 & 1.04 & 1.066\
19.18 & 16.27 & 0.86 & 1.063\
19.19 & 16.68 & 2.45 & 1.074\
19.20 & - & - & 1.067\
\
123.1 & 15.32 & 1.07 & 1.094\
123.2 & 16.45 & 1.09 & 1.093\
123.3 & 16.68 & 1.10 & 1.094\
123.4 & 16.99 & 1.11 & 1.107\
123.5 & 16.64 & 0.93 & 1.094\
123.6 & 17.35 & 0.99 & 1.091\
\
17.1 & 16.94 & 2.74 & 1.1120\
17.2 & 17.35 & 1.03 & 1.1108\
17.3 & 16.48 & 1.07 & 1.111\
17.4 & 15.97 & 1.05 & 1.115\
17.5 & 16.31 & 1.04 & 1.111\
17.6 & 15.85 & 1.09 & 1.110\
17.7 & 16.04 & 1.35 & 1.121\
17.8 & 17.57 & 1.20 & 1.116\
17.9 & 16.87 & 1.00 & 1.11\
17.10 & 17.31 & 1.04 & 1.1086\
17.11 & 17.01 & 1.14 & 1.098\
17.12 & 17.43 & 1.12 & 1.105\
17.13 & 17.51 & 0.98 & 1.109\
17.14 & 17.32 & 0.91 & 1.112\
17.15 & 16.59 & 1.07 & 1.104\
17.16 & 16.92 & 1.19 & 1.119\
17.17 & 17.42 & 1.09 & 1.115\
17.18 & 17.33 & 1.11 & 1.115\
17.19 & 17.29 & 1.06 & 1.118\
17.20 & 16.90 & 1.18 & 1.107\
17.21 & 16.38 & 1.26 & 1.115\
17.22 & - & - & 1.114\
17.23 & - & - & 1.110\
\
34.1 & 16.35 & 1.03 & 1.1439\
34.2 & 15.75 & 1.04 & 1.1301\
34.3 & 16.01 & 1.11 & 1.1271\
34.4 & 15.69 & 1.00 & 1.1328\
34.5 & 16.12 & 1.18 & 1.134\
34.6 & 16.90 & 0.91 & 1.13\
34.7 & 17.24 & 0.89 & 1.144\
\
14.1 & 16.87 & 1.10 & 1.16\
14.2 & 15.96 & 1.05 & 1.157\
14.3 & 16.44 & 1.19 & 1.1631\
14.4 & 16.64 & 1.17 & 1.1634\
14.5 & 16.88 & 1.04 & 1.1637\
\
342.1 & 16.76 & 1.21 & 1.240\
342.2 & 17.71 & 1.21 & 1.251\
342.3 & - & - & 1.256\
\
30.1 & 16.65 & 1.18 & 1.245\
30.2 & 16.39 & 1.13 & 1.26\
30.3 & 17.24 & 1.15 & 1.2576\
30.4 & 17.37 & 1.07 & 1.263\
30.5 & 17.41 & 1.29 & 1.2611\
30.6 & 16.86 & 1.08 & 1.2583\
30.7 & 16.97 & 1.31 & 1.2582\
30.8 & 15.19 & 0.97 & 1.2632\
30.9 & - & - & 1.2546\
\
29.1 & 17.63 & 1.26 & 1.3559\
29.2 & 17.61 & 1.25 & 1.35\
29.3 & 16.39 & 1.32 & 1.35\
29.4 & 16.06 & 1.21 & 1.3320\
29.5 & 17.55 & 1.40 & 1.347\
29.6 & 16.77 & 1.29 & 1.347\
29.7 & 16.86 & 1.30 & 1.34\
29.8 & 16.14 & 1.26 & 1.353\
\
25.1 & 17.52 & 3.33 & 1.37\
25.2 & 16.98 & 1.53 & 1.372\
25.3 & 16.28 & 1.34 & 1.37\
25.4 & 15.68 & 1.33 & 1.374\
25.5 & - & - & 1.380\
\
22.1 & 15.86 & 1.25 & 1.411\
22.2 & 15.79 & 1.38 & 1.418\
22.3 & 16.91 & 1.38 & 1.412\
22.4 & 16.76 & 1.35 & 1.414\
22.5 & - & - & 1.412\
[^1]: There is no generally agreed upon definition for galaxy clusters: see e.g. @Abell1958, @Postman1996, and @Rosati2002. We define our criteria for candidate galaxy clusters and groups in §\[sec:detn\], and for spectroscopically confirmed candidates in §\[clusterz\].
[^2]: http://www.noao.edu/noao/noaodeep
[^3]: These numbers extend beyond 335 for continuity with an earlier, preliminary version of the catalog used to plan spectroscopic and other followup observations.
[^4]: [http://astro.berkeley.edu/\$\\sim\$cooper/deep/spec2d/](http://astro.berkeley.edu/$\sim$cooper/deep/spec2d/)
|
---
abstract: 'The plane wave decomposition method (PWDM) is one of the most popular strategies for numerical solution of the quantum billiard problem. The method is based on the assumption that each eigenstate in a billiard can be approximated by a superposition of plane waves at a given energy. By the classical results on the theory of differential operators this can indeed be justified for billiards in convex domains. On the contrary, in the present work we demonstrate that eigenstates of non-convex billiards, in general, cannot be approximated by any solution of the Helmholtz equation regular everywhere in $\R^2$ (in particular, by linear combinations of a finite number of plane waves having the same energy). From this we infer that PWDM cannot be applied to billiards in non-convex domains. Furthermore, it follows from our results that unlike the properties of integrable billiards, where each eigenstate can be extended into the billiard exterior as a regular solution of the Helmholtz equation, the eigenstates of non-convex billiards, in general, do not admit such an extension.'
---
¶[ [P]{}]{} §[ [S]{}]{}
[H]{}
0.5 cm
[**Can billiard eigenstates be approximated by superpositions of plane waves?**]{}
1 cm
[**Boris Gutkin** ]{}
0.5 cm
[CEA-Saclay, Service de Physique Théorique\
Gif-sur-Yvette Cedex, France]{}\
[E-mail: [email protected] ]{}
1.0 cm
1.5 cm
Introduction
============
The quantum billiard problem in a domain $\Omega\subset \R^2$ is defined (in units m=1) by the Helmholtz equation $$(-\Delta-\k^2)\varphi(x)=0, \qquad E=\hbar^2\k^2/2 \label{1.1}$$ with Dirichlet boundary conditions $$\varphi(x)|_{\partial \Omega}=0. \label{1.2}$$ The solutions $E_n$, $\varphi_n$ of these equations determine the energy spectrum and the set of eigenstates of $\Omega$. Studying the properties of $(E_n, \varphi_n)$ in quantum billiards has became a prototype problem in “quantum chaos”. A simple form of eqs. \[1.1\], \[1.2\] suggests a natural way to solve them. First, for a given energy $E$ one looks for a set of solutions $\{\psi^{(n)}(\k), \, n\in\N\}$ of the Helmholtz equation (\[1.1\]) in the entire plane (without any boundary conditions). For example, $\{\psi^{(n)}(\k)\}$ can be chosen as a set of plane waves: $\{ \exp(i k_n x), \,\, |k_n|=\k,\,k_n\in\Real^2\}$, or as a set of radial waves: $\{J_n(\k r)\exp(i n \theta)\}$. Then regarding $\{\psi^{(n)}(\k)\}$ as a basis one can search for solutions of eqs. \[1.1\], \[1.2\] using the ansatz $$\varphi(x)=\sum a_i \psi^{(i)}(\k, x). \label{1.3}$$ As a result, solving eqs. \[1.1\], \[1.2\] is reduced to the algebraic problem of finding the coefficients $a_i$ such that the linear combination (\[1.3\]) vanishes whenever $ x\in \partial\Omega $.
The above approach has been widely used both in analytical and numerical studies of quantum billiards. In particular, it has been suggested by Berry in [@berry1] to use the expansion (\[1.3\]) with a Gaussian amplitude distribution to represent eigenfunctions of quantum systems with fully chaotic dynamics. This idea has been applied in numerous works to calculate various quantities associated with eigenfunctions, e.g., autocorrelation functions [@berry1], amplitude distributions [@romanbacker], statistics of nodal domains [@uzy2] etc. The same strategy can be also used for a numerical solution of eqs. \[1.1\], \[1.2\]. In this context it has been first introduced by Heller [@heller] with the application to the Bunimovich stadium. Since that several modification of the method have been considered in [@kitajzi1], [@kitajzi2] and in [@doronheller]. Depending on the choice of the basis in the decomposition (\[1.3\]) one gets, in general, different numerical methods for solving eqs. \[1.1\], \[1.2\]. Here we will single out the basis of plane waves (PW), most often used in applications. For the sake of briefness we will refer to the corresponding numerical method as plane wave decomposition method (PWDM).
As a matter of fact, the whole strategy described above is based on the assumption that the set $\{\psi^{(n)}(\k)\}$ furnishes an appropriate basis for the expansion of solutions of eqs. \[1.1\], \[1.2\]. In other words, one can use PWDM only if billiard eigenstates can be approximated by linear combinations of plane waves. That means $$||\varphi_n - \psi^{[N]}||_{L^2(\Omega)} \to 0, \mbox{ as } N \to \infty \label{1.35}$$ for some sequence of the states $\psi^{[N]}$ which are of the form $$\psi^{[N]} = \sum_{i=1}^N a_i e^{ik_i x},\qquad k_i\in \Real^2, \,\,\, |k_i|=\k. \label{1.45}$$ We will say that the plane wave approximation holds for a state $\varphi_n$ if the limit (\[1.35\]) exists.
Up to now it has been often assumed that the PWDM can be applied to billiards of arbitrary shape. From the results of Malgrange [@malgrange] (see also [@hormander]) on the theory of differential operators it is known that any solution of eq. \[1.1\] regular in a convex open domain can be approximated by superpositions of plane waves with $k_i\in\mathds{C}^2$. Moreover, since each evanescent plane wave ($\im \,k_i\neq 0$) can be approximated in a bounded domain by plane waves with real wavenumbers [@berry2], one immediately gets:
[**Proposition 1.**]{} [*Let $\Omega\subset\R^2$ be a convex bounded domain, then any solution of eq. \[1.1\] regular in $\Omega$ can be approximated by plane waves.*]{}
This shows that the eigenstates of a quantum billiard $\Omega$ admit PW approximation inside any convex domain $\Omega_1\subset\Omega$, see fig. 1a. Hence, PW approximation always holds for billiard eigenstates in a local sense. Furthermore, if $\Omega$ is a convex domain one can choose $\Omega_1$ in such a way that $\partial\Omega_1$ is arbitrary close to $\partial\Omega$. Consequently, as a simple corollary of Proposition 1 one gets:
[**Corollary 1.**]{} [*Eigenstates of a convex billiard $\Omega$ can be approximated by superpositions of plane waves.*]{}
The question naturally arises whether the same property holds for eigenstates of non-convex billiards, and thus, whether the PWDM can be actually applied to the class of non-convex billiards.
Note that there exists an important link between the PWDM and the problem of eigenstate extension in quantum billiards. Suppose $\varphi_n$ is an eigenstate of $\Omega$ which can be extended (as a regular solution of eq. \[1.1\]) from $\Omega$ to a convex domain $\Omega_{2}\supset\Omega$. Then it follows immediately by Proposition 1 that PW approximation holds for $\varphi_n$. The example of a billiard where each eigenstate can be continued in a convex domain is shown in fig 1.b. This is the “cake” billiard whose boundary consists of two concentric circle arcs connected by two segments of radii at an angle $\alpha <\pi$. In the polar coordinates $x=(r,\theta)$ the eigenstates of the “cake” billiard can be written explicitly as a sum of Bessel and Neumann functions: $$\varphi^{(m)}_n(x)= \left( a^m_n J_{\nu_m}(\k^{(m)}_{n} r) + b^m_n Y_{\nu_m}(\k^{(m)}_n r)\right)\sin\left(\nu_m(\theta -\theta_0)\right), \qquad \nu_m=\frac{\pi m}{\alpha}.$$ Since the singularity point of $\varphi^{(m)}_n(x)$ is always at the center $O$ of the circle arcs it is possible to extend each eigenstate into a convex domain $\Omega_2$, see fig 1.b. Accordingly, any eigenstate of the “cake” billiard can be approximated by superpositions of PW.
On the other hand, assume that for a billiard $\Omega$ an eigenstate $\varphi_n$ can be expanded in a basis $\{\psi^{(n)}\}$ (see eq. \[1.3\]), where $\psi^{(i)}$’s are solutions of the Helmholtz equation regular in $\R^2$ (e.g., plane waves). If furthermore, the corresponding sum (\[1.3\]) converges everywhere in $\R^2$ it makes sense to consider $\varphi_n(x)$ both inside and outside $\Omega$. Such extension of $\varphi_n(x)$ into $\R^2$ provides simultaneously solutions for the interior Dirichlet problem (when $x\in\Omega$) and for the exterior Dirichlet problem (when $x\in\Omega^c\equiv\R^2/\Omega$). Based on this observation a connection ([*spectral duality*]{}) between the interior Dirichlet and the exterior scattering problems has been suggested by Doron and Smilansky in [@uzydoron]. The rigorous result has been established by Eckmann and Pillet [@ep]. In most general form ([*weak spectral duality*]{}) it could be stated as follows: $E_n$ is an eigenvalue of the interior problem if and only if there exists an eigenvalue $e^{-i\vartheta_n}$ of the exterior scattering matrix $S(E)$ such that $\vartheta_n (E) \to 2\pi$ whenever $E\to E_n$. Moreover, if $\vartheta_n (E_n)=2\pi$ ([*strong spectral duality*]{}) then the corresponding interior eigenstate $\varphi_n$ could be extended into $\R^2$ as $L^2$ functions. Therefore if strong form of spectral duality holds for some eigenenergy $E_n$ then PW approximation holds for the corresponding eigenstate $\varphi_{n}$. It has been explicitly shown that strong form of spectral duality holds for convex integrable billiards [@dis]. However, as has been pointed out in [@ep], strong spectral duality cannot hold for billiards in general.
[**Remark.**]{} It should be pointed out that the approximability by PW is much weaker property then strong spectral duality. As has been explained above, strong spectral duality implies PW approximation for the corresponding eigenstate. The opposite, however, is not true: PW approximation for an eigenstate does not imply, in general, strong spectral duality. In fact, in [@berry2; @ep] the examples of convex billiards (in this case the approximation by PW is possible) have been constructed where the eigenstates extension into the exterior domain as $L^2$ functions is not possible.
Main results
============
Let $\Omega$ be a simply connected bounded domain in $\R^2$ with a piecewise smooth boundary $\partial\Omega$. Two different billiard maps can be associated with $\Omega$. First, the standard billiard map $\Psi$ corresponding to the motion of a pointlike particle in the interior domain. Second, the exterior map $\Psi^c$ which corresponds to the scattering off $\Omega$ as an obstacle, see e.g., [@uzy]. In order to define the exterior map one can place $\Omega$ on a sphere $\Sf^2$ of “infinite” radius. Then $\Psi^c$ is a standard billiard map corresponding to the motion of a pointlike particle in the domain $\Sf^2/\Omega$. It should be noted that there is an essential difference between convex and non-convex billiards. Whenever $\Omega$ is a convex domain the interior map $\Psi$ determines the same dynamics as the exterior map $\Psi^c$. For any interior trajectory inside $\Omega$ there is a [*dual*]{} trajectory in $\Omega^c$ which travels through the same set of points on the boundary $\partial \Omega$, see fig. 2a. We will refer to this property as [*interior-exterior duality*]{}. In particular, for convex billiards there is one to one correspondence between the interior and exterior periodic trajectories. For each periodic trajectory $\gamma$ its continuation $\gamma^c$ into the exterior domain will be the dual periodic trajectory of the exterior map. On the other hand, it is straightforward to see that in non-convex billiards interior-exterior duality breaks down. Generally, in a non-convex billiard $\Omega$ there exist interior periodic trajectories whose extension into the exterior domain intersects $\Omega$ again, see fig. 2b. Let $\gamma$ be such a trajectory and let $\gamma^c$ be its extension in the exterior. Note that $\gamma\cup\gamma^c$ is a union of straight lines in $\R^2$. Take $l\subset\gamma\cup\gamma^c$ to be a line which intersects the boundary $\partial\Omega$ at $2n$, $n>1$ points (for the sake of simplicity we will always assume that $n=2$). Then the intersection $\Omega\cap l$ is the union of two disconnected segments: $\gamma_1\subset\gamma$ and $\bar{\gamma}_1\subset\gamma^c$. If $\bar{\gamma}_1 $ does not belong to any periodic trajectory in $\Omega$, we will refer to $\gamma$ as [*single periodic trajectory*]{} (). By definition any has no dual periodic trajectory in the exterior domain. In what follows we call a non-convex billiard $\Omega$ as [*generic*]{} if it contains at least one stable (elliptic) or unstable (hyperbolic) . According to this terminology the “cake” billiard in fig. 1b is non-generic, since all its periodic trajectories are of neutral (parabolic) type.
We call a smooth function $\psi(x)$ as a [*regular solution*]{} of the Helmholtz equation if it solves eq. \[1.1\] everywhere in $\R^2$. For a given energy $E$ we will denote by $\M(E)$ the set of all regular solutions of eq. 1 and by $\M_{\mathtt{PW}}(E)\subset\M(E)$ the subset of functions which can be represented as linear combinations of finite number of plane waves with real wavenumbers $k_i$, $|k_i|^2=2E/\hbar^2$. In particular, $\M(E)$ includes convergent superpositions of plane waves (also with complex wavenumbers i.e., evanescent modes) and radial waves with the energy $E$. In its crudest form the main result of the present paper can be formulated in the following way. Based on the breaking of interior-exterior duality we demonstrate that eigenstates of a generic non-convex billiard (in general) cannot be approximated by regular solutions of eq. \[1.1\]. To illustrate the main idea of our approach it is instructive to consider a non-convex billiard $\Omega$ with an elliptic $\gamma$. It is well known that a sequence of quasimodes $(\tilde{\varphi}_i ,\tilde{\k}_i)$ associated with $\gamma$ can be constructed (see e.g., [@paul], [@fed]). Each pair $(\tilde{\varphi}_n ,\tilde{\k}_n)$ represents an approximate solution of eqs. \[1.1\], \[1.2\] such that $\tilde{\varphi}_n$ is localized along $\gamma$. Furthermore, in the absence of systematic degeneracies in the spectrum of $\Omega$ the quasimodes $(\tilde{\varphi}_n ,\tilde{\k}_n)$ approximate (in $L^2$ sense) a sequence of real solutions $( \varphi_n ,\k_n)$ of eqs. \[1.1\], \[1.2\]. For each such eigenstate ${\varphi}_n$ let us consider the corresponding Husimi function
$$H_{\varphi_n}(\x)=|\langle \x|\varphi_n\rangle|^2, \qquad \x=(q,p): \,\,\,\,\, q\in \Omega\label{2.1},\,\,\,\,\, |p|=\hbar\k_n ,$$
where $\langle\x|$ denotes a coherent state localized at the point $\x$ of the phase space of $\Omega$. By the definition $H_{\varphi_n}(\x)$ is localized along $\gamma$ and exponentially small everywhere else. On the other hand, assume that $\varphi_n$ could be approximated by regular solutions of eq. \[1.1\]. That means for any $\epsilon>0$ there is $\psi_{\epsilon}\in\M(E_n)$ such that $||\varphi_n- \psi_{\epsilon}||<{\epsilon}$, where $||\cdot||$ denotes the $L^2(\Omega)$ norm. Set $q$ be a point at $\gamma_1$ and set $p$ be directed along $\gamma_1$. Then for $\x=(q,p)$ we have
$$H_{\varphi_n}(\x)=\lim_{\epsilon\to 0}|\langle \x|\psi_{\epsilon}\rangle|^2 =
\lim_{\epsilon\to 0}|\langle \x|e^{-it\Delta/\hbar}\psi_{\epsilon}\rangle|^2,\label{2.2}$$
where $e^{-it\Delta/\hbar}$ is the free evolution operator in $\R^2$. Furthermore, in the semiclassical limit the quantum evolution of coherent states is governed by the corresponding classical evolution $$e^{-it\Delta/\hbar}|\x\rangle=e^{it E/\hbar}|\x(t)\rangle +O(\hbar^{\infty}), \qquad \x(t)=(q(t),p). \label{2.3}$$ Plugging (\[2.3\]) into eq. \[2.2\] and taking time $t$ to be such that $q(t)=q'\in \bar{\gamma}_1$ one gets
$$H_{\varphi_n}(\x) - H_{\varphi_n}(\x')=O(\hbar^{\infty}), \qquad \x'=(q',p). \label{2.4}$$
This, however, contradicts the fact that the Husimi function $ H_{\varphi_n}(\x)$ should be exponentially decaying outside $\gamma$.
The above argument can be extended to the case of hyperbolic $\gamma$ as follows. Contrary to the elliptic case it is not possible to construct quasimodes concentrated on hyperbolic periodic orbits. Instead, one can use a statistical approach in that case. By the results of Paul and Uribe [@paul] it is known that the average of the Husimi functions (\[2.1\])
$$\big\langle H_{\varphi_n}(\x)\big\rangle=\frac{1}{\#\P_{c\hbar}}\sum_{E_n\in \P_{c\hbar}}|\langle \x|{\varphi}_n\rangle|^2$$
over the energy interval $\P_{c\hbar}= [E-c\hbar, E+c\hbar]$, $c>0$ depends in the semiclassical limit $\hbar\to 0$ on whether $\x$ belongs to a periodic trajectory or not. On the other hand, as has been explained above, if each $\varphi_n$ could be approximated by a regular solution of eq. \[1.1\] then each $H_{\varphi_n}(\x)$ (and therefore the average $\big\langle H_{\varphi_n}(\x)\big\rangle$) would be (semiclassically) invariant along ${\gamma}_1\cup\bar{\gamma}_1$.
The preceding discussion provides an intuitive explanation why it is impossible to approximate eigenstates of a generic non-convex billiard by a superposition of plane waves. Speaking informally our argument says that contrary to the real eigenstates of non-convex billiard $\Omega$, any regular solution of eq. \[1.1\] always “preserves” interior-exterior duality. In what follows we consider the $L^2(\Omega)$ norm $$\eta_n(\psi)= ||\varphi_n-\psi||,$$ for a solution $(\varphi_n, E_n)$ of eqs. \[1.1\], \[1.2\] in $\Omega$ and an arbitrary $\psi\in \M(E_n)$. By the definition $\eta_n(\psi)$ measures approximability of $\varphi_n$ by regular solutions of the Helmholtz equation. Recall that a state $\varphi_n$ is approximable by PW if $$\inf_{\psi\in\M_{\mathtt{PW}}(E_n) } \eta_n(\psi)=0.$$
[**Remark.**]{} Note that by Proposition 1 for any $\psi\in\M(E_n)$ and any $\epsilon>0$ one can always find $\psi_{\epsilon}\in\M_{\mathtt{PW}}(E_n)$ such that $ |\eta_n(\psi)-\eta_n(\psi_{\epsilon})|<\epsilon$. In particular this implies
$$\eta^{\mathsf{min}}_n\equiv\inf_{\psi\in\M(E_n) } \eta_n(\psi) = \inf_{\psi\in\M_{\mathsf{PW}}(E_n) } \eta_n(\psi).\label{2.75}$$
In other words, an eigenstate $\varphi_n$ can be approximated by $\psi\in\M(E_n)$ if and only if it can be approximated by PW. Therefore, in what follows one can always assume without lost of generality that $\psi$ belongs to $\M_{\mathtt{PW}}(E_n)$ rather than to the set $\M(E_n)$.
By Corollary 1, $\eta^{\mathsf{min}}_n=0$ for any eigenstate of a convex billiard. On the contrary, in the body of the paper we show that for a generic non-convex billiard the average of $\eta^{\mathsf{min}}_n$ over an energy interval is bounded from below by a strictly positive constant:
[**Proposition 2.**]{}
*Let $\Omega$ be a non-convex billiard with at least one stable or unstable and let ($\varphi_n,E_n$), $n=1,2, ...\,\infty$ denote the eigenstates and eigenenergies of the corresponding quantum billiard. For any set of approximating functions $\{\psi_i\in\M(E_i), \,\, i\in \N\}$ the average of $\eta_n=\eta_n (\psi_n)$ over the energy interval $\P_{c\hbar}= [E-c\hbar, E+c\hbar]$, satisfies*
$$\big\langle\eta_n\big\rangle > \C(\hbar),\qquad \mbox{ where }\qquad \B=\lim_{\hbar\to 0}\C(\hbar)/\hbar \label{2.7}$$
is strictly positive and independent of $\psi_i$’s. Moreover, if $\Omega$ contains a $\gamma$ of elliptic type then (provided the spectrum of $\Omega$ has no systematic degeneracies) there exists an infinite subsequence $\S_{\gamma}=\{(\varphi_{j_m},E_{j_m}), \,\, m\in \N\}$ (of a positive density, i.e., $\lim_{N\to\infty}\frac{\#\{j_m|j_m<N\}}{N}>0$) such that for any $(\varphi_n, E_n) \in \S_{\gamma}$ and any regular solution $\psi\in \M(E_n)$
$$\eta_n(\psi) > \C_{\gamma}+O(\hbar^{1/2}),\label{2.8}$$
where $\C_{\gamma}$ is a strictly positive constant independent of $\psi$ and $\hbar$.
From (\[2.7\],\[2.8\]) one immediately obtains the corollary:
[**Corollary 2.**]{} [*For a generic non-convex billiard $\Omega$ there exists an infinite subsequence of eigenstates $\{\varphi_{j_n}, \,\, n\in\N\}$ such that: 1) $\eta^{\mathsf{min}}_{j_n}>0$; 2) $\varphi_{j_n}$ cannot be extended into the domain $\Omega^c$ (as a regular solution of eq. \[1.1\]).*]{}
Obviously, this implies the following properties of a generic non-convex billiard:
- In general, eigenstates of non-convex billiards do not admit approximation by PW and PWDM cannot be used in that case;
- The spectral duality for a generic non-convex billiard holds only in the weak form.
The paper is organized as follows. In the next section we collect several necessary facts about coherent states. In Sec. 4 the case of elliptic ’s is considered. First, using the coherent states we construct a family of quasimodes $(\tilde{\varphi}_n,\tilde{E}_n)$ associated with such trajectories. Then, we show that the lower bound (\[2.8\]) holds for the eigenstates $\varphi_n$ approximated by $\tilde{\varphi}_n$. The case of hyperbolic ’s is considered in Sec. 5. Here we use the results of Paul and Uribe to estimate the average $\big\langle \eta_n\big\rangle $ over an energy interval. Finally in Sec. 6 we discuss our results and consider possible generalizations.
Coherent states
===============
[**Definition of coherent states.**]{} The coherent states have been introduced already in the beginning of quantum mechanics and have been used in many areas since then. The basic idea is to built a complete set of vectors of Hilbert space localized in the phase space both in $q$ and $p$ directions at the scale $\sqrt{\hbar}$. The standard example of such states in $\R^d$ is given by the Gaussians:
$$u_{\x}^{\sigma}(x)=\left({\det \im \sigma}\right)^{\frac{1}{4}}\left({\hbar\pi}\right)^{-\frac{d}{4}}e^{\frac{i}{\hbar}[\langle p,x-q\rangle+\frac{1}{2}\langle x-q, \sigma\, (x-q)\rangle]}, \,\,\,\, \x=(q,p), \,\,\,\, \im \sigma>0. \label{3.1}$$
In the present work we will consider a slightly more general class of coherent states. (For a more general definition of coherent states see e.g., [@paul].) Let $\rho^{\varepsilon}_q(\cdot)$ be a $C_{0}^{\infty}$ function in $\R^d$ equal to one in a neighborhood of the point $q$ and zero outside the sphere of radius $\varepsilon$ centered at $q$. A coherent state at $\x=(q,p)$ is the vector
$$\phi^{\sigma}_{\x} (x)=\rho^{\varepsilon}_q(x) u_{\x}^{\sigma}(x).\label{3.2}$$
It is easy to see that the coherent states (\[3.2\]) are semiclassicaly orthogonal: $$||\phi^{\sigma}_{\x}||^2=1+O(\hbar), \,\,\, \langle\phi^{\sigma}_{\x}|\phi^{\sigma}_{\x'}\rangle=O(\hbar^{\infty})\mbox{ if }\x\neq\x'.\label{3.4}$$ The role of the cut-off $\rho^{\varepsilon}_q(x)$ is rather technical, it allows to define coherent states inside compact domains. To use the vectors (\[3.2\]) as coherent states inside a billiard domain $\Omega$ one needs that $$\mbox{supp}[\rho^{\varepsilon}_q(x)]\subset\Omega. \label{3.3}$$
[**Propagation of coherent states.**]{} An important property of coherent states is that their quantum evolution in the semiclassical limit is completely determined by the corresponding classical evolution. Let $\H=-\hbar^2\Delta/2+v(x)$ be the operator of symbol $\mathcal{H}=p^2/2 + v(x)$ inducing the flow $\Psi^t:V\to V$ on the phase space $V$. Then, as it is well known, for any time $t$ the propagation of the coherent state $\phi^{\sigma}_{\x}$ localized at $\x\in V$ is given by $$e^{-i t \H/\hbar}\phi^{\sigma}_{\x} = e^{i\left({ S(t)/\hbar}+\mu(t)\right)} \phi^{\sigma(t)}_{\x(t)} +O(\hbar^{1/2}), \label{3.5}$$ where $S(t)=\int_0^t \left(p \dot{q}-\mathcal{H}(p,q)\right) dt$ is the classical action along the path $\x(t)$ and $\mu(t)$ is the Maslov index. The parameters $ \x(t)=\Psi^t\cdot\x$, $\sigma{(t)}=D\Psi^t \cdot\sigma$ in eq. \[3.5\] are determined by the evolution of the initial data $ \x$, $\sigma$ under the flow $\Psi^t:\x\to\x(t)$ and its derivative $$D\Psi^t :\sigma\to\sigma(t)=\frac{a\sigma+b}{c\sigma+d} ,$$ where $d\times d$ matrices $a,b,c,d$ are the components of $D\Psi^t$ in a given coordinate system: $$D\Psi^t =\left(\begin{array}{cc}
a & b \\
c & d \\ \end{array}\right).$$ It is convenient to chose two of $2d$ coordinates in the phase space $V$ to be along the flow and along the line orthogonal to the energy surface. Then the matrix $\sigma$ can be decomposed into $\sigma=\sigma^{0}\oplus\sigma^{1}$, where the scalar part $\sigma^{0}$ corresponds to the above two directions and $(d-1)\times(d-1)$ matrix $\sigma^{1}$ corresponds to the orthogonal subspace. It is straightforward to see that in such a basis $ D\Psi^t$ acts separately on $\sigma^{1}$ and $\sigma^{0}$. In particular, $D\Psi^t \cdot\sigma^0$ is given by a linear transformation:
$$D\Psi^t \cdot\sigma^0=\frac{\sigma^0}{u\sigma^0+1}.$$
In the present paper we will use the above results for two types of two-dimensional flows: free evolution on $\R^2$ under the Hamiltonian $\H_0$ ($v(x)=0$) and the evolution induced by the billiard Hamiltonian $\H_{\Omega}$ ($v(x)=0$ if $x\in\Omega$ and $v(x)=\infty$ otherwise). Let us consider in some detail the evolution of coherent states in billiards. Set $\Omega$ be the billiard domain. We will denote by $\Psi_{\Omega}^t: V\to V$ the billiard flow, whose action is on the standard phase space $V$ of $\Omega$. It should be pointed out that one can use the coherent states (\[3.2\]) for the point $\x=(q,p)\in V$ only if $q$ is sufficiently far away from the boundary $\partial\Omega$. Indeed, to satisfy the condition (\[3.3\]) $q$ has to be at the distance larger than $\varepsilon$ from the boundary. For the sake of simplicity, we will not consider a generalized class of coherent states defined in the whole domain $\Omega$, rather we will use the states (\[3.2\]) but only for the interior points of $\Omega$. For this purpose let us define the inner domain $\Omega_{\varepsilon}\subset\Omega$ which contains all the points $q$ of $\Omega$ such that the distance between $q$ and $\partial\Omega$ is larger then $\varepsilon$: $\mbox{dist}(q,\partial\Omega)\geq \varepsilon$, see fig. 3. In what follows we will fix $\varepsilon$ to be a small compare to linear sizes of the billiard (but large compare to $\hbar^{1/2}$) and consider the coherent states propagation under the condition that at the initial moment $t_1=0$ and the final moment $t_2=t$ the points $\x(0)$, $\x(t)$ belong to the domain $\Omega_{\varepsilon}$. Whenever this condition is fulfilled one can use the formula (\[3.5\]), where the states $\phi^{\sigma}_{\x}, \phi^{\sigma(t)}_{\x(t)}$ are both of the form (\[3.2\]). Furthermore, if $q(t)\in\Omega_{\varepsilon}$ for all $t\in [t_1,t_2]$ (i.e., there is no collisions with the boundary between the times $t_1$ and $t_2$) then the reminder term in (\[3.5\]) is of the order $O(\hbar^{\infty})$.
[**Husimi functions.**]{} Let $\varphi_n$ be an eigenstate of $\H$ with the eigenenergy $E_n$. Given a coherent state $\phi^{\sigma}_{\x}$ one can construct the corresponding Husimi function:
$$H_{n}(\x)=|\langle\phi^{\sigma}_{\x}|\varphi_n\rangle|^2\qquad \x=(q,p); \,\,\, \sigma=(\sigma^0,\sigma^1), \,\, -i\sigma^0=\beta >0.\label{3.6}$$
Based on the propagation formula (\[3.5\]) the following average over Husimi functions $$\sum_n f (\omega_n)| \langle \varphi_n|\phi^{\sigma}_{\x} \rangle|^2
=\sum_{l=0}^{\infty} d_l \, \hbar^{\frac{1}{2}+l}, \,\,\, \omega_n=\frac{E_n -E}{\hbar}, \,\,\, E=p^2/2 \label{3.7}$$ has been calculated to the leading order by Paul and Uribe [@paul]. It turns out that the result depends on whether the classical trajectory through $\x$ is periodic or not. With the application to the Hamiltonian $\H_{\Omega}$ the results in [@paul] read as follows. Let $\tilde{f}(\cdot)$ be the Fourier transform of $f$. If $\x$ is not periodic under the flow $\Psi_{\Omega}^t$ then
$$d_{0}=\left(\frac{1}{\beta E}\right)^{1/2} \tilde{f}(0).\label{3.8}$$
Alternatively, if $\x$ belongs to a periodic trajectory the additional terms (of the same order in $\hbar$) arise. In particular, for a hyperbolic periodic trajectory $\gamma$ with the period $T_{\gamma}$ the leading term in (\[3.7\]) is given by $$d_{0}= \left(\frac{1}{\beta E}\right)^{1/2}\left( \sum_{l=-\infty}^{+\infty} \tilde{f}(l T_{\gamma})
\frac{e^{il( S_{\gamma}/\hbar +\mu_{\gamma} )} }{\cosh^{1/2}(l\lambda_{\gamma})} \right
) , \label{3.9}$$ where $$S_{\gamma}=2ET_{\gamma}, \,\, \,\, \mu_{\gamma}, \,\, \,\, \lambda_{\gamma}$$ are the action, Maslov index and Laypunov exponent of $\gamma$.
PW approximation for eigenstates of non-convex billiards (elliptic case)
========================================================================
Let $\gamma$ be a periodic orbit in the billiard $\Omega$ and let $\Gamma(E)$ be the “lift” of $\gamma$ to the phase space $V$ at the energy $E$. This means $\Gamma(E)$ is a set of the points $\x=(q,p)\in V$ such that $q\in\gamma$, $p^2=2E$ and the vector $p$ is directed along $\gamma$. Obviously, for any $\x \in\Gamma(E)$, $\Psi^{ T_{\gamma}}_{\Omega}\cdot\x =\x$, where $T_{\gamma}$ is the period of the trajectory. We will make use of the letter $\varepsilon$ to denote the restriction of $\gamma$, $\Gamma(E)$ to the domain $\Omega_{\varepsilon}$ i.e., $\gamma^{\varepsilon}=\{q\in\gamma\cap\Omega_{\varepsilon}\}$, $\Gamma^{\varepsilon}(E)=\{\x=(q,p)\in\Gamma(E): \,\, q\in\Omega_{\varepsilon}\}$. Provided that $\gamma$ is elliptic a set of approximate solutions ([*quasimodes*]{}) $ \tilde{\varphi}_n(x)$ of eqs. \[1.1\], \[1.2\] associated with $\gamma$ can be constructed. The possibility of quasimode construction on elliptic periodic orbits is well known. In the following we will follow the approach developed in [@paul'], [@paul] (see also [@roman], [@paul''] and the references there).
Before we turn to the construction of the states $ \tilde{\varphi}_n(x)$ in billiards let us recall a general definition for quasimodes.
[**Definition.**]{} Let $H$ be a Hilbert space and $\H$ be a self adjoint operator with the domain $D(H)$. A pair $(\tilde{\varphi}_n,\widetilde{E}_n)$ with $\tilde{\varphi}_n\in D(H)$, $||\tilde{\varphi}_n||=1$ and $\widetilde{E}_n\in \R$ is called a quasimode with the discrepancy $\delta_n$, if
$$(\H - \widetilde{E}_n)\tilde{\varphi}_n=r_n, \,\,\,\mbox{ with }\,\,\, ||r_n||=\delta_n.\label{4.0.1}$$
By a general theory (see e.g., [@laz]) the quasimodes $(\tilde{\varphi}_n,\widetilde{E}_n)$ should be close to an exact solution $(\varphi_n,E_n)$ of the equation $$(\H - {E})\varphi=0 \label{4.0.2}$$ in the following sense. If $(\tilde{\varphi},\widetilde{E})$ is a quasimode with the discrepancy $\delta$ then there exists at least one eigenvalue of $\H$ in the interval $$\P_{\delta}=[\widetilde{E}-\delta, \widetilde{E}+\delta].\label{4.0.3}$$ Furthermore, let $\nu$ be the distance between $\widetilde{E}$ and an eigenvalue $E_i$ of $\H$ outside $\P_{\delta}$, then $$||\tilde{\varphi}-\pi_{\nu}\, \tilde{\varphi}||\leq \frac{\delta}{\nu},\label{4.0.4}$$ where $\pi_{\nu}$ denotes the spectral projection operator on the part of the spectrum $\{E_n\}$ inside the interval $(\widetilde{E}-\nu, \widetilde{E}+\nu)$.
[**Remark.**]{} In general, the formula (\[4.0.4\]) implies that any state $\tilde{\varphi}_n$ approximates a superposition of eigenstates $\varphi_n$. In order to approximate individual eigenstates of $\H$, $\delta_n$ should be much less than the energy intervals: $\Delta E_{n}=|E_n -E_{n+1}|$, $\Delta E_{n-1} =|E_n -E_{n-1}|$. For two dimensional billiards $\big\langle\Delta E_{n}\big\rangle \sim \hbar^2 $, so the approximation of ${\varphi}_n$ by $\tilde{\varphi}_n$ becomes semiclassically ($\hbar\to 0$) meaningful only if the spectrum of $\Omega$ has no systematic degeneracies and quasimodes with discrepancy $\delta \sim \hbar^{\alpha}$, $\alpha > 2$ can be constructed. For the quantum billiard problem a quasimode construction providing $\delta=O(\hbar^{\infty})$ is known to exist [@popov] and for the rest of this section we will assume that the billiard spectrum has no systematic degeneracies.
Quasimode construction
----------------------
We will now schematically describe the construction of quasimodes concentrated on elliptic periodic orbits. The basic idea is to lunch a coherent state along the orbit and average over the time. As it can be shown, this procedure yields an approximately invariant state if the initial state is chosen in the right way, see e.g., [@paul; @roman]. Let $\phi^{\sigma}_{\x }$, $\x=(q, p)\in\Gamma^{\varepsilon}(E)$ be a coherent state localized on the periodic orbit $\gamma$. We will associate with $\gamma$ the state
$$|\Phi^{\sigma}_{\Gamma(E)}\rangle= \frac{1}{C}\int_{0}^{T_{\gamma}} e^{i t (E-\H_{\Omega}) /\hbar}|\phi^{\sigma}_{\x }\rangle \,dt, \label{4.1.1}$$
where $C$ is fixed by the normalization condition $||\Phi^{\sigma}_{\Gamma(E)}||=1$ and $T_{\gamma}$ is the period of the classical evolution along $\gamma$: $\x(T_{\gamma})=\x$. The propagation formula (\[3.5\]) yields
$$\begin{aligned}
(E-\H_{\Omega})\Phi^{\sigma}_{\Gamma(E)}&=&{r_{\gamma}}, \nonumber \\
C r_{\gamma}
&=&i\hbar\left(e^{i(S_{\gamma}/\hbar+\mu_{\gamma})}\phi^{\sigma(T_{\gamma})}_{\x }-\phi^{\sigma}_{\x }\right) +O(\hbar^{3/2}),
\label{4.1.2}\end{aligned}$$
where $S_{\gamma}$, $\mu_{\gamma}$ are the classical action and Maslov index after one period. Therefore, $C r_{\gamma}=O(\hbar^{3/2})$ provided that the following conditions are satisfied:
[**Condition 1:**]{} $\sigma(T_{\gamma})=\sigma$; [**Condition 2:**]{} $S_{\gamma}/\hbar+\mu_{\gamma}= 2\pi n$ for some integer $n$.
For each $n$ let $\E_n$, $\sigma_n=(\sigma^0_n,\sigma^1_n)$ denote solutions of Conditions 1, 2. It is possible to show (see e.g., [@paul]) that the first condition can be satisfied if and only if $\sigma^0_n=0$ and $\gamma$ is an elliptic periodic orbit. The second condition impose the Bohr-Sommerfeld quantization on the quasienergy $\E_n$. When both conditions are satisfied the corresponding pair $(\E_n,\Phi^{\sigma_n}_{\Gamma(\E_n)} )$ provides the quasimode with the discrepancy $\delta_{\gamma}=O(\hbar^{3/2})/C$.
[**Remark.**]{} It should be noted that a much wider class of quasimodes concentrated on $\gamma$ can be constructed by this method if one uses in (\[4.1.1\]) coherent states with transverse excitations [@paul; @paul'']. For simplicity of exposition, we restrict our consideration only to the quasimodes without transverse excitations, whose leading order is determined by eq. \[4.1.1\].
To construct quasimodes with discrepancies of higher order in $\hbar$ one has to consider the time evolution of coherent states of a more general type. This leads to transport equations whose solvability pose additional conditions on the quasienergies, see [@roman]. From the results of Cardoso and Popov [@popov] the prossibillity to construct quasimodes $(\widetilde{E}_n,\tilde{\varphi}_n)$ in billiards having discrepancy $\delta_{\gamma}= O(\hbar^{\infty})$ is known to exist. Let $(s, y)$ be a coordinate system in a neighborhood of $\gamma$ such that $s$ is a coordinate along the trajectory and $y$ is a coordinate in the orthogonal direction. Using these coordinates the leading order of $(\widetilde{E}_n,\tilde{\varphi}_n)$ can be written as follows [@fed; @roman]: $$\widetilde{E}_n={\E}_{n}+O(\hbar^2), \qquad
\tilde{\varphi}_n(x)=e^{iv(x)/\hbar}u(x)+O(\hbar),\label{4.1.4}$$ where $$v(s,y)=v_0(s)y^2+O(y^3), \qquad
u(s,y)=u_0(s)+O(y^2)$$ and the parameters $v_0(s)$, $u_0(s)$ are determined by Conditions 1, 2: $$\Phi^{\sigma_n}_{\Gamma(\E_n)}(x)=e^{iv_0(s)y^2/\hbar}u_0(s),\qquad x=(s, y).\label{4.1.45}$$
As has been explained before, in the absence of systematic degeneracies in the billiard spectrum one can expect that, in general, a state $\tilde{\varphi}_n$ approximates an individual eigenstate of the billiard $\Omega$. In what follows we will denote by $\widetilde{\S}_{\gamma}$ the set of quasimodes for which $\tilde{\varphi}_n$ approximates some eigenstate $ \varphi_n$ (rather than a linear combination of $ \varphi_n$’s) and by ${\S}_{\gamma}$ the set of true solutions of eqs. \[1.1\], \[1.2\] corresponding to $\widetilde{\S}_{\gamma}$. Then by eq. \[4.0.4\] for each $(\tilde{\varphi}_i,\widetilde{E}_i)\in\widetilde{\S}_{\gamma} $ and $({\varphi}_i,E_i)\in{\S}_{\gamma} $ we have
$$\C^1_i=||\tilde{\varphi}_i-\varphi_i||=O(\hbar^{\infty}), \qquad |\widetilde{E}_i -E_i|=O(\hbar^{\infty}) .\label{4.1.5}$$
A lower bound for the approximation of eigenstates
--------------------------------------------------
The quasimode construction described in the previous section is quit general and can be applied to an arbitrary elliptic periodic trajectory. In the present section we will consider eigenstates of the billiard $\Omega$ from the subset ${\S}_{\gamma}$, where $\gamma$ is an elliptic . We show that for $(\varphi_n, E_n)\in {\S}_{\gamma}$ and any regular solution $\psi\in\M(E_n)$ of eq. \[1.1\] in $\R^2$ the norm
$$\eta_n(\psi)=||\varphi_n -\psi|| \label{4.2.1}$$
is bounded from below by $$\eta_n(\psi) \geq {\C}_{\gamma} + {\mathcal C}^1_n +
O(\hbar^{1/2}),\label{4.2.2}$$ where ${\mathcal C}_{\gamma}$ is a positive constant determined only by geometrical parameters of the periodic orbit. Since ${\C}^1_n = O(\hbar^{\infty})$, this implies the inequality ($\ref{2.8}$) holds for any $(\varphi_n, E_n)\in {\S}_{\gamma}$.
Let $\gamma$ be an elliptic and let $\gamma_1$, $\bar{\gamma}_1$ be as defined in Sec. 2, see fig. 3. Now fix the parameter $\varepsilon$ to be sufficiently small such that $\gamma^{\varepsilon}_1\equiv\gamma_1\cap\Omega_{\varepsilon}\neq\emptyset$, $\bar{\gamma}^{\varepsilon}_1\equiv\bar{\gamma}_1\cap\Omega_{\varepsilon}\neq\emptyset$. We will denote by the capital letters $\Gamma_1(E)$, $\bar{\Gamma}_1(E)$ (resp. $\Gamma^{\varepsilon}_1(E)$, $\bar{\Gamma}^{\varepsilon}_1(E)$) the corresponding “lifts” of $\gamma_1$, $\bar{\gamma}_1$ (resp. $\gamma^{\varepsilon}_1$, $\bar{\gamma}^{\varepsilon}_1$) into the phase space $V$ at the energy shell $E$. Recall that the main idea behind the quasimode construction (\[4.1.1\]) is to use coherent states propagating along a periodic orbit. By analogy, one can construct states localized on $\gamma_1$ and $\bar{\gamma}_1$. Let $\x(0)=\x \in \Gamma_1(E)$. Consider the classical evolution (both for positive and negative time) of $\x$ under the free flow $\Psi^t_0:\x\to\x(t)=(q(t),p(t))$ in $\R^2$. Obviously, as time evolves, the point $q(t)$ successively crosses the boundary of $\Omega_{\varepsilon}$ at the sequence of points $q_1, q_2,\bar{q}_1,\bar{q}_2$, see fig. 3. We will denote by $t_1, t_2, \bar{t}_1,\bar{t}_2$ the corresponding time moments: $q_1=q(t_1), q_2=q(t_2),\bar{q}_1=q(\bar{t}_1),\bar{q}_2=q(\bar{t}_2)$. Then the states localized along $\gamma_1$ and $\bar{\gamma}_1$ are given by
$$|\Phi^{\sigma}_{\Gamma_1(E)}\rangle= \int^{t_1}_{t_2} e^{it (E-\H_0) /\hbar}|\phi^{\sigma}_{\x }\rangle \,dt, \label{4.2.3}$$
$$|\Phi^{\sigma}_{\bar{\Gamma}_1(E)}\rangle= \int^{\bar{t}_1}_{\bar{t}_2} e^{it (E-\H_0) /\hbar}|\phi^{\sigma}_{\x }\rangle \,dt. \label{4.2.4}$$
Note, that under the free evolution $e^{-it \H_0 /\hbar}$ the support of $\phi^{\sigma}_{\x }$ is not preserved inside $\Omega$, and therefore the supports of $\Phi^{\sigma}_{\Gamma_1}, \Phi^{\sigma}_{\bar{\Gamma}_1}$ do not belong to the billiard domain. However, one can slightly modify the definition of the states $\Phi^{\sigma}_{\Gamma_1}, \Phi^{\sigma}_{\bar{\Gamma}_1}$ to make them admissible as billiard states in $\Omega$. Let $\x=\x_1$, $\sigma=\sigma_1$ be as before and set $\tau$ be such that under the classical evolution $\Psi^{\tau}_0:\x_1\to\x(\tau)$ the point $\x(\tau)=\x_2$ belongs to ${\bar{\Gamma}^{\varepsilon}_1}$. Set $\phi^{\sigma_2}_{\x_2 }(x)= e^{-i\tau \H_0 /\hbar}\phi^{\sigma}_{\x }(x)+O(\hbar^{\infty})$ be the coherent state in $\Omega$, whose parameters are given by: $(\sigma_2, \x_2)= (D\Psi^{\tau}_0 \cdot\sigma_1, \Psi^{\tau}_0\cdot\x_1)$. Then the states $$\begin{aligned}
|\bar{\Phi}^{\sigma}_{\Gamma_1(E)}\rangle&=& \int^{t_1}_{t_2} e^{it (E-\H_{\Omega}) /\hbar}|\phi^{\sigma_1}_{\x_1 }\rangle \,dt, \label{4.2.5}\\
|\bar{\Phi}^{\sigma}_{\bar{\Gamma}_1(E)}\rangle
&=& \int^{\bar{t}_1-\tau}_{\bar{t}_2-\tau} e^{it (E-\H_{\Omega}) /\hbar}|\phi^{\sigma_2}_{\x_2 }\rangle \,dt, \label{4.2.6}\end{aligned}$$ have their supports in $\Omega$ and satisfy $$|\bar{\Phi}^{\sigma}_{\Gamma_1(E)}\rangle=
|\Phi^{\sigma}_{\Gamma_1(E)}\rangle
+O(\hbar^{\infty}), \qquad
|\bar{\Phi}^{\sigma}_{\bar{\Gamma}_1(E)}\rangle
=|\Phi^{\sigma}_{\bar{\Gamma}_1(E)}\rangle
+O(\hbar^{\infty}).\label{4.2.7}$$
To get the lower bound (\[4.2.2\]) we are going first to construct a state ${\Phi}$ with the property
$$\langle \psi |{\Phi}\rangle=0+ O(\hbar^\infty),\label{4.2.9}$$
for any $\psi\in\M(E')$. Let us show how such a state can be constructed using $\bar{\Phi}^{\sigma}_{\Gamma_1}$, $\bar{\Phi}^{\sigma}_{\bar{\Gamma}_1}$. Set $\langle\cdot|\cdot\rangle_{\R^2}$, $\langle\cdot|\cdot\rangle$ be the scalar products in $L^2(\R^2)$ and $L^2(\Omega)$ respectively. From the definitions (\[4.2.3\],\[4.2.4\]) one has
$$\langle \psi |\Phi^{\sigma}_{\Gamma_1(E)}\rangle_{\R^2}
=\int^{t_1}_{t_2} e^{it (E-E') /\hbar}\langle \psi |\phi^{\sigma}_{\x }\rangle \,dt
={C_1(\omega)}\langle \psi |\phi^{\sigma}_{\x }\rangle, \label{4.2.10}$$
where $$C_{1}(\omega) = \exp{\left( \frac{i(t_1+t_2) \omega}{2} \right) }\, \frac{2 \sin(\omega T_{\gamma_1}/2)} { \omega }, \,\, T_{\gamma_1}=|t_1-t_2|, \label{4.2.11}$$ and $ \omega = {(E - E')}/{\hbar}$. Analogously:
$$\langle \psi |\Phi^{\sigma}_{\bar{\Gamma}_1(E)}\rangle_{\R^2}={C_2(\omega)}\langle \psi |\phi^{\sigma}_{\x }\rangle,\label{4.2.12}$$
with
$$C_{2}(\omega) = \exp{\left( \frac{i(\bar{t}_1+\bar{t}_2) \omega}{2} \right) }\, \frac{2 \sin(\omega T_{\bar{\gamma}_1}/2)} { \omega }, \,\, T_{\bar{\gamma}_1}=|\bar{t}_1-\bar{t}_2| .\label{4.2.13}$$
Furthermore, let us introduce the states
$$|\Phi^{\sigma}_{1}(E,E')\rangle=\frac{1}{C_1(\omega)}|\bar{\Phi}^{\sigma}_{\Gamma_1(E)}\rangle, \qquad |\Phi^{\sigma}_{2}(E,E')\rangle=\frac{1}{C_2(\omega)}|\bar{\Phi}^{\sigma}_{\bar{\Gamma}_1(E)}\rangle.\label{4.2.135}$$
Then it follows immediately from eqs. \[4.2.10\], \[4.2.12\] that the state $\Phi ={\Phi}^{\sigma}(E,E')$,
$$|{\Phi}^{\sigma}(E,E')\rangle=|\Phi^{\sigma}_{1}(E,E')\rangle -|\Phi^{\sigma}_{2}(E,E')\rangle
\label{4.2.14}$$
satisfies orthogonality condition (\[4.2.9\]).
Let $({\varphi}_n, E_n)\in\S_{\gamma}$ be a solution of eqs. \[1.1\], \[1.2\] and let $(\tilde{\varphi}_n,\tilde{E}_n)\in\widetilde{\S}_{\gamma}$ be the corresponding quasimode, whose leading order parameters $\E_n$, $\sigma_n=(\sigma^0_n,\sigma^1_n)$ are determined by Conditions 1, 2, see eqs. \[4.1.4\], \[4.1.45\]. Now fix the energy parameters in eq. \[4.2.14\] by $E=\E_n$, $E'=E_n$ and put $\sigma=\bar{\sigma}_n$, where $\bar{\sigma}_n=(i\beta, \sigma^1_n)$ and $\beta$ is an arbitrary real positive number. We will make use of the state $$|{\Phi_n}\rangle=|\Phi^{\bar{\sigma}_n}(\E_n,E_n)\rangle$$ in order to get a lower bound on $\eta_n$. For any $\psi\in \M(E_n)$ we have
$$||\tilde{\varphi}_n-\psi|| \, ||{\Phi}_n || \geq
|\langle\tilde{\varphi}_n-\psi|{\Phi}_n\rangle| =
|\langle\tilde{\varphi}_n|{\Phi}_n\rangle|+O(\hbar^{\infty}).
\label{4.2.15}$$
Using the triangle inequality
$$||\tilde{\varphi}_n-\varphi_n || + ||\varphi_n-\psi ||\geq ||\tilde{\varphi}_n-\varphi_n +\varphi_n-\psi || =||\tilde{\varphi}_n-\psi||\label{4.2.16}$$
one gets immediately from (\[4.2.15\])
$$\eta_n(\psi)=||\varphi_n-\psi|| \geq \frac{ |\langle\tilde{\varphi}_n|\Phi_n \rangle|}{ || \Phi_n || } - {\mathcal C}^1_n+O(\hbar^{\infty}). \label{4.2.17}$$
It remains to estimate the scalar product $|\langle\tilde{\varphi}_n|\Phi_n \rangle|$ and the norm of the state ${\Phi_n}$. First, consider the norm $||{\Phi_n}||$. Since $ {\gamma}_1\cap{\bar{\gamma}_1}=\emptyset$ one has from the definition of $\Phi_n$
$$\langle\Phi_n |\Phi_n\rangle=
\frac{1}{|C_1(\omega_n)|^2}\langle{\Phi}^{\bar{\sigma}_n}_{\Gamma_1(\E_n)} |{\Phi}^{\bar{\sigma}_n}_{\Gamma_1(\E_n)}\rangle+\frac{1}{|C_1(\omega_n)|^2}
\langle{\Phi}^{\bar{\sigma}_n}_{\bar{\Gamma}_1(\E_n)}|
{\Phi}^{\bar{\sigma}_n}_{\bar{\Gamma}_1(\E_n)}\rangle+O(\hbar^{\infty}),\label{4.2.18}$$
with $\omega_n=(E_n -\E_n)/ \hbar$. The calculations of the scalar products performed in Appendix give: $$\begin{aligned}
\langle{\Phi}^{\bar{\sigma}_n}_{\Gamma_1(\E_n)} |{\Phi}^{\bar{\sigma}_n}_{\Gamma_1(\E_n)}\rangle&=& T_{\gamma_1} \left(\frac{2\pi\hbar}{\beta E_n}\right)^{1/2} +O(\hbar) , \nonumber \\
\langle{\Phi}^{\bar{\sigma}_n}_{\bar{\Gamma}_1(\E_n)}|
{\Phi}^{\bar{\sigma}_n}_{\bar{\Gamma}_1(\E_n)}\rangle
&=& T_{\bar{\gamma}_1} \left(\frac{2\pi\hbar}{\beta E_n}\right)^{1/2}+O(\hbar)\label{4.2.19} \end{aligned}$$ and for the leading order of $C_1(\omega_n)$, $C_2(\omega_n)$ one has from eqs. \[4.2.11\], \[4.2.13\] $$|C_2(\omega_n)| = T_{\bar{\gamma}_1} +O(\hbar),\qquad |C_1(\omega_n)|= T_{\gamma_1} +O(\hbar).\label{4.2.20}$$ Combining (\[4.2.19\]) and (\[4.2.20\]) together one finally gets $$\langle \Phi_n |\Phi_n\rangle= \left(\frac{2\pi\hbar}{\beta E_n}\right)^{1/2}
\left(
\frac{1}{ T_{\gamma_1}}
+ \frac{1}{ T_{\bar{\gamma}_1}}
\right) +O(\hbar). \label{4.2.21}$$ In the same way for the scalar product $\langle\tilde{\varphi}_{n} |\Phi_{n}\rangle$ we have by (\[4.1.4\],\[4.1.45\])
$$\begin{aligned}
|\langle\tilde{\varphi}_{n} |\Phi_n\rangle|=
| \langle\Phi^{\sigma_n}_{\Gamma(\E_n)} |\Phi^{\bar{\sigma}_n}_{\Gamma_1(\E_n)}\rangle|+O(\hbar)
&=& \frac{T_{\gamma_1}}{ T_{\gamma} }|\langle\Phi^{\sigma_n}_{\Gamma(\E_n)} |\Phi^{\sigma_n}_{\Gamma(\E_n)}\rangle|^{1/2} | \langle\Phi^{\bar{\sigma}_n}_{\Gamma_1(\E_n)}|\Phi^{\bar{\sigma}_n}_{\Gamma_1(\E_n)}
\rangle|^{1/2} \nonumber\\
+ O(\hbar)
&=& \frac{1}{T_{\gamma}}\left(\frac{2\pi\hbar}{\beta E_n}\right)^{1/2}+O(\hbar).
\label{4.2.22} \end{aligned}$$
The estimation (\[4.2.2\]) follows now immediately after inserting eqs. \[4.2.21\], \[4.2.22\] into (\[4.2.17\]). The resulting constant $\C_{\gamma}$, which determines the lower bound on $\eta_n$ in the semiclassical limit reads as $${\C}_{\gamma}=\sqrt{ \frac{ T_{\bar{\gamma}_1} T_{\gamma_1} }
{(T_{\bar{\gamma}_1}+ T_{\gamma_1}) T_{\gamma} } }=
\sqrt{ \frac{ \ell_{\bar{\gamma}_1} \ell_{\gamma_1} }
{(\ell_{\bar{\gamma}_1}+ \ell_{\gamma_1}) \ell_{\gamma} } } +O(\varepsilon),\label{4.2.23}$$ where $\ell_{\bar{\gamma}_1}, \ell_{\gamma_1}, \ell_{\gamma} $ are the lengths of ${\bar{\gamma}_1}, {\gamma_1}$ and ${\gamma} $ respectively.
PW approximation for eigenstates of non-convex billiards (hyperbolic case)
==========================================================================
In the present section we consider the case of a hyperbolic $\gamma$. Let as before $\{\varphi_n(x)\}$ be the set of eigenfunctions in ${\Omega}$ approximated by regular solutions $\{\psi_n(x)\}$ of eq. \[1.1\]. For an arbitrary set of $\psi_n(x)\in\M(E_n)$, $n=1,2,...\,\infty$ we will estimate the average of $$\eta_n\equiv\eta_n(\psi_n)=||\varphi_n -\psi_n||\label{5.1}$$ over an energy interval. Our objective is to show that independently of the choice of $\psi_n$’s, in the limit $\hbar\to 0$ the average $\big\langle\eta_n\big\rangle$ is bounded from below by a strictly positive constant.
Let $\Phi^{\sigma}_{1}(E,E')$, $\Phi^{\sigma}_{2}(E,E')$, $\Phi^{\sigma}(E,E')$ be as in the previous section with the parameter $\sigma$ of the form $\sigma=(i\beta, \sigma^1)$, $\beta>0$. For each integer $n$ we will consider the states $$|\Phi_{n,1}\rangle=|\Phi^{\sigma}_{1}(E,E_n)\rangle, \qquad
|\Phi_{n,2}\rangle=|\Phi^{\sigma}_{2}(E,E_n)\rangle \label{5.2}$$ and their difference $$|\widetilde{\Phi}_{n}\rangle=|\Phi_{n,1}\rangle-|\Phi_{n,2}\rangle=|\Phi^{\sigma}(E,E_n)\rangle, \label{5.25}$$ which is orthogonal to any $\psi\in\M(E_n)$ up to the term $O(\hbar^{\infty})$ (see eq. \[4.2.9\]). In addition, it will be also useful to introduce the state $$|\widetilde{\Phi}'_{n}\rangle=|\Phi_{n,1}\rangle+|\Phi_{n,2}\rangle.\label{5.3}$$ Note that $\widetilde{\Phi}'_{n}$ is orthogonal to $\widetilde{\Phi}_{n}$ in the semiclassical limit.
Similarly to the case of elliptic ’s, one can make use of the state $\widetilde{\Phi}_n$ to get a lower bound on $\eta_n$:
$$\eta_n \geq \frac{|\langle\widetilde{\Phi}_n|
\,\varphi_n-\psi_n \rangle|} {||\widetilde{\Phi}_n||}= \frac{|\langle\widetilde{\Phi}_n|\varphi_n \rangle|} {||\widetilde{\Phi}_n||}+O(\hbar^{\infty}).\label{5.4}$$
In order to estimate the right side of this inequality let us consider the following difference
$$\D=
|\langle{\Phi}_{n,1}|
\varphi_n \rangle|^2 - |\langle{\Phi}_{n,2}|
\varphi_n \rangle |^2. \label{5.5}$$
Using the states $\widetilde{\Phi}_n$, $ \widetilde{\Phi}'_n$ one can rewrite $\D$ as
$$\D=\re\left( \langle\widetilde{\Phi}_n|\varphi_n \rangle \langle\widetilde{\Phi}'_n|\varphi_n \rangle^{*}\right).\label{5.6}$$
Hence, the following inequality follows immediately
$$|\D| \leq |\langle\widetilde{\Phi}_n|\varphi_n \rangle| \, | \langle\widetilde{\Phi}'_n|\varphi_n \rangle| \leq ||\widetilde{\Phi}'_n||\,\, |\langle\widetilde{\Phi}_n|\varphi_n \rangle|. \label{5.7}$$
Finally, since $||\widetilde{\Phi}_n||-||\widetilde{\Phi}'_n||=O(\hbar^{\infty})$, we get by (\[5.4\]) and (\[5.7\])
$$\eta_n \geq \frac{|\D|}{||\widetilde{\Phi}_n||\,||\widetilde{\Phi}'_n||}+O(\hbar^{\infty}) = \left| \frac{ |\langle{\Phi}_{n,1}|
\varphi_n \rangle|^2 - |\langle{\Phi}_{n,2}|
\varphi_n \rangle |^2 }{\langle\widetilde{\Phi}_n|\widetilde{\Phi}_n\rangle}\right|+O(\hbar^{\infty}) .\label{5.8}$$
We will now use this inequality to get a lower bound for the sum of $ \eta_n $ over the energy interval $\P_{c\hbar}=[E-c\hbar, E+c\hbar]$, where $c$ is a positive constant. One has straightforwardly from (\[5.8\])
$$\sum_{E_n \in\P_{c\hbar} } \eta_n > \left| \sum_{E_n \in \P_{c\hbar}} \frac{ |\langle{\Phi}_{n,1}|
\varphi_n \rangle|^2 }{\langle\widetilde{\Phi}_n|\widetilde{\Phi}_n\rangle}
- \sum_{E_n \in\P_{c\hbar} } \frac{ |\langle{\Phi}_{n,2}|
\varphi_n \rangle|^2 }{\langle\widetilde{\Phi}_n|\widetilde{\Phi}_n\rangle}
\right|+O(\hbar^{\infty}).\label{5.9}$$
Furthermore, the definition of the states ${\Phi}_{n,1}$, ${\Phi}_{n,2}$ implies $$|\langle{\Phi}_{n,1}|\varphi_n \rangle|^2= |\langle\phi^{\sigma_1}_{\x_1}|\varphi_n \rangle|^2, \,\, \x_1\in\Gamma^{\varepsilon}_1; \,\,\,\,
|\langle{\Phi}_{n,2}|\varphi_n \rangle|^2= |\langle\phi^{\sigma_2}_{\x_2}|\varphi_n \rangle|^2, \,\, \x_2\in\bar{\Gamma}^{\varepsilon}_1, \label{5.10}$$ where $(\x_1, \sigma_1)=(\x, \sigma)$ and $(\x_2, \sigma_2)=(\x(\tau), \sigma(\tau))$ are related by the free classical evolution as in the previous section. As a result, the inequality (\[5.9\]) reads as
$$\sum_{E_n \in\P_{c\hbar} } \eta_n >
\left|\sum_n f (\omega_n)| \langle \varphi_n|\phi^{\sigma_1}_{\x_1} \rangle|^2
-\sum_n f (\omega_n)| \langle \varphi_n|\phi^{\sigma_2}_{\x_2} \rangle|^2\right|+O(\hbar^\infty), \label{5.11}$$
with $ \omega_n={(E-E_n)}/{\hbar}$ and $$f(\omega_n)=\left\{\begin{array}{cl}
{1/\langle\widetilde{\Phi}_n|\widetilde{\Phi}_n\rangle} & \mbox{if $\omega_n \in [-c,c]$} \\
0 & \mbox{otherwise}.
\end{array} \right.$$ The elementary calculations (see Appendix) provide the leading order of the function $f(\omega_n)$, $\omega_n \in [-c,c]$: $$\begin{aligned}
f(\omega_n)&=&\frac{1}{{\langle{\Phi_{n,1}}|{\Phi_{n,1}}\rangle} + {\langle{\Phi_{n,2}}|{\Phi_{n,2}}\rangle} }+O(\hbar^{\infty})\nonumber \\
&=&\frac{2|p|}{(\pi\hbar\beta)^{\frac{1}{2}} } \left( \frac{\omega^2_n T_{\gamma_1}}{\sin^2(\omega_n T_{\gamma_1}/2)}+
\frac{\omega^2_n T_{\bar{\gamma}_1}}{\sin^2(\omega_n T_{\bar{\gamma}_1}/2)}
\right)^{-1}+O(\hbar^{0}).
\label{5.12}\end{aligned}$$
Now we can apply to (\[5.11\]) the results of Paul and Uribe (see Sec. 3). Taking into account that $\x_1\in\Gamma$ while $\x_2$ does not belong to any periodic trajectory, we get by eqs. \[3.8\], \[3.9\] the following estimation for the average of $\eta_n$:
$$\big\langle\eta_n\big\rangle\equiv\frac{1}{\#\P_{c\hbar} }\sum_{E_n \in \P_{c\hbar} } \eta_n >
\frac{1}{\#\P_{c\hbar} } \left| \sum_{l\neq 0} \widetilde{F}(l T_{\gamma})
\frac{e^{il( S_{\gamma}/\hbar +\mu_{\gamma} )} }{\cosh^{1/2}(l\lambda_{\gamma})} \right| +O(\hbar^{3/2}), \label{5.13}$$
where $ \widetilde{F}(\cdot)$ is the Fourier transform of the function
$$F(x)=\left\{\begin{array}{cl} \left(\frac{8}{\pi}\right)^{\frac{1}{2}}
\left( \frac{x^2 T_{\gamma_1}}{\sin^2(x T_{\gamma_1}/2)}+
\frac{x^2 T_{\bar{\gamma}_1}}{\sin^2(x T_{\bar{\gamma}_1}/2)}
\right)^{-1} & \mbox{if $x \in [-c,c]$} \\
0 & \mbox{otherwise}
\end{array} \right.$$ and $\#\P_{c\hbar}$ is the number of eigenstates in the interval $\P_{c\hbar}$ whose leading order for a billiard of area ${\mathcal A}$ is given by the Weyl formula: $$\#\P_{c\hbar}={\mathcal A} c/2\pi\hbar+O(\hbar^0).$$ Consequently, if $$Y=\left| \sum_{l\neq 0} \widetilde{F}(l T_{\gamma})
\frac{e^{il( S_{\gamma}/\hbar +\mu_{\gamma} )} }{\cosh^{1/2}(l\lambda_{\gamma})} \right| \neq 0 \label{5.14}$$ one has from (\[5.13\]) $$\big\langle\eta_n\big\rangle \,\, > \B \hbar +O(\hbar^{3/2}), \,\,\,\, \,\, \,\, \B=2\pi Y/c{\mathcal A}>0. \label{5.15}$$ If moreover one assumes that $T_{\bar{\gamma}_1}c, \, T_{{\gamma}_1} c<<1$, the function $F(x)$ takes a simple form:
$$F(x)\approx\left\{\begin{array}{cl} \left(\frac{1}{2\pi}\right)^{\frac{1}{2}}
\left( \frac{T_{\bar{\gamma}_1}T_{\gamma_1} }{T_{\bar{\gamma}_1}+ T_{\gamma_1} } \right) & \mbox{if $x \in [-c,c]$} \\
0 & \mbox{otherwise}
\end{array} \right.$$ and the constant $\B$ can be written explicitly:
$$\B\approx\frac{ \sqrt{2\pi} }{\mathcal A}
\left( \frac{T_{\bar{\gamma}_1}T_{\gamma_1} }{T_{\bar{\gamma}_1}+ T_{\gamma_1} } \right)
\left| \sum_{l\neq 0} \frac{\sin(l c T_{\gamma})}{l c T_{\gamma}} \,\,
\frac{e^{il( S_{\gamma}/\hbar +\mu_{\gamma} )} }{\cosh^{1/2}(l\lambda_{\gamma})}
\right| . \label{5.16}$$
Note, that the lower bound (\[5.15\]) has been obtained using only one . In the case of hyperbolic dynamics, however, the periodic orbits (and, in particular, ’s) proliferate exponentially. Therefore, one can improve the estimation (\[5.15\]) making use of a state $\widetilde{\Phi}_n^{{\sf sum}}$ which is concentrated on a set of ’s $\{\gamma\}$ and satisfies eq. \[4.2.9\]. A simple way to construct such a state is to define it as the superposition: $$\widetilde{\Phi}_n^{{\sf sum}}=\sum_{\{\gamma\}}\widetilde{\Phi}_n({\gamma}),\label{5.17}$$ where $\widetilde{\Phi}_n({\gamma})$ stands for the state (\[5.25\]) associated with a $\gamma$.
Finally, let us mention that the statistical estimation (\[5.15\]) can be straightforwardly generalized to the case of elliptic ’s. In that case one should use the analogs of eqs. \[3.8\], \[3.9\] (which are known to exist [@paul]) for stable periodic trajectories.
Discussion and conclusions
==========================
Speaking informally, Proposition 2 implies that there is no on-shell basis of regular solutions of the Helmholtz equation which can be used to approximate all eigenstates of a generic non-convex billiard. That means any linear combination of plane waves, radial waves etc., with the same energy fails to approximate real eigenstates of non-convex billiards. In fact, a stronger result can be shown. Let $\Omega$ be a generic non-convex billiard and let $\Omega'$ be a domain (not necessarily convex) which properly contains $\Omega$: $\Omega'\supset\Omega$, $\partial\Omega'\cap\partial\Omega=\emptyset$. Denote by $\M_{\Omega'}(E)$ the set of all solutions of eq. \[1.1\] regular in $\Omega'$ (note, that $\M_{\Omega'}(E)\supseteq\M(E)$). Let us argue that the eigenstates of $\Omega$ cannot be approximated, in general, by states belonging to $\M_{\Omega'}(E)$. Let $\gamma$ be a and let $l,\gamma_1,\bar{\gamma}_1$ be as defined before. Furthermore, assume that the segment of the line $l$ between $\gamma_1$ and $\bar{\gamma}_1$ is entirely in $\Omega'$, see fig. 4. (It seems to be a natural assumption that in a generic case one can always fined such a , provided $\Omega'$ properly contains $\Omega$). Then take $\Omega_0\subset\Omega$ to be a convex domain satisfying: $\Omega_0\cap\gamma_1\neq\emptyset$, $\Omega_0\cap\bar{\gamma}_1\neq\emptyset$. Now, suppose an eigenstate $\varphi_n$ of $\Omega$ can be approximated by states $\psi'(x)$ from $\M_{\Omega'}(E_n)$. According to Proposition 1 $\psi'(x)$ can be approximated in ${\Omega_0}$ by regular solutions of eq. \[1.1\] and thus for any ${\epsilon}>0$ there exists $\psi_{\epsilon}\in\M(E_n)$ such that $||\varphi_n(x)-\psi_{\epsilon}(x)||_{L^2(\Omega_0)}<{\epsilon}$. Therefore, applying the same arguments as in Sec. 2 we get $$H_{\varphi_n}(\x_{1})-H_{\varphi_n}(\x_{2}) =\lim_{\epsilon\to 0}|\langle\x_{1}|\psi_{\epsilon}\rangle|^2 -\lim_{\epsilon\to 0}|\langle\x_{2}|\psi_{\epsilon}\rangle|^2=O(\hbar^{\infty}),$$ where $\x_{1}=(q_1,p)\in\Gamma_1(E_n)$, $q_1\in\gamma_1\cap\Omega_0$ and $\x_{2}=(q_2,p)\in\bar{\Gamma}_1(E_n)$, $q_2\in\bar{\gamma}_1\cap\Omega_0$. However, as has been pointed out before, this cannot be true for each $n$ since $\x_{2}\notin\Gamma$.
The two properties of generic non-convex billiards follows immediately from the above analysis. First, it is not possible to approximate eigenstates of a generic non-convex billiard $\Omega$ also if one includes in the basis $\{\psi^{(n)}(\k)\}$ singular solutions of eq. \[1.1\], e.g., the Hankel functions $$\{ H_{n}^{\pm}(k|x-x_i|)e^{in\theta(x,x_i)}, \, n\in \N \},$$ with a finite number of singularity points $x_i$. Second, there exists an infinite sequence of eigenstates which do not admit extension into any large domain $\Omega'$ properly containing $\Omega$. That means the continuation of the interior eigenstates of a generic non-convex billiard into the exterior domain should be (in general) impossible because of singularities which occur arbitrary close to the billiard’s boundary. It remains as an open problem what is the exact nature of such singularities. (For example, whether one can, in principal, extend eigenstates beyond the boundary of a generic non-convex billiard.) It should be also mentioned that the problem of the eigenstates extension in convex billiards is beyond the scope of the present paper. It would become a natural question to inquire about the relation between the billiard shape and the type of singularities arising for the extended eigenstates. In particular, it would be interesting to know whether the strong form of spectral duality (when it is possible to extend eigenfunctions in $\R^2$ as regular solutions of the Helmholtz equation) holds exclusively for integrable billiards.
Further, let us stress an important difference between the cases of elliptic and hyperbolic dynamics. The counting function $\mathcal{N}^*(\k)=\#\{\tilde{\k}_n<\k\}$ for quasimodes $(\tilde{\varphi}_n,\tilde{\k}_n )$ which can be constructed on an elliptic periodic trajectory is known to be of the same asymptotic form $\mathcal{N}^*(\k)=\alpha\k^{2}+ O(\k)$, $\alpha>0$ as the counting function $\mathcal{N}={\mathcal A}\k^{2}/4\pi+ O(\k)$ for the real spectrum $\{\k_n\}$, see [@popov]. Therefore, in a generic case, if an elliptic $\gamma$ exists the subsequence $\{\varphi_{j_n}, n\in \N\}$ of billiard eigenstates approximated by the quasimodes concentrated on $\gamma$ should be of the positive density: $$\lim_{N\to\infty}\frac{1}{N}\#\{j_n|j_n\leq N\}=\lim_{\k\to\infty}\frac{\mathcal{N}^*(\k)}{\mathcal{N}(\k)}>0.$$ Since for each $ \varphi_{j_n}$ the estimation (\[2.8\]) holds, that means there exists a subsequence of eigenstates with a positive density which do not admit approximation by plane waves. In the case of hyperbolic dynamics the statistical lower bound (\[2.7\]) implies, in fact, only a weaker result. It says that an infinite sequence (possibly of zero density) of such states exists. However, if one assumes that all eigenstates of fully chaotic billiards have “uniform properties” the inequality (\[2.7\]) suggests a natural conjecture:
[**Conjecture.**]{} [*For a non-convex billiard with fully chaotic dynamics the set of states which can be approximated by PW is of density zero.*]{}
Note, that it is impossible to exclude the possibility of existence of “exceptional” eigenstates (the eigenstates which can be approximated by PW) in non-convex billiards. Indeed, one can take a finite superposition of plane waves $\psi^{[N]}$ and set a (non-convex) nodal domain of $\psi^{[N]}$ to be the billiard’s boundary. Then $\psi^{[N]}$ itself is the eigenstate of this billiard which can be approximated by PW.
Finally, the study of the present paper is restricted to the two-dimensional simply connected domains with Dirichlet boundary conditions. However, it is easy to see that presented results allow several rather straightforward generalizations. First, higher dimensional billiards and different types of boundary conditions can be treated in the same way. Second, billiards in multiply connected domains (fig. 5) have the same properties as non-convex billiards. Consequently, all the results obtained for non-convex billiards hold for multiply connected billiards as well. Third, we conjecture that our results can be generalized to the billiards on non-compact manifolds with non-trivial metrics (also in the presence of a potential) e.g., billiards on the hyperbolic plane. In such a case, one needs to adjust the notion of domain’s “convexity” to the corresponding classical dynamics. In other words, a domain should be defined as “convex” if the interior-exterior duality holds and defined as “non-convex” if it breaks dawn.
Acknowledgments {#acknowledgments .unnumbered}
===============
I am grateful to R. Schubert, U. Smilansky, A. Voros and S. Nonnenmacher for useful discussions.
Appendix {#appendix .unnumbered}
========
[**Proposition 3.**]{} [*Let $\Phi^{\sigma}_{\Gamma}$, $\Phi^{\bar{\sigma}}_{\Gamma}$ be the states: $$\begin{aligned}
|\Phi^{\sigma}_{\Gamma}\rangle &=& \frac{1}{C_1}\int_{0}^{T}e^{i(E-\H_0)t/\hbar} |\phi^{\sigma}_{\x}\rangle dt, \qquad \sigma=(\sigma^0,\sigma^1), \nonumber \\
|\Phi^{\bar{\sigma}}_{\Gamma}\rangle &=& \frac{1}{C_2}\int_{0}^{T}e^{i(E-\H_0)t/\hbar} |\phi^{ \bar{\sigma}}_{\x}\rangle dt, \qquad \bar{\sigma}=( \bar{\sigma}^0, \bar{\sigma}^1)
\label{A.1}\end{aligned}$$ localized along the path $\Gamma=\Gamma(E)$, $\Gamma(E)=\{ \Psi^t \cdot \x=(q(t),p(t)),\, t\in [0, T] , E=p^2/2\}$ with $\sigma^0=i\beta_1$, $\bar{\sigma}^0=i\beta_2$; $\beta_1,\beta_2 >0$ and $\sigma^1=\bar{\sigma}^1$. Then $$\langle \Phi^{\sigma}_{\Gamma} |\Phi^{\sigma}_{\Gamma}\rangle =\frac{T}{C_1^2}\left(\frac{2\pi\hbar}{\beta_1 E}\right)^{1/2} +O(\hbar); \,\,\,\,
\langle \Phi^{\bar{\sigma}}_{\Gamma} |\Phi^{\bar{\sigma}}_{\Gamma}\rangle =\frac{T}{C_2^2}\left(\frac{2\pi\hbar}{\beta_2 E}\right)^{1/2} +O(\hbar),
\label{A.2}$$ $$\langle \Phi^{\sigma}_{\Gamma} |\Phi^{\bar{\sigma}}_{\Gamma}\rangle = \langle \Phi^{\sigma}_{\Gamma} |\Phi^{\sigma}_{\Gamma}\rangle^{1/2} \langle \Phi^{\bar{\sigma}}_{\Gamma} |\Phi^{\bar{\sigma}}_{\Gamma}\rangle^{1/2} +O(\hbar). \label{A.25}$$* ]{}
[*Proof.*]{} The inner product
$$\langle \Phi^{\sigma}_{\Gamma} |\Phi^{\bar{\sigma}}_{\Gamma}\rangle =\frac{1}{ C_1 C_2}\int_{0}^{T} \int_{0}^{T} \langle\phi^{\sigma}_{\x}| e^{i(E-H_0)(t_1-t_2)/\hbar} |\phi^{\bar{\sigma}}_{\x} \rangle dt_{1} dt_{2} \label{A.3}$$
can be written as
$$\begin{aligned}
\langle \Phi^{\sigma}_{\Gamma} |\Phi^{\bar{\sigma}}_{\Gamma}\rangle&=& \frac{1}{2 C_1 C_2 }\left(\int_{0}^{T}(T-t)H(t) dt
+ \int_{0}^{T}(T-t)H(-t) dt \right) \nonumber \\
&=&\frac{1}{2 C_1 C_2 }\int_{-T}^{T}(T-|t|)H(t) dt, \label{A.4} \end{aligned}$$
where $$H(t)=\langle\phi^{\sigma}_{\x}| e^{i(E-H_0)t/\hbar} |\phi^{\bar{\sigma}}_{\x} \rangle.\label{A.5}$$ By the propogation formula (\[3.5\]) we get for (\[A.5\])
$$\begin{aligned}
H(t)&=& e^{i(S(t)+E)/\hbar +i\mu(t)}\langle\phi^{\sigma}_{\x}|\phi^{\bar{\sigma}(t)}_{\x(t)} \rangle+O(\hbar)\nonumber \\
&=&\det\left( \frac{4\im \sigma\im \bar{\sigma}^*(t)}{(\sigma-\bar{\sigma}^*(t))^2}\right)^{1/4}\exp\left(-\frac{i t^2}{2\hbar} \langle p , \bar{\sigma}^*(t)\frac{1}{\sigma-\bar{\sigma}^*(t)}\sigma \, p \rangle \right) +O(\hbar) \nonumber \\
&=&\left(\frac{(\beta_2 \beta_1)^{1/4}}{(\beta_2+ \beta_1)^{1/2}}+O(t)\right)\exp\left(-\frac{ t^2 p^2\beta_2 \beta_1}{2\hbar(\beta_2 +\beta_1)}
+O(t^3)\right)+O(\hbar).\end{aligned}$$
After inserting this expresion into eq. \[A.4\] and applying the stationary phase approximation to the integral one gets (\[A.2\],\[A.25\]). Finaly, let us note that eq. \[A.25\] remains true also when $\beta_1$ or $\beta_2$ equals zero.
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abstract: 'Recent scattering-type scanning near-field optical spectroscopy (s-SNOM) experiments on single-layer graphene have reported Dirac plasmon lifetimes that are substantially shorter than the dc transport scattering time $\tau_{\rm tr}$. We highlight that the plasmon lifetime is fundamentally different from $\tau_{\rm tr}$ since it is controlled by the imaginary part of the current-current linear response function at [*finite*]{} momentum and frequency. We first present the minimal theory of the extrinsic lifetime of Dirac plasmons due to scattering against impurities. We then show that a very reasonable concentration of charged impurities yields a plasmon damping rate which is in good agreement with s-SNOM experimental results.'
author:
- Alessandro Principi
- Giovanni Vignale
- Matteo Carrega
- Marco Polini
title: The impact of disorder on Dirac plasmon losses
---
[*Introduction.—*]{}Graphene plasmonics [@koppens_nanolett_2011; @grigorenko_naturephoton_2012] is a rapidly growing branch of research which aims at exploiting the interaction of infrared light with the so-called “Dirac plasmons" (DPs) [@DiracplasmonsRPA; @principi_prb_2009; @orlita_njp_2012; @abedinpour_prb_2011] for a variety of applications ranging from photodetectors [@freitag_naturecommun_2013; @vicarelli_naturemater_2012] to biosensors [@grigorenko_naturemater_2013]. DPs, the self-sustained density oscillations of the two-dimensional (2D) electron liquid in a doped graphene sheet [@novoselov_naturemater_2007; @kotov_rmp_2012], have been studied experimentally by a variety of spectroscopic methods [@grigorenko_naturephoton_2012].
Fei [*et al.*]{} [@fei_nature_2012] and Chen [*et al.*]{} [@chen_nature_2012] have carried out seminal scattering-type scanning near-field optical microscopy (s-SNOM) experiments in which DPs are launched and imaged in real space. They showed that the plasmon wavelength $\lambda_{\rm p}$ can be $\sim 40$-$60$ times smaller than the illumination wavelength, allowing an extreme concentration of electromagnetic energy, and that DP properties are easily gate tunable. They have also presented an experimental analysis of DP losses. By comparing theory to experimental data, Fei [*et al.*]{} [@fei_nature_2012] have reported a DP damping rate $\gamma_{\rm p} \simeq 0.08$ (subtracting here the damping due to the complex dielectric constant of the substrate), which is four times larger than that estimated by means of the Drude transport time of their samples. Similarly, Chen [*et al.*]{} [@chen_nature_2012] showed that theoretical calculations of the local density of optical states compare well with measurements when rather low values, $\sim 1200~{\rm cm}^2/({\rm V} {\rm s})$, of the sample mobility are used as inputs in the numerics.
The plasmon damping rate (or inverse quality factor), which is a key figure of merit of nanoplasmonics, is defined as [@principi_arxiv_2013] $$\label{eq:dampingrate}
\gamma_{\rm p}(q) \equiv 2\frac{\Gamma_{\rm p}(q)}{\omega_{\rm p}(q)}~,$$ where $\omega_{\rm p}(q)$ is the DP dispersion [@DiracplasmonsRPA; @principi_prb_2009; @orlita_njp_2012; @abedinpour_prb_2011] and $\Gamma_{\rm p}(q)$ is its linewidth [@Giuliani_and_Vignale]. The factor two on the right-hand side of Eq. (\[eq:dampingrate\]) has been deliberately added to make direct contact with the results of Ref. . The DP damping rate is controlled by several mechanisms, such as electron-electron (e-e), electron-phonon (e-phon), and electron-impurity (e-imp) scattering. The rates associated to the first two scattering mechanism have been theoretically studied in Refs. and , respectively. To the best of our knowledge, the impact of e-imp scattering on the DP lifetime has not yet been quantitatively analyzed. We believe that this is because the plasmon lifetime associated with e-imp scattering is typically [@koppens_nanolett_2011; @fei_nature_2012; @chen_nature_2012] believed to be close to the dc Drude transport time $\tau_{\rm tr}$. It is well known [@belitz_prb_1986] that this is not the case and that the plasmon lifetime $\tau_{\rm p}(q)\equiv [2\Gamma_{\rm p}(q)]^{-1}$ can be substantially smaller than $\tau_{\rm tr}$.
A microscopic understanding of DP losses is central to the success of graphene as a novel platform for nanoplasmonics. As a step towards the elucidation of the mechanisms that contribute to the DP damping rate, in this Rapid Communication we present the simplest theory of the DP plasmon damping rate due to e-imp scattering. Although our theory is general, our numerical calculations focus, for the sake of concreteness, on the role of [*charged*]{} impurities. Long-range Coulomb disorder is indeed the most “popular" [@dassarma_rmp_2011] candidate for the main scattering mechanism limiting mobility in graphene sheets deposited on substrates like ${\rm SiO}_2$. Other important sources of disorder, such as corrugations [@gibertini_prb_2012] and resonant scatterers [@Katsnelsonbook] can also affect the DP lifetime. In this respect, we note that Yuan [*et al.*]{} [@yuan_prb_2011] have shown that even a small amount of resonant scatterers such as lattice defects or adsorbates can account for the observed [@li_natphys_2008] background of optical absorption below the single-particle threshold. A comparative study of the impact of various disorder models on the DP lifetime is beyond the scope of the present work.
In this Rapid Communication we demonstrate that neglecting the dependence of the “memory kernel" $1/\tau(q,\omega)$ on wavevector $q$ and frequency $\omega$—for example by approximating $\tau(q,\omega)$ with $\tau_{\rm tr}$—results in a severe underestimation of disorder-induced DP losses. We find that a very reasonable concentration of charged impurities is enough to explain the experimental findings of Refs. . Our results, in combination with those reported in Ref. , strongly suggest that current s-SNOM experiments [@fei_nature_2012; @chen_nature_2012] are dominated by disorder. As suggested in Ref. , the “intrinsic" regime where many-body effects dominate DP losses can be reached in suspended graphene sheets or in graphene flakes deposited on hexagonal Boron Nitride.
[*Theoretical formulation.—*]{}DP losses are quantified by the plasmon damping rate [@footnotecomplex] $
\gamma_{\rm p}(q) = {\cal R}\big(q,\omega_{\rm p}(q)\big)
$, where $$\label{eq:sigma_ratio}
{\cal R}(q,\omega) \equiv \frac{\Re e [\sigma(q, \omega)]}{\Im m [\sigma(q,\omega)]}~,$$ and $\omega_{\rm p}(q) = \sqrt{2 {\cal D}_0 q/\epsilon}$ is the long-wavelength plasmon dispersion calculated at the level of the random phase approximation (RPA) [@DiracplasmonsRPA]. Here ${\cal D}_0 = 4 \varepsilon_{\rm F} \sigma_{\rm uni}/\hbar$ is the Drude weight [@grigorenko_naturephoton_2012] of non-interacting massless Dirac fermions (MDFs), $\sigma_{\rm uni} = N_{\rm f} e^2/(16\hbar)$ is the universal conductivity [@grigorenko_naturephoton_2012], and $N_{\rm f}=4$ is the number of fermion flavors in graphene stemming from spin and valley degrees of freedom. We also introduced $\epsilon = (\epsilon_{\rm air} + \epsilon_{\rm sub})/2$, [*i.e.*]{} the average of the dielectric constants of the media above ($\epsilon_{\rm air} = 1$) and below ($\epsilon_{\rm sub}$) the graphene flake. Finally, $\varepsilon_{\rm F} = \hbar v_{\rm F} k_{\rm F}$ is the Fermi energy [@electronholesymmetry], where $k_{\rm F} = \sqrt{4\pi n/N_{\rm f}}$ is the Fermi wavevector. The Fermi velocity $v_{\rm F}$ is $\sim 10^6~{\rm m}/{\rm s}$.
It is crucial to note that the relation between $\gamma_{\rm p}(q)$ and the conductivity differs from that given in Ref. . In deriving a similar relation, the authors of Ref. have [*neglected*]{} the dependence of the conductivity $\sigma(q,\omega)$ on wavevector $q$. This quantity is related to the density-density response function $\chi_{nn}(q,\omega)$ by [@Giuliani_and_Vignale] $
\sigma(q,\omega) = i e^2 \omega \chi_{nn} (q,\omega)/q^2
$. Below we calculate $\Im m[\chi_{nn}(q,\omega)]$ at finite $q$ and $\omega$ to the lowest non-vanishing order ([*i.e.*]{} second order) in the e-imp potential.
The homogeneous optical conductivity of non-interacting MDFs in a doped graphene sheet is given by [@abedinpour_prb_2011] $
\sigma_{\rm c}(q=0,\omega) = i{\cal D}_0[\pi (\omega + i 0^+)]^{-1}
$. The previous result, which expresses the intraband contribution to the long-wavelength conductivity, is valid for $\hbar \omega < 2\varepsilon_{\rm F}$ and in the absence of disorder. Note that $\Re e[\sigma_{\rm c}(q=0,\omega)]$ has a delta function peak (the so-called “Drude peak”) at $\omega=0$, whose strength is given by ${\cal D}_0$.
In the presence of weak disorder and in the spirit of the Drude transport theory, it is natural to define a wavevector- and frequency-dependent scattering time $\tau(q,\omega)$ as follows [@Giuliani_and_Vignale]: $$\label{eq:sigma_omega_def}
\sigma(q,\omega) \equiv \frac{i {\cal D}_0/\pi}{\omega + i \tau^{-1}(q,\omega)}~.$$ This expression is valid in the limit of $v_{\rm F} q \ll \omega \ll 2\varepsilon_{\rm F}/\hbar$: this is precisely the region of the $(q,\omega)$ plane where the DP lives [@DiracplasmonsRPA]. The function $\tau(q,\omega)$ has been assumed real. The usual dc Drude transport time is given by $
\tau_{\rm tr} \equiv \lim_{\omega \to 0} \tau(q=0,\omega)
$. As expected, a finite transport time broadens the zero-frequency Drude peak into a Lorentzian. It is easy to prove that, in the weak scattering limit $\omega \tau(q,\omega) \gg 1$, the DP lifetime is given by $\tau\big(q,\omega_{\rm p}(q)\big)$ and that ${\cal R}(q,\omega) = [\omega\tau(q,\omega)]^{-1}$. In the same limit one gets [@Giuliani_and_Vignale] $$\label{eq:lifetime_def}
\frac{1}{\tau(q,\omega)} = -\frac{\pi e^2 \omega^3}{{\cal D}_0 q^2} \Im m[\chi_{nn} (q,\omega)]
~.$$ In deriving Eq. (\[eq:lifetime\_def\]) we have used that $
\Re e[\chi_{nn} (q,\omega)] \to {\cal D}_0 q^2/(\pi e^2 \omega^2)
$ in the limit $1, v_{\rm F} q \tau(q,\omega) \ll \omega\tau(q,\omega)\ll 2\varepsilon_{\rm F}\tau(q,\omega)/\hbar$.
Following Ref. , we describe the electron system in a doped graphene sheet in a tight-binding framework which takes into account only the $\pi$ and $\pi^\star$ bands of graphene. We neglect all the other bands. This approach is sufficient to describe graphene at low energies and eliminates spurious problems associated with the short-distance physics of the MDF model [@abedinpour_prb_2011; @sabio_prb_2008]. We take the low-energy MDF limit only [*after*]{} carrying out all the necessary commutators and algebraic steps briefly sketched in the following.
Our calculations of the role of e-imp scattering on the DP lifetime are based on the following Hamiltonian: ${\hat {\cal H}} = {\hat {\cal H}}_0 + {\hat {\cal H}}_{\rm ei}$, where ${\hat {\cal H}}_0$ is the aforementioned tight-binding Hamiltonian (see Ref. for more details). The e-imp Hamiltonian reads $
{\hat {\cal H}}_{\rm ei} = {\cal A}_{\rm BZ}^{-1} \sum_{{\bm q}} u_{\bm q} {\hat n}_{\bm q} n^{({\rm i})}_{-{\bm q}}
$, where ${\bm q}$ is restricted inside the first Brillouin zone and ${\cal A}_{\rm BZ}$ is its area. Here ${\hat n}_{\bm q}$ is the electron density operator [@principi_arxiv_2013] and $n^{({\rm i})}_{\bm q} = \sum_{{\bm R}_i} e^{i {\bm q}\cdot {\bm R}_i}$ is the impurity density. The vector ${\bm R}_i$ labels the random position of the $i$-th impurity. Finally, $u_{\bm q}$ is the [*discrete*]{} Fourier transform of the e-imp potential. We emphasize that e-e interactions, which, for the sake of simplicity, have not been explicitly added to ${\hat {\cal H}}$, play a twofold role: they enable the existence of plasmons [@Giuliani_and_Vignale] and weaken the bare e-imp potential $u_{\bm q}$ through screening. Both effects are taken into account below at the RPA level.
[*Elimination of the e-imp potential via a canonical transformation.—*]{}We now calculate $\tau(q,\omega)$ as defined in Eq. (\[eq:lifetime\_def\]) and in the presence of weak disorder. To this aim, we evaluate $\chi_{nn}(q,\omega)$ on the right-hand side of Eq. (\[eq:lifetime\_def\]) to second order in the strength of e-imp interactions. Within the tight-binding model, the density operator ${\hat n}_{\bm q}$ can be related to the longitudinal component of the current density operator $\hat{\bm j}_{\bm q}$ by the continuity equation (from now on we set $\hbar =1$): $i\partial_t {\hat n}_{\bm q} =[{\hat {\cal H}}, {\hat n}_{\bm q}] = -{\bm q} \cdot {\hat {\bm j}}_{\bm q}$. The continuity equation does not show any anomalous commutator [@abedinpour_prb_2011; @sabio_prb_2008] and allows us to express [@principi_arxiv_2013; @Giuliani_and_Vignale] $\Im m[\chi_{nn}(q,\omega)]$ in terms of the imaginary part of the longitudinal current-current response function $\chi_{\rm L}(q,\omega)$. In view of the low-energy MDF limit and disorder average taken at the end of the calculation (which restore isotropy), without any lack of generality we can take ${\bm q} = q {\hat {\bm x}}$ and arrive at the desired result: \[eq:continuity\_equation\] m\[\_[nn]{}(q,)\] = m\[\_[L]{}(q,)\] . Eq. (\[eq:continuity\_equation\]) is the usual relation between density-density and longitudinal current-current response functions, which holds for an isotropic, rotationally-invariant electron liquid.
To proceed further, we adopt the same strategy detailed in Ref. . We introduce a unitary transformation generated by an Hermitian operator ${\hat F}$ that cancels exactly the e-imp interaction from ${\hat {\cal H}}$, [*i.e.*]{} $
{\hat {\cal H}}' = e^{i {\hat F}} {\hat {\cal H}} e^{-i {\hat F}} \equiv {\hat {\cal H}}_0
$. This equation can be solved order by order in perturbation theory, by expanding ${\hat F} = \openone + {\hat F}_1 + {\hat F}_2 + \ldots$, where $\openone$ is the identity operator and ${\hat F}_n$ denotes the $n$-th order term in powers of the strength of e-imp interactions. We obtain a chain of equations connecting ${\hat F}_n$ to ${\hat {\cal H}}_{\rm ei}$. As an example, ${\hat F}_1$ solves the equality $[{\hat {\cal H}}_0, i {\hat F}_1] = {\hat {\cal H}}_{\rm ei}$.
We then calculate the “rotated” current operator, which can be expanded in powers of the e-imp interaction as $
{\bm q}\cdot{\hat {\bm j}}_{\bm q}' = {\bm q}\cdot{\hat {\bm j}}_{\bm q} + {\bm q}\cdot{\hat {\bm j}}_{1,{\bm q}} + {\bm q}\cdot{\hat {\bm j}}_{2,{\bm q}} + \ldots
$, where ${\hat {\bm j}}_{n,{\bm q}}$ is of $n$-th order in the e-imp potential $u_{\bm q}$. Note that only the zeroth-order contribution to ${\bm q}\cdot{\hat {\bm j}}_{\bm q}'$ ([*i.e.*]{} ${\bm q}\cdot{\hat {\bm j}}_{{\bm q}}$) does not break momentum conservation by transferring part of the momentum ${\bm q}$ to the impurity subsystem. Indeed, it can only generate single-pair excitations of total momentum ${\bm q}$ which lie inside the particle-hole continuum. This in turn implies that in the limit $v_{\rm F} q \ll \omega \ll 2\varepsilon_{\rm F}$ the only non-vanishing second-order contribution in the strength of e-imp interactions to $\Im m[\chi_{\rm L}(q,\omega)]$ is $\Im m[\chi_{j_{1,x} j_{1,x}}(q{\hat {\bm x}},\omega)]$. We find $$\begin{aligned}
\label{eq:SM_j_1_element}
{\bm q}\cdot{\hat {\bm j}}_{1,{\bm q}} = [i {\hat F}_1, {\bm q}\cdot{\hat {\bm j}}_{\bm q}] = {\cal A}_{\rm BZ}^{-1} \sum_{{\bm q}'} u_{{\bm q}'}
{\hat \Upsilon}_{{\bm q}, {\bm q}'} n^{({\rm i})}_{-{\bm q}'}
~.\end{aligned}$$ It is clear from Eq. (\[eq:SM\_j\_1\_element\]) that ${\bm q}\cdot{\hat {\bm j}}_{1,{\bm q}}$ breaks momentum conservation, since an amount $-{\bm q}'$ is transferred to impurities. In the limit $v_{\rm F} q \ll \omega \ll 2\varepsilon_{\rm F}$, the operator ${\hat \Upsilon}_{{\bm q}, {\bm q}'}$ reads $$\begin{aligned}
\label{eq:Upsilon_approx}
{\hat \Upsilon}_{{\bm q},{\bm q}'} &=&
- \sum_{\alpha=x,y} \left\{
\frac{v_{\rm F}}{\omega^2 k_{\rm F}} q'_x q'_\alpha + \left[ \frac{v_{\rm F}}{\omega^2} \frac{{\bm q}\cdot{\bm q}'}{k_{\rm F}} \left( 3 - \frac{q'^2}{2 k_{\rm F}^2} \right)
\right.
\right.
\nonumber\\
&-&
\left.
\left. \frac{q'^2}{4 v_{\rm F} k_{\rm F}^3} \right] \delta_{\alpha,x}
\right\}
{\hat j}_{{\bm q}+{\bm q}', \alpha}
\nonumber\\
&\equiv&
- \sum_{\alpha=x,y} \Gamma^{({\rm dis})}_\alpha({\bm q},{\bm q}',\omega) {\hat j}_{{\bm q}+{\bm q}', \alpha}
~.\end{aligned}$$ Taking the low-energy MDF limit we finally obtain $$\begin{aligned}
\label{eq:transporttime_final}
\frac{1}{\tau(q,\omega)} &=& -\frac{\pi e^2 n_{\rm imp} \omega}{{\cal D}_0} \sum_{\alpha,\beta} \int\frac{d^2{\bm q}}{(2\pi)^2} u_{{\bm q}'}^2 \Gamma^{({\rm dis})}_\alpha({\bm q},{\bm q}',\omega)
\nonumber\\
&\times&
\Gamma^{({\rm dis})}_\beta({\bm q},{\bm q}',\omega)\Im m[\chi_{j_\alpha j_\beta}^{(0)} ({\bm q}+{\bm q}',\omega)]~,\end{aligned}$$ where the average over disorder $\langle n^{({\rm i})}_{\bm q} n^{({\rm i})}_{{\bm q}'}\rangle_{\rm dis}/{\cal A} = n_{\rm imp} \delta_{{\bm q}+{\bm q}', 0}$ has been taken. Here $n_{\rm imp}$ is the average impurity density and ${\cal A}$ is the sample area. Eq. (\[eq:transporttime\_final\]) overestimates the effect of disorder on the electronic system. Indeed, when e-e interactions are taken into account, the bare e-imp potential is weakened. We take into account [*screening*]{} by replacing in Eq. (\[eq:transporttime\_final\]) the longitudinal and transverse components of the non-interacting current-current response function $\chi_{j_\alpha j_\beta}^{(0)} ({\bm q}+{\bm q}',\omega)$ with the RPA current-current response $\chi_{j_\alpha j_\beta}^{({\rm RPA})} ({\bm q}+{\bm q}',\omega)$. We remind the reader that the transverse RPA current-current response function coincides with the non-interacting one [@Giuliani_and_Vignale].
[*Numerical results.—*]{}We now turn to present our main numerical results for $\tau_{\rm p}(q)\equiv \tau(q,\omega_{\rm p}(q))$ as calculated from Eq. (\[eq:transporttime\_final\]) with $\chi_{j_\alpha j_\beta}^{(0)} ({\bm q}+{\bm q}',\omega) \to \chi_{j_\alpha j_\beta}^{({\rm RPA})} ({\bm q}+{\bm q}',\omega)$ and for ${\cal R}_{\rm p}(q) \equiv {\cal R}(q,\omega_{\rm p}(q))$—see Eq. (\[eq:sigma\_ratio\]). For the sake of definiteness, we choose $u_{\bm q}$ to be the long-range potential generated by impurities of unitary charge located on the graphene sheet, [*i.e.*]{} $u_{\bm q} = 2\pi e^2/(\epsilon q)$. The impurity density $n_{\rm imp}$ is obtained by making sure that the calculated transport time $\tau_{\rm tr}$ equals the experimental value given in Ref. , [*i.e.*]{} $\tau_{\rm exp} = 260~{\rm fs}$, corresponding to a mobility $\mu \sim 8.000~{\rm cm}^2/({\rm V} {\rm s})$, at a carrier density $n=8.0 \times 10^{12}~{\rm cm}^{-2}$. We remind the reader that in this experiment $\epsilon = 2.52$. This constraint is satisfied [@exponential] with an impurity concentration $n_{\rm imp} \simeq 5.8 \times 10^{11}~{\rm cm}^{-2}$.
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![(Color online) The disorder-induced DP lifetime $\tau_{\rm p}(q_{\rm p})$ and the transport time $\tau_{\rm tr}$ are plotted as functions of the electron density $n$ and for a fixed photon energy $\hbar\omega_{\rm ph}$ and impurity concentration $n_{\rm imp}$. The solid line refers to $\tau_{\rm p}(q_{\rm p})$, while the dashed line refers to $\tau_{\rm tr}$. Different panels refer to different values of the photon energy and impurity concentration: in panel a) we have set $\hbar \omega_{\rm ph} = 112~{\rm meV}$, corresponding to mid-infrared plasmons, and $n_{\rm imp}=5.8 \times 10^{11}~{\rm cm}^{-2}$; in panel b) $\hbar\omega_{\rm ph} = 11.2~{\rm meV}$, corresponding to Terahertz plasmons, and $n_{\rm imp} = 5.8 \times 10^{10}~{\rm cm}^{-2}$. Note the difference in the scales of horizontal and vertical axes between the two panels. In both panels we have set $\epsilon=2.52$, corresponding [@fei_nature_2012] to graphene on ${\rm SiO}_2$. \[fig:one\]](fig1a "fig:"){width="1.0\linewidth"}
![(Color online) The disorder-induced DP lifetime $\tau_{\rm p}(q_{\rm p})$ and the transport time $\tau_{\rm tr}$ are plotted as functions of the electron density $n$ and for a fixed photon energy $\hbar\omega_{\rm ph}$ and impurity concentration $n_{\rm imp}$. The solid line refers to $\tau_{\rm p}(q_{\rm p})$, while the dashed line refers to $\tau_{\rm tr}$. Different panels refer to different values of the photon energy and impurity concentration: in panel a) we have set $\hbar \omega_{\rm ph} = 112~{\rm meV}$, corresponding to mid-infrared plasmons, and $n_{\rm imp}=5.8 \times 10^{11}~{\rm cm}^{-2}$; in panel b) $\hbar\omega_{\rm ph} = 11.2~{\rm meV}$, corresponding to Terahertz plasmons, and $n_{\rm imp} = 5.8 \times 10^{10}~{\rm cm}^{-2}$. Note the difference in the scales of horizontal and vertical axes between the two panels. In both panels we have set $\epsilon=2.52$, corresponding [@fei_nature_2012] to graphene on ${\rm SiO}_2$. \[fig:one\]](fig1b "fig:"){width="1.0\linewidth"}
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----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![(Color online) The quantities ${\cal R}_{\rm p}(q_{\rm p})$ and ${\cal R}_{\rm tr}$ are plotted as functions of $n$ and for a fixed $\hbar\omega_{\rm ph}$ and $n_{\rm imp}$. In panel a) the solid (short-dashed) line refers to ${\cal R}_{\rm p}(q_{\rm p})$ (${\cal R}_{\rm tr}$) calculated for $\hbar\omega_{\rm ph} = 112~{\rm meV}$ and $n_{\rm imp} = 5.8\times 10^{11}~{\rm cm}^{-2}$. For the sake of comparison, we show the density-independent experimental value ${\cal R}_{\rm exp} = 0.08$ (dotted line) and ${\cal R}_{\rm p}(q_{\rm p})$ (long-dashed line) calculated for a larger impurity concentration, $n_{\rm imp} = 1.0 \times 10^{12}~{\rm cm}^{-2}$, which matches ${\cal R}_{\rm exp}$ at $n=8.0 \times 10^{12}~{\rm cm}^{-2}$. Panel b) illustrates the ratio ${\cal R}_{\rm p}(q_{\rm p})/{\cal R}_{\rm tr}$ as a function of $n$ for $\hbar\omega_{\rm ph} = 112~{\rm meV}$. Note that this ratio is independent of the impurity concentration and always larger than unity in the explored range of densities.\[fig:two\]](fig2a "fig:"){width="1.0\linewidth"}
![(Color online) The quantities ${\cal R}_{\rm p}(q_{\rm p})$ and ${\cal R}_{\rm tr}$ are plotted as functions of $n$ and for a fixed $\hbar\omega_{\rm ph}$ and $n_{\rm imp}$. In panel a) the solid (short-dashed) line refers to ${\cal R}_{\rm p}(q_{\rm p})$ (${\cal R}_{\rm tr}$) calculated for $\hbar\omega_{\rm ph} = 112~{\rm meV}$ and $n_{\rm imp} = 5.8\times 10^{11}~{\rm cm}^{-2}$. For the sake of comparison, we show the density-independent experimental value ${\cal R}_{\rm exp} = 0.08$ (dotted line) and ${\cal R}_{\rm p}(q_{\rm p})$ (long-dashed line) calculated for a larger impurity concentration, $n_{\rm imp} = 1.0 \times 10^{12}~{\rm cm}^{-2}$, which matches ${\cal R}_{\rm exp}$ at $n=8.0 \times 10^{12}~{\rm cm}^{-2}$. Panel b) illustrates the ratio ${\cal R}_{\rm p}(q_{\rm p})/{\cal R}_{\rm tr}$ as a function of $n$ for $\hbar\omega_{\rm ph} = 112~{\rm meV}$. Note that this ratio is independent of the impurity concentration and always larger than unity in the explored range of densities.\[fig:two\]](fig2b "fig:"){width="1.0\linewidth"}
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
In Fig. \[fig:one\]a) we plot the DP lifetime $\tau_{\rm p} (q_{\rm p})$, where the plasmon wavevector is given by [@fei_nature_2012; @principi_arxiv_2013] $q_{\rm p}/k_{\rm F} = [\hbar v_{\rm F} \epsilon/(2 e^2)](\hbar \omega_{\rm ph}/\varepsilon_{\rm F})^2$. The photon energy $\hbar \omega_{\rm ph}$ is kept fixed, [*i.e.*]{} $\hbar \omega_{\rm ph}=112~{\rm meV}$, as in Ref. . As density decreases $q_{\rm p}/k_{\rm F}$ increases: filled circles in Fig. \[fig:one\] refer to the value of doping such that $(q_{\rm p}/k_{\rm F})_{\rm max} = 0.2$. From this figure we clearly see that the disorder-induced DP lifetime is of the order of $100~{\rm fs}$ for mid-infrared plasmons. As a comparison, we plot also the calculated transport time $\tau_{\rm tr}$, which is clearly larger than $\tau_{\rm p}$. Identifying $\tau_{\rm p}(q_{\rm p})$ with $\tau_{\rm tr}$ leads to an error of factor $\sim 2-3$ in the experimentally relevant range of densities. In Fig. \[fig:one\]b) we show our predictions for the DP lifetime for a photon energy of $\hbar \omega_{\rm ph} = 11.2~{\rm meV}$ (Terahertz plasmons). In this case we have fixed $n_{\rm imp} = 5.8\times 10^{10}~{\rm cm}^{-2}$.
In Fig. \[fig:two\] we plot the quantity ${\cal R}_{\rm p}(q)$ evaluated at $q = q_{\rm p}$ (at fixed $\omega = \omega_{\rm ph}$). In the same figure we have also plotted the dc value defined as ${\cal R}_{\rm tr} \equiv \lim_{\omega\to 0} {\cal R}(q=0,\omega)$. This figure refers to a photon frequency in the mid-infrared. From panel a), we note that, in the range of densities explored in Refs. , the dependence of ${\cal R}_{\rm p}$ on doping is weak. This is in perfect agreement with the findings of Refs. . Note that even at carrier densities as large as $10^{13}~{\rm cm}^{-2}$, ${\cal R}_{\rm p}$ is a factor of two larger than ${\cal R}_{\rm tr}$—see panel b). For $n=8.0 \times 10^{12}~{\rm cm}^{-2}$ we find ${\cal R}_{\rm tr} \simeq 0.02$ and ${\cal R}_{\rm p} \simeq 0.05$. The experimentally measured damping rate is represented by a density-independent value [@fei_nature_2012], ${\cal R}_{\rm exp}=0.08$, which is matched (at $n=8.0 \times 10^{12}~{\rm cm}^{-2}$) by ${\cal R}_{\rm p}$ calculated for an impurity concentration of $1.0 \times 10^{12}~{\rm cm}^{-2}$. Note also that, since the memory kernel $1/\tau(q,\omega)$ scales linearly with impurity concentration, the ratio ${\cal R}_{\rm p}/{\cal R}_{\rm tr}$ is independent of $n_{\rm imp}$.
[*Summary and discussion.—*]{}In summary, we have presented a theory of disorder-induced Dirac plasmon losses. We have carried out numerical calculations for a specific disorder model, [*i.e.*]{} charged impurities located on graphene. We have shown that the plasmon lifetime is substantially shorter than the dc Drude transport time, even at high carrier densities. The calculated damping rate qualitatively agrees with the experimental findings of Refs. and differs by less than a factor of two with respect to the measured value. We stress that the damping rate calculated on the basis of the dc Drude transport time is a factor of four smaller than the measured value.
We stress that the remaining discrepancy between theory and experiments may stem from a number of issues. First, the disorder model we have used (charged impurities) may not be sufficient. Other important sources of extrinsic scattering, such as corrugations, resonant scatterers, and disorder at the edges, may well explain the difference. This remains to be studied. We highlight that scattering of electrons from optical phonons in the ${\rm SiO}_2$ substrate gives a damping rate [@low_private] $\simeq 0.02$, for $\hbar\omega_{\rm p} = 112~{\rm meV}$ and $n=8\times 10^{12}~{\rm cm}^{-1}$. This number, added to the disorder-induced damping rate at the same photon energy and carrier density, gives $\gamma_{\rm p} \simeq 0.07$, in very good agreement with the experimental result. Second, our numerical results heavily rely on an input parameter: the mobility of the samples employed in Ref. . This quantity has not been directly measured in Refs. but has been inferred from previous measurements on similarly prepared samples. A concentration of charged impurities equal to $n_{\rm imp} = 1.0 \times 10^{12}~{\rm cm}^{-2}$, corresponding to a mobility $\mu \sim 4.000~{\rm cm}^2/({\rm V} {\rm s})$, gives ${\cal R}_{\rm p} = 0.08$, in perfect agreement with the measured value—see Fig. \[fig:two\]a).
Our calculations strongly suggest that plasmon losses in current graphene samples [@fei_nature_2012; @chen_nature_2012] are dominated by disorder and that there is plenty of room to increase the sample purity to reach the intrinsic regime of ultra long Dirac plasmon lifetimes [@principi_arxiv_2013].
[*Acknowledgements.—*]{}We thank Frank Koppens and Tony Low for useful discussions. A.P. and G.V. were supported by the BES Grant DE-FG02-05ER46203. M.C. and M.P. acknowledge support by MIUR through the program “FIRB - Futuro in Ricerca 2010" - Project PLASMOGRAPH (Grant No. RBFR10M5BT).
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\epsilon_{\rm R} + i\epsilon_{\rm I}$, the relation between $\gamma_{\rm p}(q)$ and ${\cal R}(q,\omega)$ reads $\gamma_{\rm p}(q) = \epsilon_{\rm I}/\epsilon_{\rm R}
+ {\cal R}(q,\omega_{\rm p}(q))$. Since this effect is trivial, the focus of this work is on ${\cal R}(q,\omega_{\rm p}(q))$. In Ref. , $\epsilon_{\rm I}/\epsilon_{\rm R} \sim 0.05$. We discuss electron doping for the sake of definiteness: for weak disorder, $\gamma_{\rm p}(q)$ is a particle-hole symmetric quantity. J. Sabio, J. Nilsson, and A.H. Castro Neto, [, 075410 (2008)](http://dx.doi.org/10.1103/PhysRevB.78.075410). Introducing a finite separation $d$ between the “impurity plane" and graphene yields a form factor $e^{- q d}$ in the e-imp interaction $u_{\bm q}$. Larger values of the impurity concentration are thus needed to match the experimental transport time. For example, for a carrier density $n = 8.0\times 10^{12}~{\rm cm}^{-2}$ and a distance [@dassarma_rmp_2011] $d = 1~{\rm nm}$ we find that the experimental transport time [@fei_nature_2012] $\tau_{\rm exp} = 260~{\rm fs}$ is reproduced by increasing the impurity concentration from $n_{\rm imp} = 5.8 \times 10^{11}~{\rm cm}^{-2}$ to $n_{\rm imp} = 1.9 \times 10^{12}~{\rm cm}^{-2}$. T. Low, private communication.
|
---
abstract: 'In this paper, some new results on the distribution of the generalized singular value decomposition (GSVD) are presented.'
author:
- 'Zhuo Chen, Zhiguo Ding, [^1]'
bibliography:
- 'IEEEfull.bib'
- 'cz.bib'
title: ' [ On the Distribution of GSVD]{}'
---
Some new results on GSVD
========================
In this section, the GSVD of two Gaussian matrices is defined first. Then, the distribution of the squared generalized singular values are presented.
Definition of GSVD {#GSVDdef}
------------------
Given two matrices $\mathbf{A} \in \mathbb{C}^{m \times n}$ and $\mathbf{C} \in \mathbb{C}^{q \times n}$, whose entries are i.i.d. complex Gaussian random variables with zero mean and unit variance. Let us define $k=\text{rank}((\mathbf{A}^H, \mathbf{C}^H)^H)=\min\{m+q,n\}$, $r=\text{rank}((\mathbf{A}^H, \mathbf{C}^H)^H)-\text{rank}(\mathbf{C})
=\min\{m+q,n\}-\min\{q,n\}$, and $s=\text{rank}(\mathbf{A})+\text{rank}(\mathbf{C})-\text{rank}((\mathbf{A}^H, \mathbf{C}^H)^H)=
\min\{m,n\}+\min\{q,n\}-\min\{m+q,n\}$. Then, the GSVD of $\mathbf{A}$ and $\mathbf{C}$ can be expressed as follows [@cz2018]: $$\begin{aligned}
\label{GSVDFORM}
\mathbf{U}\mathbf{A} \mathbf{Q}=\left(\mathbf{\Sigma}_\mathbf{A}, \mathbf{O} \right) \quad \text{and} \quad \mathbf{V}\mathbf{C}\mathbf{Q}=\left(\mathbf{\Sigma}_\mathbf{C}, \mathbf{O} \right),\end{aligned}$$ where $\mathbf{\Sigma}_\mathbf{A} \in \mathbb{C}^{m \times k} $ and $\mathbf{\Sigma}_\mathbf{C} \in \mathbb{C}^{q \times k}$ are two nonnegative diagonal matrices, $\mathbf{U} \in \mathbb{C}^{m \times m}$ and $\mathbf{V} \in \mathbb{C}^{q \times q}$ are two unitary matrices, and $\mathbf{Q} \in \mathbb{C}^{n \times n}$ can be expressed as in .
Moreover, $\mathbf{\Sigma}_\mathbf{A}$ and $\mathbf{\Sigma}_\mathbf{C}$ have the following form: $$\begin{aligned}
\label{Diagform}
\mathbf{\Sigma}_\mathbf{A}= \left(\begin{array}{ccc}
\mathbf{I}_r&& \\
&\mathbf{S}_\mathbf{A}& \\
&&\mathbf{O}_\mathbf{A}
\end{array}\right)
\quad \text{and} \quad \mathbf{\Sigma}_\mathbf{C}= \left(\begin{array}{ccc}
\mathbf{O}_\mathbf{C}&& \\
&\mathbf{S}_\mathbf{C}& \\
&&\mathbf{I}_{k-r-s}
\end{array}\right),\end{aligned}$$ where $\mathbf{S}_\mathbf{A}= \diag(\alpha_1,\cdots,\alpha_s)$ and $\mathbf{S}_\mathbf{C}=\diag(\beta_1,\cdots,\beta_s)$ are two $s \times s$ nonnegative diagonal matrices, satisfying $\mathbf{S}_\mathbf{A}^2 + \mathbf{S}_\mathbf{C}^2 = \mathbf{I}_s$. Then, the squared generalized singular values can be defined as $w_i=\alpha_i^2/\beta_i^2$, $i \in \{1,\cdots,s\}$.
Distribution of the squared generalized singular values {#GSVDdistri}
-------------------------------------------------------
To characterize the distribution of the squared generalized singular value $w_i$, a relationship between $w_i$ and the eigenvalue of a common matrix model is established first as in the following theorem.
\[GSVDrelation\] Suppose that $\mathbf{A} \in \mathbb{C}^{m \times n}$ and $\mathbf{C} \in \mathbb{C}^{q \times n}$ are two Gaussian matrices whose elements are i.i.d. complex Gaussian random variables with zero mean and unit variance and their GSVD is defined as in . Without loss of generality, it is assumed that $q \geq m$. Then, the distribution of their squared generalized singular values, $w_i, i \in \{1,\cdots,s\}$, is identical to that of the nonzero eigenvalues of $\mathbf{L}$, where $$\begin{aligned}
\label{expressionL}
\mathbf{L}=\mathbf{X}^H \left(\mathbf{Y}\mathbf{Y}^H \right)^{-1}\mathbf{X}.\end{aligned}$$ $\mathbf{X}\in \mathbb{C}^{m' \times p}$ and $\mathbf{Y}\in \mathbb{C}^{m' \times n'}$ are two independent Gaussian matrices whose elements are are i.i.d. complex Gaussian random variables with zero mean and unit variance. Moreover, $m'$, $p$ and $n'$ can be expressed as follows: $$\begin{aligned}
\label{weidu}
(m',p,n')=
\left\{ \begin{array}{ll}
(n,m,q) & q \geq n \\
(q,s,n) & q<n<(q+m)
\end{array} \right..\end{aligned}$$ When $(q+m)\leq n$, $s=0$ and $\mathbf{\Sigma}_\mathbf{A}$ and $\mathbf{\Sigma}_\mathbf{C}$ are deterministic.
See Appendix A.
The distribution of the nonzero eigenvalues of $\mathbf{L}$ can be characterized as in the following corollary.
\[Corro1\] Suppose that $\mathbf{L}=\mathbf{X}^H \left(\mathbf{Y}\mathbf{Y}^H \right)^{-1}\mathbf{X}$, and $\mathbf{X}\in \mathbb{C}^{m' \times p}$ and $\mathbf{Y}\in \mathbb{C}^{m' \times n'}$ are two independent Gaussian matrices whose elements are are i.i.d. complex Gaussian random variables with zero mean and unit variance. And, it is assumed that $m' \leq n'$. Then, the joint probability density function (p.d.f.) of the nonzero eigenvalues of $\mathbf{L}$ can be characterized as follows: $$\begin{aligned}
\label{jointpdf}
f_{m',p,n'}(w_1,\cdots,w_{l})= \mathbf{M}_{m',p,n'} \frac{ \prod_{i=1}^{l} w_i^{t_1} }
{\prod_{i=1}^{l} (1+w_i)^{t_2} }
\prod_{i<j}^{l} (w_i-w_j)^2,\end{aligned}$$ where $l=\min\{p,m'\}$, $t_1=|m'-p|$, $t_2=p+n'$, and $\mathbf{M}_{m',p,n'}$ can be expressed as follows: $$\begin{aligned}
\label{Mexpress}
\mathbf{M}_{m',p,n'}=\left\{ \begin{array}{ll}
\frac{\pi^{m'(m'-1)} \widetilde{\Gamma}_{m'}(p+n') }
{m'! \widetilde{\Gamma}_{m'}(p) \widetilde{\Gamma}_{m'}(n') \widetilde{\Gamma}_{m'}(m') } & p \geq m' \\
\frac{\pi^{p(p-1)} \widetilde{\Gamma}_{p}(p+n') }
{p! \widetilde{\Gamma}_{p}(m') \widetilde{\Gamma}_{p}(p+n'-m') \widetilde{\Gamma}_{p}(p) } & p < m'
\end{array} \right.,\end{aligned}$$ where $m'!=1\times 2 \times \cdots m'$ and $\widetilde{\Gamma}_{m'}(p)$ is the complex multivariate gamma function[@james1964distributions].
Following steps similar to those in [@2cz2018 Appendix A], the distribution of the nonzero eigenvalues of $\mathbf{L}$ can be obtained.
Moreover, the marginal p.d.f. of $w_i$, $i \in \{1,\cdots,l\}$, can be characterized as in the following lemma.
\[lemma1\] The marginal p.d.f. derived from can be expressed as follows: $$\begin{aligned}
\label{pdayumpianmargin}
g_{m',p,n'}(w_l)=
\mathbf{M}_{m',p,n'} g'_{l,t_1,t_2}(w_l) \quad \text{or} \quad
\frac{1}{w_l^2} \mathbf{M}_{m',p,n'} g'_{l,t'_1,t_2}(1/w_l),\end{aligned}$$ where $t'_1=n'-m'$ and $g'_{l,t_1,t_2}(w_l)$ can be expressed as follows: $$\begin{aligned}
\label{gpianeq}
g'_{l,t_1,t_2}(w_l)&=&\sum_{\sigma_1, \sigma_2 \in S_l} \mathbf{sign} (\sigma_1) \mathbf{sign} (\sigma_2)
\frac{w_l^{t_1+2l-\sigma_1(l)-\sigma_2(l)}}{(1+w_l)^{t_2}}
\\ \nonumber && \times
\prod_{i=1}^{l-1}B(t_1+2l-\sigma_1(i)-\sigma_2(i)+1,t_2-t_1-2l-1+\sigma_1(i)+\sigma_2(i)),\end{aligned}$$ where $\sigma_1=\left(\sigma_1(1), \sigma_1(2) \cdots \sigma_1(l)\right)$, $\sigma_2=\left(\sigma_2(1), \sigma_2(2) \cdots \sigma_2(l)\right) \in S_l$ are permutations of length $l$, $\mathbf{sign} (*)$, $(*)\in\{\sigma_1,\sigma_2\}$, is $1$ if the permutation is even and $-1$ if it is odd, and $B(t_1+2l-\sigma_1(i)-\sigma_2(i)+1,t_2-t_1-2l-1+\sigma_1(i)+\sigma_2(i))$ is the Beta function [@IntegralTable].
See Appendix B.
Some properties about $\mathbf{Q}$
----------------------------------
As shown in [@cz2018], the GSVD decomposition matrix $\mathbf{Q}$ is often used to construct the precoding matrix at the transmitting end. In this section, some properties about $\mathbf{Q}$ are discussed.
First, define $\mathbf{B}=(\mathbf{A}^H, \mathbf{C}^H)^H$. As shown in [@paige1981 eq.(2.2)], $\mathbf{Q}$ can be expressed as $$\begin{aligned}
\label{expressQ}
\mathbf{Q}=\mathbf{Q}'\left(\begin{array}{cc}
\left(\mathbf{W}^H\mathbf{R} \right)^{-1}&\\
& \mathbf{O}_{n-k}
\end{array}\right),\end{aligned}$$ where $\mathbf{Q}'\in \mathbb{C}^{n \times n}$ and $\mathbf{W} \in \mathbb{C}^{k \times k}$ are two unitary matrices, $\mathbf{R} \in \mathbb{C}^{k \times k}$ is a nonnegative diagonal matrix and has the same singular values as the nonzero singular values of $\mathbf{B}$. Thus, the power of $\mathbf{Q}$ can be expressed as $$\begin{aligned}
\label{powerQ}
\text{trace} \{ \mathbf{Q}\mathbf{Q}^H \}&=&
\text{trace} \left\{ \mathbf{Q}'\left(\begin{array}{cc}
\mathbf{R}^{-1}\mathbf{W}&\\
& \mathbf{O}_{n-k}
\end{array}\right)
\left(\begin{array}{cc}
\mathbf{W}^H\mathbf{R}^{-H} &\\
& \mathbf{O}_{n-k}
\end{array}\right) \mathbf{Q}'^H \right\}
\\ \nonumber &=&
\text{trace} \left\{ \mathbf{Q}'^H \mathbf{Q}'\left(\begin{array}{cc}
\mathbf{R}^{-1}\mathbf{R}^{-H}&\\
& \mathbf{O}_{n-k}
\end{array}\right)
\right\} \\ \nonumber &=&
\text{trace} \left\{ \left(\mathbf{R}^{H}\mathbf{R}\right)^{-1}
\right\}
\\ \nonumber &=&
\sum_i^k \lambda_{B,i}^{-1},\end{aligned}$$ where $\lambda_{B,i}$, $i \in \{1,\cdots,k\}$, are the nonzero eigenvalues of $\mathbf{B}\mathbf{B}^H$.
Note that $\mathbf{A}$ and $\mathbf{C}$ are two independent Gaussian matrices. Then, it is easy to know that $\mathbf{B}\mathbf{B}^H$ is a Wishart matrix. Finally, directly from [@tulino2004foundations Lemma 2.10], the following corollary can be derived.
\[Corro2\] Suppose that $\mathbf{A} \in \mathbb{C}^{m \times n}$ and $\mathbf{C} \in \mathbb{C}^{q \times n}$ are two Gaussian matrices whose elements are i.i.d. complex Gaussian random variables with zero mean and unit variance and their GSVD is defined as in . The average power of the GSVD decomposition matrix $\mathbf{Q}$ can be expressed as follows: $$\begin{aligned}
\label{AverpowerQ}
\mathcal{E} \{ \text{trace} \{ \mathbf{Q}\mathbf{Q}^H \} \}= \frac{\min \{m+q,n \}}{ |m+q-n|}.\end{aligned}$$
Appendix A: Proof of Theorem \[GSVDrelation\] {#appendix-a-proof-of-theorem-gsvdrelation .unnumbered}
=============================================
1. The case when $q \geq n$ {#the-case-when-q-geq-n .unnumbered}
---------------------------
When $q \geq n$, $k=\text{rank}((\mathbf{A}^H, \mathbf{C}^H)^H)=\min\{m+q,n\}=n$, $r=\text{rank}((\mathbf{A}^H, \mathbf{C}^H)^H)-\text{rank}(\mathbf{C})
=\min\{m+q,n\}-\min\{q,n\}=0$, and $s=\text{rank}(\mathbf{A})+\text{rank}(\mathbf{C})-\text{rank}((\mathbf{A}^H, \mathbf{C}^H)^H)=
\min\{m,n\}+\min\{q,n\}-\min\{m+q,n\}=\min\{m,n\}$. Thus, from , the GSVD of $\mathbf{A}$ and $\mathbf{C}$ can be further expressed as follows: $$\begin{aligned}
\label{GSVDFORMq>n}
\mathbf{U}\mathbf{A} \mathbf{Q}=\mathbf{\Sigma}_\mathbf{A} \quad \text{and} \quad \mathbf{V}\mathbf{C}\mathbf{Q}=\mathbf{\Sigma}_\mathbf{C} .\end{aligned}$$ Moreover, when $k=n$, as shown in , $\mathbf{Q}$ is nonsingular and it can be shown that $$\begin{aligned}
\mathbf{A}^H\mathbf{A}= \mathbf{Q}^{-H} \mathbf{\Sigma}_\mathbf{A}^H \mathbf{\Sigma}_\mathbf{A} \mathbf{Q}^{-1}
\quad \text{and} \quad
\mathbf{C}^H\mathbf{C}= \mathbf{Q}^{-H} \mathbf{\Sigma}_\mathbf{C}^H \mathbf{\Sigma}_\mathbf{C} \mathbf{Q}^{-1}.\end{aligned}$$ Furthermore, from , it can be shown that $$\begin{aligned}
\mathbf{\Sigma}_\mathbf{A}^H \mathbf{\Sigma}_\mathbf{A}=
\left(\begin{array}{cc}
\mathbf{S}_\mathbf{A}^H\mathbf{S}_\mathbf{A}&\\
& \mathbf{O}_{n-s}
\end{array}\right)
\quad \text{and} \quad
\mathbf{\Sigma}_\mathbf{C}^H \mathbf{\Sigma}_\mathbf{C}=
\left(\begin{array}{cc}
\mathbf{S}_\mathbf{C}^H\mathbf{S}_\mathbf{C}&\\
& \mathbf{I}_{n-s}
\end{array}\right).\end{aligned}$$ Thus, $\left(\mathbf{C}^H\mathbf{C}\right)^{-1}\mathbf{A}^H\mathbf{A}$ can be expressed as $$\begin{aligned}
\left(\mathbf{C}^H\mathbf{C}\right)^{-1}\mathbf{A}^H\mathbf{A}= \mathbf{Q} \left(\begin{array}{cc}
\mathbf{S}_\mathbf{A}^H\mathbf{S}_\mathbf{A}\left[\mathbf{S}_\mathbf{C}^H\mathbf{S}_\mathbf{C}\right]^{-1}&\\
& \mathbf{O}_{n-s}
\end{array}\right) \mathbf{Q}^{-1}.\end{aligned}$$ Recall that $\mathbf{S}_\mathbf{A}= \diag(\alpha_1,\cdots,\alpha_s)$ and $\mathbf{S}_\mathbf{C}=\diag(\beta_1,\cdots,\beta_s)$. Then, it can be shown that $\mathbf{S}_\mathbf{A}^H\mathbf{S}_\mathbf{A}\left[\mathbf{S}_\mathbf{C}^H\mathbf{S}_\mathbf{C}\right]^{-1}
=\diag(\alpha_1^2/\beta_1^2,\cdots,\alpha_s^2/\beta_s^2)
=\diag(w_1,\cdots,w_s)$. Finally, it is easy to see that the distribution of $w_i, i \in \{1,\cdots,s\}$, is identical to that of the nonzero eigenvalues of $\mathbf{L}$, where $$\begin{aligned}
\mathbf{L}&=& \mathbf{A} \left(\mathbf{C}^H\mathbf{C} \right)^{-1}\mathbf{A}^H
\\ \nonumber
&=&\mathbf{X}^H \left(\mathbf{Y}\mathbf{Y}^H \right)^{-1}\mathbf{X}.\end{aligned}$$ $\mathbf{X}\in \mathbb{C}^{m' \times p}=\mathbf{A}^H$ and $\mathbf{Y}\in \mathbb{C}^{m' \times n'}=\mathbf{C}^H$ are two independent Gaussian matrices whose elements are are i.i.d. complex Gaussian random variables with zero mean and unit variance. Moreover, $(m',p,n')=(n,m,q)$.
2. The case when $ q<n<(q+m)$ {#the-case-when-qnqm .unnumbered}
-----------------------------
When $q<n<(q+m)$, $k=\text{rank}((\mathbf{A}^H, \mathbf{C}^H)^H)=\min\{m+q,n\}=n$, $r=\text{rank}((\mathbf{A}^H, \mathbf{C}^H)^H)-\text{rank}(\mathbf{C})
=\min\{m+q,n\}-\min\{q,n\}=n-q$, and $s=\text{rank}(\mathbf{A})+\text{rank}(\mathbf{C})-\text{rank}((\mathbf{A}^H, \mathbf{C}^H)^H)=
\min\{m,n\}+\min\{q,n\}-\min\{m+q,n\}=m+q-n$. As shown in , $w_1, \cdots, w_s$ are the squared generalized singular values of $\mathbf{A}$ and $\mathbf{C}$, and $\alpha_i^2=\frac{w_i}{1+w_i}, i \in \{1,\cdots,s\}$.
On the other hand, define $\mathbf{B}=(\mathbf{A}^H, \mathbf{C}^H)^H$ and the SVD of $\mathbf{B}$ as $\mathbf{B}=\mathbf{P}\mathbf{\Sigma}_\mathbf{B}\mathbf{R}$. Note that $\mathbf{P}\in \mathbb{C}^{(m+q) \times (m+q)}$ is a Haar matrix. Divide $\mathbf{P}$ into the following four blocks: $$\begin{aligned}
\label{Pblocks}
\mathbf{P}=\left(\begin{array}{cc}
\mathbf{P}_{11} & \mathbf{P}_{12}\\ \mathbf{P}_{21} & \mathbf{P}_{22}
\end{array}\right),\end{aligned}$$ where $\mathbf{P}_{11} \in \mathbb{C}^{m \times n} $ and $\mathbf{P}_{22} \in \mathbb{C}^{q \times (m+q-n)} $. From [@paige1981 eq.(2.7)], it is easy to see that $\alpha_1^2, \cdots, \alpha_s^2$ equal the non-one eigenvalues of $\mathbf{P}_{11}\mathbf{P}_{11}^H$. Moreover, from the fact that $\mathbf{P}$ is a Haar matrix, it can be shown that $$\begin{aligned}
\label{Punitary}
\mathbf{P}\mathbf{P}^H=\mathbf{P}^H\mathbf{P}=\mathbf{I}_{m+q}.\end{aligned}$$ Thus, the following equations can be derived: $$\begin{aligned}
\label{eqsP1}
\mathbf{P}_{11}\mathbf{P}_{11}^H+\mathbf{P}_{12}\mathbf{P}_{12}^H=\mathbf{I}_{m}
\quad \text{and} \quad
\mathbf{P}_{22}^H\mathbf{P}_{22}+\mathbf{P}_{12}^H\mathbf{P}_{12}=\mathbf{I}_{m+q-n}.\end{aligned}$$ Note that $\mathbf{P}_{12}\mathbf{P}_{12}^H$ and $\mathbf{P}_{12}^H\mathbf{P}_{12}$ have the same non-zero eigenvalues. Thus, $\mathbf{P}_{11}\mathbf{P}_{11}^H$ and $\mathbf{P}_{22}^H\mathbf{P}_{22}$ have the same non-one eigenvalues. Therefore, $\alpha_1^2, \cdots, \alpha_s^2$ equal the non-one eigenvalues of $\mathbf{P}_{22}^H\mathbf{P}_{22}$. Define $\mathbf{P}'$ as $$\begin{aligned}
\label{Ppian}
\mathbf{P}'=\left(\begin{array}{cc}
\mathbf{P}_{22}^H & \mathbf{P}_{12}^H\\ \mathbf{P}_{21}^H & \mathbf{P}_{11}^H
\end{array}\right).\end{aligned}$$ It is easy to see that $\mathbf{P}'\mathbf{P}'^H=\mathbf{P}'^H\mathbf{P}'=\mathbf{I}_{m+q}$ and $\mathbf{P}'$ is also a Haar matrix.
Then, from the above discussions, it can be concluded that the distribution of $\alpha_1, \cdots, \alpha_s$ is identical to the distribution of the non-one singular valus of the $m \times n$ or $(m+q-n) \times q$ truncated sub-matrix of a $(m+q) \times (m+q)$ Haar matrix. Thus, define $\mathbf{A}' \in \mathbb{C}^{(m+q-n) \times q}$ and $\mathbf{C}' \in \mathbb{C}^{n \times q}$, whose entries are i.i.d. complex Gaussian random variables with zero mean and unit variance. The distribution of the squared squared generalized singular values of $\mathbf{A}$ and $\mathbf{C}$, is identical to that of the squared squared generalized singular values of $\mathbf{A}'$ and $\mathbf{C}'$. Since $n>q$, from Appendix A-1, it can be known that the distribution of the squared squared generalized singular values of $\mathbf{A}'$ and $\mathbf{C}'$, is identical to that of the eigenvalues of $\mathbf{L}
=\mathbf{X}^H \left(\mathbf{Y}\mathbf{Y}^H \right)^{-1}\mathbf{X}$, where $\mathbf{X}\in \mathbb{C}^{q \times (m+q-n)}$ and $\mathbf{Y}\in \mathbb{C}^{q \times n}$ are two independent Gaussian matrices whose elements are are i.i.d. complex Gaussian random variables with zero mean and unit variance.
This completes the proof of the theorem. $\blacksquare$
Appendix B: Proof of Lemma \[lemma1\] {#appendix-b-proof-of-lemma-lemma1 .unnumbered}
=====================================
First, the marginal p.d.f. derived from can be expressed as follows: $$\begin{aligned}
\label{marging1}
g_{m',p,n'}(w_l)&=&\int_0^{\infty}\cdots\int_0^{\infty}
\mathbf{M}_{m',p,n'} \frac{ \prod_{i=1}^{l} w_i^{t_1} }
{\prod_{i=1}^{l} (1+w_i)^{t_2} }
\prod_{i<j}^{l} (w_i-w_j)^2 dw_1\cdots dw_{l-1}
\\ \nonumber
&=& \mathbf{M}_{m',p,n'} \int_0^{\infty}\cdots\int_0^{\infty}
f'_{l,t_1,t_2}(w_1,\cdots,w_{l}) dw_1\cdots dw_{l-1}
\\ \nonumber
&=& \mathbf{M}_{m',p,n'} g'_{l,t_1,t_2}(w_l),\end{aligned}$$ where $f'_{l,t_1,t_2}(w_1,\cdots,w_{l}) dw_1\cdots dw_{l-1}
=\frac{ \prod_{i=1}^{l} w_i^{t_1} }
{\prod_{i=1}^{l} (1+w_i)^{t_2} }
\prod_{i<j}^{l} (w_i-w_j)^2$, and $$\begin{aligned}
\label{marging12}
g'_{l,t_1,t_2}(w_l)&=&\int_0^{\infty}\cdots\int_0^{\infty}
f'_{l,t_1,t_2}(w_1,\cdots,w_{l}) dw_1\cdots dw_{l-1}
\\ \nonumber
&=&\int_0^{\infty}\cdots\int_0^{\infty}
\frac{ \prod_{i=1}^{l} w_i^{t_1} }
{\prod_{i=1}^{l} (1+w_i)^{t_2} }
\prod_{i<j}^{l} (w_i-w_j)^2 dw_1\cdots dw_{l-1}.\end{aligned}$$ Note that $\prod_{i<j}^{l} (w_i-w_j)^2$ can be expressed as $$\begin{aligned}
\label{W1}
\prod_{i<j}^{l} (w_i-w_j)^2&=&|\mathbf{W}|^2
\\ \nonumber &=&\left| \begin{array}{cccc}
w_1^{l-1} & w_1^{l-2} & \ldots & 1\\
w_2^{l-1} & \ldots & \ldots & 1\\
\vdots & \ddots & \vdots& \vdots\\
w_{l}^{l-1} & \ldots & \ldots & 1
\end{array} \right|^2.\end{aligned}$$ Thus, $\prod_{i<j}^{l} (w_i-w_j)^2$ can be further expressed as $$\begin{aligned}
\label{W2}
\prod_{i<j}^{l} (w_i-w_j)^2=|\mathbf{W}|^2
=\sum_{\sigma_1, \sigma_2 \in S_l} \mathbf{sign} (\sigma_1) \mathbf{sign} (\sigma_2)
\prod_{i=1}^{l} w_i^{2l-\sigma_1(i)-\sigma_2(i)}.\end{aligned}$$ Therefore, $g'_{l,t_1,t_2}(w_l)$ can be expressed as $$\begin{aligned}
\label{marging133}
g'_{l,t_1,t_2}(w_l)&=&
\sum_{\sigma_1, \sigma_2 \in S_l} \mathbf{sign} (\sigma_1) \mathbf{sign} (\sigma_2)
\int_0^{\infty}\cdots\int_0^{\infty}
\prod_{i=1}^{l} \frac{ w_i^{t_1+2l-\sigma_1(i)-\sigma_2(i)} }
{ (1+w_i)^{t_2} }
dw_1\cdots dw_{l-1}.
\\ \nonumber &=&
\sum_{\sigma_1, \sigma_2 \in S_l} \mathbf{sign} (\sigma_1) \mathbf{sign} (\sigma_2)
\frac{ w_l^{t_1+2l-\sigma_1(l)-\sigma_2(l)} }
{ (1+w_l)^{t_2} }
\prod_{i=1}^{l-1}\int_0^{\infty} \frac{ w_i^{t_1+2l-\sigma_1(i)-\sigma_2(i)} }
{ (1+w_i)^{t_2} }
dw_i.\end{aligned}$$ Moreover, from Eq.(3.194.3) [@IntegralTable], it can be shown that $\int_0^{\infty} \frac{ w_i^{t_1+2l-\sigma_1(i)-\sigma_2(i)} }
{ (1+w_i)^{t_2} }
dw_i=B(t_1+2l-\sigma_1(i)-\sigma_2(i)+1,t_2-t_1-2l-1+\sigma_1(i)+\sigma_2(i)).$ Then, $g'_{l,t_1,t_2}(w_l)$ can be expressed as $$\begin{aligned}
g'_{l,t_1,t_2}(w_l)&=&\sum_{\sigma_1, \sigma_2 \in S_l} \mathbf{sign} (\sigma_1) \mathbf{sign} (\sigma_2)
\frac{w_l^{t_1+2l-\sigma_1(l)-\sigma_2(l)}}{(1+w_l)^{t_2}}
\\ \nonumber && \times
\prod_{i=1}^{l-1}B(t_1+2l-\sigma_1(i)-\sigma_2(i)+1,t_2-t_1-2l-1+\sigma_1(i)+\sigma_2(i)).\end{aligned}$$
On the other hand, as shown in , $g'_{l,t_1,t_2}(1/w_l)$ can be be expressed as $$\begin{aligned}
\label{margingfan12}
g'_{l,t_1,t_2}(1/w_l)
&=&\frac{ w_l^{t_2-t_1} }
{ (1+w_l)^{t_2} }
\int_0^{\infty}\cdots\int_0^{\infty}
\frac{ \prod_{i-1}^{l-1} w_i^{t_1} }
{\prod_{i=1}^{l-1} (1+w_i)^{t_2} }
\prod_{i=1}^{l-1} \left(w_i-\frac{1}{w_l}\right)^{2}
\\ \nonumber && \times
\prod_{i<j}^{l-1} (w_i-w_j)^2 dw_1\cdots dw_{l-1}.\end{aligned}$$ Moreover, define $w'_i=\frac{1}{w_i}$, $i \in \{1,\cdots,l-1\}$. Then, $g'_{l,t_1,t_2}(1/w_l)$ can be be further expressed as $$\begin{aligned}
\label{margingfan1233}
g'_{l,t_1,t_2}(1/w_l)
&=&\frac{ w_l^{t_2-t_1} }
{ (1+w_l)^{t_2} }
\int_0^{\infty}\cdots\int_0^{\infty}
w_l^{-2(l-1)}
\frac{ \prod_{i-1}^{l-1} w_i'^{t_2-t_1-2l} }{\prod_{i=1}^{l-1} (1+w'_i)^{t_2} }
\prod_{i=1}^{l-1} \left(w_i'-w_l\right)^{2}
\\ \nonumber && \times
\prod_{i<j}^{l-1} (w_i'-w_j')^2 dw_1'\cdots dw_{l-1}'
\\ \nonumber &=&
w_l^2 \frac{ w_l^{t_2-t_1-2l} }
{ (1+w_l)^{t_2} }
\int_0^{\infty}\cdots\int_0^{\infty}
\frac{ \prod_{i-1}^{l-1} w_i'^{t_2-t_1-2l} }{\prod_{i=1}^{l-1} (1+w'_i)^{t_2} }
\prod_{i=1}^{l-1} \left(w_i'-w_l\right)^{2}
\\ \nonumber && \times
\prod_{i<j}^{l-1} (w_i'-w_j')^2 dw_1'\cdots dw_{l-1}'
\\ \nonumber &=&
w_l^2 g'_{l,t_1',t_2}(w_l),\end{aligned}$$ where $t_1'=t_2-t_1-2l=n'-m'$. Thus, it can be known that $g_{m',p,n'}(w_l)
=\mathbf{M}_{m',p,n'} g'_{l,t_1,t_2}(w_l)
=\mathbf{M}_{m',p,n'} (1/w_l)^2 g'_{l,t_1',t_2}(1/w_l)$.
This completes the proof of the lemma. $\blacksquare$
[^1]: Zhuo Chen is with Key Lab of Wireless-Optical Commun., Chinese Acad. of Sciences, Sch. Info Science & Tech., Univ. Science & Tech. China, Hefei, Anhui, 230027, P.R.China (E-mail: [email protected], [email protected]).
Zhiguo Ding is with the School of Electrical and Electronic Engineering, the University of Manchester, Manchester, UK (E-mail: [email protected]).
|
---
abstract: 'The three–dimensional incompressible viscous Boussinesq equations, under the assumption of hydrostatic balance, govern the large scale dynamics of atmospheric and oceanic motion, and are commonly called the primitive equations. To overcome the turbulence mixing a partial vertical diffusion is usually added to the temperature advection (or density stratification) equation. In this paper we prove the global regularity of strong solutions to this model in a three-dimensional infinite horizontal channel, subject to periodic boundary conditions in the horizontal directions, and with no-penetration and stress-free boundary conditions on the solid, top and bottom, boundaries. Specifically, we show that short time strong solutions to the above problem exist globally in time, and that they depend continuously on the initial data.'
address:
- |
Department of Mathematics\
Florida International University\
University Park\
Miami, FL 33199, USA.
- |
Department of Mathematics\
and Department of Mechanical and Aerospace Engineering\
University of California\
Irvine, CA 92697-3875, USA. [**Also:**]{} Department of Computer Science and Applied Mathematics\
Weizmann Institute of Science\
Rehovot 76100, Israel.
author:
- Chongsheng Cao
- 'Edriss S. Titi'
date: 'October 25, 2010'
title: 'Global Well–posedness of the $3D$ Primitive Equations With Partial Vertical Turbulence Mixing Heat Diffusion'
---
0.125in
[^1]
[**MSC Subject Classifications:**]{} 35Q35, 65M70, 86-08,86A10.
[**Keywords:**]{} Primitive equations, Boussinesq equations, Navier–Stokes equations, turbulence mixing model, global regularity.
Introduction {#S-1}
============
The partial differential equation model that describes convective flow in ocean dynamics is known to be the Boussinesq equations, which are the Navier–Stokes equations (NSE) of incompressible flows with rotation coupled to the heat (or density stratification) and salinity transport equations. The questions of the global well–posedness of the $3D$ Navier–Stokes equations are considered to be among the most challenging mathematical problems. In the context of the atmosphere and the ocean circulation dynamics geophysicists take advantage of the shallowness of the oceans and the atmosphere to simplify the Boussinesq equations by modeling the vertical motion with the hydrostatic balance. This leads to the well-known primitive equations for ocean and atmosphere dynamics (see, e.g., [@LTW92], [@LTW92A], [@PJ87], [@Richardson], [@SALMON], [@TZ04], [@VG06] and references therein). A vertical heat diffusivity is usually added as a leading order approximation to the effect of micro-scale turbulence mixing (cf., e.g., [@GA84], [@GC93], [@LTW92], [@Richardson]). As a result one arrives to the following dimensionless $3D$ variant of the primitive equations (Boussinesq equations): $$\begin{aligned}
&&\hskip-.8in \frac{\pp v}{\pp t} + (v\cdot {\nabla}_{H}) v + w
\frac{\pp v}{\pp z} + f_0 \vec{k} \times v +
{\nabla}_{H} p + L_1v = 0 \label{EQ-1} \\
&&\hskip-.8in
\pp_z p + T =0, \label{EQ-2} \\
&&\hskip-.8in
{\nabla}_{H} \cdot v +\pp_z w =0, \label{EQ-3} \\
&&\hskip-.8in \frac{\pp T}{\pp t} + v \cdot {\nabla}_{H} T + w
\frac{\pp T}{\pp z} + L_2 T = Q, \label{EQ-4}\end{aligned}$$ where the horizontal velocity vector field $v=(v_1, v_2)$, the velocity vector field $(v_1, v_2, w)$, the temperature $T$ and the pressure $p$ are the unknowns. $f_0$ is the Coriolis parameter, $Q$ is a given heat source. For simplicity, we drop the coupling with the salinity equation, which is an advection diffusion equation, but the results reported here will be equally valid with the addition of the coupling with the salinity. Moreover, we also assume for simplicity that $Q$ is time independent. The viscosity and the heat vertical diffusion operators $L_1$ and $L_2$, respectively, are given by $$\begin{aligned}
&&\hskip-.8in
L_1 = -\frac{1}{ R_1} {\Dd}_{H} - \frac{1}{ R_2 } \;
\frac{\pp^2}{\pp z^2}, \label{L-1} \\
&&\hskip-.8in
L_2 = - \frac{1}{ R_3} \;
\frac{\pp^2}{\pp z^2}, \label{L-2}\end{aligned}$$ where $R_1, R_2$ are positive constants representing the horizontal and vertical dimensionless Reynolds numbers, respectively, and $ R_3$ is positive constant which stands for the vertical dimensionless eddy heat diffusivity turbulence mixing coefficient (cf., e.g., [@GA84], [@GC93]). We set ${\nabla}_{H} =
(\pp_x, \pp_y)$ to be the horizontal gradient operator and ${\Dd}_{H} = \pp_x^2 +\pp_y^2$ to be the horizontal Laplacian. We denote by $$\begin{aligned}
&&\hskip-.8in
\Gg_u = \{ (x,y,0) \in \mathbb{R}^3 \}, \\
&&\hskip-.8in
\Gg_b = \{ (x,y, -h) \in \mathbb{R}^3 \},\end{aligned}$$ the upper and lower solid boundaries, respectively. We equip system (\[EQ-1\])–(\[EQ-4\]), on the physical top and bottom boundaries, with the following no–normal flow and stress free boundary conditions for the flow velocity vector field $(v, w)$, namely, $$\begin{aligned}
&&\hskip-.8in \mbox{on } \Gg_u: \frac{\pp v }{\pp z} = 0, \; w=0,
\label{B-1}\\
&&\hskip-.8in \mbox{on } \Gg_b: \frac{\pp v }{\pp z} = 0, \; w=0, \label{B-2}\end{aligned}$$ and for simplicity, we set the Dirichlet boundary condition for $T$: $$\begin{aligned}
&&\hskip-.8in \left. T\right|_{z=0} =0, \quad \left. T\right|_{z=-h} =1. \label{B-333}\end{aligned}$$ Horizontally, we set $(v, w)$ and $T$ to satisfy periodic boundary conditions: $$\begin{aligned}
&&\hskip-.8in v(x+1,y,z)=v(x,y+1,z)=v(x,y,z); \label{B-31} \\
&&\hskip-.8in w(x+1,y,z)=w(x,y+1,z)=w(x,y,z); \label{B-32} \\
&&\hskip-.8in T(x+1,y,z)=T(x,y+1,z)=T(x,y,z). \label{B-33}\end{aligned}$$ We will denote by $$M=(0, 1)^2
\qquad
\text{and} \quad \Om= M \times (-h, 0).$$ In addition, we supply the system with the initial condition: $$\begin{aligned}
&&\hskip-.8in
v(x,y,z,0) = v_0 (x,y,z), \label{INIT-1}\\
&&\hskip-.8in T(x,y,z,0) = T_0 (x,y,z). \label{INIT-2}\end{aligned}$$ System (\[EQ-1\])–(\[INIT-2\]) is a modified form of the rotational Rayleigh–Bénard convection problem taking into consideration the geophysical situation of the shallowness of oceans and atmosphere. The original three-dimensional Rayleigh–Bénard convection model (which is identical, in the absence of heat diffusion, to the Boussinesq model of stratified fluid) has been a subject to study for many years, numerically, experimentally and analytically (see, e.g., [@BH01], [@CC89], [@ES94], [@GE98], [@NSSD], [@PJ87], [@RL16], [@TZ04], and references therein). However, the question of global regularity is still open and is as challenging as the $3D$ NSE. Recently, the authors of [@CH05] and [@HL05] have shown the global well-posedness to the $2D$ Boussinesq equations without diffusivity in the heat transport equation (see also, [@DP08], for recent improvement). In [@CT05] it is observed that thanks to hydrostatic balance (\[EQ-2\]) the unknown pressure is essentially a function of the two–dimensional horizontal variables. We take advantage of this observation in [@CT05] to establish the $L^6$ estimates for the velocity vector field which allows us to prove the global well–posedness of the $3D$ primitive equations under the geophysical boundary conditions. In [@KZ07] the authors take advantage of this observation as well, and proved the global well–posedness of the $3D$ primitive equations with the Dirichlet boundary conditions by dealing directly with the “pressure" which is a function of two variables. In this paper we study system (\[EQ-1\])–(\[INIT-2\]), exploring again the hydrostatic balance which leads to an unknown “pressure" that is a function of only two variables, and use the techniques and ideas introduced in [@CT05], [@CH05] and [@HL05], to show in section \[S-3\] that strong solutions exist globally in time provided they exist for a short interval of time. Furthermore, we show in section \[S-4\] the uniqueness and continuous dependence on initial data of these strong solutions. The short time existence of strong solutions to this model will be reported in a forthcoming paper.
This paper is organized as follows. In section \[S-2\], we reformulate system (\[EQ-1\])–(\[INIT-2\]) and introduce our notations and recall some well-known inequalities. Section \[S-3\] is the main section in which we establish the required estimates for proving the global existence in time for any initial data. In section \[S-4\] we prove the uniqueness of the solutions and their continuous dependence on initial data.
Functional setting and Formulation {#S-2}
==================================
Equivalent Formulation
----------------------
We denote by $$\overline{\phi} (x, y)= \frac{1}{h} \int_{-h}^0 \phi(x,y, z)
dz, \qquad \forall \; (x,y) \in M; \label{VBAR}$$ and denote the fluctuation by $$\widetilde{\phi} = \phi - \overline{\phi}. \label{V--T}$$ Notice that $$\begin{aligned}
&&\hskip-.68in \overline{\widetilde{\phi}} = 0. \label{ZERO}\end{aligned}$$ Similar to [@CT05], by integrating (\[EQ-2\]) and (\[EQ-3\]) vertically, we get $$w(x,y,z,t) = - \int_{-h}^z \nabla_H \cdot v(x,y, \xi,t) d\xi,
\label{DIV-1}$$ and $$p(x,y,z,t) = - \int_{-h}^z T(x,y,\xi,t) d\xi + p_s(x,y,t), \label{PPP}$$ where $p_s$ is the pressure on the bottom $z=-h$. Essentially, $p_s(x,y,t)$ is the unknown pressure, and we observe, as before, that it is a function of two spatial variables $(x,y).$ As we mentioned in the introduction we explore this property as in [@CT05] (see also [@KZ07]) to prove our global regularity result.
Replacing $T$ by $T + \frac{z}{h}$, we have the following equivalent formulation for system (\[EQ-1\])–(\[INIT-2\]): $$\begin{aligned}
&&\hskip-.68in \frac{\pp v}{\pp t} + L_1 v+ (v\cdot {\nabla}_{H}) v
- \left( \int_{-h}^z {\nabla}_{H} \cdot v(x,y, \xi,t) d\xi
\right) \frac{\pp v}{\pp z} \nonumber \\
&&\hskip-.5in + f_0 \vec{k} \times v + {\nabla}_{H} p_s(x,y,t) - {\nabla}_{H} \int_{-h}^z
T(x,y,\xi,t) d\xi = 0,
\label{EQV} \\
&&\hskip-.68in {\nabla}_{H} \cdot \overline{v} = 0, \label{EQ22} \\
&&\hskip-.68in
\frac{\pp T}{\pp t} + L_2 T + v \cdot {\nabla}_{H} T
- \left( \int_{-h}^z {\nabla}_{H} \cdot v (x,y, \xi,t)
d\xi
\right) \left( \frac{\pp T}{\pp z}+ \frac{1}{h}\right) = Q,
\label{EQ5} \\
&&\hskip-.68in \left. \frac{\pp v }{\pp z} \right|_{z=0} = \left.
\frac{\pp v }{\pp z} \right|_{z=-h} = 0, \quad v(x+1,y,z)=v(x,y+1,z)=v(x,y,z),
\label{EQ6} \\
&&\hskip-.68in \left. T \right|_{z=0}= \left. T \right|_{z=-h}=
0, \quad T(x+1,y,z)=T(x,y+1,z)=T(x,y,z),
\label{EQ7} \\
&&\hskip-.68in v (x,y,z,0) = v_0 (x,y,z),
\label{EQ8} \\
&&\hskip-.68in T(x,y,z,0) = T_0 (x,y,z)-\frac{z}{h}.
\label{EQ9}\end{aligned}$$ In addition, $\overline{v}$ and $\widetilde{v}$ satisfy the following coupled system of equations: $$\begin{aligned}
&&\hskip-.68in \frac{\pp \overline{v}}{\pp t} - \frac{1}{R_1} {\Dd}_{H}
\overline{v} + (\overline{v} \cdot {\nabla}_{H} ) \overline{v} +
\overline{ \left[ (\widetilde{v} \cdot {\nabla}_{H}) \widetilde{v}
+ ({\nabla}_{H} \cdot \widetilde{v}) \; \widetilde{v}\right]}
\nonumber \\
&&\hskip-.28in + f_0 \vec{k} \times \overline{v} + {\nabla}_{H} \left[ p_s(x,y,t) - \frac{1}{h}
\int_{-h}^0 \int_{-h}^z \, T (x,y,\xi,t) \; d\xi \; dz \right]
= 0, \label{EQ1} \\
&&\hskip-.68in {\nabla}_{H} \cdot \overline{v} = 0, \label{EQ2}
\\
&&\hskip-.68in \frac{\pp \widetilde{v}}{\pp t} + L_1 \widetilde{v} +
(\widetilde{v} \cdot {\nabla}_{H}) \widetilde{v} - \left(
\int_{-h}^z {\nabla}_{H} \cdot \widetilde{v}(x,y, \xi,t) d\xi
\right) \frac{\pp \widetilde{v}}{\pp z} +(\widetilde{v} \cdot
{\nabla}_{H} )
\overline{v}+ (\overline{v} \cdot {\nabla}_{H}) \widetilde{v} \nonumber \\
&&\hskip-.58in - \overline{ \left[(\widetilde{v} \cdot
{\nabla}_{H}) \widetilde{v} + ({\nabla}_{H} \cdot \widetilde{v}) \;
\widetilde{v}\right]} + f_0 \vec{k} \times \widetilde{v} - {\nabla}_{H} \left( \int_{-h}^z
T(x,y,\xi,t) d\xi -\frac{1}{h} \int_{-h}^0 \int_{-h}^z T(x,y,\xi,t)
d\xi dz \right) =0. \label{EQ4}\end{aligned}$$
0.1in
Functional spaces and inequalities
----------------------------------
Let us denote by $L^q(\Om), L^q(M)$ and $W^{m, q}(\Om), W^{m,
q}(M)$, and $H^m(\Om) =: W^{m, 2}(\Om), H^m(M) =: W^{m, 2}(M)$, the usual $L^q-$Lebesgue and Sobolev spaces, respectively ([@AR75]). We denote by $$\| \phi\|_q = \left\{ \begin{array}{ll} \left( \int_{\Om} |\phi
(x,y,z)|^q \; dxdydz
\right)^{\frac{1}{q}}, \qquad & \mbox{ for every $\phi \in L^q(\Om)$} \\
\left( \int_{M} |\phi (x,y)|^q \; dxdy \right)^{\frac{1}{q}}, &
\mbox{ for every $\phi \in L^q(M)$}.
\end{array} \right.
\label{L2}$$
For convenience, we recall the following Sobolev and Ladyzhenskaya type inequalities in $(\mathbb{R}/\mathbb{Z})^2$ and in $\Om$ (see, e.g., [@AR75], [@CF88], [@GA94], [@LADY]) $$\begin{aligned}
&&\hskip-.68in
\| \phi \|_{L^4(M)} \leq C_0 \| \phi \|_{L^2}^{1/2} \| \phi \|_{H^1(M)}^{1/2},
\qquad \forall \phi \in H^1(M), \label{SI-1}\\
&&\hskip-.68in \| \phi\|_{L^8(M)} \leq C_0 \| \phi
\|_{L^6(M)}^{3/4} \| \phi \|_{H^1(M)}^{1/4}, \qquad \forall \phi \in H^1(M),
\label{SI-2} \\
&&\hskip-.68in \|{\nabla}_{H} \phi \|_{L^4(M)} \leq C_0 \| \phi
\|_{\infty}^{1/2} \| \phi \|_{H^2(M)}^{1/2}, \qquad \forall \phi
\in H^2(M),
\label{SI-11}\\
&&\hskip-.68in \|{\nabla}_{H} \phi \|_{L^4(M)} \leq C_0 \| \phi
\|_{L^2(M)}^{1/2} \| {\nabla}_{H} \phi \|_{\infty}^{1/2} +\| \phi
\|_{L^2(M)}, \qquad \forall \; \phi \; \mbox{such that } \; {\nabla}_{H} \phi \in L^{\infty}(M),
\label{SI-111}\end{aligned}$$ and $$\begin{aligned}
&&\hskip-.68in
\| \psi \|_{L^3(\Om)} \leq C_0 \| \psi \|_{L^2(\Om)}^{1/2} \| \psi \|_{H^1(\Om)}^{1/2},
\label{SI1}\\
&&\hskip-.68in \| \psi \|_{L^6(\Om)} \leq C_0 \| \psi
\|_{H^1(\Om)}, \label{SI2}\end{aligned}$$ for every $\psi\in H^1(\Om).$ Here $C_0$ is a positive scale invariant constant. Also, we recall the following version of Helmholtz-–Weyl decomposition Theorem (cf. for example, [@BM02], [@GA94], [@YV63]) $$\begin{aligned}
&& \|{\nabla}_{H} \phi \|_{W^{m,q}(M)} \leq C \left(
\|{\nabla}_{H} \cdot \phi \|_{W^{m,q}(M)} + \|{\nabla}_{H}^{\perp}
\cdot \phi \|_{W^{m,q}(M)} \right), \label{DIV-CUR}\end{aligned}$$ for every $\vec{\phi}\in \left( W^{m, q}(M)\right)^2.$ Moreover, we recall the following Brezis–Gallouet or, Brezis–Wainger inequality (see, e.g., [@BM02], [@BG80], [@BW80], [@EN89]) $$\begin{aligned}
&& \|\phi \|_{L^{\infty}(M)} \leq C \|\phi \|_{H^1(M)}\; \left(1+
\log^+ \|\phi \|_{H^2(M)} \right)^{1/2}, \label{BW-1}\end{aligned}$$ for every $\phi \in H^2(M),$ where $\log^+ r = \log r,$ when $r \geq
1$, and $\log^+ r = 0,$ when $r \leq 1$. Also, we recall the following inequality (see, e.g., [@BKM84] and [@KATO83]) $$\begin{aligned}
&& \|\nabla_H \phi \|_{L^{\infty}(M)} \leq C
(\|\nabla_H \cdot \phi \|_{L^{\infty}(M)} + \|\nabla_H \times \phi \|_{L^{\infty}(M)}) \;
\left(1+ \log^+ \|\nabla_H \phi \|_{H^2(M)} \right), \label{BW-2}\end{aligned}$$ for every $\nabla_H \phi \in H^2(M)$. Moreover, by (\[SI-1\]) we get $$\begin{aligned}
&&\hskip-.68in \| \phi \|_{L^{4q}(M)}^{4q} = \|\;
|\phi|^q\;\|_{L^{4}(M)}^{4}
\leq C \|\; |\phi|^q \;\|_{L^2(M)}^{2} \| \;|\phi|^q \;\|_{H^1(M)}^{2} \nonumber \\
&&\hskip-.68in \leq C_q \| \phi \|_{2q}^{2q} \; \left( \int_M
|\phi|^{2q-2} \left| {\nabla}_{H} \phi \right|^2 \; dxdy \right) +
\| \phi \|_{2q}^{4q}, \label{TWE}\end{aligned}$$ for every $\phi$ satisfying $\int_M |\phi|^{2q-2} \left| {\nabla}_{H}
\phi \right|^2 \; dxdy < \infty$ and $q\geq 1$. Also, we recall the integral version of Minkowsky inequality for the $L^p$ spaces, $p\geq 1$. Let $\Om_1 \subset \mathbb{R}^{m_1}$ and $\Om_2 \subset \mathbb{R}^{m_2}$ be two measurable sets, where $m_1$ and $m_2$ are two positive integers. Suppose that $f(\xi,\eta)$ is a measurable function over $\Om_1 \times \Om_2$. Then, $$\hskip0.35in \left[ { \int_{\Om_1} \left( \int_{\Om_2}
|f(\xi,\eta)| d\eta \right)^p d\xi } \right]^{1/p} \leq
\int_{\Om_2} \left( \int_{\Om_1} |f(\xi,\eta)|^p d\xi
\right)^{1/p} d\eta. \label{MKY}$$ Finally, we recall the following inequality from Proposition 2.2 in [**[@CT03]**]{} $$\begin{aligned}
&&\hskip-.68in
\left| \int_M \left( \int_{-h}^0 \psi_1 (x, y, z)\; dz
\right)\; \left(\int_{-h}^0
\psi_2 (x, y, z) \, \psi_3 (x, y, z)\; dz \right)\; dxdy \right| \nonumber \\
&&\hskip-.68in \leq C \| \psi_1 \|_2^{1/2} \| {\nabla}_{H} \psi_1
\|_2^{1/2}\| \psi_2 \|_2^{1/2}\|{\nabla}_{H} \psi_2 \|_2^{1/2}\|
\psi_3 \|_2 +\| \psi_1 \|_2\| \psi_2 \|_2 \| \psi_3 \|_2,
\label{MAIN-1}\end{aligned}$$ for every $\psi_1, \psi_2 \in H^1(\Om)$ and $\psi_3 \in L^2(\Om),$ and $$\begin{aligned}
&&\hskip-.68in
\left| \int_M \left( \int_{-h}^0 \psi_1 (x, y, z)\; dz
\right) \, \left( \int_{-h}^0
|{\nabla}_{H} \psi_2(x, y, z)| \, \psi_3(x, y, z)\; dz \right)\; dxdy \right| \nonumber \\
&&\hskip-.68in \leq C \| \psi_1 \|_2^{1/2} \| {\nabla}_{H} \psi_1
\|_2^{1/2}\| \psi_2 \|_{\infty}^{1/2}\|{\nabla}_{H}{\nabla}_{H} \psi_2
\|_2^{1/2}\| \psi_3 \|_2 +\| \psi_1 \|_2\| \psi_2 \|_2 \| \psi_3
\|_2, \label{MAIN-2}\end{aligned}$$ for every $\psi_1 \in H^1(\Om)$, ${\nabla}_{H} \psi_2 \in H^1(\Om)$ and $\psi_3 \in L^2(\Om).$
Global existence of strong solutions {#S-3}
====================================
In the previous section we have reformulated system (\[EQ-1\])–(\[INIT-2\]) to be equivalent to system (\[EQV\])–(\[EQ9\]). In this section we will show that strong solutions to system (\[EQV\])–(\[EQ9\]) exist globally in time provided they exist in short time intervals.
\[T-MAIN\] Let $Q \in H^2(\Om), v_0 \in H^4(\Om), T_0\in
H^2(\Om)$ and $\mathcal{T} > 0.$ Suppose that there exists a strong solution $(v(t), T(t))$ of system [*(\[EQV\])–(\[EQ9\])*]{} on $[0,
\mathcal{T}]$ corresponding to the initial data $(v_0, T_0)$ such that $$\begin{aligned}
&& {\Dd}_{H} v_z, \;\; \nabla_H T \in C([0,\mathcal{T}], H^1(\Om)), \\
&& v_{zz}, {\Dd}_{H} \nabla_H v_z, \;\;
{\nabla}_{H} T_z \in L^2 ([0,\mathcal{T}], H^1(\Om)).\end{aligned}$$ Then this strong solution $(v(t), T(t))$ exists globally in time.
Notice that one can recover the pressure $p_s$ from system (\[EQ1\])–(\[EQ2\]) in the same way as in $2D$ NSE (see, e.g., [@CF88], [@SO01A], [@TT84]).
Let $[0,\mathcal{T}_*)$ be the maximal interval of existence of a strong solution $(v(t), T(t))$. In order to establish the global existence, we need to show That $\mathcal{T}_* = \infty.$ If $\mathcal{T}_* < \infty$ we will show $\|{\Dd}_{H}
v_z(t)\|_{H^1(\Om)}, \;\; \|\nabla_H T
(t)\|_{H^1(\Om)},$ $ \int_0^t\|{\Dd}_{H} \nabla_H
v_z(s)\|_{H^1(\Om)}^2 \; ds $, $ \int_0^t
\|\nabla_H T_z (s)\|_{H^1(\Om)}^2 \; ds$, and $
\int_0^t \| v_{zz} (s)\|_{H^1(\Om)}^2 \; ds$ are all bounded uniformly in time, for $t \in
[0,\mathcal{T}_{*})$. As a result the interval $[0,
\mathcal{T}_*)$ can not be a maximal interval of existence, and consequently the strong solution $(v(t), T(t))$ exists globally in time.
Therefore, we focus our discussion below on the interval $[0,\mathcal{T}_*).$
$\|v\|_2^2+\|T\|_{2}^2$ estimates
---------------------------------
By taking the inner product of equation (\[EQ5\]) with $T$, in $L^2(\Om)$, we get $$\begin{aligned}
&&\hskip-.68in \frac{1}{2} \frac{d \|T\|_{2}^{2}}{dt}
+ \frac{1}{R_3}\;\|T_z\|_2^2 \nonumber \\
&&\hskip-.65in = \int_{\Om} Q T \; dxdydz -\int_{\Om} \left[ v
\cdot {\nabla}_{H} T - \left( \int_{-h}^z {\nabla}_{H} \cdot v(x,y,
\xi,t) d\xi \right) \left(\frac{\pp T}{\pp z}+\frac{1}{h} \right)
\right]\; T \; dxdydz. \label{EST-1}\end{aligned}$$ Integrating by parts and using the boundary condition (\[EQ7\]), we get $$\begin{aligned}
&&\hskip-.065in -\int_{\Om} \left( v \cdot {\nabla}_{H} T - \left(
\int_{-h}^z {\nabla}_{H} \cdot v(x,y, \xi,t) d\xi \right) \frac{\pp
T}{\pp z}\right) T \; dxdydz =0. \label{EST-2}\end{aligned}$$ As a result of the above we conclude that $$\begin{aligned}
&&\hskip-.68in \frac{1}{2} \frac{d \|T\|_{2}^{2}}{dt} +
\frac{1}{R_3}\;\|T_z\|_2^2
\nonumber \\
&&\hskip-.65in =\int_{\Om} \left[ Q - \frac{1}{h} \left(
\int_{-h}^z {\nabla}_{H} \cdot v(x,y, \xi,t) d\xi \right) \right] \;
T \; dxdydz \leq \|Q\|_{2} \; \|T\|_{2} + \|{\nabla}_{H} v\|_2
\|T\|_2. \label{T-INT}\end{aligned}$$ Moreover, by taking the inner product of equation (\[EQV\]) with $v$, in $L^2(\Om)$, we reach $$\begin{aligned}
&&\hskip-.28in \frac{1}{2} \frac{d \|v\|_2^2}{dt} + \frac{1}{R_1}
\|{\nabla}_{H} v\|_2^2
+ \frac{1}{R_2}\|v_z\|_2^2 \nonumber \\
&&\hskip-.265in = -\int_{\Om} \left[ (v \cdot {\nabla}_{H}) v -
\left( \int_{-h}^z {\nabla}_{H} \cdot v(x,y, \xi,t) d\xi \right)
\frac{\pp v}{\pp z} \right] \cdot v \;dxdydz
\nonumber \\
&&\hskip-.1065in
- \int_{\Om} \left( f_0 \vec{k} \times v \right) \cdot v \;dxdydz
- \int_{\Om} \left(
{\nabla}_{H} p_s - {\nabla}_{H} \left( \int_{-h}^z T(x,y,\xi,t)
d\xi \right) \right) \cdot v \;dxdydz. \label{EST-3}\end{aligned}$$ First, we notice that $$\begin{aligned}
&&\hskip-.065in \left( f_0 \vec{k} \times v\right) \cdot v =0. \label{DT-1}\end{aligned}$$ Next, by integration by parts and using the boundary conditions (\[EQ6\]), in particular, the horizontal periodic boundary conditions, we get $$\begin{aligned}
&&\hskip-.065in \int_{\Om} \left[ (v \cdot {\nabla}_{H}) v - \left(
\int_{-h}^z {\nabla}_{H} \cdot v(x,y,\xi,t) d\xi \right) \frac{\pp
v}{\pp z} \right] \cdot v \;dxdydz =0. \label{DT-11}\end{aligned}$$ Thanks to (\[EQ2\]) and, again, the horizontal periodic boundary conditions, we also have $$\begin{aligned}
&&\hskip-.065in \int_{\Om} {\nabla}_{H} p_s(x, y, t) \cdot v(x, y,
z, t) \;dxdydz = h \int_M {\nabla}_{H} p_s \cdot \overline{v} \;dxdy
= -h \int_{\Om} p_s ({\nabla}_{H} \cdot \overline{v}) \;dxdy =0.
\label{DT-44}\end{aligned}$$ By integration by parts, the periodic boundary conditions (\[EQ6\]), and Cauchy–Schwarz inequality, we obtain $$\begin{aligned}
&&\hskip-.065in
\left| \int_{\Om} {\nabla}_{H} \left( \int_{-h}^z T(x,y,\xi,t)
d\xi \right) \cdot v \;dxdydz \right| \leq h \, \|T\|_2 \; \|\nabla_H v\|_2.
\label{DT-4}\end{aligned}$$ Thus, by (\[T-INT\])–(\[DT-4\]) we have $$\begin{aligned}
&&\hskip-.68in \frac{1}{2} \frac{d (\|v \|_2^2+\|T \|_2^2) }{dt} +
\frac{1}{R_1} \|{\nabla}_{H} v\|_2^2
+ \frac{1}{R_2}\|v_z\|_2^2+ \frac{1}{R_3}\|T_z\|_2^2 \nonumber \\
&&\hskip-.68in \leq \|Q\|_2 \|T\|_2+ (1+h) \|T\|_2\; \|
{\nabla}_{H} v \|_2. \label{EST-4}\end{aligned}$$ By Cauchy–Schwarz inequality, we obtain $$\begin{aligned}
&&\hskip-.68in \frac{d (\|v \|_2^2+\|T \|_2^2) }{dt}+
\frac{1}{R_1} \|{\nabla}_{H} v\|_2^2
+ \frac{1}{R_2}\|v_z\|_2^2 + \frac{1}{R_3}\|T_z\|_2^2 \\
&&\hskip-.68in \leq \|Q\|_{2}^2 + (1+R_1)(1+h)^2 \|T\|_2^2.
\label{EST-44}\end{aligned}$$ Thanks to Gronwall’s inequality we get $$\begin{aligned}
&&\hskip-.68in \| v(t)\|_2^2 +\|T(t) \|_2^2 \leq C \left( \|v_0\|_2^2+
\|T_0\|_2^2 \right) e^{(1+R_1)(1+h)^2 \, t} + C \|Q\|^2_2; \label{V-2}
\end{aligned}$$ and $$\begin{aligned}
&&\hskip-.68in \int_0^t \left[ \frac{1}{R_1} \| {\nabla}_{H} v(s)
\|_2^2 + \frac{1}{R_2} \| v_z (s)\|_2^2 + \frac{1}{R_3}\|T_z(s)\|_2^2
\right]\; ds
\leq C \left[
\left( \|v_0\|_2^2+ \|T_0\|_2^2\right)e^{(1+R_1)(1+h)^2 \, t} +
\|Q\|^2_2\; t \right]. \label{VEE}\end{aligned}$$ Therefore, for every $t \in [0,\mathcal{T}_*),$ we have $$\begin{aligned}
&&\hskip-.168in \| v (t)\|_2^2 +\|T (t)\|_2^2+ \int_0^t \left[ \|
{\nabla}_{H} v (s)\|_2^2 +
\| v_z (s)\|_2^2 +\| T_z (s)\|_2^2 \right]\; ds \leq K_1, \label{K-1}\end{aligned}$$ where $$\begin{aligned}
&&\hskip-.168in K_1 = C \left[ \left( \|v_0\|_2^2+ \|T_0\|_2^2 \right)
e^{(1+R_1)(1+h)^2 \, t}+ \|Q\|^2_2\; t \right]. \label{K1}\end{aligned}$$
$\|T\|_{\infty}$ estimates
--------------------------
We follow here the idea of Stampaccia for proving the Maximum Principle. The proof we present here is also similar to the one in [@FMT86] (see also [@TT88]). Denote by $\tau (t)= T(t)-(1+\|T_0\|_{\infty} + \|Q\|_{\infty} \; t).$ It is clear that $\tau$ satisfies: $$\begin{aligned}
&&\hskip-.168in
\frac{\pp \tau}{\pp t} + v \cdot {\nabla}_{H} \tau + w
\frac{\pp \tau}{\pp z} + L_2 \tau = Q-\|Q\|_{\infty}. \label{TAU}\end{aligned}$$ Let $\tau^+ = \max\{0, \tau\}$ which belongs to $H^1(\Om)$ and satisfies $$\begin{aligned}
&&\hskip-.8in \tau^+(z=0)=\tau^+(z=-h)=0.
\label{TB-1}\end{aligned}$$ Taking the inner product of the equation (\[TAU\]) with $\tau^+$ in $L^2(\Om)$ and applying the boundary conditions (\[TB-1\]), we get $$\begin{aligned}
&&\hskip-.68in \frac{1}{2} \frac{d \|\tau^+\|_{2}^{2}}{dt}
+ \frac{1}{R_3}\|\pp_z\tau^+\|_2^2
= \int_{\Om} (Q-\|Q\|_{\infty}) \tau^+ \; dxdydz -\int_{\Om} \left[ v
\cdot {\nabla}_{H} \tau +w \,\pp_z \tau
\right]\; \tau^+ \; dxdydz. \label{TEST-1}\end{aligned}$$ By integration by parts and using the boundary conditions (\[EQ7\]) and (\[TB-1\]), we get $$\begin{aligned}
&&\hskip-.065in \int_{\Om} \left[ v
\cdot {\nabla}_{H} \tau +w \,\pp_z \tau
\right]\; \tau^+ \; dxdydz =0. \label{TEST-2}\end{aligned}$$ Thus, $$\begin{aligned}
&&\hskip-.68in \frac{1}{2} \frac{d \|\tau^+\|_{2}^{2}}{dt}
+ \frac{1}{R_3}\|\pp_z\tau^+\|_2^2 \; dxdydz
= \int_{\Om} (Q-\|Q\|_{\infty}) \tau^+ \; dxdydz \leq 0. \label{TEST-3}\end{aligned}$$ Therefore, we obtain $$\begin{aligned}
&&\hskip-.68in \|\tau^+(t)\|_{2}^{2} \leq \|\tau^+(t=0)\|_{2}^{2}=0. \label{TEST-4}\end{aligned}$$ Thus, $\tau^+(t)\equiv 0.$ As a result, we have $$\begin{aligned}
&&\hskip-.68in T(t) \leq 1+\|T_0\|_{\infty} + \|Q\|_{\infty} \; t. \label{TEST-5}\end{aligned}$$ By applying similar arguments, we also have $$\begin{aligned}
&&\hskip-.68in T(t) \geq -(1+\|T_0\|_{\infty} + \|Q\|_{\infty} \; t). \label{TEST-6}\end{aligned}$$ Therefore, $T$ satisfies the following $L^{\infty}-$estimate: $$\begin{aligned}
&&\hskip-.68in \|T (t)\|_{\infty}
\leq K_2 = 1+\|T_0\|_{\infty} + \|Q\|_{\infty} \; t. \label{K-2}\end{aligned}$$
0.1in
$\|\widetilde{v} \|_6$ estimates
--------------------------------
Taking the inner product of the equation (\[EQ4\]) with $|\widetilde{v}|^4 \widetilde{v}$ in $L^2(\Om)$ and using the boundary conditions (\[EQ6\]), we get $$\begin{aligned}
&&\hskip-.168in \frac{1}{6} \frac{d \| \widetilde{v} \|_{6}^{6} }{d
t} + \frac{1}{R_1} \int_{\Om} \left(|{\nabla}_{H} \widetilde{v}|^2
|\widetilde{v}|^{4} + \left|{\nabla}_{H} |\widetilde{v}|^2 \right|^2
|\widetilde{v}|^{2} \right) \; dxdydz + \frac{1}{R_2} \int_{\Om}
\left(|\widetilde{v}_z|^2 |\widetilde{v}|^{4}
+ \left|\pp_z |\widetilde{v}|^2 \right|^2 |\widetilde{v}|^{2} \right) \; dxdydz \\
&&\hskip-.165in = - \int_{\Om} \left\{ (\widetilde{v} \cdot
{\nabla}_{H}) \widetilde{v} - \left( \int_{-h}^z {\nabla}_{H} \cdot
\widetilde{v}(x,y, \xi,t) d\xi \right) \frac{\pp \widetilde{v}}{\pp
z} +(\widetilde{v} \cdot {\nabla}_{H} ) \overline{v}+ (\overline{v}
\cdot {\nabla}_{H}) \widetilde{v} - \overline{ \left[(\widetilde{v}
\cdot {\nabla}_{H}) \widetilde{v}
+ ({\nabla}_{H} \cdot \widetilde{v}) \; \widetilde{v}\right]} \right. \\
&&\hskip-.06in
\left. + f_0 \vec{k} \times \widetilde{v}
- {\nabla}_{H} \left( \int_{-h}^z
T(x,y,\xi,t) d\xi -\frac{1}{h} \int_{-h}^0 \int_{-h}^z
T(x,y,\xi,t) d\xi dz \right) \right\} \cdot
|\widetilde{v}|^{4} \widetilde{v} \; dxdydz.\end{aligned}$$ Observe, again, that $$\begin{aligned}
&&\hskip-.065in
\left( f_0 \vec{k} \times \widetilde{v} \right) \cdot
|\widetilde{v}|^{4} \widetilde{v} =0.
\label{D6-1}\end{aligned}$$ Moreover, by integration by parts and the boundary conditions (\[EQ6\]), we also get $$\begin{aligned}
&&\hskip-.065in - \int_{\Om} \left[ (\widetilde{v} \cdot
{\nabla}_{H}) \widetilde{v} - \left( \int_{-h}^z {\nabla}_{H} \cdot
\widetilde{v}(x,y, \xi,t) d\xi \right) \frac{\pp \widetilde{v}}{\pp
z} \right] \cdot |\widetilde{v}|^{4} \widetilde{v} \; dxdydz =0.
\label{D6-11}\end{aligned}$$ Furthermore, by virtue of (\[EQ2\]) and by the boundary conditions (\[EQ6\]), in particular the horizontal periodic boundary conditions, we have $$\begin{aligned}
&&\hskip-.065in \int_{\Om} (\overline{v} (x, y, t) \cdot
{\nabla}_{H}) \widetilde{v} (x, y, z, t) \cdot |\widetilde{v}(x, y,
z, t)|^{4} \widetilde{v}(x, y, z, t) \;dxdydz =0. \label{D6-3}\end{aligned}$$ Thus, (\[D6-1\])–(\[D6-3\]) imply $$\begin{aligned}
&&\hskip-.168in \frac{1}{6} \frac{d \| \widetilde{v} \|_{6}^{6} }{d
t} + \frac{1}{R_1} \int_{\Om} \left(|{\nabla}_{H} \widetilde{v}|^2
|\widetilde{v}|^{4} + \left|{\nabla}_{H} |\widetilde{v}|^2 \right|^2
|\widetilde{v}|^{2} \right) \; dxdydz + \frac{1}{R_2} \int_{\Om}
\left(|\widetilde{v}_z|^2 |\widetilde{v}|^{4}
+ \left|\pp_z |\widetilde{v}|^2 \right|^2 |\widetilde{v}|^{2} \right) \; dxdydz
\nonumber \\
&&\hskip-.165in
= - \int_{\Om} \left\{
(\widetilde{v} \cdot {\nabla}_{H} ) \overline{v} - \overline{
(\widetilde{v} \cdot {\nabla}_{H}) \widetilde{v}
+ ({\nabla}_{H} \cdot \widetilde{v}) \; \widetilde{v}} \right.
\nonumber \\
&&\hskip-.06in
\left.
- {\nabla}_{H} \left( \int_{-h}^z T(x,y,\xi,t) d\xi -\frac{1}{h}
\int_{-h}^0 \int_{-h}^z T(x,y,\xi,t) d\xi dz \right\} \right)
\cdot |\widetilde{v}|^{4} \widetilde{v} \; dxdydz.
\label{D6-33}\end{aligned}$$ Notice that by integration by parts and using the boundary conditions (\[EQ6\]), in particular the horizontal periodic boundary conditions, we have $$\begin{aligned}
&&\hskip-.165in
- \int_{\Om} \left[
(\widetilde{v} \cdot {\nabla}_{H} ) \overline{v} - \overline{\left[
(\widetilde{v} \cdot {\nabla}_{H}) \widetilde{v}
+ ({\nabla}_{H} \cdot \widetilde{v}) \; \widetilde{v} \right]} \right. \\
&&\hskip-.06in \left. - {\nabla}_{H} \left( \int_{-h}^z
T(x,y,\xi,t) d\xi -\frac{1}{h} \int_{-h}^0 \int_{-h}^z T(x,y,\xi,t)
d\xi dz \right) \right] \cdot |\widetilde{v}|^{4}
\widetilde{v} \; dxdydz
\nonumber \\
&&\hskip-.165in
= \int_{\Om} \left[
({\nabla}_{H} \cdot \widetilde{v}) \; \overline{v} \cdot
|\widetilde{v}|^{4} \widetilde{v} + (\widetilde{v} \cdot
{\nabla}_{H} ) (|\widetilde{v}|^{4} \widetilde{v}) \cdot
\overline{v}
- \sum_{k=1}^2 \sum_{j=1}^3 \overline{ \widetilde{v}^k \widetilde{v}^j} \;
\pp_{x_k} (|\widetilde{v}|^{4}
\widetilde{v}^j) \right. \nonumber \\
&&\hskip-.06in
\left.
- \left( \int_{-h}^z T(x,y,\xi,t) d\xi -\frac{1}{h} \int_{-h}^0
\int_{-h}^z T(x,y,\xi,t) d\xi dz \right) {\nabla}_{H} \cdot
(|\widetilde{v}|^{4} \widetilde{v}) \right] \; dxdydz.
\label{D6-333}\end{aligned}$$ As a result, we obtain $$\begin{aligned}
&&\hskip-.168in \frac{1}{6} \frac{d \| \widetilde{v} \|_{6}^{6} }{d
t} + \frac{1}{R_1} \int_{\Om} \left(|{\nabla}_{H} \widetilde{v}|^2
|\widetilde{v}|^{4} + \left|{\nabla}_{H} |\widetilde{v}|^2 \right|^2
|\widetilde{v}|^{2} \right) \; dxdydz + \frac{1}{R_2} \int_{\Om}
\left(|\widetilde{v}_z|^2 |\widetilde{v}|^{4}
+ \left|\pp_z |\widetilde{v}|^2 \right|^2 |\widetilde{v}|^{2} \right) \; dxdydz
\nonumber \\
&&\hskip-.165in \leq C \int_{M} \left[ |\overline{v}| \int_{-h}^0
|{\nabla}_{H} \widetilde{v}| \; | \widetilde{v}|^5 \; dz \right]
\; dxdy +C \int_{M} \left[ \left( \int_{-h}^0
|\widetilde{v}|^2 \; dz \right) \left( \int_{-h}^0 |{\nabla}_{H}
\widetilde{v}| \;
| \widetilde{v}|^4 \; dz \right) \right] \; dxdy \nonumber \\
&&\hskip-.065in +C \int_{M} \left[ \overline{|T|} \int_{-h}^0
|{\nabla}_{H} \widetilde{v}| \; | \widetilde{v}|^4 \; dz \right]
\; dxdy.
\label{D6-3333}\end{aligned}$$ Therefore, by the Cauchy–Schwarz inequality and Hölder inequality we reach $$\begin{aligned}
&&\hskip-.168in \frac{1}{6} \frac{d \| \widetilde{v} \|_{6}^{6} }{d
t} + \frac{1}{R_1} \int_{\Om} \left(|{\nabla}_{H} \widetilde{v}|^2
|\widetilde{v}|^{4} + \left|{\nabla}_{H} |\widetilde{v}|^2 \right|^2
|\widetilde{v}|^{2} \right) \; dxdydz + \frac{1}{R_2} \int_{\Om}
\left(|\widetilde{v}_z|^2 |\widetilde{v}|^{4}
+ \left|\pp_z |\widetilde{v}|^2 \right|^2 |\widetilde{v}|^{2} \right) \; dxdydz \nonumber \\
&&\hskip-.165in \leq C \int_{M} \left[ |\overline{v}| \left(
\int_{-h}^0 |{\nabla}_{H} \widetilde{v}|^2 \; | \widetilde{v}|^4 \;
dz \right)^{1/2} \left( \int_{-h}^0 | \widetilde{v}|^6 \; dz
\right)^{1/2} \right]
\; dxdy \nonumber \\
&&\hskip-.06in +C \int_{M} \left[ \left( \int_{-h}^0
|\widetilde{v}|^2 \; dz \right) \left( \int_{-h}^0 |{\nabla}_{H}
\widetilde{v}|^2 \; | \widetilde{v}|^4 \; dz \right)^{1/2}
\left( \int_{-h}^0 | \widetilde{v}|^4 \; dz \right)^{1/2} \right] \; dxdy \nonumber \\
&&\hskip-.06in +C \int_{M} \left[ \overline{|T|} \left( \int_{-h}^0
|{\nabla}_{H} \widetilde{v}|^2 \; | \widetilde{v}|^4 \; dz
\right)^{1/2} \left( \int_{-h}^0 | \widetilde{v}|^4 \; dz
\right)^{1/2} \right]
\; dxdy\end{aligned}$$ Moreover, $$\begin{aligned}
&&\hskip-.168in \frac{1}{6} \frac{d \| \widetilde{v} \|_{6}^{6} }{d
t} + \frac{1}{R_1} \int_{\Om} \left(|{\nabla}_{H} \widetilde{v}|^2
|\widetilde{v}|^{4} + \left|{\nabla}_{H} |\widetilde{v}|^2 \right|^2
|\widetilde{v}|^{2} \right) \; dxdydz + \frac{1}{R_2} \int_{\Om}
\left(|\widetilde{v}_z|^2 |\widetilde{v}|^{4}
+ \left|\pp_z |\widetilde{v}|^2 \right|^2 |\widetilde{v}|^{2} \right) \; dxdydz \nonumber \\
&&\hskip-.165in \leq C \|\overline{v}\|_{L^4(M)}
\left( \int_{\Om} |{\nabla}_{H} \widetilde{v}|^2 \; | \widetilde{v}|^4 \; dxdydz \right)^{1/2}
\left( \int_M \left( \int_{-h}^0 | \widetilde{v}|^6 \; dz
\right)^{2}
\; dxdy \right)^{1/4} \nonumber \\
&&\hskip-.06in +C \left( \int_M \left( \int_{-h}^0 |
\widetilde{v}|^2 \; dz \right)^{4} \; dxdy \right)^{1/4} \left(
\int_{\Om} |{\nabla}_{H} \widetilde{v}|^2 \; | \widetilde{v}|^4 \;
dxdydz \right)^{1/2} \left( \int_M \left( \int_{-h}^0 |
\widetilde{v}|^4 \; dz \right)^{2}
\; dxdy \right)^{1/4} \nonumber \\
&&\hskip-.06in +C \|\,\overline{|T|}\,\|_{L^4(M)} \left( \int_{\Om}
|{\nabla}_{H} \widetilde{v}|^2 \; | \widetilde{v}|^4 \; dxdydz
\right)^{1/2} \left( \int_M \left( \int_{-h}^0 | \widetilde{v}|^4
\; dz \right)^{2} \; dxdy \right)^{1/4}. \label{D6_22}\end{aligned}$$ Using the Minkowsky inequality (\[MKY\]) we get $$\begin{aligned}
&&\hskip-.168in \left( \int_M \left( \int_{-h}^0 |
\widetilde{v}|^6 \; dz \right)^{2} \; dxdy \right)^{1/2} \leq C
\int_{-h}^0 \left( \int_M | \widetilde{v}|^{12} \; dxdy
\right)^{1/2} \; dz. \label{D6_2}\end{aligned}$$ Thanks to (\[TWE\]), $$\begin{aligned}
&&\hskip-.168in \int_M | \widetilde{v}|^{12} \; dxdy \leq C_0
\left( \int_M | \widetilde{v}|^{6} \; dxdy \right) \left( \int_M |
\widetilde{v}|^{4} |{\nabla}_{H} \widetilde{v} |^2 \; dxdy \right)
+ \left( \int_M | \widetilde{v}|^{6} \; dxdy \right)^2.
\label{D6_3}\end{aligned}$$ Thus, by the Cauchy–Schwarz inequality we obtain $$\begin{aligned}
&&\hskip-.168in \left( \int_M \left( \int_{-h}^0 | \widetilde{v}|^6
\; dz \right)^{2} \; dxdy \right)^{1/2} \leq C \|
\widetilde{v}\|_{L^6(\Om)}^{3} \left( \int_{\Om}
|\widetilde{v}|^{4} |{\nabla}_{H} \widetilde{v} |^2 \; dxdydz
\right)^{1/2} + \| \widetilde{v}\|_{L^6(\Om)}^{6}. \label{M1}\end{aligned}$$ Similarly, by (\[MKY\]) and (\[SI-2\]), we also obtain $$\begin{aligned}
&&\hskip-.168in \left( \int_M \left( \int_{-h}^0 |
\widetilde{v}|^4 \; dz \right)^{2} \; dxdy \right)^{1/2} \leq C
\int_{-h}^0 \left( \int_M | \widetilde{v}|^{8}
\; dxdy \right)^{1/2} \; dz \nonumber \\
&&\hskip-.168in \leq C \int_{-h}^0 \|
\widetilde{v}\|_{L^6(M)}^{3} \left( \|{\nabla}_{H}
\widetilde{v}\|_{L^2(M)} + \| \widetilde{v}\|_{L^2(M)} \right) \;
dz \leq C \| \widetilde{v}\|_6^{3} \left( \|{\nabla}_{H}
\widetilde{v}\|_2 + \| \widetilde{v}\|_2 \right), \label{M2}\end{aligned}$$ and $$\begin{aligned}
&&\hskip-.168in \left( \int_M \left( \int_{-h}^0 |
\widetilde{v}|^2 \; dz \right)^{4} \; dxdy \right)^{1/4} \leq C
\int_{-h}^0 \left( \int_M | \widetilde{v}|^{8}
\; dxdy \right)^{1/4} \; dz \nonumber \\
&&\hskip-.168in \leq C \int_{-h}^0 \|
\widetilde{v}\|_{L^6(M)}^{3/2} \left(
\|{\nabla}_{H} \widetilde{v}\|_{L^2(M)}^{1/2} + \| \widetilde{v}\|_{L^2(M)}^{1/2} \right) \; dz
\leq C \| \widetilde{v}\|_6^{3/2} \left( \|{\nabla}_{H}
\widetilde{v}\|_2 + \| \widetilde{v}\|_2 \right)^{1/2}.
\label{M3}\end{aligned}$$ Therefore, using (\[M1\])–(\[M3\]) and (\[SI-1\]), we reach to $$\begin{aligned}
&&\hskip-.168in \frac{1}{6} \frac{d \| \widetilde{v} \|_{6}^{6} }{d
t} + \frac{1}{R_1} \int_{\Om} \left(|{\nabla}_{H} \widetilde{v}|^2
|\widetilde{v}|^{4} + \left|{\nabla}_{H} |\widetilde{v}|^2 \right|^2
|\widetilde{v}|^{2} \right) \; dxdydz + \frac{1}{R_2} \int_{\Om}
\left(|\widetilde{v}_z|^2 |\widetilde{v}|^{4}
+ \left|\pp_z |\widetilde{v}|^2 \right|^2 |\widetilde{v}|^{2} \right) \; dxdydz \\
&&\hskip-.165in \leq C \|\overline{v}\|_2^{1/2} \; \|{\nabla}_{H}
\overline{v}\|_2^{1/2} \| \widetilde{v}\|_6^{3/2}
\left( \int_{\Om} |{\nabla}_{H} \widetilde{v}|^2 \; | \widetilde{v}|^4 \; dxdydz \right)^{3/4}
+C \| \widetilde{v}\|_6^{3} \left( \|{\nabla}_{H}
\widetilde{v}\|_2 + \| \widetilde{v}\|_2 \right) \left( \int_{\Om}
|{\nabla}_{H} \widetilde{v}|^2 \; | \widetilde{v}|^4 \; dxdydz
\right)^{1/2}
\\
&&\hskip-.065in +C \|\overline{v}\|_2^{1/2} \; \|{\nabla}_{H} \overline{v}\|_2^{1/2}
\| \widetilde{v}\|_6^{6} +C \|T\|_{\infty} \; \| \widetilde{v}\|_6^{3/2}
\left( \|{\nabla}_{H} \widetilde{v}\|_2^{1/2} + \|
\widetilde{v}\|_2^{1/2} \right) \left( \int_{\Om} |{\nabla}_{H}
\widetilde{v}|^2 \; | \widetilde{v}|^4 \; dxdydz \right)^{1/2}.\end{aligned}$$ By Young’s inequality and Cauchy–Schwarz inequality we have $$\begin{aligned}
&&\hskip-.168in
\frac{d \| \widetilde{v} \|_{6}^{6} }{d t} +
\frac{1}{R_1} \int_{\Om} \left(|{\nabla}_{H} \widetilde{v}|^2
|\widetilde{v}|^{4} + \left|{\nabla}_{H} |\widetilde{v}|^2 \right|^2
|\widetilde{v}|^{2} \right) \; dxdydz + \frac{1}{R_2} \int_{\Om}
\left(|\widetilde{v}_z|^2 |\widetilde{v}|^{4}
+ \left|\pp_z |\widetilde{v}|^2 \right|^2 |\widetilde{v}|^{2} \right) \; dxdydz \\
&&\hskip-.165in \leq C \|\overline{v}\|_2^{2} \; \|{\nabla}_{H}
\overline{v}\|_2^{2} \| \widetilde{v}\|_6^{6} +C \|
\widetilde{v}\|_6^{6} \|{\nabla}_{H} \widetilde{v}\|_2^2 +C
\|T\|_{\infty}^4 +C \| \widetilde{v}\|_2^2\| \widetilde{v}\|_6^{6}.\end{aligned}$$ Thanks to (\[K-1\]), (\[K-2\]) and Gronwall inequality, we get $$\begin{aligned}
&&\hskip-.68in \| \widetilde{v} (t)\|^6_6 + \int_0^t \left(
\frac{1}{R_1} \int_{\Om} |{\nabla}_{H} \widetilde{v}|^2
|\widetilde{v}|^{4}
\; dxdydz +
\frac{1}{R_2} \int_{\Om} |\widetilde{v}_z|^2
|\widetilde{v}_z|^{4} \; dxdydz \right) \leq K_3, \label{K-3}\end{aligned}$$ where $$\begin{aligned}
&&\hskip-.68in K_3 = e^{K_1^2 t} \left[ \|v_0\|_{H^1(\Om)}^6 +
K_2^4 \; t \right]. \label{K3}\end{aligned}$$
$\|{\nabla}_{H} \overline{v}\|_2$ estimates
-------------------------------------------
By taking the inner product of equation (\[EQ1\]) with $-
{\Dd}_{H} \overline{v}$ in $L^2(M)$, and applying (\[EQ2\]), and using the boundary conditions (\[EQ6\]), we reach $$\begin{aligned}
&&\hskip-.68in \frac{1}{2} \frac{d \| {\nabla}_{H} \overline{v}
\|_2^2 }{d t} + \frac{1}{R_1} \|{\Dd}_{H} \overline{v}\|_2^2 =
\int_{M} \left\{ (\overline{v} \cdot {\nabla}_{H} ) \overline{v} +
\overline{ \left[ (\widetilde{v} \cdot {\nabla}_{H}) \widetilde{v} +
({\nabla}_{H} \cdot \widetilde{v}) \; \widetilde{v}\right]} +f_0
\vec{k}\times \overline{v}\right\} \cdot {\Dd}_{H} \overline{v} \;
dxdy. \label{DH-1}\end{aligned}$$ Following the situation for the $2D$ Navier–Stokes equations (cf. e.g., [@CF88], [@SALMON]) we have $$\begin{aligned}
&&\hskip-.68in \left| \int_{M} (\overline{v} \cdot {\nabla}_{H} )
\overline{v} \cdot {\Dd}_{H} \overline{v} \; dxdy \right|
\leq C \|\overline{v}\|_2^{1/2} \|{\nabla}_{H} \overline{v}\|_2
\; \|{\Dd}_{H} \overline{v}\|_2^{3/2}. \label{DH-2}\end{aligned}$$ By the Cauchy–Schwarz and the Hölder inequalities, we have $$\begin{aligned}
&&\hskip-.68in \left|\int_{M} \; \overline{ (\widetilde{v} \cdot
{\nabla}_{H}) \widetilde{v} + ({\nabla}_{H} \cdot \widetilde{v}) \;
\widetilde{v}} \cdot {\Dd}_{H} \overline{v} \; dxdy \right| \leq C
\int_M \int_{-h}^0 |\widetilde{v}| \; |{\nabla}_{H} \widetilde{v}|
\; dz \; |{\Dd}_{H} \overline{v}| \; dxdy \nonumber
\\
&&\hskip-.68in \leq C \int_M \left[ \left( \int_{-h}^0
|\widetilde{v}|^2 \; |{\nabla}_{H} \widetilde{v}| \; dz
\right)^{1/2} \left( \int_{-h}^0 |{\nabla}_{H} \widetilde{v}| \; dz
\right)^{1/2} \; |{\Dd}_{H} \overline{v}| \right] \; dxdy \nonumber
\\
&&\hskip-.68in \leq C \left[ \int_M \left( \int_{-h}^0
|\widetilde{v}|^2 \; |{\nabla}_{H} \widetilde{v}| \; dz \right)^{2}
\; dxdy \right]^{1/4} \; \left[ \int_M \left( \int_{-h}^0
|{\nabla}_{H} \widetilde{v}| \; dz \right)^{2} \; dxdy \right]^{1/4}
\; \left[ \int_M
|{\Dd}_{H} \overline{v}|^2 \; dxdy \right]^{1/2} \nonumber \\
&&\hskip-.68in \leq C \|{\nabla}_{H} \widetilde{v}\|_2^{1/2} \left(
\int_{\Om} |\widetilde{v}|^4 |{\nabla}_{H} \widetilde{v}|^2 \;
dxdydz \right)^{1/4}\|{\Dd}_{H} \overline{v}\|_2, \label{DH-3}\end{aligned}$$ and $$\begin{aligned}
&&\hskip-.68in \left|\int_{M} \; f_0 \times \overline{v} \cdot
{\Dd}_{H} \overline{v} \; dxdy \right| \leq C \| \overline{v}\|_2
\|{\Dd}_{H} \overline{v}\|_2. \label{DH-4}\end{aligned}$$ Thus, by Young’s inequality and the Cauchy–Schwarz inequality, we have $$\begin{aligned}
&&\hskip-.68in \frac{d \| {\nabla}_{H} \overline{v} \|_2^2 }{d t} +
\frac{1}{R_1} \|{\Dd}_{H} \overline{v}\|_2^2 \leq C
\|\overline{v}\|_2^2 \|{\nabla}_{H} \overline{v}\|_2^4 + C
\|{\nabla}_{H} \widetilde{v}\|_2^{2} + C \int_{\Om}
|\widetilde{v}|^4 |{\nabla}_{H} \widetilde{v}|^2 \; dxdydz + C
\|\overline{v}\|_2^2. \label{DH-5}\end{aligned}$$ By (\[K-1\]), (\[K-3\]) and thanks to Gronwall inequality we obtain $$\begin{aligned}
&&\hskip-.68in \| {\nabla}_{H} \overline{v} \|_2^2 + \frac{1}{R_1}
\int_0^t |{\Dd}_{H} \overline{v}|_2^2 \; ds \leq K_4, \label{K-4}\end{aligned}$$ where $$\begin{aligned}
&&\hskip-.68in K_4 = e^{K_2^2 t} \left[ \|v_0\|_{H^1(\Om)}^2 + K_2
+K_3 \right]. \label{K4}\end{aligned}$$
$\|v_z\|_6$ estimates
---------------------
The [*a priori*]{} estimates (\[K-1\])–(\[K-4\]) are essentially similar to those obtained in [@CT05]. From now on, we will get new [*a priori*]{} estimates of various norms.
Denote by $u=v_z.$ It is clear that $u$ satisfies $$\begin{aligned}
&&\hskip-.68in
\frac{\pp u }{\pp t} + L_1 u +
(v \cdot {\nabla}_{H}) u - \left( \int_{-h}^z {\nabla}_{H} \cdot v
(x,y, \xi,t)
d\xi \right) \frac{\pp u}{\pp z} \nonumber \\
&&\hskip-.46in + (u \cdot {\nabla}_{H} ) v - ({\nabla}_{H} \cdot v) u
+ f_0 \vec{k} \times u - {\nabla}_{H} T
= 0. \label{UU} \\
&&\hskip-.68in \left. u \right|_{z=0}= \left. u \right|_{z=-h}=
0. \label{UU-B}\end{aligned}$$ Taking the inner product of the equation (\[UU\]) with $u |u|^4$ in $L^2$, we get $$\begin{aligned}
&&\hskip-.68in \frac{1}{6} \frac{d \| u \|_6^6 }{d t} +
\frac{5}{R_1} \| \; |u|^2 \; |{\nabla}_{H} u| \; \|_2^2 +
\frac{5}{R_2} \|\; |u|^2 \; |\pp_z u|\; \|_2^2 \nonumber \\
&&\hskip-.65in =- \int_{\Om} \left( (v \cdot {\nabla}_{H}) u -
\left( \int_{-h}^z {\nabla}_{H} \cdot v (x,y, \xi,t)
d\xi \right) \frac{\pp u}{\pp z} \right) \cdot u |u|^4 \; dxdydz \nonumber \\
&&\hskip-.58in - \int_{\Om} \left( (u \cdot {\nabla}_{H} ) v -
({\nabla}_{H} \cdot v) u + f_0 \vec{k} \times u -{\nabla}_{H} T
\right) \cdot u |u|^4 \; dxdydz. \label{DZ_1}\end{aligned}$$ Notice again that $$\begin{aligned}
&&\hskip-.065in f_0 \vec{k} \times u \cdot u |u|^4 =0.
\label{DZ-1}\end{aligned}$$ Integrating by parts and using the boundary conditions, in particular (\[UU-B\]), give $$\begin{aligned}
&&\hskip-.065in - \int_{\Om} \left( (v \cdot {\nabla}_{H}) u -
\left( \int_{-h}^z {\nabla}_{H} \cdot v(x,y, \xi,t) d\xi \right)
\frac{\pp u}{\pp z} \right) \cdot u |u|^4\; dxdydz =0.
\label{DZ-11}\end{aligned}$$ Thus, by (\[DZ-1\]), (\[DZ-11\]) and Hölder inequality, we have $$\begin{aligned}
&&\hskip-.68in \frac{1}{6} \frac{d \| u \|_6^6 }{d t} +
\frac{5}{R_1} \| \; |u|^2 \; |{\nabla}_{H} u| \; \|_2^2 +
\frac{5}{R_2} \|\; |u|^2 \; |\pp_z u|\; \|_2^2 \nonumber \\
&&\hskip-.65in =- \int_{\Om} \left( (u \cdot {\nabla}_{H} ) v -
({\nabla}_{H} \cdot v) u
-{\nabla}_{H} T \right) \cdot u |u|^4\; dxdydz \nonumber \\
&&\hskip-.65in \leq C \int_{\Om} |v| \,|u|^5 \, |{\nabla}_{H} u|
\; dxdydz +C \int_{\Om} |T| \,|u|^4 \,
|{\nabla}_{H} u| \; dxdydz \nonumber \\
&&\hskip-.65in \leq C \left( \|v\|_6 \|u^3\|_3 \|{\nabla}_{H}
u^2\|_2 +
\|T\|_6 \; \| u^2 \|_3 \| {\nabla}_{H} u^3 \|_2 \right) \nonumber \\
&&\hskip-.65in \leq C \left( \|v\|_6 \|u\|_6^{3/2} \|{\nabla}_{H}
u^3\|_2^{3/2} + \|T\|_6 \; \| u \|_6^2 \| {\nabla}_{H} u^3 \|_2
\right). \label{DZ_2}\end{aligned}$$ Thanks to the Cauchy–Schwarz inequality, we have $$\begin{aligned}
&&\hskip-.68in \frac{d \| u \|_6^6 }{d t} + \frac{1}{R_1} \| \;
|u|^2 \; |{\nabla}_{H} u| \; \|_2^2 +
\frac{1}{R_2} \|\; |u|^2 \; |\pp_z u|\; \|_2^2 \nonumber \\
&&\hskip-.65in \leq C \left( 1+\|v\|_6^4 \right) \; \| u \|_6^6 + \|T\|_6^6 \\
&&\hskip-.65in \leq C \left(1+ \|{\nabla}_{H} \overline{v}\|_2^4
+\|\widetilde{v}\|_6^4 \right) \; \| u \|_6^6 + \|T\|_6^6. \label{DZ_3}\end{aligned}$$ Using (\[K-1\]), (\[K-3\]), (\[K-4\]), and Gronwall inequality, we get $$\begin{aligned}
&&\hskip-.68in \| u \|_6^6 + \int_0^t \left[ \; \frac{1}{R_1} \| \;
|u|^2 \; |{\nabla}_{H} u| \; \|_2^2 + \frac{1}{R_2} \|\; |u|^2 \;
|\pp_z u|\; \|_2^2 \right] \; ds \leq K_5, \label{K-5}\end{aligned}$$ where $$\begin{aligned}
&&\hskip-.68in K_5 = e^{(1+K_3^{2/3}+K_4^{2}) t} \left[ \|\pp_z
v_0\|_{H^1(\Om)}^6 + K_2^6 \; t \right]. \label{K5}\end{aligned}$$
$\|v_{zz}\|_2$ estimates
------------------------
Taking the inner product of the equation (\[UU\]) with $-
u_{zz}$ in $L^2(\Om)$ and recalling that $u=v_z,$ which satisfies the boundary condition (\[UU-B\]), we get $$\begin{aligned}
&&\hskip-.168in \frac{1}{2} \frac{d \| u_z \|_2^2 }{d t} +
\frac{1}{R_1} \| {\nabla}_{H} u_z \|_2^2 +
\frac{1}{R_2} \|u_{zz} \|_2^2 \\
&&\hskip-.065in =\int_{\Om} \left( (v \cdot {\nabla}_{H}) u -
\left( \int_{-h}^z {\nabla}_{H} \cdot v (x,y, \xi,t)
d\xi \right) \frac{\pp u}{\pp z} \right) \cdot u_{zz} \; dxdydz \\
&&\hskip-.058in + \int_{\Om} \left( (u \cdot {\nabla}_{H} ) v -
({\nabla}_{H} \cdot v) u + f_0 \vec{k} \times u -{\nabla}_{H} T
\right) \cdot u_{zz}\;
dxdydz \\
&&\hskip-.065in = -\int_{\Om} \left[ (u \cdot {\nabla}_{H}) u +
(u_z \cdot {\nabla}_{H} ) v +(u \cdot {\nabla}_{H} ) u - ({\nabla}_{H}
\cdot u) u - ({\nabla}_{H} \cdot v) u_z \right] \cdot u_{z}\;
dxdydz - \int_{\Om} T_z ({\nabla}_{H} \cdot u_z) \;
dxdydz \\
&&\hskip-.065in \leq C \|u\|_6 \|{\nabla}_{H} u \|_2 \|u_z\|_3 +C
\|v\|_6 \|{\nabla}_{H} u_z \|_2 \|u_z\|_3
+ \|T_z\|_2 \| {\nabla}_{H} u_z \|_2 \\
&&\hskip-.065in \leq C \left[ \|u\|_6 \|{\nabla}_{H} u \|_2 +
\|v\|_6 \|{\nabla}_{H} u_z \|_2 \right] \|u_z\|_2^{1/2} \left(
\|{\nabla}_{H} u_z\|_2^{1/2} +\| u_{zz} \|_2^{1/2}\right)
+ \|T_z\|_2 \| {\nabla}_{H} u_z \|_2.\end{aligned}$$ By the Cauchy–Schwarz and Young’s inequalities, we have $$\begin{aligned}
&&\hskip-.68in
\frac{1}{2} \frac{d \| u_z \|_2^2 }{d t} +
\frac{1}{R_1} \| {\nabla}_{H} u_z \|_2^2 +
\frac{1}{R_2} \|u_{zz} \|_2^2 \\
&&\hskip-.65in \leq C \left( \|v\|_6^4 + \|u\|_6^4 \right) \; \|
u_z\|_2^2 + C \|{\nabla}_{H} u \|_2^2 + C \|T_z\|_2^2.\end{aligned}$$ Applying (\[K-1\]), (\[K-3\]), (\[K-5\]), and Gronwall inequality yield $$\begin{aligned}
&&\hskip-.68in \| u_z \|_2^2+ \int_0^t \left[ \; \frac{1}{R_1} \|
{\nabla}_{H} u_z \|_2^2 + \frac{1}{R_2} \|u_{zz} \|_2^2 \right]
\; ds \leq K_6, \label{K-6}\end{aligned}$$ where $$\begin{aligned}
&&\hskip-.68in K_6 = C e^{(K_3^{2/3}+K_5^{2/3}) t} \left[
\|v_0\|_{H^1(\Om)}^2 + K_1 \right]. \label{K6}\end{aligned}$$
$\|{\nabla}_{H} \times v_z\|_2^2 + \| {\nabla}_{H} \cdot v_z + R_1 T
\|_2^2$ estimates
--------------------------------------------------------------------
Let $\beta$ be the solution of the following two–dimensional elliptic problem with periodic boundary conditions: $$\begin{aligned}
{\Dd}_{H} \beta = {\nabla}_{H} T, \quad \int_M \beta \; dxdy =0, \label{BETA-1}\end{aligned}$$ where $z$ is considered as a parameter. Roughly speaking, $\beta$ is like the potential vorticity. Notice that $$\begin{aligned}
{\nabla}_{H} \cdot \beta =T, \qquad {\nabla}_{H} \times \beta =0. \label{BETA-2}\end{aligned}$$ Recall that $u=v_z.$ We denote by $$\begin{aligned}
&& \zeta=u+R_1 \beta, \label{ZZZ} \\
&& \eta=\left({\nabla}_H^{\perp
} \cdot \zeta \right)= {\nabla}_H^{\perp} \cdot u
= \pp_x u_2 - \pp_y u_1, \label{EEE} \\
&& \tt = \left( {\nabla}_{H} \cdot \zeta \right) = {\nabla}_{H}
\cdot u + R_1 T = \pp_x u_1 + \pp_y u_2 + R_1 T. \label{TTT}\end{aligned}$$ Applying (\[DIV-CUR\]) for $\zeta$ and $\beta$ with $m \geq 0, 1< q <\infty$, using (\[BETA-2\])–(\[TTT\]), we obtain $$\begin{aligned}
&& \|{\nabla}_{H} u\|_{W^{m,q}(M)} \leq C\left( \|{\nabla}_{H}
\zeta\|_{W^{m,q}(M)}
+ R_1 \|{\nabla}_{H} \beta\|_{W^{m,q}(M)} \right) \nonumber \\
&& \leq C \left( \|\eta \|_{W^{m,q}(M)} +\|\tt \|_{W^{m,q}(M)}
+ \| T \|_{W^{m,q}(M)} \right), \label{INQ}\end{aligned}$$ where, again, we consider $z$ as a parameter; consequently, the constant $C$ is independent of $z$. By applying the operator ${\nabla}_H^{\perp}
\cdot$ to equation (\[UU\]) we obtain $$\begin{aligned}
&&\hskip-.68in
\frac{\pp \eta }{\pp t} + L_1 \eta + {\nabla}_H^{\perp} \cdot \left[
(v \cdot {\nabla}_{H}) u - \left( \int_{-h}^z {\nabla}_{H} \cdot v
(x,y, \xi,t) d\xi \right) \frac{\pp u}{\pp z} + (u \cdot
{\nabla}_{H} ) v - ({\nabla}_{H} \cdot v) u \right] \nonumber \\
&&\hskip-.6in - f_0 \left(R_1\, T- \tt\right)
=0. \label{ETA}\end{aligned}$$ Then, multiplying equation (\[EQ5\]) by $R_1$ and adding to the above equation we reach $$\begin{aligned}
&&\hskip-.68in
\frac{\pp \tt }{\pp t} + L_1 \tt + {\nabla}_{H} \cdot \left[
(v \cdot {\nabla}_{H}) u - \left( \int_{-h}^z {\nabla}_{H} \cdot v
(x,y, \xi,t) d\xi \right) \frac{\pp u}{\pp z} + (u \cdot
{\nabla}_{H} ) v -
({\nabla}_{H} \cdot v) u \right] - f_0\, \eta \nonumber \\
&&\hskip-.6in + R_1\left[ v \cdot {\nabla}_{H} T - \left(
\int_{-h}^z {\nabla}_{H}
\cdot v (x,y, \xi,t) d\xi \right) \left( \frac{\pp T}{\pp
z}+\frac{1}{h}\right)\right]
=R_1Q +R_1\, \left(\frac{1}{R_3}-\frac{1}{R_2}\right) T_{zz}. \label{TT}\end{aligned}$$ Taking the inner product of equation (\[ETA\]) with $\eta$ in $L^2(\Om)$ and the equation (\[TT\]) with $\tt$ in $L^2(\Om)$, integrating by parts and observing that $\left. \eta \right|_{z=0}= \left. \eta
\right|_{z=-h}=\left. \tt \right|_{z=0}= \left. \tt
\right|_{z=-h}= 0$, we get $$\begin{aligned}
&&\hskip-.168in \frac{1}{2} \frac{d \left( \| \eta \|_2^2+ \| \tt
\|_2^2 \right) }{d t} + \frac{1}{R_1} \left( \| {\nabla}_{H} \eta
\|_2^2+ \| {\nabla}_{H} \tt \|_2^2 \right) +
\frac{1}{R_2} \left( \|\pp_z \eta \|_2^2 + \|\pp_z \tt \|_2^2 \right) \\
&&\hskip-.158in = - \int_{\Om} \left\{ {\nabla}_H^{\perp} \cdot \left[
(v \cdot {\nabla}_{H}) u - \left( \int_{-h}^z {\nabla}_{H} \cdot v
(x,y, \xi,t) d\xi \right) \frac{\pp u}{\pp z} + (u \cdot
{\nabla}_{H} ) v - ({\nabla}_{H} \cdot v) u \right] \; \eta -f_0 \,
R_1 T \, \eta\right\}
\; dxdydz \\
&&\hskip-.05in - \int_{\Om} \left\{ {\nabla}_{H} \cdot \left[ (v
\cdot {\nabla}_{H}) u - \left( \int_{-h}^z {\nabla}_{H} \cdot v
(x,y, \xi,t) d\xi \right) \frac{\pp u}{\pp z} + (u \cdot
{\nabla}_{H} ) v - ({\nabla}_{H} \cdot v) u \right]\, \tt\right\}
\; dxdydz \\
&&\hskip-.05in - \int_{\Om} \left\{ R_1\left[ v \cdot {\nabla}_{H} T
- \left( \int_{-h}^z {\nabla}_{H}
\cdot v (x,y, \xi,t) d\xi \right) \left( \frac{\pp T}{\pp
z}+\frac{1}{h}\right)
- Q +\, \left(\frac{1}{R_3}-\frac{1}{R_2}\right) T_{zz} \right] \, \tt\right\}
\; dxdydz
\\
&&\hskip-.165in \leq C \|Q\|_2 \|\tt\|_2 + C \int_{\Om} \left(|v|
\,|{\nabla}_{H} u| \,+|u| \,|{\nabla}_{H} v| +|u_z|
\,\int_{-h}^0|{\nabla}_{H} \cdot v| \; d\xi \right)
\left(|{\nabla}_{H} \eta|+|{\nabla}_{H}
\tt|\right) \; dxdydz +C \|T\|_2 \|\eta\|_2 \\
&&\hskip-.06in + C \int_{\Om} \left[ |{\nabla}_{H} v| |T| |\tt|
+|v| |T| |{\nabla}_{H} \tt| + |T_z| \; \left( \int_{-h}^0
|{\nabla}_{H} \cdot v| \; d\xi\right) \;|\tt| \right] \; dxdydz +C
\|v\|_2 \|{\nabla}_{H} \tt\|_2 + C \|T_z\|_2 \|\tt_z\|_2
\\
&&\hskip-.165in \leq C \|Q\|_2 \|\tt\|_2 +C \int_{\Om} \left(|v|
\,|{\nabla}_{H} u| \,+|u| \,|{\nabla}_{H} v| +|u_z|
\,\int_{-h}^0(|\tt|+ |T|) \; d\xi \right) \left(|{\nabla}_{H}
\eta|+|{\nabla}_{H} \tt|\right)
\; dxdydz +C \|T\|_2 \|\eta\|_2 \\
&&\hskip-.06in + C \int_{\Om} \left[ |{\nabla}_{H} v| |T| |\tt| +
|v| |T| |{\nabla}_{H} \tt| + |T_z| \; \left( \int_{-h}^0
(|\tt|+|T|) \; d\xi \right) | \tt| \right] \; dxdydz +C \|v\|_2
\|{\nabla}_{H} \tt\|_2 + C \|T_z\|_2 \|\tt_z\|_2.\end{aligned}$$ Using Hölder inequality, and inequalities (\[MAIN-1\]) and (\[INQ\]), we obtain $$\begin{aligned}
&&\hskip-.168in \frac{1}{2} \frac{d \left( \| \eta \|_2^2+ \| \tt
\|_2^2 \right) }{d t} + \frac{1}{R_1} \left( \| {\nabla}_{H} \eta
\|_2^2+ \| {\nabla}_{H} \tt \|_2^2 \right) + \frac{1}{R_2}
\left( \|\pp_z \eta \|_2^2 + \|\pp_z \tt \|_2^2 \right)
\\
&&\hskip-.165in \leq C \left(\|v\|_6 \| {\nabla}_{H} u \|_3 +
\|u\|_6 \| {\nabla}_{H} v \|_3 + \|u_z\|_2^{\frac{1}{2}} \|
{\nabla}_{H} u_z \|_2^{\frac{1}{2}} \| \tt \|_2^{\frac{1}{2}} \|
{\nabla}_{H} \tt \|_2^{\frac{1}{2}} + \|T\|_{\infty} \| u_z \|_2
\right) \left( \| {\nabla}_{H}
\eta \|_2 +\| {\nabla}_{H} \tt \|_2\right) \\
&&\hskip-.06in +C \|T\|_2 \|\eta\|_2 +C \|{\nabla}_{H} v\|_2 \|T
\|_{\infty} \|\tt\|_2 + C \|v\|_2 \|T \|_{\infty} \|{\nabla}_{H}
\tt\|_2
+ C \|T_z\|_2 \| \tt \|_2 \| {\nabla}_{H} \tt \|_2
+ C \|T\|_{\infty} \| T_z \|_2 \| \tt \|_2 \\
&&\hskip-.06in
+C \|Q\|_2 \|\tt\|_2
+C \|v\|_2 \|{\nabla}_{H} \tt\|_2 + C \|T_z\|_2 \|\tt_z\|_2 \\
&&\hskip-.165in \leq C (\|T\|_3 + \| \eta \|_2^{\frac{1}{2}} \|
{\nabla}_{H} \eta \|_2^{\frac{1}{2}}+ \| \tt \|_2^{\frac{1}{2}} \|
{\nabla}_{H} \tt \|_2^{\frac{1}{2}}) \; \left(\|v\|_6 +\|u\|_6
\right) \;
\left(\| {\nabla}_{H} \eta \|_2 +\| {\nabla}_{H} \tt\|_2\right) + \\
&&\hskip-.06in +C \left( \|u_z\|_2^{1/2} \| {\nabla}_{H} u_z
\|_2^{1/2} \| \tt \|_2^{1/2} \| {\nabla}_{H} \tt \|_2^{1/2} +
\|T\|_{\infty} \| u_z \|_2 + \|T_z\|_2 \| \tt \|_2 +\|v\|_2 \|T
\|_{\infty} \right) \left(\| {\nabla}_{H} \eta \|_2
+\| {\nabla}_{H} \tt\|_2\right) \\
&&\hskip-.06in +C \|T\|_2 \|\eta\|_2 + C \left( \|{\nabla}_{H} v
\|_2 \; \|T \|_{\infty} + \|T\|_{\infty} \| T_z \|_2 \right) \|
\tt \|_2+C \|Q\|_2 \|\tt\|_2+C \|v\|_2 \|{\nabla}_{H} \tt\|_2 + C
\|T_z\|_2 \|\tt_z\|_2.\end{aligned}$$ By Young’s and the Cauchy–Schwarz inequalities, we have $$\begin{aligned}
&&\hskip-.68in \frac{d \left( \| \eta \|_2^2+ \| \tt \|_2^2 \right)
}{d t} + \frac{1}{R_1} \left( \| {\nabla}_{H} \eta \|_2^2+ \|
{\nabla}_{H} \tt \|_2^2 \right) +
\frac{1}{R_2} \left( \|\pp_z \eta \|_2^2 + \|\pp_z \tt \|_2^2 \right) \\
&&\hskip-.65in \leq C \left(1+ \|v\|_6^4 + \|u\|_6^4+ \|T_z\|_2^2 +
\|u_z\|_2^2\|{\nabla}_{H} u_z\|_2^2 \right) \; \left(
\| \eta \|_2^2+ \| \tt \|_2^2 \right) \\
&&\hskip-.5in
+ C \|T\|_2^2 + C \|Q\|_2^2 +C \left( \|v\|_6^2 +
\|u\|_6^2 + \|u_z\|_2^2 +\|v\|_2^2+\|{\nabla}_{H}
v\|_2^2+\|T_z\|_2^2 \right) \left( 1+\|T\|_{\infty}^2\right).\end{aligned}$$ Thanks to (\[K-1\]), (\[K-2\]), (\[K-3\]), (\[K-5\]), (\[K-6\]), and Gronwall inequality, we have $$\begin{aligned}
&&\hskip-.68in \| \eta \|_2^2+ \| \tt \|_2^2 +
\int_0^t \left[ \frac{1}{R_1} \left( \| {\nabla}_{H} \eta \|_2^2+ \| {\nabla}_{H} \tt \|_2^2
\right) +
\frac{1}{R_2} \left( \|\pp_z \eta \|_2^2 + \|\pp_z \tt \|_2^2
\right) \right] \; ds \leq K_7, \label{K-7}\end{aligned}$$ where $$\begin{aligned}
&&\hskip-.68in K_7= C e^{(K_1+K_3^{2/3} +K_5^{2/3} +K_6^2) t} \left[
\|v_0\|_{H^1(\Om)}^2 + K_1 + \|Q\|_2^2+ K_2^2\; (K_2+K_3^{1/3}
+K_5^{1/3} +K_6) \right]. \label{K7}\end{aligned}$$
$\|{\Dd}_{H} \overline{v}\|_{H^1(M)}^2 +
\|{\nabla}_{H}({\nabla}_H^{\perp} \cdot v_z)\|_{H^1(\Om)}^2 + \|
{\nabla}_{H}\left({\nabla}_{H} \cdot v_z + R_1 T\right)
\|_{H^1(\Om)}^2 + \|{\nabla}_{H} T\|_{H^1(\Om)}^2$ [**estimates**]{}
--------------------------------------------------------------------
First, let us observe that $$\begin{aligned}
&&\hskip-.68in
|{\nabla}_{H} v(x,y,z)| \leq |{\nabla}_{H}
\overline{v}(x,y)| + \int_{-h}^0 |{\nabla}_{H} v_z(x,y,z)| \; dz.\end{aligned}$$ Therefore, from the above and (\[ZZZ\]), we have $$\begin{aligned}
&&\hskip-.68in \|{\nabla}_{H} v\|_{\infty} \leq \|{\nabla}_{H}
\overline{v}\|_{\infty} + \left\|\int_{-h}^0 |{\nabla}_{H} u| \; dz
\right\|_{\infty} \\
&&\hskip-.68in
\leq \|{\nabla}_{H} \overline{v}\|_{\infty} + R_1 \int_{-h}^0 \|{\nabla}_{H} \beta\|_{\infty} \;
dz + \left\|\int_{-h}^0 |{\nabla}_{H} \zeta| \; dz
\right\|_{\infty}.\end{aligned}$$ By applying inequality (\[BW-1\]) to ${\nabla}_{H} \overline{v}$ and $\int_{-h}^0
|{\nabla}_{H} \zeta| \; dz$ we reach $$\begin{aligned}
&&\hskip-.68in
\|{\nabla}_{H} \overline{v}\|_{\infty}
\leq C \|{\nabla}_{H} \overline{v}\|_{H^1(M)} \left(1+\log^+ \|{\nabla}_{H}
\overline{v}\|_{H^2(M)}\right)^{1/2}, \label{EE-1} \\
&&\hskip-.58in
\left\|\int_{-h}^0 |{\nabla}_{H} \zeta| \; dz
\right\|_{\infty} \leq C \left\|\int_{-h}^0 |{\nabla}_{H} \zeta|
\; dz \right\|_{H^1(M)}\left(1+ \log^+ \left\|\int_{-h}^0
|{\nabla}_{H} \zeta| \; dz \right\|_{H^2(M)}\right)^{1/2}
\label{EE-2} \\\end{aligned}$$ Applying inequality (\[BW-2\]) to ${\nabla}_{H}
\beta$, also by (\[BETA-1\]) and (\[BETA-2\]), we reach $$\begin{aligned}
&&\hskip-.68in \|{\nabla}_{H} \beta\|_{\infty}
\leq C \left( \|{\nabla}_{H} \cdot \beta \|_{\infty} + \|{\nabla}_H^{\perp} \cdot \beta \|_{\infty} \right)
\left(1+\log^+ \|{\nabla}_{H} \beta\|_{H^2(M)}\right)\leq C
\|T \|_{\infty} \left(1+\log^+ \|T\|_{H^2(M)}\right) \label{EE-3}\end{aligned}$$ Therefore, by (\[EE-1\])–(\[EE-3\]), we infer that $$\begin{aligned}
&&\hskip-.68in \|{\nabla}_{H} v\|_{\infty}
\leq C \|{\nabla}_{H} \overline{v}\|_{H^1(M)} \left(1+\log^+ \|{\nabla}_{H}
\overline{v}\|_{H^2(M)}\right)^{1/2} + C\int_{-h}^0 \left[
\|T \|_{\infty} \left(1+\log^+ \|T\|_{H^2(M)}\right) \right]\; dz
\nonumber \\
&&\hskip-.58in
+ C \left\|\int_{-h}^0 |{\nabla}_{H} \zeta|
\; dz \right\|_{H^1(M)}\left(1+ \log^+ \left\|\int_{-h}^0
|{\nabla}_{H} \zeta| \; dz \right\|_{H^2(M)}\right)^{1/2}
\nonumber \\
&&\hskip-.68in
\leq C \|{\nabla}_{H} \overline{v}\|_{H^1(M)} \left(1+\log^+ \|{\nabla}_{H}
\overline{v}\|_{H^2(M)}\right)^{1/2} + C \|T
\|_{\infty} \left(1+\log^+ \int_{-h}^0 \|T\|_{H^2(M)} \; dz \right)
\nonumber \\
&&\hskip-.58in
+ C \left\|\int_{-h}^0 |{\nabla}_{H} \zeta|
\; dz \right\|_{H^1(M)}\left(1+ \log^+ \left\|\int_{-h}^0
|{\nabla}_{H} \zeta| \; dz \right\|_{H^2(M)}\right)^{1/2}
\nonumber \\
&&\hskip-.68in
\leq C \|{\nabla}_{H} \overline{v}\|_{H^1(M)} \left(1+\log^+ \|{\nabla}_{H}
\overline{v}\|_{H^2(M)}\right)^{1/2} + C \left\|T\right\|_{\infty} \left(1+\log^+
\|{\Dd}_{H} T\|_{L^2(\Om)}\right)
\nonumber \\
&&\hskip-.58in
+ C \left( \| \eta \|_{H^1(\Om)}+\| \tt \|_{H^1(\Om)}\right)
\left[1+ \log^+ \left( \| {\Dd}_{H} \eta \|_{L^2(\Om)}+\|{\Dd}_{H} \tt
\|_{L^2(\Om)}\right)\right]^{1/2}. \label{LOGG}\end{aligned}$$
0.1in
### $\|{\nabla}_{H} {\Dd}_{H} \overline{v}\|_2^2 $ [**estimates**]{}
By taking the ${\Dd}_{H}$ to equation (\[EQ1\]) and then taking the inner product of equation (\[EQ1\]) with $ - {\Dd}_{H}^2 \overline{v}$ in $L^2(M)$, we reach $$\begin{aligned}
&&\hskip-.68in \frac{1}{2} \frac{d \|{\nabla}_{H} {\Dd}_{H}
\overline{v}
\|_{2}^{2} }{d t} + \frac{1}{R_1} \|{\Dd}_{H}^2 \overline{v} \|_2^2 \\
&&\hskip-.68in
= \int_{M} {\Dd}_{H} \left\{
(\overline{v} \cdot {\nabla}_{H} ) \overline{v} + \overline{ \left[
(\widetilde{v} \cdot {\nabla}_{H}) \widetilde{v} + ({\nabla}_{H}
\cdot \widetilde{v}) \; \widetilde{v}\right]} +f_0 \vec{k}\times
\overline{v} \right\} \cdot
{\Dd}_{H}^2 \overline{v} \; dxdy.\end{aligned}$$ Integrating by parts and applying (\[EQ2\]), we obtain $$\begin{aligned}
&&\hskip-.68in \frac{1}{2} \frac{d \|{\nabla}_{H} {\Dd}_{H}
\overline{v}
\|_{2}^{2} }{d t} + \frac{1}{R_1} \|{\Dd}_{H}^2 \overline{v} \|_2^2 \\
&&\hskip-.68in \leq C \int_{M} \left\{|\overline{v}|\;
|{\nabla}_{H}^3 \overline{v}|+ |{\nabla}_{H} \overline{v}|\;
|{\nabla}_{H}^2 \overline{v}| + \int_{-h}^0 \left(
|\widetilde{v}|\;|{\nabla}_{H}^3 \widetilde{v}|+ |{\nabla}_{H}
\widetilde{v}|\; |{\nabla}_{H}^2 \widetilde{v}| \right)\; dz +
|{\Dd}_{H} \overline{v} | \right\}
|{\Dd}_{H}^2 \overline{v}| \; dxdy \\
&&\hskip-.68in \leq C \int_{M} \left\{|\overline{v}|\;
|{\nabla}_{H}^3 \overline{v}|+ |{\nabla}_{H} \overline{v}|\;
|{\nabla}_{H}^2 \overline{v}| + \left( \int_{-h}^0
|\widetilde{v}|\;dz \right) \left( \int_{-h}^0 (|{\nabla}_{H}^3
\zeta| + |{\nabla}_{H}^3 \beta| ) \; dz\right)
\right. \\
&&\hskip-.58in \left. + \left( \int_{-h}^0 (|{\nabla}_{H}
\zeta|+|{\nabla}_{H} \beta|) \; dz \right) \left( \int_{-h}^0 (
|{\nabla}_{H}^2 \zeta|+|{\nabla}_{H}^2 \beta| ) \; dz \right) +
|{\Dd}_{H} \overline{v} | \right\} |{\Dd}_{H}^2 \overline{v}| \;
dxdy.\end{aligned}$$ By applying Hölder inequality, (\[SI-1\]), (\[SI-111\]), \[DIV-CUR\], (\[MAIN-1\]) and (\[INQ\]) to the above estimate, we obtain $$\begin{aligned}
&&\hskip-.168in \frac{1}{2} \frac{d \|{\nabla}_{H} {\Dd}_{H}
\overline{v}
\|_{2}^{2} }{d t} + \frac{1}{R_1} \|{\Dd}_{H}^2 \overline{v} \|_2^2 \\
&&\hskip-.268in \leq C \left\{\|\overline{v}\|_4\; \|{\nabla}_{H}^3
\overline{v}\|_4+ \|{\nabla}_{H} \overline{v}\|_4\; \|{\nabla}_{H}^2
\overline{v}\|_4 + \int_{-h}^0\|\widetilde{v}\|_{\infty}\; dz
\left\|\int_{-h}^0
(|{\nabla}_{H}^3 \zeta| + |{\nabla}_{H}^3 \beta| ) \; dz \right\|_2 \right. \\
&&\hskip-.058in \left. + \left\| \int_{-h}^0 (|{\nabla}_{H}
\zeta|+|{\nabla}_{H} \beta|) \; dz \right\|_4 \left\| \int_{-h}^0 (
|{\nabla}_{H}^2 \zeta|+|{\nabla}_{H}^2 \beta| ) \; dz \right\|_4 +
\|{\Dd}_{H} \overline{v}\|_2 \right\}
\|{\Dd}_{H}^2 \overline{v}\|_2 \\
&&\hskip-.268in \leq C
\left\{\|\overline{v}\|_2^{\frac{1}{2}}\|{\nabla}_{H}
\overline{v}\|_2^{\frac{1}{2}}\; \|{\nabla}_{H}^3
\overline{v}\|_2^{\frac{1}{2}} \|{\Dd}_{H}^2
\overline{v}\|_2^{\frac{1}{2}} + \|{\nabla}_{H} \overline{v}\|_2\;
\|{\nabla}_{H}^3 \overline{v}\|_2 \right. \\
&&\hskip-.058in \left. + \|\widetilde{v}\|_2^{\frac{1}{2}}
\|{\nabla}_{H} \widetilde{v}\|_{\infty}^{\frac{1}{2}} \left(
\|{\Dd}_{H} \eta\|_2
+\|{\Dd}_{H} \tt\|_2 + \|{\Dd}_{H} T\|_2 \right) \right. \\
&&\hskip-.058in \left. + \left( \|\eta\|_2+ \|\tt\|_2+\|T\|_{2}
\right) \left( \|{\nabla}_{H}^2 \eta\|_2+\|{\nabla}_{H}^2
\tt\|_2+\|{\nabla}_{H}^2 T\|_2 \right) +\|{\Dd}_{H} \tt\|_2 \;\;
\right\} \|{\Dd}_{H}^2 \overline{v}\|_2.\end{aligned}$$ Thus, by Young’s and the Cauchy–Schwarz inequalities, we have $$\begin{aligned}
&&\hskip-.68in \frac{d \|{\nabla}_{H} {\Dd}_{H} \overline{v}
\|_{2}^{2} }{d t} + \frac{1}{R_1} \|{\Dd}_{H}^2 \overline{v} \|_2^2 \nonumber
\\
&&\hskip-.58in
\leq C \left( \|\overline{v}\|_2^2\|{\nabla}_{H} \overline{v}\|_2^2
+ \|{\nabla}_{H} \overline{v}\|_2^2 \right) \; \|{\nabla}_{H}
{\Dd}_{H} \overline{v}\|_2^2 \nonumber
\\
&&\hskip-.58in + \left( \|\widetilde{v}\|_2 \|{\nabla}_{H}
\widetilde{v}\|_{\infty} + \|\eta\|_2^2+ \|\tt\|_2^2+\|T\|_{2}^2
\right) \left( \|{\Dd}_{H} \eta\|_2^2+\|{\Dd}_{H}
\tt\|_2^2+\|{\Dd}_{H} T\|_2^2 \right). \label{VB-EST}\end{aligned}$$
### $\|{\Dd}_{H} T\|_2+\|{\nabla}_{H} T_z\|_2$ [**estimates**]{}
By applying the operator ${\Dd}_{H}$ to equation (\[EQ5\]), and then taking the inner product of equation (\[EQ5\]) with $ {\Dd}_{H} T+T_{zz}$ in $L^2(\Om)$, we get $$\begin{aligned}
&&\hskip-.168in \frac{1}{2} \frac{d (\|{\Dd}_{H}
T\|_2^2+\|{\nabla}_{H} T_z\|_2^2)}{dt} +
\frac{1}{R_3}\left(\|{\Dd}_{H} T_z\|_2^2 +\|{\nabla}_{H} T_{zz}\|_2^2 \right) \\
&&\hskip-.165in = - \int_{\Om} {\Dd}_{H} \left[ v \cdot {\nabla}_{H}
T - \left( \int_{-h}^z {\nabla}_{H} \cdot v(x,y, \xi,t) d\xi
\right) \left(\frac{\pp T}{\pp z}+\frac{1}{h}\right) -Q \right] \; {\Dd}_{H} T \; dxdydz \\
&&\hskip-.05in - \int_{\Om} {\nabla}_{H} \left[ v \cdot {\nabla}_{H}
T_z - \left( \int_{-h}^z {\nabla}_{H} \cdot v(x,y, \xi,t) d\xi
\right) \frac{\pp^2 T}{\pp z^2}+ u \cdot {\nabla}_{H} T \right. \\
&&\hskip-.05in \left. - \left(
{\nabla}_{H} \cdot v \right) \left(\frac{\pp T}{\pp
z}+\frac{1}{h}\right)
-Q_z \right] \cdot {\nabla}_{H} T_z \; dxdydz \\
&&\hskip-.165in = - \int_{\Om} \left[ {\Dd}_{H} v \cdot
{\nabla}_{H} T +2 {\nabla}_{H} v \cdot {\nabla}_{H}^2 T - \left(
\int_{-h}^z {\Dd}_{H}({\nabla}_{H} \cdot v(x,y, \xi,t)) d\xi \right)
\left(\frac{\pp T}{\pp
z}+\frac{1}{h}\right) \right. \\
&&\hskip-.05in \left. - 2 \left( \int_{-h}^z {\nabla}_{H}
({\nabla}_{H} \cdot v(x,y, \xi,t)) d\xi
\right) {\nabla}_{H} T_z -{\Dd}_{H} Q \right] \; {\Dd}_{H} T \; dxdydz \\
&&\hskip-.05in - \int_{\Om} \left[ {\nabla}_{H} v \cdot
{\nabla}_{H} T_z - \left( \int_{-h}^z {\nabla}_{H}({\nabla}_{H}
\cdot v(x,y, \xi,t)) d\xi \right)
\frac{\pp^2 T}{\pp z^2} \right. \\
&&\hskip-.05in \left. + {\nabla}_{H} u \cdot {\nabla}_{H} T +u \cdot
{\nabla}_{H}^2 T - {\nabla}_{H} \left( {\nabla}_{H} \cdot v \right)
\left(\frac{\pp T}{\pp z}+\frac{1}{h}\right)-\left( {\nabla}_{H}
\cdot v \right) {\nabla}_{H} T_z
-{\nabla}_{H} Q_z \right] \cdot {\nabla}_{H} T_z \; dxdydz \\
&&\hskip-.165in = - \int_{\Om} \left[ {\Dd}_{H} v \cdot
{\nabla}_{H} T +2 {\nabla}_{H} v \cdot {\nabla}_{H}^2 T -
\frac{1}{h} \left( \int_{-h}^z {\Dd}_{H} ({\nabla}_{H} \cdot v(x,y,
\xi,t)) d\xi \right) + {\Dd}_{H}({\nabla}_{H} \cdot v) T
\right. \\
&&\hskip-.05in \left. + 2 {\nabla}_{H} ({\nabla}_{H} \cdot v) \cdot
{\nabla}_{H} T -{\Dd}_{H} Q \right] \; {\Dd}_{H} T \;
dxdydz \\
&&\hskip-.05in - \int_{\Om} \left[ \int_{-h}^z
{\Dd}_{H}({\nabla}_{H} \cdot v(x,y, \xi,t)) d\xi \; T + 2
\int_{-h}^z {\nabla}_{H} ({\nabla}_{H} \cdot v(x,y, \xi,t)) d\xi
\cdot {\nabla}_{H} T \right] \cdot {\Dd}_{H} T_z \; dxdydz
\\
&&\hskip-.05in - \int_{\Om} \left[ {\nabla}_{H} v \cdot
{\nabla}_{H} T_z + \left( {\nabla}_{H}({\nabla}_{H} \cdot v
)\right)
\frac{\pp T}{\pp z} + {\nabla}_{H} u \cdot {\nabla}_{H} T +u \cdot
{\nabla}_{H}^2 T \right. \\
&&\hskip-.05in \left. - \frac{1}{h} {\nabla}_{H} \left( {\nabla}_{H}
\cdot v \right) + {\nabla}_{H} \left( {\nabla}_{H} \cdot u \right) T
+ \left( {\nabla}_{H} \cdot u \right) {\nabla}_{H} T
-{\nabla}_{H} Q_z \right] \cdot {\nabla}_{H} T_z \; dxdydz \\
&&\hskip-.05in - \int_{\Om} \left[ \left( \int_{-h}^z
{\nabla}_{H}({\nabla}_{H} \cdot v(x,y, \xi,t)) d\xi \right)
\frac{\pp T}{\pp z}+ {\nabla}_{H} \left( {\nabla}_{H} \cdot v
\right) T + \left( {\nabla}_{H} \cdot v \right) {\nabla}_{H} T
\right] \cdot {\nabla}_{H} T_{zz} \; dxdydz.\end{aligned}$$ Thus, $$\begin{aligned}
&&\hskip-.168in \frac{1}{2} \frac{d (\|{\Dd}_{H}
T\|_2^2+\|{\nabla}_{H} T_z\|_2^2)}{dt} +
\frac{1}{R_3}\left(\|{\Dd}_{H} T_z\|_2^2 +\|{\nabla}_{H} T_{zz}\|_2^2 \right)
\\
&&\hskip-.168in \leq C \int_{\Om} \left\{ |{\nabla}_{H} v| |
{\nabla}_{H}^2 T|+ |{\Dd}_{H} v|\, |{\nabla}_{H}
T|+\overline{|{\Dd}_{H} ({\nabla}_{H} \cdot v)|}
+|{\nabla}_{H}^2({\nabla}_{H} \cdot v)|\; |T | + |{\nabla}_{H}
({\nabla}_{H} \cdot v)|\; |{\nabla}_{H} T | \right. \\
&&\hskip-.004in \left. +|{\Dd}_{H} Q| \right\} \; |{\Dd}_{H} T| \;
dxdydz
\\
&&\hskip-.05in + C \int_{\Om}\left\{ \left( \int_{-h}^0 |{\Dd}_{H}
({\nabla}_{H} \cdot v)|\; dz |T |+\int_{-h}^0
|{\nabla}_{H}({\nabla}_{H} \cdot v)|\; dz |{\nabla}_{H} T |
\right) |{\Dd}_{H} T_z| \right\} \; dxdydz \\
&&\hskip-.05in + C \int_{\Om} \left\{ |{\nabla}_{H} v| |
{\nabla}_{H} T_z|+ | {\nabla}_{H}({\nabla}_{H} \cdot v )|\, |T_z| +
| {\nabla}_{H} u |\, |{\nabla}_{H} T|+ |u|\;|{\nabla}_{H}^2 T|
\right. \\
&&\hskip-.005in \left. +|{\nabla}_{H} ({\nabla}_{H} \cdot v)| +|{\nabla}_{H} ( {\nabla}_{H}
\cdot u )|\, |T| +|{\nabla}_{H} Q_z| \right\} \; |{\nabla}_{H}
T_z| \; dxdydz
\\
&&\hskip-.05in + C \int_{\Om} \left[ \left( \int_{-h}^0
|{\nabla}_{H} ({\nabla}_{H}\cdot v)|\; dz \right) |T_z | +
|{\nabla}_{H}({\nabla}_{H} \cdot v)| |T | + |{\nabla}_{H} \cdot v|
|{\nabla}_{H} T | \right] |{\nabla}_{H} T_{zz}| \; dxdydz.\end{aligned}$$ Thanks to (\[ZZZ\])–(\[TTT\]), we obtain $$\begin{aligned}
&&\hskip-.168in \frac{1}{2} \frac{d (\|{\Dd}_{H}
T\|_2^2+\|{\nabla}_{H} T_z\|_2^2)}{dt} +
\frac{1}{R_3}\left(\|{\Dd}_{H} T_z\|_2^2 +\|{\nabla}_{H} T_{zz}\|_2^2 \right) \\
&&\hskip-.168in \leq C \int_{\Om} \left\{ |{\nabla}_{H} v| |
{\nabla}_{H}^2 T|+ \int_{-h}^0 (|{\Dd}_{H} \zeta|+|{\nabla}_{H}
T|)\;dz \, |{\nabla}_{H} T|+\int_{-h}^0 (|{\Dd}_{H}
\tt|+|{\Dd}_{H} T|)\; dz \right. \\
&&\hskip-.05in \left. +\int_{-h}^0 (|{\nabla}_{H}^2 \tt|+|{\nabla}_{H}^2 T|)\, dz \; |T |
+\int_{-h}^0 (|{\nabla}_{H} \tt|
+|{\nabla}_{H} T|)\;dz |{\nabla}_{H} T | +|{\Dd}_{H} Q| \right\} \; |{\Dd}_{H} T| \;
dxdydz
\\
&&\hskip-.05in + C \int_{\Om}\left\{ \left( \int_{-h}^0
(|{\nabla}_{H}^2 \tt|+|{\nabla}_{H}^2 T|) \; dz |T |+\int_{-h}^0
(|{\nabla}_{H} \tt|+|{\nabla}_{H} T|)\; dz |{\nabla}_{H} T |
\right) |{\Dd}_{H} T_z| \right\} \; dxdydz \\
&&\hskip-.05in + C \int_{\Om} \left\{ |{\nabla}_{H} v| |
{\nabla}_{H} T_z|+ \int_{-h}^0 (|{\nabla}_{H} \tt|+|{\nabla}_{H}
T|)\,dz\; |T_z| + | {\nabla}_{H} u |\, |{\nabla}_{H} T| \right. \\
&&\hskip-.004in \left. + |u|\;|{\nabla}_{H}^2 T| +
\int_{-h}^0(|{\nabla}_{H} \tt|+|{\nabla}_{H} T|)\; dz
+(|{\nabla}_{H} \tt|+|{\nabla}_{H} T|)\, |T| +|{\nabla}_{H} Q_z|
\right\} \; |{\nabla}_{H} T_z| \; dxdydz
\\
&&\hskip-.05in + C \int_{\Om} \left[ \left( \int_{-h}^0
(|{\nabla}_{H} \tt|+|{\nabla}_{H} T|)\; dz \right) |T_z | +
(|{\nabla}_{H} \tt|+|{\nabla}_{H} T|) |T | \right. \\
&&\hskip-.004in \left.
+ \int_{-h}^0
(|\tt|+|T|)\; dz |{\nabla}_{H} T | \right] |{\nabla}_{H} T_{zz}| \;
dxdydz.\end{aligned}$$ Using Hölder inequality, and inequalities (\[MAIN-1\]), (\[MAIN-2\]) and (\[INQ\]), we obtain $$\begin{aligned}
&&\hskip-.168in \frac{1}{2} \frac{d (\|{\Dd}_{H}
T\|_2^2+\|{\nabla}_{H} T_z\|_2^2)}{dt} +
\frac{1}{R_3}\left(\|{\Dd}_{H} T_z\|_2^2 +\|{\nabla}_{H}
T_{zz}\|_2^2 \right)
\\
&&\hskip-.168in \leq C \|{\nabla}_{H} v\|_{\infty} \| {\Dd}_{H}
T\|_2^2 + C \|{\Dd}_{H} \zeta\|_2^{1/2}\|{\nabla}_{H} {\Dd}_{H}
\zeta\|_2^{1/2} \|T\|_{\infty}^{1/2} \|{\nabla}_{H}^2 T\|_2^{3/2} +
C \|T\|_{\infty}
\|{\nabla}_{H}^2 T\|_2^2 \\
&&\hskip-.05in + C \|{\nabla}_{H}^2 \tt\|_2\, (1+\|T \|_{\infty})
\| {\Dd}_{H} T\|_2
+ C \|{\nabla}_{H} \tt\|_2^{1/2}\|{\nabla}_{H}^2 \tt\|_2^{1/2}
\|T\|_{\infty}^{1/2} \|{\nabla}_{H}^2 T\|_2^{3/2} + C \|{\Dd}_{H} Q\|_2 \; \|{\Dd}_{H} T\|_2 \\
&&\hskip-.05in
+ C (\|{\nabla}_{H}^2 \tt\|_2+\|{\nabla}_{H}^2 T\|_2) \; \|T \|_{\infty} \|{\Dd}_{H} T_z\|_2
+C \|{\nabla}_{H} \tt\|_2^{1/2}\|{\nabla}_{H}^2 \tt\|_2^{1/2}
\|T\|_{\infty}^{1/2} \|{\nabla}_{H}^2 T\|_2^{1/2} \|{\Dd}_{H}
T_z\|_2
\\
&&\hskip-.05in + C \|{\nabla}_{H} v\|_{\infty} \| {\nabla}_{H}
T_z\|_2^2 +
C \|{\nabla}_{H} \tt\|_2^{1/2}\|{\nabla}_{H}^2 \tt\|_2^{1/2} \|T_z\|_2^{1/2}\|{\nabla}_{H} T_z\|_2^{3/2} \\
&&\hskip-.05in + C \|T\|_{\infty}^{1/2} \|{\nabla}_{H}^2 T\|_2^{1/2}
\|T_z\|_2^{1/2}\|{\nabla}_{H} T_z\|_2^{3/2}
+ C \| {\nabla}_{H} u \|_3\,
\|{\nabla}_{H} T\|_6 \|{\nabla}_{H} T_z\|_2 +
C \|u\|_{\infty} \;\|{\nabla}_{H}^2 T\|_2 \|{\nabla}_{H} T_z\|_2 \\
&&\hskip-.05in +C (\|{\nabla}_{H} \tt\|_2+\|{\nabla}_{H} T\|_2 )(1+ \|T\|_{\infty}) \\
&&\hskip-.05in
+ C \left( \|{\nabla}_{H} \tt\|_2^{1/2}\|{\nabla}_{H}^2 \tt\|_2^{1/2}
+C \|T\|_{\infty}^{1/2} \|{\nabla}_{H}^2 T\|_2^{1/2} \right)
\|T_z\|_2^{1/2}\|{\nabla}_{H} T_z\|_2^{1/2}
\|{\nabla}_{H} T_{zz}\|_2 \\
&&\hskip-.05in
+\|{\nabla}_{H} Q_z\|_2 \;
\|{\nabla}_{H} T_z\|_2 + C (\|{\nabla}_{H} \tt\|_2+\|{\nabla}_{H}
T\|_2)
\|T\|_{\infty} \|{\nabla}_{H} T_{zz}\|_2 \\
&&\hskip-.05in +C \|\tt\|_2^{1/2}\|{\nabla}_{H} \tt\|_2^{1/2}
\|T\|_{\infty}^{1/2} \|{\nabla}_{H}^2 T\|_2^{1/2} \|{\nabla}_{H} T_{zz}\|_2 \\
&&\hskip-.168in \leq C \|{\nabla}_{H} v\|_{\infty} (\| {\Dd}_{H}
T\|_2^2+\| {\nabla}_{H} T_z\|_2^2)
+ C \left( \|{\nabla}_{H} \eta\|_2\ +\|{\nabla}_{H} \tt \|_2\right)^{1/2}
\left( \|{\Dd}_{H} \eta\|_2\ +\|{\Dd}_{H} \tt \|_2\right)^{1/2}
\|T\|_{\infty}^{1/2} \|{\nabla}_{H}^2 T\|_2^{3/2} \\
&&\hskip-.05in +C \left(\|{\Dd}_{H} \eta\|_2 +\|{\Dd}_{H} \tt\|_2
+\|{\Dd}_{H} T\|_2 \right) \left(1+\|T\|_{\infty}\right) \;
\|{\Dd}_{H} T\|_2
+C \|{\Dd}_{H} Q\|_2 \; \|{\Dd}_{H} T\|_2 +C \|{\nabla}_{H} Q_z\|_2 \; \|{\nabla}_{H} T_z\|_2 \\
&&\hskip-.05in
+\left( \|{\nabla}_{H} \eta\|_2^{1/2}\|{\nabla}_{H}^2
\eta\|_2^{\frac{1}{2}}+\|{\nabla}_{H}
\tt\|_2^{\frac{1}{2}}\|{\nabla}_{H}^2
\tt\|_2^{\frac{1}{2}}+\|T\|_{\infty}^{\frac{1}{2}} \|{\Dd}_{H}
T\|_2^{\frac{1}{2}} \right) \; \|T\|_{\infty}^{\frac{1}{2}}
\|{\Dd}_{H} T\|_2^{\frac{1}{2}} \left(\|{\Dd}_{H} T\|_2+\|{\Dd}_{H}
T_z\|_2\right) \\
&&\hskip-.05in +C \|u\|_6 \|{\nabla}_{H}
T_z\|_2^{\frac{1}{2}}\|{\nabla}_{H}^2 T_z\|_2^{\frac{1}{2}}
\|{\Dd}_{H} T \| +C \left[ \|{\nabla}_{H}
\tt\|_2^{\frac{1}{2}}\|{\nabla}_{H}^2 \tt\|_2^{\frac{1}{2}}
\|T_z\|_2^{\frac{1}{2}} \|{\nabla}_{H}
T_z\|_2^{\frac{1}{2}} \right. \\
&&\hskip-.05in + \left. \left(
\|T\|_4+\|\tt\|_2^{1/2}\|{\nabla}_{H} \tt\|_2^{1/2}+ \|T_z\|_2^{1/2}
\|{\nabla}_{H} T_z\|_2^{1/2} \right) \; \|T\|_{\infty}^{1/2}
\|{\Dd}_{H} T\|_2^{1/2} \right] \|{\nabla}_{H} T_{zz}\|_2.\end{aligned}$$ By the Cauchy–Schwarz and Young’s inequalities, we reach $$\begin{aligned}
&&\hskip-.68in \frac{d (\|{\Dd}_{H} T\|_2^2+\|{\nabla}_{H}
T_z\|_2^2)}{dt} +
\frac{1}{R_3}\left(\|{\Dd}_{H} T_z\|_2^2 +\|{\nabla}_{H} T_{zz}\|_2^2 \right) \nonumber \\
&&\hskip-.68in \leq C\|{\nabla}_{H} v\|_{\infty} \left( \|{\Dd}_{H}
T\|_2^2+\|{\nabla}_{H} T_z\|_2^2 \right) + C \left( \|{\nabla}_{H}
\eta\|_2\ +\|{\nabla}_{H} \tt \|_2\right)^{1/2}\left( \|{\Dd}_{H}
\eta\|_2\ +\|{\Dd}_{H} \tt \|_2\right)^{1/2}
\|T\|_{\infty}^{1/2} \|{\Dd}_{H} T\|_2^{3/2} \\
&&\hskip-.58in +C \left(\|{\Dd}_{H} \eta\|_2 +\|{\Dd}_{H} \tt\|_2
+\|{\Dd}_{H} T\|_2 \right) \|T\|_{\infty} \; \|{\Dd}_{H} T\|_2
+C \|{\Dd}_{H} Q\|_2 \; \|{\Dd}_{H} T\|_2 +C \|{\nabla}_{H} Q_z\|_2 \; \|{\nabla}_{H} T_z\|_2 \\
&&\hskip-.58in
+\left( \|{\nabla}_{H} \eta\|_2^{1/2}\|{\Dd}_{H}
\eta\|_2^{\frac{1}{2}}+\|{\nabla}_{H}
\tt\|_2^{\frac{1}{2}}\|{\Dd}_{H}
\tt\|_2^{\frac{1}{2}}+\|T\|_{\infty}^{\frac{1}{2}} \|{\Dd}_{H}
T\|_2^{\frac{1}{2}} \right) \; \|T\|_{\infty}^{\frac{1}{2}}
\|{\Dd}_{H}
T\|_2^{\frac{1}{2}} \|{\Dd}_{H} T\|_2 \\
&&\hskip-.58in
+\left( \|{\nabla}_{H} \eta\|_2\|{\Dd}_{H}
\eta\|_2+\|{\nabla}_{H} \tt\|_2\|{\Dd}_{H} \tt\|_2+\|T\|_{\infty}
\|{\Dd}_{H} T\|_2 \right) \; \|T\|_{\infty} \|{\Dd}_{H} T\|_2 +C
\|u\|_6^3 \left(\|{\nabla}_{H} T_z\|_2^2+ \|{\Dd}_{H} T
\|^2\right) \\
&&\hskip-.58in +C \left[ \|{\nabla}_{H} \tt\|_2\|{\Dd}_{H}
\tt\|_2 \|T_z\|_2 \|{\nabla}_{H} T_z\|_2 + \left(
\|T\|_4^2+\|\tt\|_2\|{\nabla}_{H} \tt\|_2+ \|T_z\|_2 \|{\nabla}_{H}
T_z\|_2 \right) \; \|T\|_{\infty} \|{\Dd}_{H} T\|_2 \right]
\\
&&\hskip-.68in \leq C \|{\Dd}_{H} Q\|_2^2 +C \|{\nabla}_{H}
Q_z\|_2^2+ C \|T\|_{\infty}^{4} + C\|{\nabla}_{H} v\|_{\infty}
\left( \|{\Dd}_{H}
T\|_2^2+\|{\nabla}_{H} T_z\|_2^2 \right) \\
&&\hskip-.58in + C \left(1+ \|{\nabla}_{H} \eta\|_2^2
+\|{\nabla}_{H} \tt \|_2^2 + \|T\|_{\infty}^{2} +\|u\|_6^3
+\|T_z\|_2^2 \right) \left( \|{\Dd}_{H} T\|_2^2+\|{\nabla}_{H}
T_z\|_2^2 +\|{\Dd}_{H} \eta\|_2^2+\|{\Dd}_{H} \tt\|_2^2 \right).\end{aligned}$$ Thus, we get $$\begin{aligned}
&&\hskip-.68in \frac{d (\|{\Dd}_{H} T\|_2^2+\|{\nabla}_{H}
T_z\|_2^2)}{dt} + \frac{1}{R_3}\left(\|{\Dd}_{H} T_z\|_2^2
+\|{\nabla}_{H} T_{zz}\|_2^2 \right) \nonumber
\\
&&\hskip-.68in \leq C \|{\Dd}_{H} Q\|_2^2 +C \|{\nabla}_{H}
Q_z\|_2^2+ C \|T\|_{\infty}^{4} + C\|{\nabla}_{H} v\|_{\infty}
\left( \|{\Dd}_{H}
T\|_2^2+\|{\nabla}_{H} T_z\|_2^2 \right) \nonumber \\
&&\hskip-.58in + C \left(1+ \|{\nabla}_{H} \eta\|_2^2
+\|{\nabla}_{H} \tt \|_2^2 + \|T\|_{\infty}^{2} +\|u\|_6^3
+\|T_z\|_2^2 \right) \left( \|{\Dd}_{H} T\|_2^2+\|{\nabla}_{H}
T_z\|_2^2 +\|{\Dd}_{H} \eta\|_2^2+\|{\Dd}_{H} \tt\|_2^2 \right).
\label{T-EST}\end{aligned}$$
### $\|{\nabla}_{H}({\nabla}_H^{\perp} \cdot v_z)\|_{H^1(\Om)}^2 + \| {\nabla}_{H}\left({\nabla}_{H}
\cdot v_z + R_1 T\right) \|_{H^1(\Om)}^2$ [**estimates**]{}
By acting with ${\Dd}_{H}$ on equation (\[ETA\]) and equation (\[TT\]), then taking the inner product of equation (\[ETA\]) with $ {\Dd}_{H} \eta +\eta_{zz}$ in $L^2$ and equation (\[TT\]) with ${\Dd}_{H} \tt+\tt_{zz}$ in $L^2$, respectively, we get $$\begin{aligned}
&&\hskip-.18in \frac{1}{2} \frac{d \left( \|{\Dd}_{H}
\eta\|_2^2+\|{\nabla}_{H} \eta_{z}\|_2^2+\|{\Dd}_{H}
\tt\|_2^2+\|{\nabla}_{H} \tt_{z}\|_2^2\right)}{dt} \\
&&\hskip-.08in + \frac{1}{R_1}
\left( \| {\nabla}_{H} {\Dd}_{H} \eta \|_2^2 +\| {\nabla}_{H}
{\Dd}_{H} \eta_z \|_2^2 + \| {\nabla}_{H} {\Dd}_{H} \tt \|_2^2+ \|
{\Dd}_{H} \tt_z \|_2^2
\right)\\
&&\hskip-.08in + \frac{1}{R_2}\left( \|{\Dd}_{H} \eta_z \|_2^2
+\|{\nabla}_{H} \eta_{zz} \|_2^2 + \|{\Dd}_{H} \tt_z \|_2^2
+ \|{\nabla}_{H} \tt_{zz} \|_2^2 \right) \\
&&\hskip-.18in = \int_{\Om} {\nabla}_{H} \left\{ {\nabla}_H^{\perp} \cdot
\left[ (v \cdot {\nabla}_{H}) u - \left( \int_{-h}^z {\nabla}_{H}
\cdot v (x,y, \xi,t) d\xi \right) \frac{\pp u}{\pp z} \right. \right. \\
&&\hskip-.08in \left. \left. +(u \cdot {\nabla}_{H} ) v -
({\nabla}_{H} \cdot v) u \right] +f_0\left( R_1\, T - \tt
\right)\right\} \cdot {\nabla}_{H} ({\Dd}_{H} \eta+\eta_{zz}) \;
dxdydz \\
&&\hskip-.08in +\int_{\Om} {\nabla}_{H} \left\{{\nabla}_{H} \cdot
\left[ (v \cdot {\nabla}_{H}) u - \left( \int_{-h}^z {\nabla}_{H}
\cdot v (x,y, \xi,t) d\xi \right) \frac{\pp u}{\pp z} \right. \right. \\
&&\hskip-.0in \left. \left. + (u \cdot {\nabla}_{H} ) v -
({\nabla}_{H} \cdot v) u \right]\, +f_0\,\eta \right\} \cdot
{\nabla}_{H} ({\Dd}_{H} \tt+\tt_{zz})
\; dxdydz \\
&&\hskip-.08in - \int_{\Om} \left\{ R_1\left[ {\nabla}_{H} v \cdot
{\nabla}_{H} T+ v \cdot {\nabla}_{H}^2 T - \left( \int_{-h}^z
{\nabla}_{H} ({\nabla}_{H}
\cdot v) (x,y, \xi,t) d\xi \right) \left( \frac{\pp T}{\pp
z}+\frac{1}{h}\right) \right. \right. \\
&&\hskip-.08in \left. \left.
- {\nabla}_{H} Q +\, \left(\frac{1}{R_3}-\frac{1}{R_2}\right) {\nabla}_{H} T_{zz} \right] \,\right\}
\cdot {\nabla}_{H} ({\Dd}_{H} \tt+\tt_{zz}) \; dxdydz
\\
&&\hskip-.08in - \int_{\Om} \left[ \left( \int_{-h}^z {\nabla}_{H}
({\nabla}_{H}
\cdot v) (x,y, \xi,t) d\xi \right) {\nabla}_{H} T_z \; {\Dd}_{H} \tt +
\left( \int_{-h}^z {\nabla}_{H}
\cdot v (x,y, \xi,t) d\xi \right) {\nabla}_{H} T_z \cdot {\nabla}_{H} \tt_{z}) \right] \; dxdydz
\\
&&\hskip-.18in \leq C \int_{\Om} \left\{ \left(
|u|\;|{\nabla}_{H}^3 v| +|{\nabla}_{H} u|\;|{\nabla}_{H}^2 v|
+|{\nabla}_{H}^2 u|\;|{\nabla}_{H} v| +|{\nabla}_{H}^3 u|
|v| +|u_z|\;\int_{-h}^0|{\nabla}_{H}^2 ({\nabla}_{H} \cdot v)|\; dz \right. \right. \\
&&\hskip-.08in \left. \left.+|{\nabla}_{H}
u_z|\;\int_{-h}^0|{\nabla}_{H} ({\nabla}_{H} \cdot v)|\; dz
+|{\nabla}_{H}^2 u_z|\;\int_{-h}^0|{\nabla}_{H} \cdot v|\; dz
\right) \; \left(|{\nabla}_{H} {\Dd}_{H} \eta|+|{\nabla}_{H}
\eta_{zz}| +|{\nabla}_{H} {\Dd}_{H} \tt| +|{\nabla}_{H} \tt_{zz}|
\right) \right. \\
&&\hskip-.08in \left. + C \left[ |{\nabla}_{H} v| |{\nabla}_{H} T| +
|v| |{\nabla}_{H}^2 T|+(1+|T_z|)\;\int_{-h}^0|{\nabla}_{H}
({\nabla}_{H} \cdot v)|\; dz
+|{\nabla}_{H} T_z|\;\int_{-h}^0|{\nabla}_{H} \cdot v|\; dz \right. \right. \\
&&\hskip-.08in \left. \left. +|{\nabla}_{H} Q| +
\left|\frac{R_1}{R_3}-\frac{R_1}{R_2}\right| |{\nabla}_{H} T_{zz}|
\right] (|{\nabla}_{H} {\Dd}_{H} \tt|+|{\nabla}_{H} \tt_{zz}|)
\right\} \;
dxdydz +C\left( \|{\Dd}_{H} T\|_2^2+\, \|{\Dd}_{H} \eta\|_2^2+\, \|{\Dd}_{H} \tt\|_2^2 \right) \\
&&\hskip-.18in \leq C \int_{\Om} \left\{ \left[ |u|\;\int_{-h}^0
(|{\nabla}_{H}^3 \zeta|+ |{\nabla}_{H}^3 \beta|) \; dz
+(|{\nabla}_{H} \zeta|+|{\nabla}_{H} \beta|) \;\int_{-h}^0
(|{\nabla}_{H}^2 \zeta|+ |{\nabla}_{H}^2 \beta|) \; dz +
(|{\nabla}_{H}^3 \zeta|+|{\nabla}_{H}^3 \beta|)|v| \right. \right. \\
&&\hskip-.08in +(|{\nabla}_{H}^2 \zeta|+|{\nabla}_{H}^2 \beta|)
\;\int_{-h}^0 (|{\nabla}_{H} \zeta|+ |{\nabla}_{H} \beta|) \; dz +
|u_z| \; \int_{-h}^0
(|{\nabla}_{H}^2 \tt|+ |{\nabla}_{H}^2 T|) \; dz \\
&&\hskip-.08in +(|{\nabla}_{H} \zeta_z|+|{\nabla}_{H} \beta_z|)
\;\int_{-h}^0(|{\nabla}_{H} \tt|+|{\nabla}_{H} T|)\; dz \\
&&\hskip-.08in \left. +(|{\nabla}_{H}^2 \zeta_z|+|{\nabla}_{H}^2
\beta_z|)\;\int_{-h}^0(|\tt|+|T|)\; dz \right] \;
\left(|{\nabla}_{H} {\Dd}_{H} \eta|+|{\nabla}_{H} \eta_{zz}|
+|{\nabla}_{H} {\Dd}_{H} \tt| +|{\nabla}_{H} \tt_{zz}|
\right) \\
&&\hskip-.08in + C \left[ |{\nabla}_{H} T| \;
\int_{-h}^0(|{\nabla}_{H}
\zeta|+|{\nabla}_{H} \beta|)\; dz + |v| |{\nabla}_{H}^2 T| \right. \\
&&\hskip-.08in +(1+|T_z|)\;\int_{-h}^0(|{\nabla}_{H}
\tt|+|{\nabla}_{H} T|)\; dz +|{\nabla}_{H}
T_z|\;\int_{-h}^0(|\tt|+|T|)\; dz \\
&&\hskip-.08in \left. \left. +|{\nabla}_{H} Q| +
\left|\frac{R_1}{R_3}-\frac{R_1}{R_2}\right| \; |{\nabla}_{H}
T_{zz}| \right] (|{\nabla}_{H} {\Dd}_{H} \tt|+|{\nabla}_{H}
\tt_{zz}|) \right\}\; dxdydz +C\left( \|{\Dd}_{H} T\|_2^2+\,
\|{\Dd}_{H} \eta\|_2^2+\, \|{\Dd}_{H} \tt\|_2^2 \right).\end{aligned}$$ Using Hölder inequality, and inequalities (\[MAIN-1\]), (\[MAIN-2\]) and (\[INQ\]), we obtain $$\begin{aligned}
&&\hskip-.18in \frac{1}{2} \frac{d \left( \|{\Dd}_{H}
\eta\|_2^2+\|{\nabla}_{H} \eta_{z}\|_2^2+\|{\Dd}_{H}
\tt\|_2^2+\|{\nabla}_{H} \tt_{z}\|_2^2\right)}{dt} \\
&&\hskip-.08in + \frac{1}{R_1}
\left( \| {\nabla}_{H} {\Dd}_{H} \eta \|_2^2 +\| {\nabla}_{H}
{\Dd}_{H} \eta_z \|_2^2 + \| {\nabla}_{H} {\Dd}_{H} \tt \|_2^2+ \|
{\Dd}_{H} \tt_z \|_2^2
\right) \\
&&\hskip-.08in + \frac{1}{R_2}\left( \|{\Dd}_{H} \eta_z \|_2^2
+\|{\nabla}_{H} \eta_{zz} \|_2^2 + \|{\Dd}_{H} \tt_z \|_2^2 +
\|{\nabla}_{H} \tt_{zz} \|_2^2 \right)
\\
&&\hskip-.18in \leq C \left(\|{\nabla}_{H} {\Dd}_{H}
\eta\|_2+\|{\nabla}_{H} \eta_{zz}\|_2 +\|{\nabla}_{H} {\Dd}_{H}
\tt\|_2 +\|{\nabla}_{H} \tt_{zz}\|_2 \right)
\left[ \left(\|u\|_{\infty} +\|v\|_{\infty}\right)\;
\left(\|{\nabla}_{H}^2 \eta\|_2+\|{\nabla}_{H}^2
\tt\|_2+\|{\nabla}_{H}^2 T\|_2\right) \right.
\\
&&\hskip-.08in + \left(\| \eta\|_2^{\frac{1}{2}}\|{\nabla}_{H}
\eta\|_2^{\frac{1}{2}}+ \| \tt\|_2^{\frac{1}{2}}\|{\nabla}_{H}
\tt\|_2^{\frac{1}{2}}+ \|T\|_{\infty} \right) \;\left(\|{\nabla}_{H}
\eta\|_2^{\frac{1}{2}}\|{\nabla}_{H}^2 \eta\|_2^{\frac{1}{2}}+
\|{\nabla}_{H} \tt\|_2^{\frac{1}{2}}\|{\nabla}_{H}^2
\tt\|_2^{\frac{1}{2}}+ \| T\|_{\infty}^{\frac{1}{2}}\|{\nabla}_{H}^2
T\|_2^{\frac{1}{2}}\right)
\\
&&\hskip-.08in +\left(\|{\nabla}_{H}
\eta\|_2^{\frac{1}{2}}\|{\nabla}_{H}^2 \eta\|_2^{\frac{1}{2}}+
\|{\nabla}_{H} \tt\|_2^{\frac{1}{2}}\|{\nabla}_{H}^2
\tt\|_2^{\frac{1}{2}}+ \|T\|_{\infty}^{\frac{1}{2}}\|{\nabla}_{H}^2
T\|_2^{\frac{1}{2}} \right) \left(\|
\eta\|_2^{\frac{1}{2}}\|{\nabla}_{H} \eta\|_2^{\frac{1}{2}}+ \|
\tt\|_2^{\frac{1}{2}}\|{\nabla}_{H} \tt\|_2^{\frac{1}{2}}+
\|T\|_{\infty} \right)
\\
&&\hskip-.08in
+ \left(\|\tt_z\|_2^{\frac{1}{2}}\|{\nabla}_{H} \tt_z\|_2^{\frac{1}{2}}
+ \|T_z\|_2^{\frac{1}{2}}\|{\nabla}_{H} T_z\|_2^{\frac{1}{2}} \right)
\left(\|{\nabla}_{H} \tt\|_2^{\frac{1}{2}}\|{\nabla}_{H}^2 \tt\|_2^{\frac{1}{2}}
+ \|T\|_{\infty}^{\frac{1}{2}}\|{\nabla}_{H}^2 T\|_2^{\frac{1}{2}} \right)
\\
&&\hskip-.08in +\|u_z\|_2^{\frac{1}{2}}\|{\nabla}_{H}
u_z\|_2^{\frac{1}{2}} \;
\left(\|{\nabla}_{H} \tt\|_2^{\frac{1}{2}}\|{\nabla}_{H}^2 \tt\|_2^{\frac{1}{2}}
+ \|T\|_{\infty}^{\frac{1}{2}}\|{\nabla}_{H}^2 T\|_2^{\frac{1}{2}} \right)
\\
&&\hskip-.08in \left. +
\left(\|{\nabla}_{H} \tt_z\|_2^{\frac{1}{2}}\|{\nabla}_{H}^2 \tt_z\|_2^{\frac{1}{2}}
+ \|{\nabla}_{H} T_z\|_2^{\frac{1}{2}}\|{\nabla}_{H}^2 T_z\|_2^{\frac{1}{2}} \right)
\left(\|\tt\|_2^{\frac{1}{2}}\|{\nabla}_{H} \tt\|_2^{\frac{1}{2}}
+ \|T\|_{\infty} \right) \right] \;
\\
&&\hskip-.08in + C \left[ \left(\|
\eta\|_2^{\frac{1}{2}}\|{\nabla}_{H} \eta\|_2^{\frac{1}{2}}+ \|
\tt\|_2^{\frac{1}{2}}\|{\nabla}_{H} \tt\|_2^{\frac{1}{2}}+ \|T\|_4
\right) \;\|T\|_{\infty}^{\frac{1}{2}}\|{\Dd}_{H}
T\|_2^{\frac{1}{2}} + \|v\|_{\infty} \|{\Dd}_{H} T\|_2
\right. \\
&&\hskip-.08in \left. +\left( \|{\nabla}_{H}
\tt\|_2^{\frac{1}{2}}\|{\nabla}_{H}^2 \tt\|_2^{\frac{1}{2}}+
\|T\|_{\infty}^{\frac{1}{2}} \|{\Dd}_{H} T\|_2^{\frac{1}{2}} \right)
\;\|T_z\|_2^{\frac{1}{2}}\|{\nabla}_{H} T_z\|_2^{\frac{1}{2}}
\right. \\
&&\hskip-.08in \left. +\left( \|\tt\|_2^{\frac{1}{2}}\|{\nabla}_{H}
\tt\|_2^{\frac{1}{2}}+ \|T\|_{\infty} \right) \;\|{\nabla}_{H}
T_z\|_2^{\frac{1}{2}}\|{\Dd}_{H}
T_z\|_2^{\frac{1}{2}}+\|{\nabla}_{H} \tt\|_2
+\|{\nabla}_{H} T\|_2 +\|{\nabla}_{H} Q\|_2 \right. \\
&&\hskip-.08in \left. + \left|\frac{R_1}{R_3}-\frac{R_1}{R_2}\right|
\|{\nabla}_{H} T_{zz}\|_2 \right] \left(\|{\nabla}_{H} {\Dd}_{H}
\tt\|_2+\|{\nabla}_{H} \tt_{zz}\|_2\right)+C\left( \|{\Dd}_{H}
T\|_2^2+\, \|{\Dd}_{H} \eta\|_2^2+\, \|{\Dd}_{H} \tt\|_2^2 \right)\end{aligned}$$ By Young’s inequality and Cauchy–Schwarz inequality, we have $$\begin{aligned}
&&\hskip-.148in \frac{d \left( \|{\Dd}_{H}
\eta\|_2^2+\|{\nabla}_{H} \eta_{z}\|_2^2+\|{\Dd}_{H}
\tt\|_2^2+\|{\nabla}_{H} \tt_{z}\|_2^2\right)}{dt} + \frac{1}{R_1}
\left( \| {\nabla}_{H} {\Dd}_{H} \eta \|_2^2 +\| {\nabla}_{H}
{\Dd}_{H} \eta_z \|_2^2 + \| {\nabla}_{H} {\Dd}_{H} \tt \|_2^2+ \|
{\Dd}_{H} \tt_z \|_2^2
\right) \nonumber \\
&&\hskip-.038in + \frac{1}{R_2}\left( \|{\Dd}_{H} \eta_z \|_2^2
+\|{\nabla}_{H} \eta_{zz} \|_2^2 + \|{\Dd}_{H} \tt_z \|_2^2
+ \|{\nabla}_{H} \tt_{zz} \|_2^2 \right) \nonumber \\
&&\hskip-.145in \leq C \left(\|{\Dd}_{H} T\|_2^2+\|{\nabla}_{H}
T_{z}\|_2^2+ \|{\Dd}_{H} \eta\|_2^2+\|{\nabla}_{H}
\eta_{z}\|_2^2+\|{\Dd}_{H} \tt\|_2^2+\|{\nabla}_{H}
\tt_{z}\|_2^2\right) \left[1+ \|T\|_{\infty}^4+ \|{\nabla}_{H}
u_z\|^2_{2}+\|u_{zz}\|^2_{2}
\right. \nonumber \\
&&\hskip-.038in \left. +\|{\nabla}_{H}
\eta\|^2_{2}+\|\eta_{z}\|^2_{2}
+\|{\nabla}_{H}\tt\|^2_{2}+\|\tt_{z}\|^2_{2}
+ \|T_z\|_2^2 +\|\eta\|_2^2 \|{\nabla}_{H} \eta\|_2^2 +\|\tt\|_2^2 \|{\nabla}_{H} \tt\|_2^2
\right]
\nonumber \\
&&\hskip-.038in + \| \eta\|_2^2\|{\nabla}_{H} \eta\|_2^2+ \|
\tt\|_2^2 \|{\nabla}_{H} \tt\|_2^2 + \|T\|_{\infty}^4
+\|u_z\|_2^2 \|{\nabla}_{H} u_z\|_2^2 + \|{\nabla}_{H} Q\|_2^2\nonumber \\
&&\hskip-.038in + \frac{R_1^2(R_1+R_2)(R_2-R_3)^2}{R_2^2R_3^2}
\|{\nabla}_{H} T_{zz}\|_2^2. \label{U-EST}\end{aligned}$$ Next, we will obtain an estimate for $$\|{\nabla}_{H} {\Dd}_{H} \overline{v}\|_2^2 +
\|{\nabla}_{H}({\nabla}_H^{\perp} \cdot v_z)\|_{H^1(\Om)}^2 + \|
{\nabla}_{H}\left({\nabla}_{H} \cdot v_z + R_1 T \right)
\|_{H^1(\Om)}^2 + C_R \|{\nabla}_{H} T\|_{H^1(\Om)}^2,$$ where $C_R = \frac{2R_1^2(R_1+R_2)(R_2-R_3)^2}{R_2^2R_3}.$ Denote by $$\begin{aligned}
&&\hskip-.68in \mathcal{X}=1+\|{\nabla}_{H} {\Dd}_{H}
\overline{v}\|_2^2+C_R \, \|{\Dd}_{H} T\|_2^2+ C_R \, \|{\nabla}_{H}
T_{z}\|_2^2+ \|{\Dd}_{H} \eta\|_2^2+\|{\nabla}_{H}
\eta_{z}\|_2^2+\|{\Dd}_{H} \tt\|_2^2+\|{\nabla}_{H} \tt_{z}\|_2^2, \\
&&\hskip-.68in \mathcal{Y}=\|{\Dd}_{H}^2
\overline{v}\|_2^2+\|{\Dd}_{H} T_z\|_2^2+\|{\nabla}_{H}
T_{zz}\|_2^2+ \|{\nabla}_{H}{\Dd}_{H} \eta\|_2^2+\|{\Dd}_{H}
\eta_{z}\|_2^2+\|{\nabla}_{H}{\Dd}_{H} \tt\|_2^2+\|{\Dd}_{H}
\tt_{z}\|_2^2.\end{aligned}$$ Thus, by (\[LOGG\]), we get $$\|{\nabla}_{H} v\|_{\infty} \leq C \left( \|{\nabla}_{H}
\overline{v}\|_{H^1(M)} + \left\|T\right\|_{\infty} +\| \eta
\|_{H^1(\Om)}+\| \tt \|_{H^1(\Om)} \right) \; \log \mathcal{X}.$$ By virtue of (\[VB-EST\]), (\[T-EST\]), (\[U-EST\]), Young’s inequality, Cauchy–Schwarz inequality and the above, we obtain $$\begin{aligned}
&&\hskip-.68in \frac{d \mathcal{X}}{dt} + C\mathcal{Y} \leq C
\|v\|_2\;\left( \|{\nabla}_{H} \overline{v}\|_{H^1(M)} +
\left\|T\right\|_{\infty} +\| \eta \|_{H^1(\Om)}+\| \tt
\|_{H^1(\Om)} \right) \; \mathcal{X} \;
\log \mathcal{X} \\
&&\hskip-.5in + \left[1+ \|T\|_{\infty}^4+ \|T_z\|_2^2+
\|\overline{v}\|_2^2\left(1+\|\overline{v}\|_{H^1(M)}^2\right) \right. \\
&&\hskip-.5in \left. +\|u_z\|^2_{H^1}
+\left(1+\|\eta\|_2^2\right) \|\eta\|_{H^1}^2 +\left(1+\|\tt\|_2^2\right)
\|\tt\|_{H^1}^2
\right] \; \mathcal{X}
\\
&&\hskip-.5in + C \left\{ \|{\Dd}_{H} Q\|_2^2+\|{\nabla}_{H}
Q_z\|_2^2 + \| \eta\|_2^2\|{\nabla}_{H} \eta\|_2^2+ \| \tt\|_2^2
\|{\nabla}_{H} \tt\|_2^2 + \|T\|_{\infty}^4
+\|u_z\|_2^2 \|{\nabla}_{H} u_z\|_2^2 \right\}.\end{aligned}$$ Let $\mathcal{X}=e^{\mathcal{Z}}.$ Then, $\frac{d \mathcal{X}}{dt}
= e^{\mathcal{Z}} \; \frac{d \mathcal{Z}}{dt}.$ As a result we have $$\begin{aligned}
&&\hskip-.68in \frac{d \mathcal{Z}}{dt} \leq C \|v\|_2\;\left(
\|{\nabla}_{H} \overline{v}\|_{H^1(M)} + \left\|T\right\|_{\infty}
+\|
\eta \|_{H^1(\Om)}+\| \tt \|_{H^1(\Om)} \right) \; \mathcal{Z} \nonumber \\
&&\hskip-.5in + \left[1+\|{\Dd}_{H} Q\|_2^2+\|{\nabla}_{H}
Q_z\|_2^2 + \|T\|_{\infty}^4+ \|T_z\|_2^2+
\|\overline{v}\|_2^2\left(1+\|\overline{v}\|_{H^1(M)}^2\right) \right. \nonumber \\
&&\hskip-.5in \left. +\left(1+\|u_z\|_2^2\right) \|u_z\|^2_{H^1}
+\left(1+\|\eta\|_2^2\right) \|\eta\|_{H^1}^2 +\left(1+\|\tt\|_2^2\right) \|\tt\|_{H^1}^2
\right]. \label{F-EST}\end{aligned}$$ Thanks to Gronwall inequality, and the estimates established in the previous subsections, we get $$\begin{aligned}
&&\hskip-.68in \mathcal{Z} \leq K, \qquad \mathcal{X} \leq e^K
\label{K-F}\end{aligned}$$ where $$\begin{aligned}
&&\hskip-.68in K= e^{C(K_1+K_2+K_7)} \left[1+ \|v_0\|_{H^4}^2+\|
T_0\|_{H^2}^2 + t+\|{\Dd}_{H} Q\|_2^2\; t +\|{\nabla}_{H}
Q_z\|_2^2\; t \right]. \label{KF}\end{aligned}$$ Moreover, we have $$\begin{aligned}
&&\hskip-.68in \int_0^t \mathcal{Y} \; ds \leq K_8,
\label{K-8}\end{aligned}$$ where $$\begin{aligned}
&&\hskip-.68in K_8= e^{C(K_1+K_2+K_7)} \left[1+ \|v_0\|_{H^4}^2+\|
T_0\|_{H^2}^2 + t+\|{\Dd}_{H} Q\|_2^2\; t +\|{\nabla}_{H}
Q_z\|_2^2\; t \right]. \label{K8}\end{aligned}$$ Thanks to (\[K-6\]) and the above we conclude that the quantities $ \int_0^t
\|v_{zz}(s)\|_{H^1(\Om)}^2 \; ds$, $\int_0^t\|{\Dd}_{H} \nabla_H v_z(s)\|_{H^1(\Om)}^2
\; ds $, $\|{\Dd}_{H} v_z(t)\|_{H^1(\Om)},
\|\nabla_H T (t)\|_{H^1(\Om)},$ and $ \int_0^t
\|\nabla_H T_z (s)\|_{H^1(\Om)}^2 \; ds$ are all bounded uniformly in time, $t$, over the interval $[0,\mathcal{T}_{*})$. Therefore, the strong solution $(v(t), T(t))$ exists globally in time.
Uniqueness of the Solutions {#S-4}
============================
In this section we state and prove the global existence and uniqueness of the strong solution of system (\[EQV\])–(\[EQ9\]).
\[T-MAIN\] Suppose that $Q \in H^2(\Om)$. Then for every $v_0
\in H^4(\Om)$, $T_0\in H^2(\Om)$ and $\mathcal{T}>0,$ there is a unique solution $(v,
p_s, T)$ of system [*(\[EQV\])–(\[EQ9\])*]{} with $$\begin{aligned}
&& {\Dd}_{H} v_z, \;\; \nabla_H T \in C([0,\mathcal{T}], H^1(\Om)), \\
&& v_{zz}, \;\;{\Dd}_{H} \nabla_H v_z, \;\; {\nabla}_{H} T_z \in L^2 ([0,\mathcal{T}], H^1(\Om)),\end{aligned}$$
0.05in
In Theorem \[T-MAIN\] of the previous section we proved that the strong solutions exist globally in time. Therefore, it remains to prove the uniqueness of strong solutions, and their continuous dependence on initial data, in the sense specified by equation (\[LAST\]) below. Let $(v_1, (p_s)_1, T_1)$ and $(v_2, (p_s)_2, T_2)$ be two strong solutions of system (\[EQV\])–(\[EQ9\]) with initial data $((v_0)_1, (T_0)_1)$ and $((v_0)_2, (T_0)_2)$, respectively. Denote by $\phi=v_1-v_2, q_s =(p_s)_1
-(p_s)_2, \psi= T_1-T_2.$ It is clear that $$\begin{aligned}
&&\hskip-.68in \frac{\pp \phi}{\pp t} + L_1 \phi + (v_1 \cdot
{\nabla}_{H}) \phi + (\phi \cdot {\nabla}_{H}) v_2 - \left(
\int_{-h}^z {\nabla}_{H} \cdot v_1(x,y, \xi,t) d\xi \right)
\frac{\pp \phi}{\pp z} - \left( \int_{-h}^z
{\nabla}_{H} \cdot \phi(x,y, \xi,t) d\xi \right) \frac{\pp v_2}{\pp z} \nonumber \\
&&\hskip-.6in +f_0 \vec{k}\times \phi + {\nabla}_{H} q_s -
{\nabla}_{H} \left( \int_{-h}^z
\psi (x,y,\xi,t) d\xi \right) =0, \label{UEQ1} \\
&&\hskip-.68in
\frac{\pp \psi}{\pp t} -\frac{1}{R_3} \, \psi_{zz} + v_1 \cdot {\nabla}_{H} \psi + \phi
\cdot {\nabla}_{H} T_2 \nonumber \\
&&\hskip-.6in - \left( \int_{-h}^z {\nabla}_{H} \cdot v_1(x,y,
\xi,t)
d\xi
\right) \frac{\pp \psi }{\pp z} - \left( \int_{-h}^z {\nabla}_{H}
\cdot \phi(x,y, \xi,t)
d\xi
\right) \left(\frac{\pp T_2}{\pp z} +\frac{1}{h}\right) = 0,
\label{UEQ2}\end{aligned}$$ with initial data $$\begin{aligned}
&&\hskip-.68in \phi(x,y,z,0) = (v_0)_1-(v_0)_2, \label{UEQ3} \\
&&\hskip-.68in \psi (x,y,z,0) = (T_0)_1-(T_0)_2. \label{UEQ4}\end{aligned}$$ Taking the inner product of equation (\[UEQ1\]) with $\phi$ in $L^2(\Om)$, and equation (\[UEQ2\]) with $\psi$ in $L^2(\Om)$, we get $$\begin{aligned}
&&\hskip-.268in \frac{1}{2} \frac{d \|\phi\|_2^2}{dt}
+ \frac{1}{R_1} \|{\nabla}_{H} \phi\|_2^2 + \frac{1}{R_2}\|\phi_z\|_2^2 \\
&&\hskip-.265in = - \int_{\Om} \left[ (v_1 \cdot {\nabla}_{H})
\phi + (\phi \cdot {\nabla}_{H}) v_2 - \left( \int_{-h}^z
{\nabla}_{H} \cdot v_1(x,y, \xi,t) d\xi \right) \frac{\pp \phi}{\pp
z} - \left( \int_{-h}^z {\nabla}_{H} \cdot \phi(x,y, \xi,t) d\xi
\right) \frac{\pp v_2}{\pp z} \right] \cdot \phi
\; dxdydz \\
&&\hskip-.16in
+ \int_{\Om} \left[ f_0 \vec{k}\times \phi + {\nabla}_{H} q_s - {\nabla}_{H} \left( \int_{-h}^z
\psi (x,y,\xi,t) d\xi \right) \right] \cdot \phi \; dxdydz,\end{aligned}$$ and $$\begin{aligned}
&&\hskip-.168in \frac{1}{2} \frac{d \|\psi\|_2^2}{dt} +
\frac{1}{R_3}\|\psi_z\|_2^2 = - \int_{\Om} \left[ v_1
\cdot {\nabla}_{H} \psi + \phi
\cdot {\nabla}_{H} T_2 \right. \\
&&\hskip-.16in \left. - \left( \int_{-h}^z {\nabla}_{H} \cdot
v_1(x,y, \xi,t)
d\xi
\right) \frac{\pp \psi }{\pp z} - \left( \int_{-h}^z {\nabla}_{H}
\cdot \phi(x,y, \xi,t)
d\xi
\right) \left(\frac{\pp T_2}{\pp z} +\frac{1}{h}\right) \right]\; \psi \; dxdydz.\end{aligned}$$ Notice that $$\begin{aligned}
&&\hskip-.65in f_0 \vec{k}\times \phi \cdot \psi =0.
\label{DUT-1}\end{aligned}$$ Integrating by parts, and using the boundary conditions (\[EQ6\]) and (\[EQ7\]), we have $$\begin{aligned}
&&\hskip-.65in - \int_{\Om} \left( (v_1 \cdot {\nabla}_{H}) \phi -
\left( \int_{-h}^z {\nabla}_{H} \cdot v_1(x,y, \xi,t) d\xi \right)
\frac{\pp \phi}{\pp z} \right) \cdot \phi \; dxdydz =0, \label{DUU-1} \\
&&\hskip-.65in - \int_{\Om} \left( v_1 \cdot {\nabla}_{H} \psi
- \left( \int_{-h}^z {\nabla}_{H} \cdot v_1(x,y, \xi,t) d\xi \right)
\frac{\pp \psi}{\pp z} \right) \cdot \psi \; dxdydz =0.
\label{DUT-11}\end{aligned}$$ Integrating by parts, and using the boundary conditions (\[EQ6\]) and (\[EQ7\]), we get $$\begin{aligned}
&&\hskip-.65in \int_{\Om} \left[ {\nabla}_{H} q_s - {\nabla}_{H}
\left( \int_{-h}^z \psi
(x,y,\xi,t) d\xi \right) \right] \cdot \phi \; dxdydz \nonumber \\
&&\hskip-.65in = \int_{\Om} \left( \int_{-h}^z \psi (x,y,\xi,t)
d\xi \right) \; \left({\nabla}_{H} \cdot \phi \right) \; dxdydz.
\label{DUU-2}\end{aligned}$$ Thus, by (\[DUT-1\])–(\[DUU-2\]) we have $$\begin{aligned}
&&\hskip-.68in \frac{1}{2} \frac{d \|\phi\|_2^2}{dt} + \frac{1}{R_1}
\|{\nabla}_{H} \phi\|_2^2 + \frac{1}{R_2}\|\phi_z\|_2^2 = -
\int_{\Om} (\phi \cdot {\nabla}_{H}) v_2 \cdot \phi \;
dxdydz \\
&&\hskip-.65in + \int_{\Om} \int_{-h}^z {\nabla}_{H} \cdot
\phi(x,y, \xi,t) d\xi \frac{\pp v_2}{\pp z} \cdot \phi \; dxdydz +
\int_{\Om} \left( \int_{-h}^z \psi (x,y,\xi,t) d\xi \right) \;
\left({\nabla}_{H} \cdot \phi \right) \; dxdydz.\end{aligned}$$ and $$\begin{aligned}
&&\hskip-.68in \frac{1}{2} \frac{d \|\psi\|_2^2}{dt}+
\frac{1}{R_3}\|\psi_z\|_2^2 = - \int_{\Om} (\phi \cdot
{\nabla}_{H}
T_2 ) \psi \; dxdydz \\
&&\hskip-.65in -\int_{\Om} \left( {\nabla}_{H} \cdot \phi \right)
T_2 \psi \; dxdydz- \int_{\Om} \int_{-h}^z {\nabla}_{H} \cdot
\phi(x,y, \xi,t) d\xi T_2 \psi_z \; dxdydz.\end{aligned}$$ Notice that by Hölder inequality and (\[MAIN-1\]) $$\begin{aligned}
&&\hskip-.68in \left| \int_{\Om} (\phi \cdot {\nabla}_{H}) v_2 \cdot
\phi \; dxdydz \right| \leq \left| \int_{\Om} |v_2|\,|\phi| \,
|{\nabla}_{H}\phi| \; dxdydz \right| \leq \| v_2\|_{6}
\|\phi\|_2^{\frac{1}{2}} \|{\nabla}_{H} \phi\|_2^{3/2}, \label{U1}
\\
&&\hskip-.68in \left | \int_{\Om} \phi \cdot {\nabla}_{H} T_2 \;
\psi \; dxdydz \right| \leq \|{\nabla}_{H} T_2\|_3 \|\psi \|_2
\|\phi\|_6\leq C \|{\nabla}_{H} T_2\|_{H^1} \|\psi \|_2
\|{\nabla}_{H} \phi\|_2; \label{U-1}
\\
&&\hskip-.68in \left| \int_{\Om} \int_{-h}^z {\nabla}_{H} \cdot
\phi(x,y, \xi,t) d\xi \frac{\pp v_2}{\pp z} \cdot \phi \; dxdydz
\right|
\nonumber \\
&&\hskip-.68in \leq C \|{\nabla}_{H} \phi\|_2 \left\|\frac{\pp
v_2}{\pp z} \right\|_6 \|\phi\|_3 \leq C \left\|\frac{\pp
v_2}{\pp z} \right\|_6 \|\phi\|_2^{\frac{1}{2}}\|{\nabla}_{H}
\phi\|_2^{3/2}; \label{U3}
\\
&&\hskip-.68in \left| \int_{\Om} \int_{-h}^z {\nabla}_{H} \cdot
\phi(x,y, \xi,t) d\xi \frac{\pp T_2}{\pp z} \psi \; dxdydz \right|
\leq \left| \int_{\Om} \int_{-h}^0 |{\nabla}_{H}\phi| dz
\int_{-h}^0 \left|\frac{\pp T_2}{\pp z} \psi\right| dz \; dxdy
\right|
\nonumber\\
&&\hskip-.68in \leq C \left| \int_{\Om} \int_{-h}^0
|{\nabla}_{H}\phi| dz \left(\int_{-h}^0 \left|\frac{\pp T_2}{\pp
z}\right|^2\, dz\right)^{1/2} \left(\int_{-h}^0 |\psi|^2\,
dz\right)^{1/2} \; dxdy \right|
\nonumber \\
&&\hskip-.68in \leq C \left\| \int_{-h}^0 \left|\frac{\pp T_2}{\pp
z}\right|^2\, dz \right\|_{\infty}^{1/2} \|{\nabla}_{H} \phi\|_2
\|\psi\|_2\leq C \left\|\frac{\pp {\Dd}_{H} T_2}{\pp z}\right\|_{2}
\|{\nabla}_{H} \phi\|_2 \|\psi\|_2. \label{U-3}\end{aligned}$$ Therefore, by estimates (\[U1\])–(\[U-3\]), we reach $$\begin{aligned}
&&\hskip-.268in \frac{1}{2} \frac{d
\left(\|\phi\|_2^2+\|\psi\|_2^2\right) }{dt} + \frac{1}{R_1}
\|{\nabla}_{H} \phi\|_2^2 + \frac{1}{R_2}\|\phi_z\|_2^2
+ \frac{1}{R_3}\|\psi_z\|_2^2 \\
&&\hskip-.265in \leq C \left( \|v_2\|_{6} \|\phi\|_2^{\frac{1}{2}}
\|{\nabla}_{H} \phi\|_2^{3/2}+\|{\nabla}_{H} T_2\|_{H^1} \|\psi \|_2
\|{\nabla}_{H} \phi\|_2 + \| \pp_z v_2\|_6 \|\phi\|_2^{\frac{1}{2}}
\|{\nabla}_{H} \phi\|_2^{3/2} \right) + C \left\|\frac{\pp {\Dd}_{H}
T_2}{\pp z}\right\|_{2} \|{\nabla}_{H} \phi\|_2 \|\psi\|_2.\end{aligned}$$ By Young’s inequality and the Cauchy–Schwarz inequality, we get $$\begin{aligned}
&&\hskip-.68in \frac{d \|\phi\|_2^2 +\|\psi(t)\|_2^2}{dt} \leq C
\left( \|v_2\|_{6}^4+\|{\nabla}_{H} T_2\|_{H^1}^2+ \| \pp_z
v_2\|_6^{4} + \left\|\frac{\pp {\Dd}_{H} T_2}{\pp z}\right\|_{2}^2
\right) \left( \|\phi\|_2^2 +\|\psi\|_2^2 \right).\end{aligned}$$ Thanks to Gronwall inequality, we obtain $$\begin{aligned}
&&\hskip-.68in \|\phi(t)\|_2^2 +\|\psi(t)\|_2^2
\leq \left( \|\phi(t=0)\|_2^2 +\|\psi(t=0)\|_2^2 \right) \times \\
&&\hskip-.6in \exp \left\{ C \int_0^t \left( \|v_2
(s)\|_{6}^4+\|{\nabla}_{H} T_2(s)\|_{H^1}^2+ \| \pp_z v_2(s)\|_6^{4}
+ \left\|\frac{\pp {\Dd}_{H} T_2}{\pp z}\right\|_{2}^2 \right) \; ds
\right\}.\end{aligned}$$ As a result of (\[K-F\]), we have $$\begin{aligned}
&&\hskip-.68in \|\phi(t)\|_2^2 +\|\psi(t)\|_2^2 \leq \left(
\|\phi(t=0)\|_2^2 +\|\psi(t=0)\|_2^2 \right) \, \exp\left\{ C \left(
(K_3^{2/3} +K + K_4^{2/3})\; t + K_8 \right) \right\}. \label{LAST}\end{aligned}$$ The above inequality proves the continuous dependence of the solutions on the initial data, and in particular, when $\phi(t=0)=0$ and $\psi(t=0)=0$, we have $\phi(t)=0$ and $\psi(t)=0,$ for all $t\ge 0$. Therefore, the strong solution is unique.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors are thankful to the kind hospitality of the Institute for Mathematics and its Applications (IMA), University of Minnesota, where part of this work was completed. This work was supported in part by the NSF grants no. DMS-0709228, DMS-0708832 and DMS-1009950. The work of C.C. was also supported in part by the Florida International University Foundation. E.S.T. also acknowledges the warm hospitality of the Freie Universität - Berlin and the partial support of the Alexander von Humboldt Stiftung/Foundation and the Minerva Stiftung/Foundation.
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[^1]: **
|
---
abstract: 'A new class of self-consistent 6-D phase space stationary distributions is constructed both analytically and numerically. The beam is then mismatched longitudinally and/or transversely, and we explore the beam stability and halo formation for the case of 3-D axisymmetric beam bunches using particle-in-cell simulations. We concentrate on beams with bunch length-to-width ratios varying from 1 to 5, which covers the typical range of the APT linac parameters. We find that the longitudinal halo forms first for comparable longitudinal and transverse mismatches. An interesting coupling phenomenon — a longitudinal or transverse halo is observed even for very small mismatches if the mismatch in the other plane is large — is discovered.'
author:
- |
A.V. Fedotov, R.L. Gluckstern, University of Maryland, College Park, MD 20742, USA\
, R.D. Ryne, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
title: 'HALO FORMATION IN SPHEROIDAL BUNCHES WITH SELF-CONSISTENT STATIONARY DISTRIBUTIONS'
---
Introduction
============
High-intensity applications of ion linacs, such as the transformation of radioactive waste, the tritium production [@APT], and drivers for spallation neutron sources [@SNS], require peak beam currents up to 100 mA with final energies about 1 GeV and beam losses below 1 ppm. Understanding mechanisms of intense-beam losses, in particular, beam instabilities and halo formation, is of primary importance to satisfy these stringent requirements.
Most efforts in halo formation study have been concentrated so far on 2-D (and often axisymmetric, essentially 1-D) beams, see [@Reiser] and references therein. While it produced some analytical results for the simplest case, the K-V distribution, for more realistic distributions particle-core model and particle-in-cell (PIC) simulations have been used, [@RLG94]-. As was recognized from these studies, an rms mismatch of the beam to the focusing channel is the main cause of the halo formation.
To single out and explore the mechanism of halo formation associated with the beam rms mismatch, it is important to start from an initial distribution that satisfies the Vlasov-Maxwell equations and, therefore, remains stationary for the matched case. A beam with some initial [*non*]{} stationary distribution will evolve from its initial state even being rms-matched to the channel, due to redistribution effects (its evolution is caused by mismatches in higher moments). For 2-D axisymmetric beams, a set of stationary distributions with a sharp beam edge was constructed and explored in : $$f_n(H) = \left \{ \begin{array}{cc}
N_n n (H_0-H)^{n-1} & \mbox{for } H \leq H_0 \ , \\
0 & \mbox{for } H > H_0 \ , \label{fn}
\end{array} \right.$$ where $H$ is the hamiltonian of the transverse motion, $H_0=const$, and $N_n$ are normalization constants. The set includes the K-V distribution as a formal limit of $n \to 0$, as well as more realistic ones, like waterbag ($n=1$) and other distributions, with higher non-linearities in space-charge forces. In this paper, we present results of a similar program in the 3-D case. More details can be found in [@HaloSt].
Stationary 3-D Distribution
===========================
Analytical Consideration
------------------------
We consider a smoothed external focusing with gradients $k_z$, $k_y$, $k_x$. In general, the beam bunch can be chosen to have an approximately ellipsoidal boundary. For simplicity, we concentrate on the axisymmetric case $(k_x = k_y)$, for which the bunch is approximately spheroidal. Our axisymmetric 6-D phase space distribution is $$\begin{aligned}
f(\mbox{\boldmath $R$},\mbox{\boldmath $p$})
= N(H_0 - H)^{-1/2} \ , \quad \mbox{ where} \label{f3d} \\
H = k_xr^2/2 + k_zz^2/2 +
e\Phi_{sc}(\mbox{\boldmath $R$}) + mv^2/2 \ . \label{H3d}\end{aligned}$$ Here $\mbox{\boldmath $p$} = m\mbox{\boldmath $v$}$, $r^2 = x^2 + y^2$, and $\Phi_{sc}(\mbox{\boldmath $R$})$ is the electrostatic potential due to the space charge. We work in the bunch Lorentz frame, where all motion is non-relativistic.
The distribution (\[f3d\]) is analogous to (\[fn\]) with $n=1/2$. Since all its dependence on the coordinates is through the hamiltonian $H=H(\mbox{\boldmath $R$},\mbox{\boldmath $p$})$, which is an integral of motion, the distribution is stationary. The same would be true for other exponents in (\[f3d\]); however, for the particular case of -1/2, the Poisson equation in 3-D case is linear. Namely, it can be written as $$\nabla^2 G(\mbox{\boldmath $R$}) = -k_s +
\kappa^2 G(\mbox{\boldmath $R$}), \label{Geq}$$ where $k_s = 2k_x + k_z$, $\kappa^2 = (eQ/\epsilon_0)/\int d\mbox{\boldmath $R$}
G(\mbox{\boldmath $R$})$, $Q$ is the bunch charge, and $$G(\mbox{\boldmath $R$}) \equiv H_0 - k_xr^2/2 -
k_zz^2/2 - e\Phi_{sc}(\mbox{\boldmath $R$}) \ .$$
The solution to Eq. (\[Geq\]) for a spheroidal shaped bunch can be written in the spherical coordinates $R$, $\theta$ ($\cos \theta \ = z/R ,\ \sin \theta = r/R$) as $G(\mbox{\boldmath $R$}) = (k_s/\kappa^2) g(\mbox{\boldmath $R$})$, where $$g(\mbox{\boldmath $R$}) = 1 + \sum^{\infty}_{\ell = 0} \alpha_{\ell}
P_{2\ell} (\cos \theta ) i_{2\ell} (\kappa R) \ . \label{gser}$$ Here $P_{2\ell} (\cos \theta )$ are the even Legendre polynomials and $i_{2\ell} (\kappa R)$ are the spherical Bessel functions (regular at $\kappa R = 0$) of imaginary argument. Since $g(\mbox{\boldmath $R$})$ is proportional to the charge density, the bunch edge is determined by the border $g(\mbox{\boldmath $x$}) = 0$, closest to the origin. We choose $\alpha_{\ell}$’s to approximate a spheroidal surface with semiaxis $a$ in the transverse direction and $c$ in the longitudinal one, $r^2/a^2 + z^2/c^2 = 1$.
From the equations of motions, we express the rms tune depressions as $$\eta^2_{x,{\rm rms}} \equiv \frac{m\langle \dot{x}^2\rangle}
{k_x\langle x^2\rangle} \ , \ \eta^2_{z,{\rm rms}} \equiv
\frac{m\langle \dot{z}^2\rangle}{k_z\langle z^2\rangle} \ .$$ Note also that $m\langle \dot{x}^2\rangle = m\langle \dot{y}^2\rangle
=m\langle \dot{z}^2\rangle = m\langle v^2\rangle/3$, because $H$ depends only on $v^2$ and $\mbox{\boldmath $R$}$. Thus our choice of the form $f(H)$ [*automatically*]{} corresponds to equipartition (equal average kinetic energy in the three spatial directions). The values of $\alpha_{\ell}$ in Eq. (\[gser\]) for given $c/a$ and $\kappa a$ are found by minimizing $\oint ds g^2(\mbox{\boldmath $R$})$ along the boundary. For a fixed bunch shape $c/a$, the rms tune depressions depend on the dimensionless parameter $\kappa a$ (see in [@HaloSt]). A contour plot of $g(\mbox{\boldmath $R$})$ for a typical case $c/a = 3$, $\kappa a = 3.0$ is shown in Fig. 1. This range of parameters corresponds to the Accelerator Production of Tritium (APT) project [@APT].
Numerical Investigation
-----------------------
A 3-D particle-in-cell (PIC) code has been developed to test the analytic model of normal modes [@HaloSt] in the distribution Eq. (\[f3d\]) and to explore halo formation. The single-particle equations of motion are integrated using a symplectic, split-operator technique. The space charge calculation uses area weighting (“Cloud-in-Cell”) and implements open boundary conditions with the Hockney convolution algorithm. The code runs on parallel computers (we mostly used T3E machine at NERSC), and in particular, the space charge calculation has been optimized for parallel platforms. Up to $2.5 \cdot 10^7$ particles have been used in our simulation runs, with $10^6$ being a typical number.
Initially, the 6-D phase space is populated according to Eq. (\[f3d\]), and then the $x,y,z$ coordinates are mismatched by factors $\mu_x = \mu_y = 1 + \delta a/a$, $\mu_z = 1 + \delta c/c$ and the corresponding momenta by $1/\mu_x = 1/\mu_y$, $1/\mu_z$. Simulations show that an initially matched distribution remains stable even for very strong space charge. Introducing some initial mismatch leads to the oscillations of the core, and later on the beam halo develops, as shown in Fig. 2. This figure shows maximal values $z_{max}$ and $x_{max}$ of the longitudinal and transverse coordinates (in units of $a$) of the bunch particles versus time, for the case $\mu_x = \mu_z=\mu$. The jumps of $z_{max}$ and $x_{max}$ correspond to the halo formation moments; after that the distribution stabilizes. One can see that the longitudinal halo develops earlier than the transverse one for equal mismatches in both directions. This is in accordance with our expectations since the longitudinal tune depression is lower for an elongated bunch.
Choosing larger mismatch either longitudinally or transversely, one can observe primarily the longitudinal or transverse halo, respectively. Results of a systematic study for different bunch shapes $c/a$ and mismatch parameters are summarized below, first for the longitudinal case.
We define the halo extent as a ratio of the halo maximal size to that of a matched distribution. The [**longitudinal halo**]{} extent is found to be approximately linearly proportional to the mismatch. In addition, the ratio $z_{max}/(\mu c)$ slightly increases for stronger space charge, from 1.2–1.3 for $\eta_z$ above 0.5 to 1.4–1.5 for $\eta_z < 0.4$. The halo intensity, defined roughly as the fraction of particles outside the bunch core, was also found depending primarily on the mismatch. Large mismatches (40% and higher) lead to several percent of the particles in the halo, which is clearly outside acceptable limits for high-current machines. Obviously, serious efforts should be made to match the beam to the channel as accurately as possible.
For a fixed mismatch, the halo starts to develop earlier for more severe tune depression. Another interesting observation is that for purely longitudinal mismatches ($\mu_x=1$) in elongated bunches ($c/a>2$) the longitudinal halo intensity shows a strong dependence on the mismatch. The number of particles in the halo drops dramatically with $\mu_z > 1$ decreasing; in fact, we see no halo for $\mu_z < 1.2$. A similar threshold behavior was observed in 2-D case .
The extent of the [**transverse halo**]{} has a similar linear dependence on the mismatch: $x_{max}/(\mu a)$ depends weakly on $\eta_x$, just slightly increasing from 1.4–1.5 for $\eta_x$ around 0.8 to 1.6–1.8 for $\eta_x < 0.4$. Again, the halo intensity is governed primarily by the mismatch. In general, the transverse halo closely duplicates all the features observed for non-linear stationary distributions in 2-D simulations . The only two differences seen are related to the moment and rate of halo development: first, in 3-D simulations it clearly starts earlier for severe tune depression, which was not the case in 2-D; and second, the transverse halo in the 3-D case develops significantly faster than in 2-D for comparable mismatches and tune depressions.
Our 3-D simulations clearly show the [**coupling**]{} between the longitudinal and transverse motion: a transverse or longitudinal halo is observed even for a very small mismatch (less than 10%) as long as there is a significant mismatch in the other plane. For example, in Fig. 3 we see a longitudinal halo for only 5% longitudinal mismatch, when $\mu_x = \mu_y = 1.5$. The coupling effect is noticeable even for modest mismatches. We mentioned above that $\mu_z \ge 1.2$ is required to observe a longitudinal halo when $\mu_x = 1$. However, when there is a mismatch in all directions, the halo develops even for $\mu_z = \mu_x =
\mu_y = 1.1$ (10% mismatch in all directions). Such a behavior clearly shows the importance of the coupling effect.
Summary and Discussion
======================
Unlike previous studies of 2-D models of long beams, this paper addresses the beam stability and halo formation in a bunched beam with the parameters in the range of new high-current linac projects [@APT; @SNS]. A new class of 6-D phase space stationary distributions for a beam bunch in the shape of a prolate spheroid has been constructed, analytically and numerically. Our choice of parameters automatically assures equipartition. We therefore study the halo development in 3-D bunches which are in thermal equilibrium, without masking effects of the initial-state redistribution. Such an approach allows us to investigate the major mechanism of halo formation associated with the beam mismatch.
Using our PIC code with smoothed linear external focusing forces, by introducing an initial mismatch in the transverse and/or longitudinal directions we find that both transverse and longitudinal halos can develop, depending on the values of tune depressions and mismatches. An interesting new result is that, due to the coupling between the $r$ and $z$ planes, a transverse or longitudinal halo is observed for a mismatch less than 10% if the mismatch in the other plane is large. Our main conclusion is that the longitudinal halo is of great importance because it develops earlier than the transverse one for elongated bunches with comparable mismatches in both planes. In addition, its control could be challenging. This conclusion agrees with the results from the particle-core model in spherical bunches.
We expect only small quantitative differences for distributions (\[f3d\]) with other exponents, not -1/2, based on results for the set (\[fn\]) in 2-D . More interesting are 3-D effects due to the phase-space redistribution of an initial non-stationary state. Our preliminary results from PIC simulations show that the redistribution process can produce the beam halo in the same fashion as a small rms mismatch [@DifDistr]. A similar conclusion was made for 2-D axysymmetric beams [@Jam; @Okam]. In 3-D, however, the effect can be amplified by the coupling, especially noticeable in the bunches with $c/a$ close to 1.
The authors would like to acknowledge support from the U.S. Department of Energy, and to thank R.A. Jameson and T.P. Wangler for useful discussions.
[99]{} APT Conceptual Design Report, LA-UR-97-1329, Los Alamos, NM, 1997. SNS Conceptual Design Report, NSNS-CDR-2/V1, Oak Ridge, TN, 1997. M. Reiser, Theory and Design of Charged Particle Beams, Wiley, New York (1994). R.L. Gluckstern, Phys. Rev. Lett. [**73**]{}, 1247 (1994). R.A. Jameson, in ‘Frontiers of Accelerator Technology’, World Scient., Singapore, 1996, p. 530. S.Y. Lee and A. Riabko, Phys. Rev. E [**51**]{}, 1609 (1995). T.P. Wangler, et al, in Proceed. of LINAC96, Geneva, Switzerland (1996). - CERN 96-07, p.372. R.L. Gluckstern, W-H. Cheng, S.S. Kurennoy and H. Ye, Phys Rev. E [**54**]{}, 6788 (1996). H. Okamoto and M. Ikegami, Phys. Rev. E [**55**]{}, 4694 (1997). R.L. Gluckstern and S.S. Kurennoy, in Proceed. of PAC97, Vancouver, BC, Canada (1997). R.L. Gluckstern, A.V. Fedotov, S.S. Kurennoy and R.D. Ryne, Univ. of Maryland, Phys. Dept. preprint 98-107, College Park, MD, 1998; submitted to Phys. Rev. E. J.J. Barnard and S.M. Lund, I & II, in Proceed. of PAC97, Vancouver, BC, Canada (1997). A.V. Fedotov, R.L. Gluckstern, S.S. Kurennoy and R.D. Ryne, Univ. of Maryland, Phys. Dept. preprint 98-108, College Park, MD, 1998; to be published.
|
---
author:
- |
Gaurav Paruthi\
\
\
Enrique Frias-Martinez\
\
\
- |
Vanessa Frias-Martinez\
\
\
bibliography:
- 'dev.bib'
title: The Role of Rating and Loan Characteristics in Online Microfunding Behaviors
---
Introduction
============
[*Microfinance institutions (MFIs)*]{} like the Grameen Bank or the Banco do Nordeste in Brazil are non-profit organizations that give small loans to low-income borrowers, typically at low interest rates. Their main aim is to contribute to the socioeconomic development of the regions where they operate while remaining financially sound. In recent years, there has been a significant growth in the number of online microlending sites that connect individuals to small businesses led by low-income citizens. This type of social lending platforms allow individuals from all over the world to explore large online databases of businesses that require small loans to succeed; and citizens without access to formal banking systems to borrow the money they need to carry out their projects. Understanding lending and borrowing activity is critical to improve the way microfinance services work. However, although there exists an important body of work regarding traditional (offline) microfinance, the research in the area of online microlending is much more limited. Previous work in traditional microlending has covered various research questions including the relationship between gender and loan reimbursement [@helena] or the impact that MFI ratings have on total investments and growth [@gutierrez; @sufi]. Nevertheless, there only exist a few studies that focus on the analysis of [*online*]{} microlending platforms [@li; @desai]. However, many aspects that have been typically addressed in traditional microlending studies such as the role that MFI ratings or teams play in the lending process, have not been fully analyzed online.
In this paper, we propose an in-depth study of lending behaviors in Kiva using a mix of quantitative and large-scale data mining techniques. Kiva is a non-profit organization that offers an online platform to connect lenders with borrowers. Their site, [*kiva.org*]{}, allows citizens to microlend small amounts of money to entrepreneurs (borrowers) from different countries. The borrowers are always affiliated with a Field Partner (FP) which can be a microfinance institution (MFI) or other type of local organization that has partnered with Kiva. Field partners give loans to selected businesses based on their local knowledge regarding the country, the business sector including agriculture, health or manufacture among others, and the borrower. Our objective is to understand the relationship between lending activity and various features offered by the online platform. Specifically, we focus on two research questions: (i) the role that MFI ratings play in driving lending activity and (ii) the role that various loan features have in the lending behavior. The first question analyzes whether there exists a relationship between the MFI ratings – that lenders can explore online – and their lending volumes. The second research question attempts to understand if certain loan features – available online at Kiva– such as the type of small business, the gender of the borrower, or the loan’s country information might affect the way lenders lend.
We carry out our analysis with a dataset collected by Schaaf [*et al.*]{} in combination with the loan-lender data snapshots provided by Kiva and a set of socioeconomic indicators provided by the World Bank Open Data website. The resulting four-month dataset contains over a million different lending actions ($1,217,627$), $47,790$ loans and $263,121$ unique lenders. We expect that this analysis will provide Kiva and other similar microlending platforms with findings and techniques to better cater to their lenders so as to facilitate and enhance lending activity.
MFI Rating and lending activity {#rq1method}
===============================
The first research question focuses on understanding the relationship between Field Partner ratings and lending activity. For that purpose, we analyze the relationship between the number of lending actions to specific loans and the rating of the FPs associated to those loans.
Kiva statistics reveal that the distribution of loans per rating is not homogeneous with a large number of loans associated to Field Partners with ratings three ($>140K$) and zero ($>80K$), while other FP ratings have considerably smaller number of loans ($ \approx 50K$, on average). Similarly, a large number of Field Partners ($90$) have ratings between $2.5$ and $3.5$ while higher and lower ratings are associated to fewer FPs. For example, there exist only $24$ FPs with ratings between $1$ and $2$ or $31$ between $4$ and $5$. In order to account for these differences and to be able to compare and correlate lending actions across ratings, we define [*normalized lending actions*]{} as the total number of lending actions to a given rating divided by the number of loans associated to FPs with that rating: $l.a_{r} = \frac{l.a_{r}}{loans_{r}}$ where $l.a_{r}$ represents the total number of individual lending actions to FPs with rating $r$ and $loans_{r}$ the total number of loans offered by FPs with rating $r$.
To test the relationship between lending activity and ratings, we compute the correlation coefficients between the sum of all individual lending actions per rating ($l.a_{r}$) and the ratings themselves. Figure \[fig:FPratings\] shows the normalized activity for each rating value. Since Kiva expresses the ratings as zero or a number between one and five in increments of $0.5$, the sample size for the correlation analysis is only ten. Given that small size, we cannot guarantee normality or linearity. For that reason we compute correlations for both parametric (Pearson’s) and non-parametric (Spearman’s rank) tests. Our analysis shows a strong positive correlation between the two with a correlation coefficient of $r(8)=0.78$ and p-value of $p=0.006$. Similarly, Spearman’s rank produced a correlation coefficient of $\rho(8)=0.74$ (with $p=0.01$). Additionally, we also performed a linear regression on the ratings to see how predictive these are of the lending activity. We obtained an $F(1,8)=10.24$ with $p=0.01$ and an adjusted $R^{2}=0.61$. Thus, the tests determine that the trend in Figure \[fig:FPratings\] approximately follows a monotonic linear trend. This means that the higher the rating of a Field Partner, the larger the number of lending actions we observe. In fact, lenders appear to be more prone to lend to loans managed by Field Partners with higher ratings. Similar results have been reported in one-to-one lending systems [@sufi].
![Normalized lending actions per FP rating.[]{data-label="fig:FPratings"}](figs/fpratingsC2.eps)
To better characterize the relationship between lending activity and ratings, we are also interested in understanding what type of lenders are more prone to lend to highly rated Field Partners. Specifically, we seek lending patterns that exclusively characterize lenders whose lending activity is mostly focused on high FP ratings. For that purpose, we will first model each individual lender in our dataset with six different lending features: (1) number of loans that the lender has lent money to, (2) invitee count, (3) number of days since she has been a member at Kiva, (4) average team size of the teams the lender is a member of, (5) number of distinct FP’s the lender has lent to and (6) entropy of the lender’s lending behavior.
Variables one to five are computed straightforward from the four-month dataset. As for the lending entropy, we compute it as the Kolmogorov Complexity of the four-month lending actions’ time series for each individual [@kolmogorov]. Higher complexity values are associated to burstier behaviors where lending patterns are harder to model as opposed to low complexity values which we associate to more stable, planned lending behaviors [*e.g.,*]{} lenders that lend approximately once every two weeks will have lower complexity values than those who lend more unpredictably.
![Box plot for median individual loan counts versus ratings. Boxes represent the 1st and 3rd quartiles and values outside the box are values within 1.5 the interquartile distance (1.5\*Q3-Q1).[]{data-label="fig:loancount"}](figs/loancountC3.eps)
1.5cm ![Box plots for median individual complexity values that characterize lending patterns.[]{data-label="fig:complexity"}](figs/complex2.eps "fig:")
To carry out this analysis, we characterize each individual in our dataset with the six lending features. For each feature and rating, we compute their median values and compare them. For simplicity purposes, we consider five different ratings: zero, one (\[1,2)), two (\[2,3)), three (\[3,4)) and four (\[4,5\]). Our results reveal differences for two features: (1) loan count and (6) complexity. Figure \[fig:loancount\] and Figure \[fig:complexity\] show the mean loan count and the mean lending complexity for lenders with a majority of lending actions on one of the five ratings, respectively. We observe that lenders whose lending activity mostly focuses on FP’s with ratings four or higher, appear to lend to a larger number of loans while showing lower lending complexity than lenders that focus their activity on lower ratings. In fact, it appears that lenders that concentrate on higher ratings might have more stable lending behaviors probably implying regularly planned lending decisions. On the other hand, lenders whose majority lending actions are mostly focused on FPs with lower ratings, appear to lend to fewer loans and their behaviors are far more complex, which might reveal burstier, more impulsive behavior. Without claiming causality, these results might suggest that offering more highly ranked Field Partners on Kiva’s website could also potentially increase lending activity. Additionally, given that planned lending decisions appear to be related to higher lending volumes, Kiva could offer planning tools to lenders such as lending calendars or lending reminders, which might also help to increase the lending activity of their users.
Loan Features
=============
Our second research question seeks to understand the relationship between lending activity and features that characterize a loan including: (1) country of the loan, (2) sector: agriculture, retail or health among others, (3) size of the loan: individual or group-based and (4) gender of the borrowers in the loan. To eliminate the bias we normalize the number of lending actions to a loan feature (country, sector, gender or size) by dividing the total number of lending actions by the number loans associated to that feature as: $l.a_{f} = \frac{l.a_{f}}{loans_{f}} \;with\; f\in\{c,s,g,z\}$ where $f$ represents one of the loan features: the country $c$, the sector $s$, the gender $g$ or the size $z$ of the loan; $l.a_{f}$ the number of lending actions to loans with feature $f$ and $loans_{f}$ the total number of loans with that feature.
To evaluate how lending actions and the country of the loan relate to each other, we first characterize each borrower country by a set of over $1000$ socioeconomic indicators extracted from the World’s Bank Open Data website. Next, we compute Pearson’s correlations between the total number of lending actions ($l.a_{c}$) to each borrower country $c$ and the values for each socioeconomic variable. Given the large number of correlations performed (over $1000$), we need to adjust the p-values [*i.e.,*]{} control for the Type I error (False Positives). Bonferroni correction is one of the most common approaches to adjust for multiple testing. However, given the large set of correlations that we perform, Bonferroni’s p-values would be too stringent [*e.g.,*]{} for a Type I error rate of 0.01, it would require a $p > 0.00001$. For that reason, we apply instead the False Discovery Rate (FDR) which controls the fraction of positive detections that are wrong. Specifically, we use the Benjamini-Yekutieli’s FDR adjustment and report the correlation results together with their p-values and their q-values [*i.e.,*]{} percentage of false discoveries accepted for that test, also known as adjusted p-values [@benjamini2].
Table \[table:loanfeatures\] shows some of the most relevant findings (the majority with $q \leq 0.05$). We observe a significant positive correlation between a country’s urban population and the lending activity it receives (recall that lending activity is normalized by the number of loans in the country). It appears that countries with a larger urban population have a higher probability of benefiting from lending than countries that are more rural. To promote more heterogeneous lending activity, Kiva could explore putting online [*lending recommendations*]{} to drive lending activity towards countries that benefit the least at each moment in time. Interestingly, we also observe a negative correlation between lending activity towards a country and its agricultural added value. This shows that Kiva lenders are lending more to countries whose agricultural production is poor and in need (the smaller the production, the larger the lending activity).
Other correlations indicate that larger lending activity is associated to countries with less manufacturing which is indirectly related to job creation. In fact, the lack of manufacturing industries can negatively impact the creation of jobs. As a result, Kiva appears to be successfully driving lending activity towards countries where job creation is harder to achieve. On the other hand, lending activity is negatively correlated to the strength of legal rights in the country, which might reveal a lending activity focused on supporting development and indirectly the improvement of freedom and rights in borrower countries.
To understand better lending actions, it is important to realize that lenders can also be influenced by the socioeconomic conditions of their own countries. In an attempt to disentangle which factors play a role, we also analyze the relationship between borrower countries and the lending activity of lender countries characterized by their socioeconomic indicators. For that purpose, we compute the total number of lending actions per lender country to each borrower country. Next, we characterize each lender country with its socioeconomic indicators extracted from the World’s Bank Open Data. For each indicator, we create three groups of lender countries depending on whether the country has a low, medium or high value for that indicator, and compute their total lending activity to each borrower country. This will allow us to refer to the lending activity of, for example, [*lender countries that have a low GDP*]{} or [*lender countries with high mobile cellular subscriptions*]{}. Next, for each lender indicator (GDP, inflation,...) and group (low, medium or high), we compute Pearson’s correlations (adjusted with FDR) between their lending activities to each borrower country and the values for each socioeconomic indicator from the borrower countries. These tests might reveal important relationships between groups of lender countries and borrower countries being able to draw statements such as [*the lower the GDP of the borrower country, the larger lending activity they attract from countries with high GDP*]{}.
Table \[table:loancountries\] shows some of the most relevant correlations between the indicators of the borrower countries previously discussed and the lending activity they receive from countries with certain low, medium or high socioeconomic values. We observe a positive correlation between lender countries that have low average interests on external debt and borrower countries with large urban populations. As discussed earlier, borrower countries with large urban populations seem to receive more lending activity which they appear to be getting from lender countries that are not strangled by their external debt payments. We also observe that lender countries where most citizens finish primary education have a lending activity that is positively correlated to the borrower’s agricultural production ($NV.AGR.TOTL.KD$). This implies that the little lending activity that borrower countries with large agricultural production manage to bring in (as shown in Table \[table:loanfeatures\]) is mostly from lender countries with high education levels. In terms of mobile penetration ($IT.CEL.SETS$), Table \[table:loanfeatures\] showed a tendency to lend more to countries with low penetration rates, and Table \[table:loancountries\] shows that it is mostly countries with high youth literacy rates, the ones who generate that lending activity. We also observe that countries with high military expenditure ($MS.MIL.XPND.
%\newline
GD.ZS$) focus their lending activity on borrower countries with low manufacturing rates ($NV.IND.MANF.ZS$). Additionally, lending actions to countries with high incidence of tuberculosis appear to be mostly driven by countries with low prevalence of overweight children ($SH.STA.OWGH.ZS$). We posit that countries that are aware of the importance of health related issues might be focusing on lending to countries who could improve their health status. To summarize, it is fair to say that the general trend is for countries with higher rates of educated citizens and larger economic activity to be more prone to lend, which is a feature that has also been found in official development assistance [@desai]. Finally, to analyze the relationship between lending activity and the sector of the loan, the size of the loan and the gender, we take a different approach to account for the discrete nature of the variables. We compute the median number of normalized lending actions and its standard deviations for: each type of loan sector (health, agriculture, retail, etc.); each group size range(1, \[2-10\], \[10-20\] and \[20-48\]) and for each gender (female, female or both for loans that go to a group of borrowers rather than an individual). In terms of sectors, we observe a larger median number of normalized lending actions in the retail sector with (M=3.2, IQR=6.2), followed by the agricultural (M=3.1,IQR=6.2) and food (M=2.9, IQR=6.1) sectors, where $M$ represents the median and $IQR$ the interquartile range. The other sectors showed considerably lower values, although these differences were not statistically significant. This shows that lenders appear to favor the retail sector which is probably far more present in urban than rural settings. In terms of group size, we observe that individuals appear to focus their lending activity on loans which are borrowed by one person (M=2.3, IQR=3) or group loans of up to ten individuals (M=4.1, IQR=6). Larger loans are not as favored which is also coherent with the findings reported in [@ghatak; @owusu]. Finally, gender presents a slight minimal advantage for female loans (M=2.4, IQR=3) versus male (M=2.1, IQR=3), but nothing conclusive, although similar results have been reported in [@li; @pitt; @mayoux]. These lending patterns might be a result of individual preferences or rather a consequence of the way Kiva presents the information on their website.
Conclusions
===========
We have presented a large-scale analysis of the role that various features might play on online microlending environments. Our results show that lenders appear to favor highly rated Field Partners that manage to drive more lending activity. Additionally, we have observed that lenders seem to lend to loans in sectors that are often times aligned with official aid donors. We believe that our work provides a better understanding of online microlending behaviors as well as a set of suggestions to improve the services that microfinance platforms currently offer to their lenders and borrowers.
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abstract: 'Let $\Mbar_{n,r}$ denote the space of isometry classes of $n$-gons in the plane with one side of length $r$ and all others of length 1, and assume that $n-r$ is not an odd integer. Using known results about the mod-2 cohomology ring, we prove that its topological complexity satisfies $\TC(\Mbar_{n,r})\ge 2n-6$. Since $\Mbar_{n,r}$ is an $(n-3)$-manifold, $\TC(\Mbar_{n,r})\le 2n-5$. So our result is within 1 of optimal.'
address: |
Department of Mathematics, Lehigh University\
Bethlehem, PA 18015, USA
author:
- 'Donald M. Davis'
date: 'July 6, 2015'
title: Topological complexity of some planar polygon spaces
---
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[^1]
Statement of results {#intro}
====================
The topological complexity, $\TC(X)$, of a topological space $X$ is, roughly, the number of rules required to specify how to move between any two points of $X$. A “rule” must be such that the choice of path varies continuously with the choice of endpoints. (See [@F §4].) We study $\TC(X)$ where $X=\Mbar_{n,r}$ is the space of isometry classes of $n$-gons in the plane with one side of length $r$ and all others of length 1. (See, e.g., [@HK §9].) Here $r$ is a real number satisfying $0<r<n-1$, and $n\ge4$. Thus $$\Mbar_{n,r}=\{(z_1,\ldots,z_n)\in (S^1)^n:z_1+\cdots+z_{n-1}+rz_n=0\}/O(2).$$ If we think of the sides of the polygon as linked arms of a robot, we might prefer the space $M_{n,r}$, in which we identify only under rotation, and not also under reflection. However, the cohomology algebra of $\Mbar_{n,r}$ is better understood than that of $M_{n,r}$, leading to better bounds on TC.
If $r$ is a positive real number, then $\Mbar_{n,r}$ is a connected $(n-3)$-manifold unless $n-r$ is an odd integer (e.g., [@HK p.314] or [@KK p.2]), and hence satisfies $$\label{bound}\TC(\Mbar_{n,r})\le 2n-5$$ by [@F Cor 4.15].[^2] By [@Hb 6.2], if, for an integer $k$, $n-2k-1<r<n-2k+1$, then $\Mbar_{n,r}$ is diffeomorphic to $\Mbar_{n,n-2k}$, and so we restrict our discussion to the latter spaces. In this paper, we obtain the following strong lower bound for $\TC(\Mbar_{n,n-2k})$.
\[thm1\] If $2<2k<n$, then $\TC(\Mbar_{n,n-2k})\ge 2n-6$.
This result is within 1 of being optimal, using (\[bound\]). The case $k=1$ is special, as $\Mbar_{n,n-2}$ is homeomorphic to real projective space $RP^{n-3}$, for which the topological complexity agrees with the immersion dimension, a much-studied concept, but not yet completely determined. See, e.g., [@FTY], [@D], or [@H]. In fact, there are often large gaps between the known upper and lower bounds for $\TC(RP^n)$.([@Da])
The proof of Theorem \[thm1\] relies on the mod 2 cohomology ring $H^*(\Mbar_{n,r};\zt)$, first described in [@HK]. Throughout the paper, all cohomology groups have coefficients in $\zt$, and all congruences are mod 2, unless specifically stated to the contrary. To prove Theorem \[thm1\], we will find $2n-7$ classes $y_i\in H^1(\Mbar_{n,n-2k})$ such that $\prod(y_i\ot1+1\ot y_i)\ne0$ in $H^{n-3}(\Mbar_{n,n-2k})\ot H^{n-4}(\Mbar_{n,n-2k})$. This implies the theorem by the basic result that if in $H^*(X\times X)$ there is an $m$-fold nonzero product of classes of the form $y_i\otimes1+1\otimes y_i$, then $\TC(X)\ge m+1$.([@F Cor 4.40]) We show at the end of the paper that our cohomology result for $\Mbar_{n,n-2k}$ is optimal, in that $(2n-6)$-fold products of $(y_1\ot1+1\ot y_i)$ are always 0. Thus we will have proved the following result. (See [@F2] for the definition.)
\[thm3\] If $2<2k<n$, the zero-divisors-cup-length of $H^*(\Mbar_{n,n-2k})$ equals $2n-7$.
Proof {#pfsec}
=====
In this section we prove Theorems \[thm1\] and \[thm3\]. We begin by stating our interpretation of the cohomology ring $H^*(\Mbar_{n,n-2k})$.
\[cohthm\] Let $k\ge1$ and $n>2k$.
1. The algebra $H^*(\Mbar_{n,n-2k})$ is generated by classes $R,V_1,\ldots,V_{n-1}$ in $H^1(\Mbar_{n,n-2k})$.
2. The product of $k$ distinct $V_i$’s is 0.
3. If $d\le n-3$ and $S\subset\{1,\ldots,n-1\}$ has $|S|<k$, then all monomials $\dstyle{R^{e_0}\prod_{i\in S}V_i^{e_i}}$ with $e_i>0$ for $i\in S$ and $\dstyle\sum_{i\ge0} e_i=d$ are equal. We denote this class by $T_{S,d}$. This includes the class $T_{\emptyset,d}=R^d$.
4. For every subset $L$ of $\{1,\ldots,n-1\}$ with $n-k\le|L|\le d+1$, there is a relation $\cR_{L,d}$ which says $$\sum_{S\subset L}T_{S,d}=0.$$ These are the only relations, in addition to those previously described.
In [@KK Theorem 1], the more general result proved in [@HK Corollary 9.2] is applied to $\Mbar_{n,n-2k}$. The first three parts of our theorem are immediate from the result stated there, although our $T_{S,d}$ notation is new. The relations stated in [@KK] are in the form of an ideal, whereas we prefer to make a listing of a basic set of relations. The result of [@KK] says that the relations in $H^*(\Mbar_{n,n-2k})$ comprise the ideal generated by $$\label{Treln}\sum_{S\subset L}T_{S,|L|-1} \text{ for }L\subset\{1,\ldots,n-1\}\text{ with }n-k\le|L|\le n-2.$$ Multiplying this relation by $R^t$ gives a relation $\dstyle{\sum_{S\subset L}T_{S,|L|-1+t}}$. This yields, in degree $d$, exactly all of our claimed relations. Additional relations in the ideal can be obtained by multiplying (\[Treln\]) by $V_\ell$. If $\ell\not\in L$, this equals our $\cR_{S\cup\{\ell\},|L|}-\cR_{S,|L|}$, while if $\ell\in L$, it equals 0.
Most of our proofs also utilize the following key result, which was proved as [@KK Theorem B].
\[KKB\] There is an isomorphism $\phi_1:H^{n-3}(\Mbar_{n,n-2k})\to\zt$ satisfying $\phi_1(T_{S,n-3})=\binom{n-2-|S|}{k-1-|S|}$.
We begin our work with a useful lemma.
\[n-4lem\] There is a homomorphism $$\phi_2:H^{n-4}(\Mbar_{n,n-2k})\to\zt$$ satisfying $\phi_2(T_{S,n-4})=\binom{n-2-|S|}{k-1-|S|}$.
We must show that $\phi_2$ sends each of the relations $\cR_{L,n-4}$ to 0. If $|L|=\ell$, then $$\phi_2(\cR_{L,n-4})=\sum_{i=0}^{k-1}{\tbinom{\ell}i\tbinom{n-2-i}{k-1-i}}=\sum_i\tbinom{\ell}i\tbinom{-n+k}{k-1-i}=\tbinom{\ell-n+k}{k-1}.$$ Since $n-k\le \ell\le n-3$, we have $0\le\ell-n+k\le k-3$, and so $\binom{\ell-n+k}{k-1}=0$.
To prove Theorem \[thm1\], we will find $2n-7$ classes $y_i\in H^1(\Mbar_{n,n-2k})$ such that $\prod(y_i\ot1+1\ot y_i)\ne0$ in $H^{n-3}(\Mbar_{n,n-2k})\ot H^{n-4}(\Mbar_{n,n-2k})$. There will be four cases, Theorems \[case1\], \[case3\], \[case4\], and \[case5\]. All of them use the following notation, which pervades the rest of the paper.
\[nk\] Let $t\ge0$ and $k=2^t+k_0$, $1\le k_0\le 2^t$, and $n=k+1+2^tB+D$ with $0\le D<2^t$ and $B\ge1$. Let $C=k_0+D-1$. Then $n=2^t(B+1)+C+2$.
Every pair $(k,n)$ with $k\ge2$ and $n>2k$ yields unique values of $t$, $k_0$, $B$, and $D$.
\[case1\] Let $B$ be odd and $$\begin{aligned}
\label{phiphi}P&=&(V_1\ot1+1\ot V_1)^{2^t(B+1)-1}\cdot\Prod^{C}(V_i\ot1+1\ot V_i)\\
&&\quad \cdot\Prod^{C}(V_i\ot1+1\ot V_i)^2\cdot(R\ot1+1\ot R)^{2^t(B+1)-C-2}.\nonumber\end{aligned}$$ If $P_1$ denotes the component of $P$ in $H^{2^t(B+1)+C-1}(\Mbar_{n,n-2k})\ot H^{2^t(B+1)+C-2}(\Mbar_{n,n-2k})$, then $(\phi_1\ot\phi_2)(P_1)\ne0\in \zt.$
The product notation here, which will be continued throughout the paper, means a product of $C$ distinct factors with subscripts distinct from other subscripts involved elsewhere in the expression. Since $P$ has $2n-7$ factors, Theorem \[case1\] implies Theorem \[thm1\] when $B$ is odd.
Since $B$ is odd, the third case of Lemma \[techlem\] applies. Note that $(V_1\ot1+1\ot V_i)^2=V_1^2\ot1+1\ot V_i^2$, and that there are $2C$ factors $F$ of this form or $V_i\ot1+1\ot V_i$ in the middle of $P$. When $P$ is expanded, the only terms $\cT$ for which $(\phi_1\ot\phi_2)(\cT)$ might possibly be nonzero are those with exactly $C$ of these factors $F$ on each side of $\ot$ accompanied by a nontrivial contribution from the $V_1$-part. (This uses Lemmas \[KKB\], \[n-4lem\], and \[techlem\].) Such a term $\cT$ which contains $j$ of the $V_i$’s ($i>1$) (and $(C-j)$ $V_i^2$’s) on the left side of $\ot$ will be of the form $$\label{is}\tbinom{2^t(B+1)-1}e\tbinom{2^t(B+1)-C-2}{2^t(B+1)-C-1+j-e}V_1^eV_{i_1}\cdots V_{i_j}V_{i_{j+1}}^2\cdots V_{i_{C}}^2R^{2^t(B+1)-C-1+j-e}\ot Q,$$ where $Q$ is the complementary factor. Here we must have $0<e<2^t(B+1)-1$, in order that there are $C+1$ distinct $V_i$ factors on both sides of $\ot$. For this choice of $(i_1,\ldots,i_{C})$, let $W$ denote the sum of all such terms as $e$ varies, with $i_1,\ldots,i_C$ fixed. Then $$\begin{aligned}
\nonumber(\phi_1\ot\phi_2)(W)&=&\sum_{e=1}^{2^t(B+1)-2}\tbinom{2^t(B+1)-1}e\tbinom{2^t(B+1)-C-2}{2^t(B+1)-C-1+j-e}\\
&\equiv&\tbinom{2^{t+1}(B+1)-C-3}{2^t(B+1)-C-1+j}+\tbinom{2^t(B+1)-C-2}{2^t(B+1)-C-1+j}+\tbinom{2^t(B+1)-C-2}{-C+j}.\label{three}\end{aligned}$$ The first of the three terms in the last line is what the sum would have been if the terms with $e=0$ and $e=2^t(B+1)-1$ were included, while the other two terms are the two omitted terms. Mod 2, the first binomial coefficient is 0 by Lemma \[bclem\], since the case with $B=1$ and $C=2^{t+1}-2$ does not satisfy $n>2k$. The second binomial coefficient in (\[three\]) is 0 because its bottom part is greater than its top, and the third is 0 unless $j=C$. Thus there is a unique[^3] $W$, namely $$W=\sum_{e=1}^{2^t(B+1)-2}c_eV_1^e\bigl(\prod^C V_i\bigr) R^{2^t(B+1)-e-1}\ot V_1^{2^t(B+1)-1-e}\bigl(\prod^C V_{i}^2\bigr)R^{e-C-1},$$ with $c_e=\tbinom{2^t(B+1)-1}e\tbinom{2^t(B+1)-C-2}{2^t(B+1)-1-e}$, for which $\phi(W)=1$, establishing the claim in this case.
The following lemmas were used above.
\[techlem\] In the notation of \[nk\], $$\binom{n-2-i}{k-1-i}\equiv\begin{cases}1&i=C+1\\
0&k_0\le i\le C\\
0&0\le i\le C\text{ if $B$ is odd.}\end{cases}$$
We have $\binom{n-2-i}{k-1-i}=\binom{n-2-i}{2^tB+D}$ with $0\le D<2^t$. If $i=C+1$, then $n-2-i=2^tB+2^t-1$, and so the binomial coefficient is odd by Lucas’s Theorem, which we will often use without comment. Decreasing $i$ by $1,\ldots,D$ increases the top of the binomial coefficient by that amount, yielding $\binom{2^t(B+1)+j}{2^tB+D}$ with $0\le j<D$. Such a binomial coefficient is even. If $B$ is odd, decreasing $i$ even more will leave the binomial coefficient even, as it will be either $\binom{B+1}B\binom jD$ with $j\ge D$ or $\binom{B+2}B\binom jD$ with $j\le C-2^t <D$.
\[bclem\] If $0\le C\le 2^{t+1}-2$ and $0\le j\le C$, then, mod 2, $$\binom{2^{t+1}(B+1)-C-3}{2^t(B+1)-C-1+j}\equiv\begin{cases}1&\text{if $B$ is a $2$-power and $C=2^{t+1}-2$}\\
0&\text{otherwise.}\end{cases}$$
If $C=2^{t+1}-2$, then the binomial coefficient is $\binom{2^{t+1}B-1}{2^tB+\Delta}$ with $|\Delta|<2^t$. For $\Delta=0$ this is odd iff $B$ is a 2-power, as is easily seen using Lucas’s Theorem. If the bottom part of the binomial coefficient is changed from $\Delta=0$ by an amount less than $2^t$, the binomial coefficient is multiplied by $p/q$ with $p$ and $q$ equally 2-divisible. If $C=2^{t+1}-3$, then the binomial coefficient is of the form $\binom{2^{t+1}B}{2^tB+\Delta}$ with $|\Delta|<2^t$. This is even for all $B$, similarly to the previous case. For smaller values of $C$, the result follows by induction on (decreasing) $C$, using Pascal’s formula. Here it is perhaps more convenient to think of the binomial coefficient as $\binom{2^{t+1}(B+1)-C-3}{2^t(B+1)-j-2}$.
The case in which $D=0$ and $B$ is even is special because then $(\phi_1\ot\phi_2)(M)=1$ for every monomial $M$ in $H^{n-3}(\Mbar_{n,n-2k})\ot H^{n-4}(\Mbar_{n,n-2k})$, and so for any appropriate product $P$, we have $(\phi_1\ot\phi_2)(P)=\binom{2n-7}{n-3}=0$ (unless $n-3$ is a 2-power.) So we modify $\phi_2$.
\[case3\] In the notation of \[nk\], let $B$ be even and $D=0$. There is a homomorphism $$\phi_3:H^{n-4}(\Mbar_{n,n-2k})\to\zt$$ defined by $$\phi_3(T_{S,n-4})=\begin{cases}1&|S|<k-1\\ 0&|S|=k-1.\end{cases}$$ If $$P=(V_1\ot1+1\ot V_1)^{n-3}\cdot\prod^{k-2}(V_i\ot1+1\ot V_i)\cdot(R\ot1+1\ot R)^{n-k-2},$$ then $(\phi_1\ot\phi_3)(P)=1\in\zt$.
To prove that $\phi_3$ is well-defined, we must show that for $n-k\le\ell\le n-3$, we have $\ds\sum_{i=0}^{k-2}\tbinom\ell i\equiv 0$. Then $2^tB<\ell\le2^tB+k-2$. Since $B$ is even and $k\le2^{t+1}$, the $2^tB$ does not affect the binomial coefficient mod 2, and the sum becomes $\ds\sum_{i=0}^{k-2}\tbinom\ell i$ for $0<\ell\le k-2$, and this equals $2^\ell$.
Since $\phi_1(M)=1$ for every monomial in $H^{n-3}(\Mbar_{n,n-2k})$, $(\phi_1\ot\phi_3)(P)$ equals the sum of coefficients in $$(1+V_1)^{n-3}\cdot\prod_{i=2}^{k-1}(1+V_i)\cdot(1+R)^{n-2-k}$$ of all monomials of degree $n-4$ which are not divisible by $V_1\cdots V_{k-1}$. This equals $S_1-(S_2-S_3)$, where $S_1$ is the sum of all coefficients in degree $n-4$, $S_2$ is the sum of coefficients of terms divisible by $V_2\cdots V_{k-1}$, and $S_3$ is the sum of coefficients of terms divisible by $V_2\cdots V_{k-1}$ but not also by $V_1$. Then $S_1=\binom{2n-7}{n-4}\equiv0$ since $n-3$ cannot be a 2-power here. Also $S_2=\binom{2n-5-k}{n-4-(k-2)}=\binom{2^{t+1}B+k-3}{2^tB-1}\equiv0$ since $k\le2^{t+1}$. Finally for $S_3$ the only monomial is $V_2\cdots V_{k-1}R^{n-2-k}$, so $S_3=1$.
Let $\lg(-)=[\log_2(-)]$.
\[case4\] Theorem \[case1\] is true if $B$ is even and $C-2^{\lg(C)}<2^{1+\lg D}$.
The first few cases of this hypothesis are ($D=1$ and $C\in\{2^e,2^e+1\}$) and ($D\in\{2,3\}$ and $C\in\{2^e,2^e+1,2^e+2,2^e+3\}$).
We consider first the portion $P_2$ of the expansion of $P$ which has $V_1^{2^t(B+1)-1}$ on the left side of $\ot$. If $j$ (resp. $g$) denotes the number of other $V_i$’s (resp. $V_i^2$’s) on the left side of $\ot$, then $(\phi_1\ot\phi_2)(P_2)$ equals $$\label{sum1}\sum_{j,g=0}^C\tbinom Cj\tbinom Cg\tbinom{2^t(B+1)-1+C-j-g}{2^tB+C+1-k_0}\tbinom{2^t(B+1)-(C-j-g)}{2^tB+C+1-k_0}\tbinom{2^t(B+1)-C-2}{C-j-2g}.$$ The third and fourth factors here are from $\phi_1(-)$ and $\phi_2(-)$, which satisfy $$\phi_1(T_{S,n-3})=\phi_2(T_{S,n-4})=\tbinom{2^t(B+1)+C-|S|}{2^t+k_0-1-|S|}=\tbinom{2^t(B+1)+C-|S|}{2^tB+C+1-k_0}.$$ These two factors in our sum are of the form $\tbinom{2^t(B+1)-1+\Delta}{2^tB+D}\tbinom{2^t(B+1)-\Delta}{2^tB+D}$ with $-C\le\Delta\le C$ and $1\le D\le 2^t-1$. In positions less than $2^t$, the top parts of these two binomial coefficients differ in every position, and so due to any position where $D$ has a 1, one of the factors will be even. Thus (\[sum1\]) is 0 in $\zt$. A similar argument works for the portion of the sum in which $V_1^{2^t(B+1)-1}$ is on the right side of $\ot$.
Arguing similarly to (\[three\]), it remains to show that the following sum is 1 mod 2. $$\begin{aligned}
\label{fh} &&\sum_{j,g=0}^C\tbinom Cj\tbinom Cg\tbinom{2^t(B+1)+C-j-g-1}{2^tB+D}\tbinom{2^t(B+1)-1-C+j+g}{2^tB+D}\\
&&\quad\cdot\bigl(\tbinom{2^{t+1}(B+1)-C-3}{2^t(B+1)+C-1-j-2g}+\tbinom{2^t(B+1)-C-2}{2^t(B+1)+C-1-j-2g}+\tbinom{2^t(B+1)-C-2}{C-j-2g}\bigr).\label{sum2}\end{aligned}$$
Let $\ell=\lg(D)$. Note that $t\ge\ell+1$. Keep in mind that $B$ is even. It is easy to check that there is a nonzero summand due to the third term of (\[sum2\]) if $(j,g)=(C,0)$, and one due to the first term of (\[sum2\]) if $t=\ell+1$, $D=2^{\ell+1}-1$, $j+g=C=2^{\ell+2}-2$ with $j$ even and $0\le j\le C$, and $B$ is a 2-power. The proof will be completed by showing that other terms are nonzero iff $C=2^{\ell+2}-1$, $t\ge\ell+2$, and $|C-j-g|=2^{\ell+1}$. The result will follow, as the total number of nonzero terms is odd in any case.
It is also easy to check that the terms of the third type give nonzero summands. For example, let $\ell=2$, so we have $D\in\{4,5,6,7\}$, $C=15$, and $j+g=7$ or 23. Then $7\le j+2g\le14$ in the first case, and, since $j,g\le 15$, we have $31\le j+2g\le 38$ in the second case. Also $t\ge4$ and $B$ is even. The latter two factors in (\[fh\]) are $\binom{2^t(B+1)-1\pm8}{2^tB+D}\equiv1$. Of the three terms in (\[sum2\]), the first will be even since it is either $\binom{2^{t+1}B+\a}{2^tB+\b}$ with $0<\a<2^{t+1}$ and $0<\b<2^t$, or $\binom{32B+14}{16B-\gamma}$ with $1\le\gamma\le8$. When $j+g=7$, the second summand in (\[sum2\]) has bottom greater than top, while the third is of the form $\binom{16A+15}b$ with $1\le b\le8$, hence is odd. When $j+g=23$, the second summand is of the form $\binom{16A+15}{16A+c}$ with $8\le c\le 15$, while the third has its bottom part negative.
Now we show that all other terms in (\[fh\])-(\[sum2\]) are 0. Let $t\ge\ell+1$, $2^e\le C<2^{e+1}$ with $e\le t$ and $C-2^e<2^{\ell+1}$. We will show
1. If $$P_0=\tbinom Cj\tbinom Cg\tbinom{2^t(B+1)+C-j-g-1}{2^tB+D}\tbinom{2^t(B+1)-1-C+j+g}{2^tB+D}$$ then $P_0$ is odd iff $\binom Cj$ and $\binom C g$ are odd and $C-j-g\equiv0$ mod $2^{\ell+1}$ and $|C-j-g|<2^t$.
2. In the cases just noted where $P_0$ is odd, $$\label{3b}\tbinom{2^{t+1}(B+1)-C-3}{2^t(B+1)+C-1-j-2g}\equiv\tbinom{2^t(B+1)-C-2}{2^t(B+1)+C-1-j-2g}\equiv\tbinom{2^t(B+1)-C-2}{C-j-2g}\equiv0$$ except in the cases noted in the paragraph following (\[fh\])-(\[sum2\]).
To prove (1), let $E=C-j-g$. Then $-C\le E\le C$, but by symmetry, it suffices to consider $0\le E\le C<2^{t+1}$. It is easy to see that $E-1$ and $-E-1$ both have a 1 in the $2^\ell$-position iff $E\equiv0$ mod $2^{\ell+1}$. Since $D$ has a 1 in the $2^\ell$-position, $P_0$ is even unless $E\equiv0$ mod $2^{\ell+1}$. Letting $E=2^{\ell+1}E'$, and removing the lower parts of the binomial coefficients, we need for $$\binom{2^t(B+1)+2^{\ell+1}(E'-1)}{2^tB}\binom{2^t(B+1)-2^{\ell+1}(E'+1)}{2^tB}$$ to be odd. If $2^{\ell+1}E'\ge 2^t$, the second binomial coefficient is 0. Otherwise, $2^{\ell+1}(E'+1)\le 2^t$, and then both binomial coefficients are odd,
For (2), we first study how the middle coefficient in (\[3b\]) can be odd. If $t=\ell+1$, then $C-j-g=0$, and the binomial coefficient is 0 since its bottom part is greater than the top. Now assume $t\ge\ell+2$. Let $C-j-g=-2^{\ell+1}K$. The binomial coefficient becomes $\ds\binom{2^t(B+1)-2-C}{2^{\ell+2}K-j-1}$. For this to be odd, in positions $<2^{\ell+2}$ the 1’s in (the binary expansion of) $(C+1$) must be contained in those of $j$. For $\binom Cj$ to be odd, the 1’s of $j$ must be contained in those of $C$. The only way that the 1’s of $(C+1)$ can be contained in those of $C$ in these positions is if $C+1\equiv0$ mod $2^{\ell+2}$. Since $C-2^{\lg C}<2^{1+\ell}$, the only such $C$ is $2^{\ell+2}-1$. Since $-2^{\ell+2}<C-j-g<0$, we must have $C-j-g=-2^{\ell+1}$. Thus the only ways the middle coefficient of (\[3b\]) can yield a nonzero value are those listed earlier.
The third coefficient in (\[3b\]) is handled similarly. If $t=\ell+1$, then $C-j-g=0$ and $g=0$, yielding a claimed condition. Now assume $t\ge\ell+2$. Let $C-j-g=2^{\ell+1}K$. The binomial coefficient becomes $\ds\binom{2^t(B+1)-C-2}{2^t(B+1)-2^{\ell+2}K-j-2}$. For this and $\binom Cj$ to both be odd, either $C+1\equiv0$ mod $2^{\ell+2}$ or $j\equiv C$ mod $2^{\ell+2}$. The former condition reduces to $C=2^{\ell+2}-1$, $C-j-g=2^{\ell+1}$ similarly to the previous case. For the latter, if $C=j$, then $g=0$ and we obtain one of the claimed conditions. Otherwise, write $C=2^e+\Delta$ with $0\le\Delta<2^{\min(e,\ell+1)}$. Then we must have $e\ge\ell+2$ and $j=\Delta$, and, since $g\equiv 0$ mod $2^{\ell+1}$ and $\binom Cg$ is odd, we must have $g=0$ or $2^e$, neither of which make $\binom{2^t(B+1)-C-2}{C-j-2g}$ odd, since $e<t$ .
For the first coefficient in (\[3b\]), we first consider the situation when $t=\ell+1$. Then $C-j-g=0$ and the coefficient equals $\ds\binom{2^{\ell+2}(B+1)-C-3}{2^{\ell+1}(B+1)-2-j}$. For this and $\binom Cj$ to both be odd, we must have $C\equiv-\eps$ mod $2^{\ell+1}$ with $\eps\in\{1,2\}$, and $j$ even. If $C=2^{\ell+1}-\eps$, then the coefficient is $\binom{2^{\ell+2}B+\a}{2^{\ell+1}B+\b}$, with $0\le\a,\b<2^{\ell+1}$, and thus is even, due to $\binom{2B}B$. If $C=2^{\ell+2}-2$ (its largest possible value) and $B$ is not a 2-power, the coefficient can be written as $\binom{2^{\nu+1}(2A+1)-1}{2^\nu(2A+1)+\Delta}$ with $A>0$ and $0\le\Delta<2^\nu$, which is even since it splits as $\binom{2^{\nu+2}A}{2^{\nu+1}A}\binom{2^{\nu+1}-1}{2^\nu+\Delta}$.
If $t\ge\ell+2$, let $C-j-g=2^{\ell+1}K$ and write the coefficient as $\binom{2^{t+1}(B+1)-C-3}{2^t(B+1)-2-2^{\ell+2}K-j}$. For both this and $\binom Cj$ to be odd, we must have $C\equiv-1$ or $-2$ mod $2^{\ell+2}$ and $j$ even. Since $C-2^{\lg C}<2^{\ell+1}$, this implies $C=2^{\ell+2}-1$ or $2^{\ell+2}-2$. The top part of the binomial coefficient splits as $2^{t+1}B+(2^{t+1}-2^{\ell+2}-\eps)$. Since $|2^{\ell+1}K|\le C$, then $|2^{\ell+1}K|\le2^{\ell+1}$. Thus the bottom of the binomial coefficient is $2^tB+\a$ with $$2^t-2^{\ell+3}\le\a\le 2^t+2^{\ell+2}-2.$$ Since $B$ is even, the binomial coefficient, mod 2, splits as $\binom{2^{t+1}B}{2^tB}\binom{2^{t+1}-2^{\ell+2}-\eps}\a\equiv0$ if $0\le\a<2^{t+1}$. This is true if $t>\ell+2$ or ($t=\ell+2$ and $\a\ge0$). If $t=\ell+2$ and $\a<0$, then the binomial coefficient is 0 by consideration of position $2^{\ell+2}$.
The final case for Theorem \[thm1\] is
\[case5\] In the notation of \[nk\], and with $\ell=\lg D$, if $B$ is even and $C-2^{\lg C}\ge 2^{\ell+1}$, let $C=2^{\ell+1}A+\g$ with $0\le\g<2^{\ell+1}$, and $m=2^t(B+1)+2^{\ell+1}A-1$. If $$\begin{aligned}
\label{phim}P&=&(V_1\ot1+1\ot V_1)^{m}\cdot\Prod^{C}(V_i\ot1+1\ot V_i)\\
&&\quad \cdot\Prod^{C}(V_i\ot1+1\ot V_i)^2\cdot(R\ot1+1\ot R)^{2n-7-m-3C},\nonumber\end{aligned}$$ then $(\phi_1\ot\phi_2)(P)\ne0\in \zt$.
Using the methods of our previous proofs, it suffices to prove that, under the hypotheses, with $\psi(i)=\binom{n-2-i}{k-1-i}$, the following mod-2 equivalences are valid.
1. For all $j$ and $g$, $$\label{c1} \tbinom Cj\tbinom Cg\psi(j+g)\psi(2C-j-g+1)\tbinom{2n-7-3C-m}{n-3-j-2g}\equiv0$$ and $$\label{c2} \tbinom Cj \tbinom Cg\psi(j+g+1)\psi(2C-j-g)\tbinom{2n-7-3C-m}{n-3-j-2g-m}\equiv0.$$
2. If $\binom Cj\binom Cg\psi(j+g+1)\psi(2C-j-g+1)\equiv1$, then
1. $\binom{2n-7-3C}{n-3-j-2g}\equiv0$;
2. $\binom{2n-7-3C-m}{n-3-j-2g}\equiv0$;
3. $\binom{2n-7-3C-m}{n-3-j-2g-m}\equiv1$ iff $g=0$ and $j=\g$, in which case $\binom Cj\binom Cg\psi(j+g+1)\psi(2C-j-g+1)\equiv1$.
The proof of (\[c1\]) and (\[c2\]) is similar to that for the corresponding terms in the proof of Theorem \[case4\]. The third and fourth factors will be of the form $\binom{2^t\a+x}{2^t\b+D}$ and $\binom{2^t\a-1-x}{2^t\b+D}$ with $0< D<2^t$. Their product is 0 mod 2 by the same reasoning as before.
The hypothesis of (2) implies $C-j-g\equiv0$ mod $2^{\ell+1}$ and $|C-j-g|<2^t$, exactly as in the proof of \[case4\]. Write $C-j-g=2^{\ell+1}K$ with $|2^{\ell+1}K|<2^t$.
Part (2a) is like the first coefficient of (\[3b\]) except that the constraint on $C-2^{\lg C}$ is different. The argument when $t=\ell+1$ is the same, since the constraint did not occur in that argument. So now assume $t\ge\ell+2$ and write the binomial coefficient as $\ds\binom{2^{t+1}(B+1)-3-C}{2^t(B+1)-2-2^{\ell+2}K-j}$. As before, for both this and $\binom Cj$ to be odd, we must have $C=2^{\ell+2}Y-\eps$ with $\eps\in\{1,2\}$. We cannot have $C=2^{t+1}-2$ (which implies $D=2^t-1$) because of the assumption that $C-2^{\lg C}\ge 2^{1+\lg D}$. Thus $2^{\ell+2}Y\le 2^{t+1}-2^{\ell+2}$. We have $2^{\ell+1}K= 2^t-2^{\ell+1}-p$ with $p\ge0$, and then $j\le 2^{\ell+2}Y-\eps-(2^t-2^{\ell+1}-p)$, and hence $$2^{\ell+2}K+j\le 2^t-2^{\ell+1}+2^{\ell+2}Y-\eps.$$ On the other hand, $2^{\ell+1}K\ge-(2^t-2^{\ell+1})$, and $j\ge-(C-j-g)=-2^{\ell+1}K$, so we have $$2^{\ell+2}K+j\ge 2^{\ell+1}K\ge -2^t+2^{\ell+1}.$$ Letting $B=2B'$, the binomial coefficient becomes $\ds\binom{2^{t+2}B'+2^{t+1}-2^{\ell+2}Y-3+\eps}{2^{t+1}B'+x}$ with $$2^{\ell+1}-2^{\ell+2}Y+\eps-2\le x\le 2^{t+1}-2^{\ell+1}-2.$$ If $x\ge0$, this binomial coefficient splits, and is 0 due to $\binom{2^{t+2}B'}{2^{t+1}B'}$. If $x<0$, the binomial coefficient splits as $$\binom{2^{t+2}B'}{2^{t+1}(B'-1)}\binom{2^{t+1}-2^{\ell+2}Y-3-\eps}{2^{t+1}+x},$$ which is 0 since the second factor has bottom part greater than the top.
To prove (2b), with $C$, $A$, and $\g$ as in the statement of the theorem, the binomial coefficient here is $\binom pq$ with $p=2^t(B+1)-2-C-2^{\ell+1}A$ and $q=2^t(B+1)-1+C-j-2g$. Then $q-p=2(C-j-g)+j+2^{\ell+1}A+1$. Since $C-j-g\ge -C$ and is a multiple of $2^{\ell+1}$, $C-j-g\ge-2^{\ell+1}A$. Similarly to the previous case, this implies $2(C-j-g)+j\ge -2^{\ell+1}A$. Thus $q-p>0$ and $\binom pq=0$.
Finally, for (2c), the binomial coefficient becomes $\binom{2^t(B+1)-2-C-2^{\ell+1}A}{\g-j-2g}$. For this to be nonzero, we must have $j+2g\le\g$. But $j+g\equiv\g$ mod $2^{\ell+1}$, and so we must have $g=0$ and $j=\g$, in which case the binomial coefficient equals 1. Clearly $\binom Cj\binom Cg\equiv1$. Also, $\psi(j+g+1)=\psi(\g+1)=\binom{2^t(B+1)+2^{\ell+1}A-1}{2^tB+D}\equiv1$ and $\psi(2C-j-g+1)=\psi(2C-\g+1)=\binom{2^t(B+1)-2^{\ell+1}A-1}{2^tB+D}\equiv1$. These use the fact that, since $D\le 2^{\ell+1}-1$, $C\le2^t+2^{\ell+1}-2$, and hence $2^{\ell+1}A\le 2^t$.
We close by showing that all $(2n-6)$-fold products $P$ of elements of the form $(y\ot1+1\ot y)$ in $H^*(\Mbar_{n,n-2k}\times \Mbar_{n,n-2k})$ are zero. This will complete the proof of Theorem \[thm3\]. First note that all such products are invariant under the involution that interchanges factors. If $m_1$ and $m_2$ are monomials of degree $n-3 $ in the $V_i$’s and $R$, then $m_1\ot m_2+m_2\ot m_1=0$ since $m_i$ equals either 0 or the unique nonzero class. Thus it suffices to show that the coefficient of any $T_{S,n-3}\ot T_{S,n-3}$ in $P$ is 0. We prove this by induction on $|S|$. Note that the factors which we must consider are not just those of the form $(V_i\ot1+1\ot V_i)$ and $(R\ot1+1\ot R)$, but also sums of these.
The coefficient of $R^{n-3}\ot R^{n-3}$ in $P$ is 0 if $P$ contains any factors which do not contain terms $(R\ot1+1\ot R)$, while if all factors contain such terms, it is $\binom{2n-6}{n-3}\equiv0$. This initiates the induction, as $|S|=0$ here. Assume that all terms $T_{S,n-3}\ot T_{S,n-3}$ in any product $P$ are 0 if $|S|<s$. Let $S$ be a subset with $|S|=s$. In $P$, we may omit all terms $(V_i\ot1+1\ot V_i)$ for which $i\not\in S$. If this omission makes any of the factors become 0, then the coefficient of $T_{S,n-3}\ot T_{S,n-3}$ is 0. Otherwise, by the induction hypothesis, the coefficient of all $T_{S',n-3}\ot T_{S',n-3}$ with $S'$ a proper subset of $S$ is 0, and since the sum of all coefficients in $H^{n-3}(\Mbar_{n,n-2k})\ot H^{n-4}(\Mbar_{n,n-2k})$ is $\binom{2n-6}{n-3}$, which is even, the coefficient of the remaining term $T_{S,n-3}\ot T_{S,n-3}$ must also be 0.
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[^1]: 2000 [*Mathematics Subject Classification*]{}: 58D29, 55R80, 70G40 .
[^2]: If $n-r$ is an odd integer, $\Mbar_{n,r}$ is often not a manifold but still satisfies $\TC(\Mbar_{n,r})\le 2n-5$, by [@F2 Theorem 4]. However, its cohomology algebra is not so well understood in this case, and so we do not study it here.
[^3]: The uniqueness refers to the choice of which squared terms appear on the left side of $\ot$ in (\[is\]), given the choice of $i$’s in (\[phiphi\]). The choice of which values of $i$ occur in (\[phiphi\]) is arbitrary, and far from unique.
|
---
abstract: |
Previous attempts to determine the worldsheet origin of the pure spinor formalism were not completely successful, but introduced important concepts that seem to be connected to its fundamental structure, *e.g.*, emergent supersymmetry and the role of reparametrization symmetry.
In this work, a new proposal towards the underlying gauge theory of the pure spinor superstring is presented, based on an extension of Berkovits’ twistor-like constraint. The gauge algebra is analyzed in detail and worldsheet reparametrization is shown to be a redundant symmetry. The master action is built with a careful account of the intrinsic gauge symmetries associated with the pure spinor constraint and a consistent gauge fixing is performed. After a field redefinition, spacetime supersymmetry emerges and the resulting action describes the pure spinor superstring.
author:
- 'Renann Lipinski Jusinskas[^1]'
title: Towards the underlying gauge theory of the pure spinor superstring
---
Overview\[sec:overview\]
========================
The pure spinor formalism of the superstring was introduced by Berkovits almost two decades ago [@Berkovits:2000fe]. Since then, it has been studied and explored in many different aspects, taking advantage of its symmetry preserving character and bosonic string-like amplitude prescription. These aspects range from the impressive $3$-loop computation of scattering amplitudes of [@Gomez:2013sla] or the recent $N$-point $1$-loop results of [@Mafra-Schlotterer], to the investigation of the quantization of the superstring in the $AdS_{5}\times S^{5}$ background (see [@Mazzucato:2011jt] for a review and references therein) or to the analysis of supersymmetry breaking effects in the superstring [@Berkovits:2014rpa].
There is abundant evidence that the pure spinor superstring is related to the spinning string [@Ramond:1971gb; @Neveu:1971rx] and to the Green-Schwarz superstring [@Green:1980zg; @Green:1981yb]. Scattering amplitudes computed in the pure spinor superstring were shown to be equivalent to the spinning string amplitudes up to two loops [@Berkovits:2005ng]. Furthermore, it has been shown in [@Berkovits:2000nn] that the pure spinor cohomology in the light-cone gauge describes the usual physical spectrum of the superstring. Later on this equivalence was explored in [@Berkovits:2004tw] and more recently in [@Berkovits:2014bra], where a combination of field redefinitions and similarity transformations helped to identify the Green-Schwarz and the pure spinor superstrings. In [@Jusinskas:2014vqa], the DDF-like structure of the pure spinor cohomology was finally made explicit. From another perspective, superembedding techniques (see [@Sorokin:1999jx] for a review and references therein) seem to provide a fertile ground for exploring the classical equivalence between the different superstring formalisms (*e.g.* [@Berkovits:1989zq; @Tonin:1991ii]). The superembedding origin of the pure spinor description of the heterotic superstring was demonstrated in [@Matone:2002ft].
However, the pure spinor formalism lacks a fundamental worldsheet description, meaning (1) a two-dimensional reparametrization invariant gauge theory which upon quantization concretely leads to its characteristic BRST structure and (2) a first principles derivation of the pure spinor constraint itself. The goal of this work is to present a possible resolution for the point (1), but still assuming that the pure spinor is a fundamental variable.
The twistor-like constraint {#the-twistor-like-constraint .unnumbered}
---------------------------
The first step to understand the gauge structure of the pure spinor formalism from a more fundamental point of view was taken in [@Berkovits:2011gh] with the introduction of the twistor-like constraint $$\begin{aligned}
H_{\alpha} & \equiv & P_{m}(\gamma^{m}\lambda)_{\alpha},\nonumber \\
& = & 0,\label{eq:TLconstraint}\end{aligned}$$ where $P_{m}$ is the canonical conjugate of the target-space coordinate $X^{m}$, with $m=0,\ldots,9$, $\gamma_{\alpha\beta}^{m}$ denotes the chiral blocks of the Dirac matrices, with $\alpha=1,\ldots,16$, and $\lambda^{\alpha}$ is a pure spinor variable satisfying $$(\lambda\gamma^{m}\lambda)=0.$$ The novel feature of this approach was that $\lambda^{\alpha}$ appeared as a fundamental variable in the worldline/worldsheet[^2] and the superpartners of $X^{m}$, denoted by $\theta^{\alpha}$, entered the formalism as ghost fields associated to the gauge symmetry generated by . In this model, supersymmetry is an emergent feature related to a ghost twisting operation on the gauge fixed action. However, the gauge symmetries related to the pure spinor constraint were not completely considered in this approach, leading to an incorrect description of the ghost fields.
A new attempt to quantize the twistor-like constraint was made in [@Berkovits:2014aia], with a different mechanism for the emergence of spacetime supersymmetry. The problem of this proposal was an overconstrained action which ultimately leads to a trivialization of the model.
This flaw was later corrected in [@Berkovits:2015yra], where a new gauge theory was proposed to explain the origin of the pure spinor formalism. Berkovits’ first order action can be cast as $$\begin{gathered}
S_{B}=\int d\tau d\sigma\{P_{m}\partial_{\tau}X^{m}+w_{\alpha}\nabla_{\tau}\lambda^{\alpha}+\hat{w}_{\hat{\alpha}}\hat{\nabla}_{\tau}\hat{\lambda}^{\hat{\alpha}}+K_{\alpha}\nabla_{\sigma}\lambda^{\alpha}+\hat{K}_{\hat{\alpha}}\hat{\nabla}_{\sigma}\hat{\lambda}^{\hat{\alpha}}\\
-\tfrac{1}{2}L^{\alpha}(\gamma^{m}\lambda)_{\alpha}(P_{m}+\partial_{\sigma}X_{m})-\tfrac{1}{2}\hat{L}^{\hat{\alpha}}(\gamma^{m}\hat{\lambda})_{\hat{\alpha}}(P_{m}-\partial_{\sigma}X_{m})\}.\label{eq:Berkovitsaction}\end{gathered}$$ Here, $\tau$ and $\sigma$ denote the worldsheet coordinates and hatted and unhatted spinors are related to the usual left and right-moving variables. The Lagrange multipliers $\{L^{\alpha},\hat{L}^{\hat{\alpha}}\}$ impose the twistor-like constraints and $\{K_{\alpha},\hat{K}_{\hat{\alpha}}\}$, as a consequence of the gauge algebra, impose the constraints $\nabla_{\sigma}\lambda^{\alpha}=\hat{\nabla}_{\sigma}\hat{\lambda}^{\hat{\alpha}}=0$. The covariant derivatives $\nabla$ and $\hat{\nabla}$ are composed by gauge fields, $A$ and $\hat{A}$, associated to local scale symmetries for $\lambda^{\alpha}$ and $\hat{\lambda}^{\hat{\alpha}}$, which effectively convert them into projective pure spinors. Although not explicitly, $S_{B}$ is invariant under worldsheet reparametrization.
One of the fundamental ideas introduced in these works [@Berkovits:2011gh; @Berkovits:2014aia; @Berkovits:2015yra] is that worldsheet reparametrization is a redundant gauge symmetry (in the sense that it can be removed by a gauge-for-gauge transformation). With this in mind, the action is already incomplete as the absence of a kinetic term for the gauge fields of the scaling symmetry prevents the existence of a gauge-for-gauge symmetry connecting the twistor-like constraints and reparametrization. Aside from this fact, the gauge symmetries due to the pure spinor constraint imply a constrained ghost system associated to the twistor-like constraint, such that the gauge fixed action cannot be spacetime supersymmetric. Observe that the action is invariant under the gauge transformation $$\delta L^{\alpha}=f\lambda^{\alpha}+f^{mn}(\gamma_{mn}\lambda)^{\alpha},$$ where $\{f,f_{mn}\}$ are the gauge parameters and $\gamma^{mn}=\tfrac{1}{2}(\gamma^{m}\gamma^{n}-\gamma^{n}\gamma^{m})$. In the gauge fixing procedure, this gauge symmetry appears as a constraint on the antighost of the twistor-like symmetry, $\pi_{\alpha}$, given by $$(\lambda\gamma^{m}\gamma^{n}\pi)=0.\label{eq:BAconstraint-1}$$ In turn, it implies that only five components of the associated ghost, $\theta^{\alpha}$, are physical. Therefore, the phase space of the action should be extended if the twistor-like constraint is to be part of a spacetime supersymmetric theory.
It is interesting to note, however, that all these features were in some sense convergent, leading to important concepts that seem to be connected to the gauge structure behind the pure spinor superstring, in particular the role of worldsheet reparametrization and the emergence of spacetime supersymmetry from the ghost sector.
The extended action {#the-extended-action .unnumbered}
-------------------
A simple way to understand the physical meaning of the twistor-like constraint is to look at its worldline version, as the massless particle can be viewed as the zero length limit of the string. Consider the constraint equation , but now with a projective pure spinor. In a Wick-rotated construction, the $SO(10)$ spinor $H_{\alpha}$ can be decomposed in terms of $U(5)$ components such that $$\bar{p}^{a}+\gamma^{ab}p_{b}=0\label{eq:TLU5}$$ correspond to the independent components of $H_{\alpha}=0$. Here, $a=1,\ldots,5$ denotes $U(5)$ vector indices, $P_{m}=\{\bar{p}^{a},p_{a}\}$ and $\gamma^{ab}=-\gamma^{ba}$ corresponds to the $U(5)$ parametrization of the projective pure spinor. Equation has a clear interpretation as any solution of the massless constraint $P^{m}P_{m}=0$ can be put in this form for a dynamical $\gamma^{ab}$.
The difference between the covariant form and is basically the scale symmetry introduced by Berkovits in [@Berkovits:2015yra]. Both of them were recently investigated by the author in [@Jusinskas:2018uci]. In order to obtain the pure spinor superparticle from first principles, Berkovits’ model was then extended with a constrained anticommuting spinor together with an additional fermionic gauge symmetry. Now, this idea will be generalized to the worldsheet with the proposal of the action $$\begin{aligned}
S_{0} & = & \int d\tau d\sigma\{P_{m}\partial_{\tau}X^{m}-\tfrac{1}{4\mathcal{T}}(L\gamma_{m}\lambda)(P^{m}+\mathcal{T}\partial_{\sigma}X^{m})-\tfrac{1}{4\mathcal{T}}(\hat{L}\gamma_{m}\hat{\lambda})(P^{m}-\mathcal{T}\partial_{\sigma}X^{m})\}\nonumber \\
& & +\int d\tau d\sigma\{w_{\alpha}^{i}\nabla_{i}\lambda^{\alpha}+B\epsilon^{ij}\partial_{i}A_{j}+p_{\alpha}^{i}\partial_{i}\xi^{\alpha}+\chi_{i}\lambda^{\alpha}p_{\alpha}^{i}-\Sigma\epsilon^{ij}\nabla_{i}\chi_{j}\}\nonumber \\
& & +\int d\tau d\sigma\{\hat{w}_{\hat{\alpha}}^{i}\hat{\nabla}_{i}\hat{\lambda}^{\hat{\alpha}}+\hat{B}\epsilon^{ij}\partial_{i}\hat{A}_{j}+\hat{p}_{\hat{\alpha}}^{i}\partial_{i}\hat{\xi}^{\hat{\alpha}}+\hat{\chi}_{i}\hat{\lambda}^{\hat{\alpha}}\hat{p}_{\hat{\alpha}}^{i}-\hat{\Sigma}\epsilon^{ij}\hat{\nabla}_{i}\hat{\chi}_{j}\}.\label{eq:Jusinskasaction}\end{aligned}$$ Here, $\{i,j\}$ denote the worldsheet directions $\tau$ and $\sigma$, and $\mathcal{T}$ is the string tension. The spinors $\xi^{\alpha}$ and $\hat{\xi}^{\hat{\alpha}}$, with conjugates $p_{\alpha}^{i}$ and $\hat{p}_{\hat{\alpha}}^{i}$, satisfy $$(\lambda\gamma^{m}\xi)=(\hat{\lambda}\gamma^{m}\hat{\xi})=0.$$ The motivation for introducing the constrained spinors $\xi^{\alpha}$ and $\hat{\xi}^{\hat{\alpha}}$ is that they should suplement the degrees of freedom from the ghosts of the twistor-like symmetry, combining into the superpartners of $X^{m}$. The fermionic symmetry in $S_{0}$ is generated by the current $\lambda^{\alpha}p_{\alpha}^{i}$, with Lagrange multiplier $\chi_{i}$. The fields $B$ and $\Sigma$ effectively work as conjugates to the gauge field of the scale symmetry ($A_{i}$) and its fermionic partner $(\chi_{i}$), respectively. They can be interpreted as Lagrange multipliers for a zero curvature condition on the gauge fields, $\epsilon^{ij}\partial_{i}A_{j}=\epsilon^{ij}\nabla_{i}\chi_{j}=0$, with $\epsilon^{ij}=-\epsilon^{ji}$ ($\epsilon_{ij}=-\epsilon_{ji}$) and $\epsilon^{\sigma\tau}=\epsilon_{\tau\sigma}=1$. The hatted variables have an analogous description.
The reparametrization invariant action has a rich gauge structure. Its gauge algebra is onshell reducible and reparametrization symmetry can be consistently overlooked in the quantization process due to the simple form of the gauge-for-gauge symmetries. Still, the Batalin-Vilkovisky formalism seems to be the most adequate for the quantization of this model, because it provides a systematic way to analyze the gauge symmetries related to the pure spinor constraints. The extra fermionic symmetry of the action does not have a clear physical interpretation but it is ultimately related to the emergence of spacetime supersymmetry in the gauge fixed action, which can be cast as $$S=\int d\tau d\sigma\{\tfrac{\mathcal{T}}{2}\partial_{+}X_{m}\partial_{-}X^{m}+w_{\alpha}\partial_{-}\lambda^{\alpha}+p_{\alpha}\partial_{-}\theta^{\alpha}+\hat{w}_{\hat{\alpha}}\partial_{+}\hat{\lambda}^{\hat{\alpha}}+\hat{p}_{\hat{\alpha}}\partial_{+}\hat{\theta}^{\hat{\alpha}}\},\label{eq:Jusinskasfixed}$$ where $\partial_{\pm}=\partial_{\tau}\pm\partial_{\sigma}$, corresponding to the usual superstring action in the pure spinor formalism. In this action, $\theta^{\alpha}$ and $\hat{\theta}^{\hat{\alpha}}$, the superpartners of the target space coordinate $X^{m}$, are composed by the ghosts associated to the twistor-like symmetry complemented with the constrained spinors $\xi^{\alpha}$ and $\hat{\xi}^{\hat{\alpha}}$. This composition is only possible due to the existence of the scalar ghosts $\gamma$ and $\hat{\gamma}$ associated to the new fermionic symmetry in . They are assumed to be everywhere non-vanishing in the worldsheet, effectively acting as ghost number twisting operators and turning all the spacetime spinors neutral under scale transformations. This is the mechanism behind the conversion of the pure spinors $\lambda^{\alpha}$ and $\hat{\lambda}^{\hat{\alpha}}$ into ghost variables.
Plan of the paper {#plan-of-the-paper .unnumbered}
-----------------
Section \[sec:PSsuperstring\] presents in detail a first principles derivation of the pure spinor superstring. Subsection \[subsec:Polyakov1st\] quickly reviews the first order formulation of the Polyakov action and subsection \[subsec:twistor1st\] describes the connection between the twistor-like constraints and worldsheet reparametrization, motivating Berkovits’ model [@Berkovits:2015yra]. In subsection \[subsec:A-new-model\], the action is introduced with a thorough analysis of its gauge and gauge-for-gauge symmetries. Subsection \[subsec:PSmasteraction\] describes the construction of the pure spinor master action, while subsection \[subsec:Gauge-fixing\] presents its gauge fixing. Subsection \[subsec:Fieldred\] closes the section with the description of the emergence of spacetime supersymmetry leading to the pure spinor superstring. Section \[sec:Final-remarks\] contains some concluding remarks and possible directions to follow.
Throughout this work, special attention has been devoted to displaying auxiliary equations in the intermediate steps, so most computations can be promptly reproduced. The appendices consist of several complements and by-products of the main text. Appendix \[sec:Partial-gauge-fixing\] shows how the partial gauge fixing of the pure spinor symmetries leads to the unconstrained spinor $\theta^{\alpha}$, the superpartner of the target space coordinates $X^{m}$. Appendix \[sec:decoupledsector\] presents some properties of the $U(1)_{R}\times U(1)_{L}$ sector constituted by the ghost fields related to the scaling and fermionic symmetries, respectively with parameters $\{\Omega,\hat{\Omega}\}$ and $\{\gamma,\hat{\gamma}\}$. Appendix \[sec:NMBC\] describes a non-minimal pure spinor superstring with fundamental $(b,c)$ ghosts, as suggested in [@Berkovits:2014aia]. Finally, appendix \[sec:sectorized\] presents the sectorized and the ambitwistor string in the pure spinor formalism as coming from a singular gauge fixing of the action .
The pure spinor superstring\[sec:PSsuperstring\]
================================================
In this section, the connection between worldsheet reparametrization and the twistor-like constraint will be investigated, with a concrete proposal for the underlying gauge theory of the pure spinor superstring.
Review of the Polyakov action in the first order formalism\[subsec:Polyakov1st\]
--------------------------------------------------------------------------------
The Polyakov action is given by $$S_{P}=\frac{\mathcal{T}}{2}\int d^{2}\sigma\sqrt{-g}\{g^{ij}\partial_{i}X^{m}\partial_{j}X_{m}\},$$ where $\mathcal{T}$ is the string tension, $g^{ij}$ is the worldsheet metric (with inverse $g_{ij}$) and $g=\det(g_{ij})$. When expanded in components ($i=\tau,\sigma$), with $\tau$ denoting the worldsheet time and $\sigma$ parametrizing the string length, $S_{P}$ is rewritten as $$S_{P}=\frac{\mathcal{T}}{2}\int d\tau d\sigma\,\sqrt{-g}\{g^{\tau\tau}\partial_{\tau}X^{m}\partial_{\tau}X_{m}+2g^{\tau\sigma}\partial_{\tau}X^{m}\partial_{\sigma}X_{m}+g^{\sigma\sigma}\partial_{\sigma}X^{m}\partial_{\sigma}X_{m}\}.$$ The canonical conjugate of the target-space coordinate $X^{m}$ is easily determined to be $$\begin{aligned}
P_{m} & \equiv & \frac{\delta S}{\delta\partial_{\tau}X^{m}},\nonumber \\
& = & \mathcal{T}\sqrt{-g}\{g^{\tau\tau}\partial_{\tau}X_{m}+g^{\tau\sigma}\partial_{\sigma}X_{m}\},\label{eq:Pmeom}\end{aligned}$$ leading to the Hamiltonian $$\begin{aligned}
H & \equiv & P_{m}\partial_{\tau}X^{m}-\tfrac{\mathcal{T}}{2}\sqrt{-g}g^{ij}\partial_{i}X^{m}\partial_{j}X_{m},\nonumber \\
& = & \frac{1}{2\mathcal{T}g^{\tau\tau}\sqrt{-g}}(P_{m}P^{m}+\mathcal{T}^{2}\partial_{\sigma}X_{m}\partial_{\sigma}X^{m})-\frac{g^{\tau\sigma}}{g^{\tau\tau}}P_{m}\partial_{\sigma}X^{m}.\end{aligned}$$
In the first order formulation, the Polyakov action takes the form $$\tilde{S}_{P}=\int d\tau d\sigma\{P_{m}\partial_{\tau}X^{m}-\frac{1}{2\mathcal{T}g^{\tau\tau}\sqrt{-g}}(P_{m}P^{m}+\mathcal{T}^{2}\partial_{\sigma}X_{m}\partial_{\sigma}X^{m})+\frac{g^{\tau\sigma}}{g^{\tau\tau}}P_{m}\partial_{\sigma}X^{m}\},$$ which is equivalent onshell to $S_{P}$, *cf*. equation . Observe that the only dependence on the worldsheet metric appears now in the form of Lagrange multipliers. In fact, by defining the Weyl invariant operators $$e_{\pm}\equiv\frac{1}{g^{\tau\tau}\sqrt{-g}}\mp\frac{g^{\tau\sigma}}{g^{\tau\tau}},\label{eq:e+-tometric}$$ the action $\tilde{S}_{P}$ is more symmetrically rewritten as $$\begin{gathered}
\tilde{S}_{P}=\int d\tau d\sigma\big\{ P_{m}\partial_{\tau}X^{m}-\tfrac{1}{4\mathcal{T}}e_{+}(P_{m}+\mathcal{T}\partial_{\sigma}X_{m})(P^{m}+\mathcal{T}\partial_{\sigma}X^{m})\\
-\tfrac{1}{4\mathcal{T}}e_{-}(P_{m}-\mathcal{T}\partial_{\sigma}X_{m})(P^{m}-\mathcal{T}\partial_{\sigma}X^{m})\big\}.\label{eq:Pol1st}\end{gathered}$$
The equations of motion for $P_{m}$, $X^{m}$ and $e_{\pm}$ are respectively given by
$$\begin{aligned}
\partial_{\tau}X^{m}-\tfrac{1}{2\mathcal{T}}[e_{+}(P^{m}+\mathcal{T}\partial_{\sigma}X^{m})+e_{-}(P^{m}-\mathcal{T}\partial_{\sigma}X^{m})] & = & 0\\
\partial_{\tau}P_{m}-\tfrac{1}{2}\partial_{\sigma}[e_{+}(P_{m}+\mathcal{T}\partial_{\sigma}X_{m})-e_{-}(P_{m}-\mathcal{T}\partial_{\sigma}X_{m})] & = & 0,\\
(P_{m}\pm\mathcal{T}\partial_{\sigma}X_{m})(P^{m}\pm\mathcal{T}\partial_{\sigma}X^{m}) & = & 0,\end{aligned}$$
and the action is invariant under the gauge transformations
\[eq:Pol1stgauge\] $$\begin{aligned}
\delta X^{m} & = & c^{i}\partial_{i}X^{m}-\tfrac{1}{2\mathcal{T}}a^{+}(P^{m}+\mathcal{T}\partial_{\sigma}X^{m})-\tfrac{1}{2\mathcal{T}}a^{-}(P^{m}-\mathcal{T}\partial_{\sigma}X^{m}),\\
\delta P_{m} & = & \partial_{i}(c^{i}P_{m})-P_{m}\partial_{\tau}c^{\tau}+\tfrac{1}{2}\partial_{\sigma}c^{\tau}[(P_{m}+\mathcal{T}\partial_{\sigma}X_{m})e_{+}-(P_{m}-\mathcal{T}\partial_{\sigma}X_{m})e_{-}]\nonumber \\
& & -\tfrac{1}{2}\partial_{\sigma}[a^{+}(P^{m}+\mathcal{T}\partial_{\sigma}X^{m})-a^{-}(P^{m}-\mathcal{T}\partial_{\sigma}X^{m})],\\
\delta e_{+} & = & \partial_{i}(c^{i}e_{+})-2(\partial_{\sigma}c^{\sigma})e_{+}+\partial_{\tau}c^{\sigma}-e_{+}^{2}\partial_{\sigma}c^{\tau}-\partial_{\tau}a^{+}-a^{+}\partial_{\sigma}e_{+}+e_{+}\partial_{\sigma}a^{+},\\
\delta e_{-} & = & \partial_{i}(c^{i}e_{-})-2(\partial_{\sigma}c^{\sigma})e_{-}-\partial_{\tau}c^{\sigma}+e_{-}^{2}\partial_{\sigma}c^{\tau}-\partial_{\tau}a^{-}+a^{-}\partial_{\sigma}e_{-}-e_{-}\partial_{\sigma}a^{-}.\end{aligned}$$
In addition to worldsheet reparametrization symmetry, with parameter $c^{i}$, the action $\tilde{S}_{P}$ is also invariant under the gauge transformations parametrized by $a^{\pm}$. However, these gauge symmetries are not irreducible. To see this, consider the gauge-for-gauge transformations
$$\begin{aligned}
\delta^{'}c^{i} & = & \phi^{i},\\
\delta^{'}a^{\pm} & = & \phi^{\tau}e_{\pm}\pm\phi^{\sigma},\end{aligned}$$
with parameter $\phi^{i}$. It is then straightforward to show that the gauge transformations are invariant up to equations of motion:
$$\begin{aligned}
\delta^{'}[\delta X^{m}] & = & \phi^{\tau}\big\{\partial_{\tau}X^{m}-\tfrac{1}{2\mathcal{T}}[e_{+}(P^{m}+\mathcal{T}\partial_{\sigma}X^{m})+e_{-}(P^{m}-\mathcal{T}\partial_{\sigma}X^{m})]\big\},\\
\delta^{'}[\delta P_{m}] & = & \phi^{\tau}\big\{\partial_{\tau}P_{m}-\tfrac{1}{2}\partial_{\sigma}[e_{+}(P_{m}+\mathcal{T}\partial_{\sigma}X_{m})-e_{-}(P_{m}-\mathcal{T}\partial_{\sigma}X_{m})]\big\},\\
\delta^{'}[\delta e_{+}] & = & 0,\\
\delta^{'}[\delta e_{-}] & = & 0.\end{aligned}$$
Therefore, worldsheet reparametrization is equivalent to the symmetries generated by $$\begin{aligned}
H^{\pm} & \equiv & (P_{m}\pm\mathcal{T}\partial_{\sigma}X_{m})(P^{m}\pm\mathcal{T}\partial_{\sigma}X^{m}).\label{eq:H+-}\end{aligned}$$
The twistor-like constraint in the first order formalism\[subsec:twistor1st\]
-----------------------------------------------------------------------------
In [@Berkovits:2015yra], Berkovits proposed the twistor-like constraints
\[eq:twistorconstraints\] $$\begin{aligned}
H_{\alpha} & \equiv & (P_{m}+\mathcal{T}\partial_{\sigma}X_{m})(\gamma^{m}\lambda)_{\alpha},\\
\hat{H}_{\hat{\alpha}} & \equiv & (P_{m}-\mathcal{T}\partial_{\sigma}X_{m})(\gamma^{m}\hat{\lambda})_{\hat{\alpha}},\end{aligned}$$
as part of the fundamental gauge algebra behind the pure spinor superstring, where $\lambda^{\alpha}$ and $\hat{\lambda}^{\hat{\alpha}}$ are bosonic spinors satisfying the pure spinor condition
\[eq:PSconstraint\] $$\begin{aligned}
(\lambda\gamma^{m}\lambda) & = & 0,\\
(\hat{\lambda}\gamma^{m}\hat{\lambda}) & = & 0.\end{aligned}$$
The key idea here is that $H^{\pm}$ in can be rewritten as
\[eq:H+-twistor\] $$\begin{aligned}
H^{+} & = & (P_{m}+\mathcal{T}\partial_{\sigma}X_{m})\frac{(\Lambda\gamma^{m})^{\alpha}}{(\Lambda\lambda)}H_{\alpha},\\
H^{-} & = & (P_{m}-\mathcal{T}\partial_{\sigma}X_{m})\frac{(\gamma^{m}\hat{\Lambda})^{\hat{\alpha}}}{(\hat{\Lambda}\hat{\lambda})}\hat{H}_{\hat{\alpha}},\end{aligned}$$
for any constant $\Lambda_{\alpha}$ and $\hat{\Lambda}_{\hat{\alpha}}$ with non-vanishing $(\Lambda\lambda)$ and $(\hat{\Lambda}\hat{\lambda})$.
The first order Polyakov’s action can be covariantly modified with the introduction of the twistor-like constraints . In order to do that, $\lambda^{\alpha}$ and $\hat{\lambda}^{\hat{\alpha}}$ have to be made dynamical. In addition, Berkovits proposed in [@Berkovits:2015yra] the use of projective pure spinors which can be achieved by endowing $\lambda^{\alpha}$ and $\hat{\lambda}^{\hat{\alpha}}$ with a scaling symmetry. The resulting action is $$\begin{gathered}
S_{B}^{'}=\int d\tau d\sigma\{P_{m}\partial_{\tau}X^{m}+w_{\alpha}^{i}\partial_{i}\lambda^{\alpha}+\hat{w}_{\hat{\alpha}}^{i}\partial_{i}\hat{\lambda}^{\hat{\alpha}}+A_{i}\lambda^{\alpha}w_{\alpha}^{i}+\hat{A}_{i}\hat{\lambda}^{\hat{\alpha}}\hat{w}_{\hat{\alpha}}^{i}\}\\
-\tfrac{1}{4\mathcal{T}}\int d\tau d\sigma\Big\{(L\gamma_{m}\lambda)(P^{m}+\mathcal{T}\partial_{\sigma}X^{m})+e_{+}(P_{m}+\mathcal{T}\partial_{\sigma}X_{m})(P^{m}+\mathcal{T}\partial_{\sigma}X^{m})\\
+(\hat{L}\gamma_{m}\hat{\lambda})(P^{m}-\mathcal{T}\partial_{\sigma}X^{m})+e_{-}(P_{m}-\mathcal{T}\partial_{\sigma}X_{m})(P^{m}-\mathcal{T}\partial_{\sigma}X^{m})\Big\},\end{gathered}$$ where $\{L^{\alpha},\hat{L}^{\hat{\alpha}}\}$ are the Lagrange multipliers for the constraints and $\{A_{i},\hat{A}_{i}\}$ are the gauge fields for the scaling symmetry generated by $\lambda^{\alpha}w_{\alpha}^{i}$ and $\hat{\lambda}^{\hat{\alpha}}\hat{w}_{\hat{\alpha}}^{i}$. Due to the identification , a field shift in $L^{\alpha}$ and $\hat{L}^{\hat{\alpha}}$ can absorb $e_{+}$ and $e_{-}$, leading to Berkovits’ action, $$\begin{aligned}
S_{B} & = & \int d\tau d\sigma\{P_{m}\partial_{\tau}X^{m}+w_{\alpha}^{i}\nabla_{i}\lambda^{\alpha}+\hat{w}_{\hat{\alpha}}^{i}\hat{\nabla}_{i}\hat{\lambda}^{\hat{\alpha}}\}\nonumber \\
& & -\tfrac{1}{4\mathcal{T}}\int d\tau d\sigma\{(L\gamma_{m}\lambda)(P^{m}+\mathcal{T}\partial_{\sigma}X^{m})+(\hat{L}\gamma_{m}\hat{\lambda})(P^{m}-\mathcal{T}\partial_{\sigma}X^{m})\}.\label{eq:Berkovitsactioncovariant}\end{aligned}$$ The gauge fields $\{A_{i},\hat{A}_{i}\}$ now appear through the covariant derivatives $\{\nabla_{i},\hat{\nabla}_{i}\}$. Observe also that all the dependence on the worldsheet metric is concentrated in the Lagrange multipliers $\{L^{\alpha},\hat{L}^{\hat{\alpha}}\}$ and this has to be taken into account during the gauge fixing process.
A new model with constrained anticommuting spinors\[subsec:A-new-model\]
------------------------------------------------------------------------
Based on the worldline results of [@Jusinskas:2018uci], it is straightforward to generalize the action to $$\begin{aligned}
S_{0} & = & \int d\tau d\sigma\{P_{m}\partial_{\tau}X^{m}+w_{\alpha}^{i}\nabla_{i}\lambda^{\alpha}+\hat{w}_{\hat{\alpha}}^{i}\hat{\nabla}_{i}\hat{\lambda}^{\hat{\alpha}}\}\nonumber \\
& & -\tfrac{1}{4\mathcal{T}}\int d\tau d\sigma\{(L\gamma_{m}\lambda)(P^{m}+\mathcal{T}\partial_{\sigma}X^{m})+(\hat{L}\gamma_{m}\hat{\lambda})(P^{m}-\mathcal{T}\partial_{\sigma}X^{m})\}\nonumber \\
& & +\int d\tau d\sigma\{p_{\alpha}^{i}\partial_{i}\xi^{\alpha}+B\epsilon^{ij}\partial_{i}A_{j}+\hat{p}_{\hat{\alpha}}^{i}\partial_{i}\hat{\xi}^{\hat{\alpha}}+\hat{B}\epsilon^{ij}\partial_{i}\hat{A}_{j}\}\nonumber \\
& & +\int d\tau d\sigma\{\chi_{i}\lambda^{\alpha}p_{\alpha}^{i}-\Sigma\epsilon^{ij}\nabla_{i}\chi_{j}+\hat{\chi}_{i}\hat{\lambda}^{\hat{\alpha}}\hat{p}_{\hat{\alpha}}^{i}-\hat{\Sigma}\epsilon^{ij}\hat{\nabla}_{i}\hat{\chi}_{j}\}.\label{eq:newaction}\end{aligned}$$ There are two guiding principles that led to the proposed action $S_{0}$, (1) the extension of the phase space with the inclusion of constrained anticommuting spinors, $\xi^{\alpha}$ and $\hat{\xi}^{\hat{\alpha}}$, satisfying
\[eq:SPSconstraint\] $$\begin{aligned}
(\lambda\gamma^{m}\xi) & = & 0,\\
(\hat{\lambda}\gamma^{m}\hat{\xi}) & = & 0,\end{aligned}$$
together with two fermionic symmetries generated by $\lambda^{\alpha}p_{\alpha}^{i}$ and $\hat{\lambda}^{\hat{\alpha}}\hat{p}_{\hat{\alpha}}^{i}$; and (2) the introduction of zero curvature conditions on the gauge fields $\{A_{i},\chi_{i},\hat{A}_{i},\hat{\chi}_{i}\}$ through the Lagrange multipliers $\{B,\Sigma,\hat{B},\hat{\Sigma}\}$, which enables the extension of the gauge algebra of the model with the inclusion of gauge-for-gauge symmetries connecting worldsheet reparametrization and the twistor-like symmetries.
The equations of motion obtained from the action can be summarized as
\[eq:eomPS\] $$\begin{aligned}
\partial_{\tau}X^{m}-\tfrac{1}{4\mathcal{T}}(L\gamma^{m}\lambda)-\tfrac{1}{4\mathcal{T}}(\hat{L}\gamma^{m}\hat{\lambda}) & = & 0,\\
\partial_{\tau}P_{m}-\tfrac{1}{4}\partial_{\sigma}(L\gamma_{m}\lambda)+\tfrac{1}{4}\partial_{\sigma}(\hat{L}\gamma_{m}\hat{\lambda}) & = & 0,\\
\nabla_{i}\lambda^{\alpha} & = & 0,\\
\nabla_{i}w_{\alpha}^{i}+\tfrac{1}{4\mathcal{T}}(P^{m}+\mathcal{T}\partial_{\sigma}X^{m})(\gamma_{m}L)_{\alpha}-\chi_{i}p_{\alpha}^{i} & = & 0,\\
\lambda^{\alpha}w_{\alpha}^{i}+\epsilon^{ij}(\Sigma\chi_{j}+\partial_{j}B) & = & 0,\\
\partial_{i}\xi^{\alpha}-\chi_{i}\lambda^{\alpha} & = & 0,\\
\partial_{i}p_{\alpha}^{i} & = & 0,\\
\epsilon^{ij}\partial_{i}A_{j} & = & 0,\\
(P^{m}+\mathcal{T}\partial_{\sigma}X^{m})(\gamma_{m}\lambda)_{\alpha} & = & 0,\\
\lambda^{\alpha}p_{\alpha}^{i}+\epsilon^{ij}\nabla_{j}\Sigma & = & 0,\\
\epsilon^{ij}\nabla_{i}\chi_{j} & = & 0,\\
\hat{\nabla}_{i}\hat{\lambda}^{\hat{\alpha}} & = & 0,\\
\hat{\nabla}_{i}\hat{w}_{\hat{\alpha}}^{i}+\tfrac{1}{4\mathcal{T}}(P^{m}-\mathcal{T}\partial_{\sigma}X^{m})(\gamma_{m}\hat{L})_{\hat{\alpha}}-\hat{\chi}_{i}\hat{p}_{\hat{\alpha}}^{i} & = & 0,\\
\hat{\lambda}^{\hat{\alpha}}\hat{w}_{\hat{\alpha}}^{i}+\epsilon^{ij}(\hat{\Sigma}\hat{\chi}_{j}+\partial_{j}\hat{B}) & = & 0,\\
\partial_{i}\hat{\xi}^{\hat{\alpha}}-\hat{\chi}_{i}\hat{\lambda}^{\hat{\alpha}} & = & 0,\\
\partial_{i}\hat{p}_{\hat{\alpha}}^{i} & = & 0,\\
\epsilon^{ij}\partial_{i}\hat{A}_{j} & = & 0,\\
(P^{m}-\mathcal{T}\partial_{\sigma}X^{m})(\gamma_{m}\hat{\lambda})_{\alpha} & = & 0,\\
\hat{\lambda}^{\hat{\alpha}}\hat{p}_{\hat{\alpha}}^{i}+\epsilon^{ij}\hat{\nabla}_{j}\hat{\Sigma} & = & 0,\\
\epsilon^{ij}\hat{\nabla}_{i}\hat{\chi}_{j} & = & 0.\end{aligned}$$
Due to the constraints and , the action $S_{0}$ is invariant under $$\begin{array}{rclcrcl}
\delta w_{\alpha}^{i} & = & d_{m}^{i}(\gamma^{m}\lambda)_{\alpha}+e_{m}^{i}(\gamma^{m}\xi)_{\alpha}, & & \delta\hat{w}_{\hat{\alpha}}^{i} & = & \hat{d}_{m}^{i}(\gamma^{m}\hat{\lambda})_{\hat{\alpha}}+\hat{e}_{m}^{i}(\gamma^{m}\hat{\xi})_{\hat{\alpha}},\\
\delta p_{\alpha}^{i} & = & e_{m}^{i}(\gamma^{m}\lambda)_{\alpha}, & & \delta\hat{p}_{\hat{\alpha}}^{i} & = & \hat{e}_{m}^{i}(\gamma^{m}\hat{\lambda})_{\hat{\alpha}},\\
\delta L^{\alpha} & = & f\lambda^{\alpha}+f_{mn}(\gamma^{mn}\lambda)^{\alpha}+g\xi^{\alpha}, & & \delta\hat{L}^{\hat{\alpha}} & = & \hat{f}\hat{\lambda}^{\hat{\alpha}}+\hat{f}_{mn}(\gamma^{mn}\hat{\lambda})^{\hat{\alpha}}+\hat{g}\xi^{\hat{\alpha}},
\end{array}\label{eq:prePSsymetries}$$ where $d_{m}$, $e_{m}$, $f$, $f_{mn}$, $g$ (hatted and unhatted) are local parameters. These gauge transformations have a special role in the formalism and will be called pure spinor symmetries. The other gauge symmetries of the model can be summarized by:
1. Worldsheet reparametrization, with parameter $c^{i}=\{c^{\tau},c^{\sigma}\}$. Although the transformations of $P_{m}$, $L^{\alpha}$ and $\hat{L}^{\hat{\alpha}}$ are nontrivial, $$\begin{array}{rcl}
\delta P_{m} & = & \partial_{\sigma}(c^{\sigma}P_{m})+c^{\tau}\partial_{\tau}P_{m}+\tfrac{1}{4}(L_{+}\gamma_{m}\lambda)(\partial_{\sigma}c^{\tau})-\tfrac{1}{4}(\hat{L}_{-}\gamma_{m}\hat{\lambda})(\partial_{\sigma}c^{\tau}),\\
\delta L^{\alpha} & = & c^{\sigma}\partial_{\sigma}L^{\alpha}+\partial_{\tau}(c^{\tau}L^{\alpha})+\partial_{\tau}c^{\sigma}(P^{m}+\mathcal{T}\partial_{\sigma}X^{m})\frac{(\gamma_{m}\Lambda)^{\alpha}}{(\Lambda\lambda)},\\
\delta\hat{L}^{\hat{\alpha}} & = & c^{\sigma}\partial_{\sigma}\hat{L}^{\hat{\alpha}}+\partial_{\tau}(c^{\tau}\hat{L}^{\hat{\alpha}})-\partial_{\tau}c^{\sigma}(P^{m}-\mathcal{T}\partial_{\sigma}X^{m})\frac{(\gamma_{m}\hat{\Lambda})^{\hat{\alpha}}}{(\hat{\Lambda}\hat{\lambda})},
\end{array}$$ all the other fields transform covariantly either as worldsheet scalars (*e.g.* $\delta X^{m}=c^{i}\partial_{i}X^{m}$), vector densities (*e.g.* $\delta w_{\alpha}^{i}=\partial_{j}(c^{j}w_{\alpha}^{i})-w_{\alpha}^{j}\partial_{j}c^{i},$) or 1-forms (*e.g.* $\delta A_{i}=c^{j}\partial_{j}A_{i}+A_{j}\partial_{i}c^{j}$).
2. Particle-like Hamiltonian symmetry, with parameter $a^{\pm}$. This symmetry is analogous to and the transformations can be cast as $$\begin{array}{rcl}
\delta X^{m} & = & a^{+}(P^{m}+\mathcal{T}\partial_{\sigma}X^{m})+a^{-}(P^{m}-\mathcal{T}\partial_{\sigma}X^{m}),\\
\delta P_{m} & = & \mathcal{T}\partial_{\sigma}[a^{+}(P^{m}+\mathcal{T}\partial_{\sigma}X^{m})]-\mathcal{T}\partial_{\sigma}[a^{-}(P^{m}-\mathcal{T}\partial_{\sigma}X^{m})],\\
\delta L^{\alpha} & = & 2\mathcal{T}a^{+}\nabla_{\sigma}L^{\alpha}+2\mathcal{T}\partial_{\tau}a^{+}(P^{m}+\mathcal{T}\partial_{\sigma}X^{m})\frac{(\gamma_{m}\Lambda)^{\alpha}}{(\Lambda\lambda)},\\
\delta w_{\alpha}^{\sigma} & = & -\tfrac{1}{2}a^{+}(P^{m}+\mathcal{T}\partial_{\sigma}X^{m})(\gamma_{m}L)_{\alpha},\\
\delta\hat{L}^{\hat{\alpha}} & = & -2\mathcal{T}a^{-}\hat{\nabla}_{\sigma}\hat{L}^{\hat{\alpha}}+2\mathcal{T}\partial_{\tau}a^{-}(P_{m}-\mathcal{T}\partial_{\sigma}X_{m})\frac{(\gamma^{m}\hat{\Lambda})^{\hat{\alpha}}}{(\hat{\lambda}\hat{\Lambda})},\\
\delta\hat{w}_{\hat{\alpha}}^{\sigma} & = & \tfrac{1}{2}a^{-}(P^{m}-\mathcal{T}\partial_{\sigma}X^{m})(\gamma_{m}\hat{L})_{\hat{\alpha}}.
\end{array}$$
3. Scaling symmetry, with parameters $\{\Omega,\hat{\Omega}\}$. The transformations are $$\begin{array}{rclcrcl}
\delta\lambda^{\alpha} & = & \Omega\lambda^{\alpha}, & & \delta\hat{\lambda}^{\hat{\alpha}} & = & \hat{\Omega}\hat{\lambda}^{\hat{\alpha}},\\
\delta w_{\alpha}^{i} & = & -\Omega w_{\alpha}^{i}, & & \delta\hat{w}_{\hat{\alpha}}^{i} & = & -\Omega\hat{w}_{\hat{\alpha}}^{i},\\
\delta A_{i} & = & -\partial_{i}\Omega, & & \delta\hat{A}_{i} & = & -\partial_{i}\hat{\Omega},\\
\delta L^{\alpha} & = & -\Omega L^{\alpha}, & & \delta\hat{L}^{\hat{\alpha}} & = & -\hat{\Omega}\hat{L}^{\hat{\alpha}},\\
\delta\chi_{i} & = & -\Omega\chi_{i}, & & \delta\hat{\chi}_{i} & = & -\hat{\Omega}\hat{\chi}_{i},\\
\delta\Sigma & = & \Omega\Sigma, & & \delta\hat{\Sigma} & = & \hat{\Omega}\hat{\Sigma}.
\end{array}\label{eq:scalingSYM}$$
4. Fermionic gauge symmetry, with gauge fields $\chi_{i}$ and $\hat{\chi}_{i}$ and parameters $\gamma$ and $\hat{\gamma}$, respectively. The action $S_{0}$ is invariant under the transformations $$\begin{array}{rclcrcl}
\delta w_{\alpha}^{i} & = & \gamma p_{\alpha}^{i}, & & \delta\hat{w}_{\hat{\alpha}}^{i} & = & \hat{\gamma}\hat{p}_{\hat{\alpha}}^{i},\\
\delta B & = & \gamma\Sigma, & & \delta\hat{B} & = & \hat{\gamma}\hat{\Sigma},\\
\delta\xi^{\alpha} & = & \gamma\lambda^{\alpha}, & & \delta\hat{\xi}^{\hat{\alpha}} & = & \hat{\gamma}\hat{\lambda}^{\hat{\alpha}},\\
\delta\chi_{i} & = & \nabla_{i}\gamma, & & \delta\chi_{i} & = & \hat{\nabla}_{i}\hat{\gamma}.
\end{array}\label{eq:fermionSYM}$$
5. Curl symmetry, with parameters $\{s_{\alpha},\epsilon_{\alpha},\hat{s}_{\hat{\alpha}},\hat{\epsilon}_{\hat{\alpha}}\}$. By construction, the reparametrization invariant form of the kinetic terms of the spinors in imply the existence of gauge transformations given by $$\begin{array}{rclcrcl}
\delta w_{\alpha}^{i} & = & \epsilon^{ij}(\nabla_{j}s_{\alpha}+\epsilon_{\alpha}\chi_{j}), & & \delta\hat{w}_{\hat{\alpha}}^{i} & = & \epsilon^{ij}(\hat{\nabla}_{j}\hat{s}_{\hat{\alpha}}+\hat{\epsilon}_{\hat{\alpha}}\hat{\chi}_{j}),\\
\delta B & = & -\lambda^{\alpha}s_{\alpha}, & & \delta\hat{B} & = & -\hat{\lambda}^{\hat{\alpha}}\hat{s}_{\hat{\alpha}},\\
\delta p_{\alpha}^{i} & = & \epsilon^{ij}\partial_{j}\epsilon_{\alpha}, & & \delta\hat{p}_{\hat{\alpha}} & = & \epsilon^{ij}\partial_{j}\hat{\epsilon}_{\hat{\alpha}},\\
\delta\Sigma & = & -\lambda^{\alpha}\epsilon_{\alpha}, & & \delta\hat{\Sigma} & = & -\hat{\lambda}^{\hat{\alpha}}\hat{\epsilon}_{\hat{\alpha}}.
\end{array}\label{eq:curlSYM}$$
6. Twistor-like symmetry, with parameters $\theta^{\alpha}$ and $\hat{\theta}^{\hat{\alpha}}$ and gauge transformations $$\begin{array}{rclcrcl}
\delta X^{m} & = & \tfrac{1}{4T}\big[(\lambda\gamma^{m}\theta)+(\hat{\lambda}\gamma^{m}\hat{\theta})\big], & & \delta w_{\alpha}^{i} & = & -\delta_{\tau}^{i}\tfrac{1}{4\mathcal{T}}(P_{m}+\mathcal{T}\partial_{\sigma}X^{m})(\theta\gamma^{m})_{\alpha}\\
\delta P_{m} & = & \tfrac{1}{4}\nabla_{\sigma}\big[(\lambda\gamma^{m}\theta)-(\hat{\lambda}\gamma^{m}\hat{\theta})\big], & & & & +\delta_{\sigma}^{i}\tfrac{1}{8\mathcal{T}}(L\gamma_{m}\lambda)(\theta\gamma^{m})_{\alpha},\\
\delta L^{\alpha} & = & \nabla_{\tau}\theta^{\alpha}, & & \delta\hat{w}_{\hat{\alpha}}^{i} & = & -\delta_{\tau}^{i}\tfrac{1}{4\mathcal{T}}(P_{m}-\mathcal{T}\partial_{\sigma}X^{m})(\hat{\theta}\gamma^{m})_{\hat{\alpha}}\\
\delta\hat{L}^{\hat{\alpha}} & = & \hat{\nabla}_{\tau}\hat{\theta}^{\hat{\alpha}}, & & & & -\delta_{\sigma}^{i}\tfrac{1}{8\mathcal{T}}(\hat{L}\gamma_{m}\hat{\lambda})(\hat{\theta}\gamma^{m})_{\hat{\alpha}}.
\end{array}\label{eq:twistorSYM}$$
To complete the analysis of the gauge structure of $S_{0}$, consider the gauge-for-gauge transformations with parameters $\phi^{i}$ and $\varphi^{\pm}$: $$\begin{array}{rclcrcl}
\delta^{'}c^{i} & = & \phi^{i}, & & \delta^{'}\Omega & = & \phi^{i}A_{i},\\
\delta^{'}a^{\pm} & = & \varphi^{\pm}\mp\phi^{\sigma}, & & \delta^{'}\hat{\Omega} & = & \phi^{i}\hat{A}_{i},\\
\delta^{'}s_{\alpha} & = & \epsilon_{ij}\phi^{j}w_{\alpha}^{i}, & & \delta^{'}\epsilon_{\alpha} & = & \epsilon_{ij}\phi^{j}p_{\alpha}^{i},\\
\delta^{'}\hat{s}_{\hat{\alpha}} & = & \epsilon_{ij}\phi^{j}\hat{w}_{\hat{\alpha}}^{i}, & & \delta^{'}\hat{\epsilon}_{\hat{\alpha}} & = & \epsilon_{ij}\phi^{j}\hat{p}_{\hat{\alpha}}^{i},\\
\delta^{'}\gamma & = & -\phi^{i}\chi_{i}, & & \delta^{'}\theta^{\alpha} & = & -\phi^{\tau}L^{\alpha}-\varphi^{+}(P_{m}+\mathcal{T}\partial_{\sigma}X_{m})\frac{(\gamma^{m}\Lambda)^{\alpha}}{(\Lambda\lambda)},\\
\delta^{'}\hat{\gamma} & = & -\phi^{i}\hat{\chi}_{i}, & & \delta^{'}\hat{\theta}^{\hat{\alpha}} & = & -\phi^{\tau}\hat{L}^{\hat{\alpha}}-\varphi^{-}(P_{m}-\mathcal{T}\partial_{\sigma}X_{m})\frac{(\gamma^{m}\hat{\Lambda})^{\hat{\alpha}}}{(\hat{\lambda}\hat{\Lambda})}.
\end{array}\label{eq:g4g}$$ Up to equations of motion and pure spinor symmetries, *cf*. equations and , the gauge transformations listed above are left invariant by :
$$\begin{aligned}
\delta^{'}[\delta X^{m}] & = & \phi^{\tau}\Big\{\partial_{\tau}X^{m}-\tfrac{1}{4\mathcal{T}}(\lambda\gamma^{m}L)-\tfrac{1}{4\mathcal{T}}(\hat{\lambda}\gamma^{m}\hat{L})\Big\}\nonumber \\
& & +\tfrac{1}{4\mathcal{T}}\varphi^{+}\big\{(P_{n}+\mathcal{T}\partial_{\sigma}X_{n})(\gamma^{n}\lambda)_{\alpha}\big\}\tfrac{(\gamma^{m}\Lambda)^{\alpha}}{(\Lambda\lambda)}\nonumber \\
& & +\tfrac{1}{4\mathcal{T}}\varphi^{-}\big\{(P_{n}-\mathcal{T}\partial_{\sigma}X_{n})(\gamma^{n}\hat{\lambda})_{\hat{\alpha}}\big\}\tfrac{(\gamma^{m}\hat{\Lambda})^{\alpha}}{(\hat{\Lambda}\hat{\lambda})},\\
\delta^{'}[\delta P_{m}] & = & \phi^{\tau}\Big\{\partial_{\tau}P_{m}-\tfrac{1}{4}\partial_{\sigma}(\lambda\gamma_{m}L)+\tfrac{1}{4}\partial_{\sigma}(\hat{\lambda}\gamma_{m}\hat{L})\Big\}\nonumber \\
& & +\tfrac{1}{4}\nabla_{\sigma}\Big[\varphi^{+}\big\{(P_{n}+\mathcal{T}\partial_{\sigma}X_{n})(\gamma^{n}\lambda)_{\alpha}\big\}\tfrac{(\gamma^{m}\Lambda)^{\alpha}}{(\Lambda\lambda)}\Big]\nonumber \\
& & -\tfrac{1}{4}\hat{\nabla}_{\sigma}\Big[\varphi^{-}\big\{(P_{n}-\mathcal{T}\partial_{\sigma}X_{n})(\gamma^{n}\hat{\lambda})_{\hat{\alpha}}\big\}\tfrac{(\gamma^{m}\hat{\Lambda})^{\hat{\alpha}}}{(\hat{\Lambda}\hat{\lambda})}\Big],\\
\delta^{'}[\delta\lambda^{\alpha}] & = & \phi^{i}\{\nabla_{i}\lambda^{\alpha}\},\\
\delta^{'}[\delta w_{\alpha}^{i}] & = & \phi^{i}\big\{\nabla_{j}w_{\alpha}^{j}+\tfrac{1}{4T}(P_{m}+\mathcal{T}\partial_{\sigma}X_{m})(\gamma^{m}L)_{\alpha}-\chi_{j}p_{\alpha}^{j}\big\}\nonumber \\
& & +\tfrac{1}{16\mathcal{T}}\delta_{\sigma}^{i}\phi^{\tau}(L\gamma_{m}L)(\gamma^{m}\lambda)_{\alpha}+\tfrac{1}{8\mathcal{T}}\delta_{\sigma}^{i}\varphi^{+}(P_{n}+\mathcal{T}\partial_{\sigma}X_{n})(L\gamma^{m}\gamma^{n}\Lambda)\tfrac{(\gamma^{m}\lambda)_{\alpha}}{(\Lambda\lambda)}\nonumber \\
& & +\tfrac{1}{4\mathcal{T}}\delta_{\tau}^{i}\varphi^{+}\big\{(P_{m}+\mathcal{T}\partial_{\sigma}X_{m})(P^{m}+\mathcal{T}\partial_{\sigma}X^{m})\big\}\tfrac{\Lambda_{\alpha}}{(\Lambda\lambda)},\nonumber \\
& & -\tfrac{1}{8\mathcal{T}}\delta_{\sigma}^{i}\varphi^{+}\big\{(P_{n}+\mathcal{T}\partial_{\sigma}X_{n})(\gamma^{n}\lambda)_{\beta}\big\}(\gamma_{m}L)_{\alpha}\tfrac{(\gamma^{m}\Lambda)^{\beta}}{(\Lambda\lambda)},\\
\delta^{'}[\delta A_{i}] & = & \phi^{j}\{\partial_{j}A_{i}-\partial_{i}A_{j}\},\\
\delta^{'}[\delta B] & = & \phi^{k}\epsilon_{ki}\{\lambda^{\alpha}w_{\alpha}^{i}+\epsilon^{ij}(\Sigma\chi_{j}+\partial_{j}B)\},\\
\delta^{'}[\delta L^{\alpha}] & = & \varphi^{+}(P_{m}+\mathcal{T}\partial_{\sigma}X_{m})\{\nabla_{\tau}\lambda^{\beta}\}\Lambda_{\beta}\tfrac{(\gamma^{m}\Lambda)^{\alpha}}{(\Lambda\lambda)^{2}}-\tfrac{1}{2}\varphi^{+}(L\gamma_{m})_{\beta}\{\nabla_{\sigma}\lambda^{\beta}\}\tfrac{(\gamma^{m}\Lambda)^{\alpha}}{(\Lambda\lambda)}\nonumber \\
& & -\varphi^{+}\Big\{\partial_{\tau}P_{m}-\tfrac{1}{4}\partial_{\sigma}(L\gamma_{m}\lambda)+\tfrac{1}{4}\partial_{\sigma}(\hat{L}\gamma_{m}\hat{\lambda})\Big\}\tfrac{(\gamma^{m}\Lambda)^{\alpha}}{(\Lambda\lambda)}\nonumber \\
& & -\mathcal{T}\varphi^{+}\partial_{\sigma}\Big\{\partial_{\tau}X_{m}-\tfrac{1}{4\mathcal{T}}(L\gamma^{m}\lambda)-\tfrac{1}{4\mathcal{T}}(\hat{L}\gamma^{m}\hat{\lambda})\Big\}\tfrac{(\gamma^{m}\Lambda)^{\alpha}}{(\Lambda\lambda)}\nonumber \\
& & -\tfrac{1}{8}\varphi^{+}(\Lambda\gamma_{mn}\nabla_{\sigma}L)\tfrac{(\gamma^{mn}\lambda)^{\alpha}}{(\Lambda\lambda)}-\tfrac{1}{4}\varphi^{+}(\Lambda\nabla_{\sigma}L)\tfrac{\lambda^{\alpha}}{(\Lambda\lambda)},\\
\delta^{'}[\delta\xi^{\alpha}] & = & \phi^{i}\{\partial_{i}\xi^{\alpha}-\chi_{i}\lambda^{\alpha}\},\\
\delta^{'}[\delta p_{\alpha}^{i}] & = & \phi^{i}\{\partial_{j}p_{\alpha}^{j}\},\\
\delta^{'}[\delta\chi_{i}] & = & \phi^{j}\{\nabla_{j}\chi_{i}-\nabla_{i}\chi_{j}\},\\
\delta^{'}[\delta\Sigma] & = & \phi^{k}\epsilon_{kj}\{\lambda^{\alpha}p_{\alpha}^{i}+\epsilon^{ij}\nabla_{j}\Sigma\}.\end{aligned}$$
Similar equations hold for the hatted sector. This confirms that worldsheet reparametrization is a redundant symmetry of the action, since the parameters $\phi^{i}$ and $\varphi^{\pm}$ can be used to set $c^{i}=a^{\pm}=0$. Furthermore, their gauge-for-gauge transformations in involve only simple field shifts, *i.e.* no derivatives of the gauge-for-gauge parameters, therefore generating no dynamical ghost-for-ghosts. Consequently, the gauge symmetries parametrized by $c^{i}$ and $a^{\pm}$ can be disregarded in the construction of the master action within the Batalin-Vilkovisky formalism. It will be demonstrated next that the quantization of the action $S_{0}$ leads to the pure spinor superstring.
The pure spinor master action\[subsec:PSmasteraction\]
------------------------------------------------------
In order to build the pure spinor master action in the Batalin-Vilkovisky formalism, the gauge parameters discussed above will be promoted to dynamical variables. The field content of the model will be collectively denoted by $\Phi^{I}$, with the index $I$ running over the set $$\begin{gathered}
\Phi^{I}=\{X^{m},P_{m},w_{\alpha}^{i},\lambda^{\alpha},L^{\alpha},A_{i},B,p_{\alpha}^{i},\xi^{\alpha},\chi_{i},\Sigma,\Omega,\theta^{\alpha},s_{\alpha},\epsilon_{\alpha},\gamma,\\
\hat{w}_{\hat{\alpha}}^{i},\hat{\lambda}^{\hat{\alpha}},\hat{L}^{\hat{\alpha}},\hat{A}_{i},\hat{B},\hat{p}_{\hat{\alpha}}^{i},\hat{\xi}^{\hat{\alpha}},\hat{\chi}_{i},\hat{\Sigma},\hat{\Omega},\hat{\theta}^{\hat{\alpha}},\hat{s}_{\hat{\alpha}},\hat{\epsilon}_{\hat{\alpha}},\hat{\gamma}\}.\end{gathered}$$ As usual, ghost fields and the correspondent gauge parameters have opposite statistics, therefore $\{\Omega,\hat{\Omega},\theta^{\alpha},\hat{\theta}^{\hat{\alpha}},s_{\alpha},\hat{s}_{\hat{\alpha}}\}$ are Grassmann odd while $\{\epsilon_{\alpha},\hat{\epsilon}_{\hat{\alpha}},\gamma,\hat{\gamma}\}$ are Grassmann even fields. Following the discussion at the end of the previous subsection, the gauge parameters $c^{i}$ and $a^{\pm}$, and the gauge-for-gauge parameters $\phi^{i}$ and $\varphi^{\pm}$ will be ignored.
For every field $\Phi^{I}$ there is an antifield $\Phi_{I}^{*}$ associated, with opposite statistics, and the antifield set is given by $$\begin{gathered}
\Phi_{I}^{*}=\{X_{m}^{*},P_{*}^{m},w_{i*}^{\alpha},\lambda_{\alpha}^{*},L_{\alpha}^{*},A_{*}^{i},B^{*},p_{i*}^{\alpha},\xi_{\alpha}^{*},\chi_{*}^{i},\Sigma^{*},\Omega^{*},\theta_{\alpha}^{*},s_{*}^{\alpha},\epsilon_{*}^{\alpha},\gamma^{*},\\
\hat{w}_{i*}^{\hat{\alpha}},\hat{\lambda}_{\hat{\alpha}}^{*},\hat{L}_{\hat{\alpha}}^{*},\hat{A}_{*}^{i},\hat{B}^{*},\hat{p}_{i*}^{\hat{\alpha}},\hat{\xi}_{\hat{\alpha}}^{*},\hat{\chi}_{*}^{i},\hat{\Sigma}^{*},\hat{\Omega}^{*},\hat{\theta}_{\hat{\alpha}}^{*},\hat{s}_{*}^{\hat{\alpha}},\hat{\epsilon}_{*}^{\hat{\alpha}},\hat{\gamma}^{*}\}.\end{gathered}$$
By definition, fields and antifields are conjugate to each other, satisfying the antibracket relation $$\{\Phi_{I}^{*},\Phi^{J}\}=\delta_{I}^{J}.\label{eq:FaFconjugates}$$ In general, the antibrackets between two operators $\mathcal{O}_{1}$ and $\mathcal{O}_{2}$ are defined as $$\{\mathcal{O}_{1},\mathcal{O}_{2}\}\equiv\sum_{I}\left\{ \mathcal{O}_{1}\left(\frac{\overleftarrow{\partial}}{\partial\Phi_{I}^{*}}\frac{\partial}{\partial\Phi^{I}}-\frac{\overleftarrow{\partial}}{\partial\Phi^{I}}\frac{\partial}{\partial\Phi_{I}^{*}}\right)\mathcal{O}_{2}\right\} ,\label{eq:antibrackets}$$ from which equation follows. However, as consequence of the constraints and , not all the components of $\Phi^{I}$ and $\Phi_{I}^{*}$ are independent. In fact, the pure spinor constraints are generalized to $$\begin{array}{rclcrcl}
(\lambda\gamma^{m}\lambda) & = & 0, & & (\lambda\gamma^{m}p_{i*})+(\xi\gamma^{m}w_{i*}) & = & 0,\\
(\lambda\gamma^{m}\xi) & = & 0, & & \lambda^{\alpha}\theta_{\alpha}^{*}+w_{\tau*}^{\alpha}L_{\alpha}^{*} & = & 0,\\
(\lambda\gamma^{m}w_{i*}) & = & 0, & & \xi^{\alpha}\theta_{\alpha}^{*}+p_{\tau*}^{\alpha}L_{\alpha}^{*} & = & 0,\\
\lambda^{\alpha}L_{\alpha}^{*} & = & 0, & & (\lambda\gamma^{mn}\theta^{*})+(w_{\tau*}\gamma^{mn}L^{*}) & = & 0,\\
(\lambda\gamma^{mn}L^{*}) & = & 0, & & (\lambda\gamma^{m}s_{*})-\tfrac{1}{2}\epsilon^{ij}(w_{i*}\gamma^{m}w_{j*}) & = & 0,\\
\xi^{\alpha}L_{\alpha}^{*} & = & 0, & & (\lambda\gamma^{m}\epsilon_{*})+(\xi\gamma^{m}s_{*})-\epsilon^{ij}(p_{i*}\gamma^{m}w_{j*}) & = & 0,
\end{array}\label{eq:PSconstraintsBV}$$ and analogous constraints on the hatted sector, and the antibracket cannot be naively computed. For example, $$\{\lambda_{\alpha}^{*},(\lambda\gamma^{m}\lambda)\}=2(\gamma^{m}\lambda)_{\alpha},$$ which is not compatible with the constraint $(\lambda\gamma^{m}\lambda)=0$. This contradiction arises because there is an intrinsic gauge freedom implied by the constraints , which only have vanishing antibrackets with operators invariant under the pure spinor gauge transformations given by
\[eq:PSsymmetriesFULL\] $$\begin{aligned}
\delta\lambda_{\alpha}^{*} & = & b_{m}(\gamma^{m}\lambda)_{\alpha}+c_{m}(\gamma^{m}\xi)_{\alpha}-d_{m}^{i}(\gamma^{m}w_{i*})_{\alpha}-e_{m}^{i}(\gamma^{m}p_{i*})_{\alpha}-fL_{\alpha}^{*}\nonumber \\
& & +f_{mn}(\gamma^{mn}L^{*})_{\alpha}-\bar{f}\theta_{\alpha}^{*}+\bar{f}_{mn}(\gamma^{mn}\theta^{*})_{\alpha}-h_{m}(\gamma^{m}s_{*})_{\alpha}-\bar{h}_{m}(\gamma^{m}\epsilon_{*})_{\alpha},\\
\delta\xi_{\alpha}^{*} & = & c_{m}(\gamma^{m}\lambda)_{\alpha}+e_{m}^{i}(\gamma^{m}w_{i*})_{\alpha}+gL_{\alpha}^{*}-\bar{g}\theta_{\alpha}^{*}-\bar{h}_{m}(\gamma^{m}s_{*})_{\alpha},\\
\delta w_{\alpha}^{i} & = & d_{m}^{i}(\gamma^{m}\lambda)_{\alpha}+e_{m}^{i}(\gamma^{m}\xi)_{\alpha}-\delta_{\tau}^{i}\bar{f}L_{\alpha}^{*}+\delta_{\tau}^{i}\bar{f}_{mn}(\gamma^{mn}L^{*})_{\alpha}+\epsilon^{ij}h_{m}(\gamma^{m}w_{j*})_{\alpha}\nonumber \\
& & +\epsilon^{ij}\bar{h}_{m}(\gamma^{m}p_{j*})_{\alpha},\\
\delta p_{\alpha}^{i} & = & e_{m}^{i}(\gamma^{m}\lambda)_{\alpha}+\delta_{\tau}^{i}\bar{g}L_{\alpha}^{*}-\epsilon^{ij}\bar{h}_{m}(\gamma^{m}w_{j*})_{\alpha},\\
\delta L^{\alpha} & = & f\lambda^{\alpha}+f_{mn}(\gamma^{mn}\lambda)^{\alpha}+g\xi^{\alpha}+\bar{f}w_{\tau*}^{\alpha}+\bar{f}_{mn}(\gamma^{mn}w_{\tau*})^{\alpha}+\bar{g}p_{\tau*}^{\alpha},\\
\delta\theta^{\alpha} & = & \bar{f}\lambda^{\alpha}+\bar{f}_{mn}(\gamma^{mn}\lambda)^{\alpha}+\bar{g}\xi^{\alpha},\\
\delta s_{\alpha} & = & h_{m}(\gamma^{m}\lambda)_{\alpha}+\bar{h}_{m}(\gamma^{m}\xi)_{\alpha},\\
\delta\epsilon_{\alpha} & = & \bar{h}_{m}(\gamma^{m}\lambda)_{\alpha},\end{aligned}$$
where $b_{m}$, $c_{m}$, $d_{m}$, $e_{m}$, $f$, $f_{mn}$, $g$, $h_{m}$, $\bar{f}$ , $\bar{f}_{mn}$, $\bar{g}$ and $\bar{h}_{m}$ are local parameters. Again, similar transformations exist for the hatted sector.
The pure spinor master action has to be concomitantly determined with the pure spinor constraints and symmetries . It can be cast as $$S=S_{0}+S_{1}+S_{2}+S_{3},\label{eq:PSMASTER}$$ where $S_{0}$ is displayed in and
$$\begin{aligned}
S_{1} & = & \int d\tau d\sigma\{\tfrac{1}{4\mathcal{T}}(\lambda\gamma^{m}\theta)X_{m}^{*}-\tfrac{1}{4}(\lambda\gamma_{m}\theta)\partial_{\sigma}P_{*}^{m}-\tfrac{1}{4\mathcal{T}}(P_{m}+T\partial_{\sigma}X_{m})(\theta\gamma^{m}w_{\tau*})\}\nonumber \\
& & +\int d\tau d\sigma\{\tfrac{1}{8\mathcal{T}}(L\gamma_{m}\lambda)(\theta\gamma^{m}w_{\sigma*})+(\nabla_{\tau}\theta^{\alpha})L_{\alpha}^{*}+\epsilon^{ij}(\nabla_{j}s_{\alpha})w_{i*}^{\alpha}-\lambda^{\alpha}s_{\alpha}B^{*}\}\nonumber \\
& & +\int d\tau d\sigma\{\Omega\lambda^{\alpha}\lambda_{\alpha}^{*}-\Omega w_{\alpha}^{i}w_{i*}^{\alpha}-(\partial_{i}\Omega)A_{*}^{i}-\Omega L^{\alpha}L_{\alpha}^{*}-\Omega\chi_{i}\chi_{*}^{i}+\Omega\Sigma\Sigma^{*}\}\nonumber \\
& & +\int d\tau d\sigma\{\gamma p_{\alpha}^{i}w_{i*}^{\alpha}+\gamma\Sigma B^{*}+\gamma\lambda^{\alpha}\xi_{\alpha}^{*}+(\nabla_{i}\gamma)\chi_{*}^{i}\}\nonumber \\
& & +\int d\tau d\sigma\{\epsilon^{ij}\epsilon_{\alpha}\chi_{j}w_{i*}^{\alpha}+\epsilon^{ij}(\partial_{j}\epsilon_{\alpha})p_{i*}^{\alpha}-\lambda^{\alpha}\epsilon_{\alpha}\Sigma^{*}\}\nonumber \\
& & +\int d\tau d\sigma\{\tfrac{1}{4\mathcal{T}}(\hat{\lambda}\gamma^{m}\hat{\theta})X_{m}^{*}+\tfrac{1}{4}(\hat{\lambda}\gamma_{m}\hat{\theta})\partial_{\sigma}P_{*}^{m}-\tfrac{1}{4\mathcal{T}}(P_{m}-\mathcal{T}\partial_{\sigma}X_{m})(\hat{\theta}\gamma^{m}\hat{w}_{\tau*})\}\nonumber \\
& & +\int d\tau d\sigma\{-\tfrac{1}{8\mathcal{T}}(\hat{L}\gamma_{m}\hat{\lambda})(\hat{\theta}\gamma^{m}\hat{w}_{\sigma*})+(\hat{\nabla}_{\tau}\hat{\theta}^{\hat{\alpha}})\hat{L}_{\hat{\alpha}}^{*}+\epsilon^{ij}(\hat{\nabla}_{j}\hat{s}_{\hat{\alpha}})\hat{w}_{i*}^{\hat{\alpha}}-\hat{\lambda}^{\hat{\alpha}}\hat{s}_{\hat{\alpha}}\hat{B}^{*}\}\nonumber \\
& & +\int d\tau d\sigma\{\hat{\Omega}\hat{\lambda}^{\hat{\alpha}}\hat{\lambda}_{\hat{\alpha}}^{*}-\hat{\Omega}\hat{w}_{\hat{\alpha}}^{i}\hat{w}_{i*}^{\hat{\alpha}}-(\partial_{i}\hat{\Omega})\hat{A}_{*}^{i}-\hat{\Omega}\hat{L}^{\hat{\alpha}}\hat{L}_{\hat{\alpha}}^{*}-\hat{\Omega}\hat{\chi}_{i}\hat{\chi}_{*}^{i}+\hat{\Omega}\hat{\Sigma}\hat{\Sigma}^{*}\}\nonumber \\
& & +\int d\tau d\sigma\{\hat{\gamma}\hat{p}_{\hat{\alpha}}^{i}\hat{w}_{i*}^{\hat{\alpha}}+\hat{\gamma}\hat{\Sigma}\hat{B}^{*}+\hat{\gamma}\hat{\lambda}^{\hat{\alpha}}\hat{\xi}_{\hat{\alpha}}^{*}+(\hat{\nabla}_{i}\hat{\gamma})\hat{\chi}_{*}^{i}\}\nonumber \\
& & +\int d\tau d\sigma\{\epsilon^{ij}\hat{\epsilon}_{\hat{\alpha}}\hat{\chi}_{j}\hat{w}_{i*}^{\hat{\alpha}}+\epsilon^{ij}(\partial_{j}\hat{\epsilon}_{\hat{\alpha}})\hat{p}_{i*}^{\hat{\alpha}}-\hat{\lambda}^{\hat{\alpha}}\hat{\epsilon}_{\hat{\alpha}}\hat{\Sigma}^{*}\},\\
S_{2} & = & \int d\tau d\sigma\{-\Omega\theta^{\alpha}\theta_{\alpha}^{*}-\Omega s_{\alpha}s_{*}^{\alpha}-\Omega\gamma\gamma^{*}-\gamma\epsilon_{\alpha}s_{*}^{\alpha}\}\nonumber \\
& & +\int d\tau d\sigma\{-\hat{\Omega}\hat{\theta}^{\hat{\alpha}}\hat{\theta}_{\hat{\alpha}}^{*}-\hat{\Omega}\hat{s}_{\hat{\alpha}}\hat{s}_{*}^{\hat{\alpha}}-\hat{\Omega}\hat{\gamma}\hat{\gamma}^{*}-\hat{\gamma}\hat{\epsilon}_{\hat{\alpha}}\hat{s}_{*}^{\hat{\alpha}}\},\\
S_{3} & = & \int d\tau d\sigma\{\tfrac{1}{16\mathcal{T}}(\theta\gamma^{m}s_{*})(\lambda\gamma_{m}\theta)+\tfrac{1}{16\mathcal{T}}(w_{\tau*}\gamma^{m}\theta)(w_{\sigma*}\gamma_{m}\theta)\}\nonumber \\
& & -\int d\tau d\sigma\{\tfrac{1}{16\mathcal{T}}(\hat{\theta}\gamma^{m}\hat{s}_{*})(\hat{\lambda}\gamma_{m}\hat{\theta})+\tfrac{1}{16\mathcal{T}}(\hat{w}_{\tau*}\gamma^{m}\hat{\theta})(\hat{w}_{\sigma*}\gamma_{m}\hat{\theta})\}.\end{aligned}$$
$S_{1}$ is connected to the gauge transformations of the action $S_{0}$, while $S_{2}$ represents the extension of the gauge algebra to the ghost fields. The last piece, $S_{3}$, is required in order for $S$ to satisfy the master equation $$\{S,S\}=0.$$
By construction, the master action is invariant under the BV transformations defined as $$\delta_{\text{\tiny{BV}}}\mathcal{O}\equiv\{S,\mathcal{O}\},$$ for any operator $\mathcal{O}$. Naturally, the BV transformations of the fundamental fields in the action $S_{0}$ have a similar structure to their gauge transformations and are given by
\[eq:BVfields\] $$\begin{aligned}
\delta_{\text{\tiny{BV}}}X^{m} & = & \tfrac{1}{4\mathcal{T}}(\lambda\gamma^{m}\theta)+\tfrac{1}{4\mathcal{T}}(\hat{\lambda}\gamma^{m}\hat{\theta}),\\
\delta_{\text{\tiny{BV}}}P_{m} & = & \tfrac{1}{4}\nabla_{\sigma}(\lambda\gamma^{m}\theta)-\tfrac{1}{4}\hat{\nabla}_{\sigma}(\hat{\lambda}\gamma^{m}\hat{\theta}),\\
\delta_{\text{\tiny{BV}}}\lambda^{\alpha} & = & \Omega\lambda^{\alpha},\\
\delta_{\text{\tiny{BV}}}w_{\alpha}^{i} & = & -\Omega w_{\alpha}^{i}+\epsilon^{ij}(\nabla_{j}s_{\alpha}+\epsilon_{\alpha}\chi_{j})+\delta_{\sigma}^{i}\tfrac{1}{8\mathcal{T}}[(L_{+}\gamma_{m}\lambda)-\tfrac{1}{2}(w_{\tau*}\gamma_{m}\theta)](\gamma^{m}\theta)_{\alpha}\nonumber \\
& & +\gamma p_{\alpha}^{i}-\delta_{\tau}^{i}\tfrac{1}{4\mathcal{T}}[P_{m}+\mathcal{T}\partial_{\sigma}X_{m}+\tfrac{1}{4}(w_{\sigma*}\gamma_{m}\theta)](\gamma^{m}\theta)_{\alpha},\\
\delta_{\text{\tiny{BV}}}A_{i} & = & -\partial_{i}\Omega,\\
\delta_{\text{\tiny{BV}}}B & = & \gamma\Sigma-\lambda^{\alpha}s_{\alpha},\\
\delta_{\text{\tiny{BV}}}L^{\alpha} & = & \nabla_{\tau}\theta^{\alpha}-\Omega L^{\alpha},\\
\delta_{\text{\tiny{BV}}}\xi^{\alpha} & = & \gamma\lambda^{\alpha},\\
\delta_{\text{\tiny{BV}}}p_{\alpha}^{i} & = & \epsilon^{ij}\partial_{j}\epsilon_{\alpha},\\
\delta_{\text{\tiny{BV}}}\chi_{i} & = & \nabla_{i}\gamma-\Omega\chi_{i},\\
\delta_{\text{\tiny{BV}}}\Sigma & = & \Omega\Sigma-\lambda^{\alpha}\epsilon_{\alpha},\\
\delta_{\text{\tiny{BV}}}\hat{\lambda}^{\hat{\alpha}} & = & \hat{\Omega}\hat{\lambda}^{\hat{\alpha}},\\
\delta_{\text{\tiny{BV}}}\hat{w}_{\hat{\alpha}}^{i} & = & -\Omega\hat{w}_{\hat{\alpha}}^{i}+\epsilon^{ij}(\hat{\nabla}_{j}\hat{s}_{\alpha}+\hat{\epsilon}_{\alpha}\hat{\chi}_{j})-\delta_{\sigma}^{i}\tfrac{1}{8\mathcal{T}}[(\hat{L}\gamma_{m}\hat{\lambda})-\tfrac{1}{2}(\hat{w}_{\tau*}\gamma_{m}\hat{\theta})](\gamma^{m}\hat{\theta})_{\hat{\alpha}}\nonumber \\
& & +\hat{\gamma}\hat{p}_{\hat{\alpha}}^{i}-\delta_{\tau}^{i}\tfrac{1}{4\mathcal{T}}[P_{m}-\mathcal{T}\partial_{\sigma}X_{m}-\tfrac{1}{4}(\hat{w}_{\sigma*}\gamma_{m}\hat{\theta})](\gamma^{m}\hat{\theta})_{\hat{\alpha}},\\
\delta_{\text{\tiny{BV}}}\hat{A}_{i} & = & -\partial_{i}\hat{\Omega},\\
\delta_{\text{\tiny{BV}}}\hat{B} & = & \hat{\gamma}\hat{\Sigma}-\hat{\lambda}^{\hat{\alpha}}\hat{s}_{\hat{\alpha}},\\
\delta_{\text{\tiny{BV}}}\hat{L}^{\hat{\alpha}} & = & \hat{\nabla}_{\tau}\hat{\theta}^{\hat{\alpha}}-\hat{\Omega}\hat{L}^{\hat{\alpha}},\\
\delta_{\text{\tiny{BV}}}\hat{\xi}^{\hat{\alpha}} & = & \hat{\gamma}\hat{\lambda}^{\hat{\alpha}},\\
\delta_{\text{\tiny{BV}}}\hat{p}_{\hat{\alpha}}^{i} & = & \epsilon^{ij}\partial_{j}\hat{\epsilon}_{\hat{\alpha}},\\
\delta_{\text{\tiny{BV}}}\hat{\chi}_{i} & = & \hat{\nabla}_{i}\hat{\gamma}-\hat{\Omega}\hat{\chi}_{i},\\
\delta_{\text{\tiny{BV}}}\hat{\Sigma} & = & \hat{\Omega}\hat{\Sigma}-\hat{\lambda}^{\hat{\alpha}}\hat{\epsilon}_{\hat{\alpha}}.\end{aligned}$$
In general, the BV transformations are nilpotent. However, due to the pure spinor constraints , the transformations above are nilpotent up to pure spinor gauge transformations. For example,
$$\begin{aligned}
\delta_{\text{\tiny{BV}}}^{2}w_{\alpha}^{i} & = & \tfrac{1}{16\mathcal{T}}\epsilon^{ij}(\theta\gamma_{m}\nabla_{j}\theta)(\gamma^{m}\lambda)_{\alpha},\\
\delta_{\text{\tiny{BV}}}^{2}\hat{w}_{\hat{\alpha}}^{i} & = & -\tfrac{1}{16\mathcal{T}}\epsilon^{ij}(\hat{\theta}\gamma^{m}\hat{\nabla}_{j}\hat{\theta})(\gamma^{m}\hat{\lambda})_{\hat{\alpha}}.\end{aligned}$$
With the promotion of gauge parameters to ghost fields, their BV transformations are nontrivial and can be cast as
\[eq:BVghosts\] $$\begin{aligned}
\delta_{\text{\tiny{BV}}}\theta^{\alpha} & = & -\Omega\theta^{\alpha},\\
\delta_{\text{\tiny{BV}}}s_{\alpha} & = & -\Omega s_{\alpha}-\gamma\epsilon_{\alpha}-\tfrac{1}{16\mathcal{T}}(\lambda\gamma_{m}\theta)(\gamma^{m}\theta)_{\alpha},\\
\delta_{\text{\tiny{BV}}}\Omega & = & 0,\\
\delta_{\text{\tiny{BV}}}\gamma & = & -\Omega\gamma,\\
\delta_{\text{\tiny{BV}}}\epsilon_{\alpha} & = & 0,\\
\delta_{\text{\tiny{BV}}}\hat{\theta}^{\hat{\alpha}} & = & -\hat{\Omega}\hat{\theta}^{\hat{\alpha}},\\
\delta_{\text{\tiny{BV}}}\hat{s}_{\hat{\alpha}} & = & -\hat{\Omega}\hat{s}_{\hat{\alpha}}-\hat{\gamma}\hat{\epsilon}_{\hat{\alpha}}+\tfrac{1}{16\mathcal{T}}(\hat{\lambda}\gamma_{m}\hat{\theta})(\gamma^{m}\hat{\theta})_{\hat{\alpha}},\\
\delta_{\text{\tiny{BV}}}\hat{\Omega} & = & 0,\\
\delta_{\text{\tiny{BV}}}\hat{\gamma} & = & -\hat{\Omega}\hat{\gamma},\\
\delta_{\text{\tiny{BV}}}\hat{\epsilon}_{\hat{\alpha}} & = & 0.\end{aligned}$$
For completeness, the BV transformations for the antifields are given by
\[eq:BVantifields\] $$\begin{aligned}
\delta_{\text{\tiny{BV}}}X_{m}^{*} & = & \partial_{\tau}P_{m}-\tfrac{1}{4}\partial_{\sigma}[(L\gamma_{m}\lambda)+(\theta\gamma_{m}w_{\tau*})-(\hat{L}\gamma_{m}\hat{\lambda})-(\hat{\theta}\gamma_{m}\hat{w}_{\tau*})],\\
\delta_{\text{\tiny{BV}}}P_{*}^{m} & = & -\partial_{\tau}X^{m}+\tfrac{1}{4\mathcal{T}}[(L\gamma^{m}\lambda)+(\theta\gamma^{m}w_{\tau*})+(\hat{L}\gamma^{m}\hat{\lambda})+(\hat{\theta}\gamma^{m}\hat{w}_{\tau*})],\\
\delta_{\text{\tiny{BV}}}\lambda_{\alpha}^{*} & = & \nabla_{i}w_{\alpha}^{i}-\chi_{i}p_{\alpha}^{i}+s_{\alpha}B^{*}+\tfrac{1}{4\mathcal{T}}[P^{m}+\mathcal{T}\partial_{\sigma}X^{m}-\tfrac{1}{2}(\theta\gamma^{m}w_{\sigma*})](\gamma_{m}L)_{\alpha}\nonumber \\
& & -\Omega\lambda_{\alpha}^{*}-\gamma\xi_{\alpha}^{*}+\epsilon_{\alpha}\Sigma^{*}+\tfrac{1}{4\mathcal{T}}[\eta^{mn}X_{n}^{*}-\mathcal{T}\partial_{\sigma}P_{*}^{m}-\tfrac{1}{4}(\theta\gamma^{m}s_{*})](\gamma_{m}\theta)_{\alpha},\\
\delta_{\text{\tiny{BV}}}w_{i*}^{\alpha} & = & -\nabla_{i}\lambda^{\alpha}+\Omega w_{i*}^{\alpha},\\
\delta_{\text{\tiny{BV}}}A_{*}^{i} & = & -\lambda^{\alpha}w_{\alpha}^{i}-\epsilon^{ij}\partial_{j}B-\Sigma\epsilon^{ij}\chi_{j}-\epsilon^{ij}s_{\alpha}w_{j*}^{\alpha}+\delta_{\tau}^{i}\theta^{\alpha}L_{\alpha}^{*}+\gamma\chi_{*}^{i},\\
\delta_{\text{\tiny{BV}}}B^{*} & = & -\epsilon^{ij}\partial_{i}A_{j},\\
\delta_{\text{\tiny{BV}}}L_{\alpha}^{*} & = & \tfrac{1}{4\mathcal{T}}[P^{m}+\mathcal{T}\partial_{\sigma}X^{m}-\tfrac{1}{2}(\theta\gamma^{m}w_{\sigma*})](\gamma_{m}\lambda)_{\alpha}+\Omega L_{\alpha}^{*},\\
\delta_{\text{\tiny{BV}}}\xi_{\alpha}^{*} & = & \partial_{i}p_{\alpha}^{i},\\
\delta_{\text{\tiny{BV}}}p_{i*}^{\alpha} & = & \partial_{i}\xi^{\alpha}-\lambda^{\alpha}\chi_{i}+\gamma w_{i*}^{\alpha},\\
\delta_{\text{\tiny{BV}}}\chi_{*}^{i} & = & \lambda^{\alpha}p_{\alpha}^{i}+\epsilon^{ij}\nabla_{j}\Sigma+\Omega\chi_{*}^{i}-\epsilon^{ij}\epsilon_{\alpha}w_{j*}^{\alpha},\\
\delta_{\text{\tiny{BV}}}\Sigma^{*} & = & -\epsilon^{ij}\nabla_{i}\chi_{j}-\Omega\Sigma^{*}+\gamma B^{*},\\
\delta_{\text{\tiny{BV}}}\hat{\lambda}_{\hat{\alpha}}^{*} & = & \hat{\nabla}_{i}\hat{w}_{\hat{\alpha}}^{i}-\hat{\chi}_{i}\hat{p}_{\hat{\alpha}}^{i}+\hat{s}_{\hat{\alpha}}\hat{B}^{*}+\tfrac{1}{4\mathcal{T}}[P^{m}-\mathcal{T}\partial_{\sigma}X^{m}+\tfrac{1}{2}(\hat{\theta}\gamma^{m}\hat{w}_{\sigma*})](\gamma_{m}\hat{L})_{\hat{\alpha}}\nonumber \\
& & -\hat{\Omega}\hat{\lambda}_{\hat{\alpha}}^{*}-\hat{\gamma}\hat{\xi}_{\hat{\alpha}}^{*}+\hat{\epsilon}_{\hat{\alpha}}\hat{\Sigma}^{*}+\tfrac{1}{4\mathcal{T}}[\eta^{mn}X_{n}^{*}+\mathcal{T}\partial_{\sigma}P_{*}^{m}+\tfrac{1}{4}(\hat{\theta}\gamma^{m}\hat{s}_{*})](\gamma_{m}\hat{\theta})_{\hat{\alpha}},\\
\delta_{\text{\tiny{BV}}}\hat{w}_{i*}^{\hat{\alpha}} & = & -\hat{\nabla}_{i}\hat{\lambda}^{\hat{\alpha}}+\hat{\Omega}\hat{w}_{i*}^{\hat{\alpha}},\\
\delta_{\text{\tiny{BV}}}\hat{A}_{*}^{i} & = & -\hat{\lambda}^{\hat{\alpha}}\hat{w}_{\hat{\alpha}}^{i}-\epsilon^{ij}\partial_{j}\hat{B}-\hat{\Sigma}\epsilon^{ij}\hat{\chi}_{j}-\epsilon^{ij}\hat{s}_{\hat{\alpha}}\hat{w}_{j*}^{\hat{\alpha}}+\delta_{\tau}^{i}\hat{\theta}^{\hat{\alpha}}\hat{L}_{\hat{\alpha}}^{*}+\hat{\gamma}\hat{\chi}_{*}^{i},\\
\delta_{\text{\tiny{BV}}}\hat{B}^{*} & = & -\epsilon^{ij}\partial_{i}\hat{A}_{j},\\
\delta_{\text{\tiny{BV}}}\hat{L}_{\hat{\alpha}}^{*} & = & \tfrac{1}{4\mathcal{T}}[P^{m}-\mathcal{T}\partial_{\sigma}X^{m}+\tfrac{1}{2}(\hat{\theta}\gamma^{m}\hat{w}_{\sigma*})](\gamma_{m}\hat{\lambda})_{\hat{\alpha}}+\hat{\Omega}\hat{L}_{\hat{\alpha}}^{*},\\
\delta_{\text{\tiny{BV}}}\hat{\xi}_{\hat{\alpha}}^{*} & = & \partial_{i}\hat{p}_{\hat{\alpha}}^{i},\\
\delta_{\text{\tiny{BV}}}\hat{p}_{i*}^{\hat{\alpha}} & = & \partial_{i}\hat{\xi}^{\hat{\alpha}}-\hat{\lambda}^{\hat{\alpha}}\hat{\chi}_{i}+\hat{\gamma}\hat{w}_{i*}^{\hat{\alpha}},\\
\delta_{\text{\tiny{BV}}}\hat{\chi}_{*}^{i} & = & \hat{\lambda}^{\hat{\alpha}}\hat{p}_{\hat{\alpha}}^{i}+\epsilon^{ij}\hat{\nabla}_{j}\hat{\Sigma}+\hat{\Omega}\hat{\chi}_{*}^{i}-\epsilon^{ij}\hat{\epsilon}_{\hat{\alpha}}\hat{w}_{j*}^{\hat{\alpha}},\\
\delta_{\text{\tiny{BV}}}\hat{\Sigma}^{*} & = & -\epsilon^{ij}\hat{\nabla}_{i}\hat{\chi}_{j}-\hat{\Omega}\hat{\Sigma}^{*}+\hat{\gamma}\hat{B}^{*},\end{aligned}$$
and for the ghost antifields,
\[eq:BVghostsantifields\] $$\begin{aligned}
\delta_{\text{\tiny{BV}}}\theta_{\alpha}^{*} & = & \tfrac{1}{4\mathcal{T}}[\eta^{mn}X_{n}^{*}-\mathcal{T}\partial_{\sigma}P_{*}^{m}-\tfrac{1}{4}(\theta\gamma^{m}s_{*})](\gamma_{m}\lambda)_{\alpha}+\Omega\theta_{\alpha}^{*}-\nabla_{\tau}L_{\alpha}^{*}\nonumber \\
& & +\tfrac{1}{8\mathcal{T}}[(L\gamma_{m}\lambda)-\tfrac{1}{2}(w_{\tau*}\gamma_{m}\theta)](\gamma^{m}w_{\sigma*})_{\alpha}+\tfrac{1}{16\mathcal{T}}(\lambda\gamma_{m}\theta)(\gamma^{m}s_{*})_{\alpha}\nonumber \\
& & -\tfrac{1}{4\mathcal{T}}[P_{m}+\mathcal{T}\partial_{\sigma}X_{m}+\tfrac{1}{4}(w_{\sigma*}\gamma_{m}\theta)](\gamma^{m}w_{\tau*})_{\alpha},\\
\delta_{\text{\tiny{BV}}}s_{*}^{\alpha} & = & \epsilon^{ij}\nabla_{i}w_{j*}^{\alpha}-\lambda^{\alpha}B^{*}+\Omega s_{*}^{\alpha},\\
\delta_{\text{\tiny{BV}}}\Omega^{*} & = & \lambda^{\alpha}\lambda_{\alpha}^{*}-w_{\alpha}^{i}w_{i*}^{\alpha}+\partial_{i}A_{*}^{i}-L^{\alpha}L_{\alpha}^{*}-\chi_{i}\chi_{*}^{i}+\Sigma\Sigma^{*}-\theta^{\alpha}\theta_{\alpha}^{*}-s_{\alpha}s_{*}^{\alpha}-\gamma\gamma^{*},\\
\delta_{\text{\tiny{BV}}}\gamma^{*} & = & -p_{\alpha}^{i}w_{i*}^{\alpha}-\Sigma B^{*}-\lambda^{\alpha}\xi_{\alpha}^{*}+\nabla_{i}\chi_{*}^{i}+\Omega\gamma^{*}+\epsilon_{\alpha}s_{*}^{\alpha},\\
\delta_{\text{\tiny{BV}}}\epsilon_{*}^{\alpha} & = & \epsilon^{ij}\chi_{i}w_{j*}^{\alpha}-\epsilon^{ij}\partial_{i}p_{j*}^{\alpha}+\lambda^{\alpha}\Sigma^{*}+\gamma s_{*}^{\alpha},\\
\delta_{\text{\tiny{BV}}}\hat{\theta}_{\hat{\alpha}}^{*} & = & \tfrac{1}{4\mathcal{T}}[\eta^{mn}X_{n}^{*}+\mathcal{T}\partial_{\sigma}P_{*}^{m}+\tfrac{1}{4}(\hat{\theta}\gamma^{m}\hat{s}_{*})](\gamma_{m}\hat{\lambda})_{\hat{\alpha}}+\hat{\Omega}\hat{\theta}_{\hat{\alpha}}^{*}-\hat{\nabla}_{\tau}\hat{L}_{\hat{\alpha}}^{*}\nonumber \\
& & -\tfrac{1}{8\mathcal{T}}[(\hat{L}\gamma_{m}\hat{\lambda})-\tfrac{1}{2}(\hat{w}_{\tau*}\gamma_{m}\hat{\theta})](\gamma^{m}\hat{w}_{\sigma*})_{\hat{\alpha}}-\tfrac{1}{16\mathcal{T}}(\hat{\lambda}\gamma_{m}\hat{\theta})(\gamma^{m}\hat{s}_{*})_{\hat{\alpha}}\nonumber \\
& & -\tfrac{1}{4\mathcal{T}}[P_{m}-\mathcal{T}\partial_{\sigma}X_{m}-\tfrac{1}{4}(\hat{w}_{\sigma*}\gamma_{m}\hat{\theta})](\gamma^{m}\hat{w}_{\tau*})_{\hat{\alpha}},\\
\delta_{\text{\tiny{BV}}}\hat{s}_{*}^{\hat{\alpha}} & = & \epsilon^{ij}\hat{\nabla}_{i}\hat{w}_{j*}^{\hat{\alpha}}-\hat{\lambda}^{\hat{\alpha}}\hat{B}^{*}+\hat{\Omega}\hat{s}_{*}^{\hat{\alpha}},\\
\delta_{\text{\tiny{BV}}}\hat{\Omega}^{*} & = & \hat{\lambda}^{\hat{\alpha}}\hat{\lambda}_{\hat{\alpha}}^{*}-\hat{w}_{\hat{\alpha}}^{i}\hat{w}_{i*}^{\hat{\alpha}}+\partial_{i}\hat{A}_{*}^{i}-\hat{L}^{\hat{\alpha}}\hat{L}_{\hat{\alpha}}^{*}-\hat{\chi}_{i}\hat{\chi}_{*}^{i}+\hat{\Sigma}\hat{\Sigma}^{*}-\hat{\theta}^{\hat{\alpha}}\hat{\theta}_{\hat{\alpha}}^{*}-\hat{s}_{\hat{\alpha}}\hat{s}_{*}^{\hat{\alpha}}-\hat{\gamma}\hat{\gamma}^{*},\\
\delta_{\text{\tiny{BV}}}\hat{\gamma}^{*} & = & -\hat{p}_{\hat{\alpha}}^{i}\hat{w}_{i*}^{\hat{\alpha}}-\hat{\Sigma}\hat{B}^{*}-\hat{\lambda}^{\hat{\alpha}}\hat{\xi}_{\hat{\alpha}}^{*}+\hat{\nabla}_{i}\hat{\chi}_{*}^{i}+\hat{\Omega}\hat{\gamma}^{*}+\hat{\epsilon}_{\hat{\alpha}}\hat{s}_{*}^{\hat{\alpha}},\\
\delta_{\text{\tiny{BV}}}\hat{\epsilon}_{*}^{\hat{\alpha}} & = & \epsilon^{ij}\hat{\chi}_{i}\hat{w}_{j*}^{\hat{\alpha}}-\epsilon^{ij}\partial_{i}\hat{p}_{j*}^{\hat{\alpha}}+\hat{\lambda}^{\hat{\alpha}}\hat{\Sigma}^{*}+\hat{\gamma}\hat{s}_{*}^{\hat{\alpha}}.\end{aligned}$$
Gauge fixing\[subsec:Gauge-fixing\]
-----------------------------------
The procedure for gauge fixing in the Batalin-Vilkovisky formalism is very straightforward. But before moving on, it is useful to discuss the particular gauge to be chosen here.
Using the scaling transformations , it is possible to choose $A_{\tau}=\hat{A}_{\tau}=0$. Similarly, the fermionic gauge transformations can be used to partially fix their gauge fields to $\chi_{\tau}=\hat{\chi}_{\tau}=0$.
For the curl symmetry , the choice will be $w_{\alpha}^{\sigma}=p_{\alpha}^{\sigma}=0$ and $\hat{w}_{\hat{\alpha}}^{\sigma}=\hat{p}_{\hat{\alpha}}^{\sigma}=0$. Observe here that there is no residual gauge transformation, since both fields ($w_{\alpha}^{\sigma}$, $p_{\alpha}^{\sigma}$, $\hat{w}_{\hat{\alpha}}^{\sigma}$ and $\hat{p}_{\hat{\alpha}}^{\sigma}$) and ghosts ($s_{\alpha}$, $\epsilon_{\alpha}$, $\hat{s}_{\hat{\alpha}}$ and $\hat{\epsilon}_{\hat{\alpha}}$) have the same number of independent components according to the pure spinor constraints and gauge transformations .
Finally, the twistor-like symmetry can be used to fix the Lagrange multipliers $L^{\alpha}$ and $\hat{L}^{\hat{\alpha}}$. Following the discussion after equation , the gauge $L^{\alpha}=\hat{L}^{\hat{\alpha}}=0$ would imply a degenerate worldsheet metric. Although worldsheet reparametrization is hidden in the twistor-like symmetry, the conformal gauge would still be the natural choice. In the Polyakov action , this gauge would be $e_{\pm}=1$. For the action $S_{0}$ in , the conformal gauge is equivalent to the choice
\[eq:conformalgauge\] $$\begin{aligned}
L^{\alpha} & = & \tfrac{(\gamma^{m}\Lambda)^{\alpha}}{(\Lambda\lambda)}(P_{m}+\mathcal{T}\partial_{\sigma}X_{m}),\\
\hat{L}^{\hat{\alpha}} & = & \tfrac{(\gamma^{m}\hat{\Lambda})^{\hat{\alpha}}}{(\hat{\Lambda}\hat{\lambda})}(P_{m}-\mathcal{T}\partial_{\sigma}X_{m}).\end{aligned}$$
The proposed gauge can be implemented through the gauge fixing fermion $\Xi$, given by $$\begin{gathered}
\Xi=\int d\tau d\sigma\{\bar{\Omega}A_{\tau}+\beta\chi_{\tau}-r^{\alpha}w_{\alpha}^{\sigma}+\eta^{\alpha}p_{\alpha}^{\sigma}+\hat{\bar{\Omega}}\hat{A}_{\tau}-\hat{r}^{\hat{\alpha}}\hat{w}_{\hat{\alpha}}^{\sigma}+\hat{\beta}\hat{\chi}_{\tau}+\hat{\eta}^{\hat{\alpha}}\hat{p}_{\hat{\alpha}}^{\sigma}\}\\
-\int d\tau d\sigma\{\pi_{\alpha}[L^{\alpha}-\tfrac{(\gamma^{m}\Lambda)^{\alpha}}{(\Lambda\lambda)}(P_{m}+\mathcal{T}\partial_{\sigma}X_{m})]+\hat{\pi}_{\hat{\alpha}}[\hat{L}^{\hat{\alpha}}-\tfrac{(\gamma^{m}\hat{\Lambda})^{\hat{\alpha}}}{(\hat{\Lambda}\hat{\lambda})}(P_{m}-\mathcal{T}\partial_{\sigma}X_{m})]\}.\label{eq:GFF}\end{gathered}$$ Here, $\bar{\Omega}$, $\beta$, $r^{\alpha}$, $\eta^{\alpha}$ and $\pi_{\alpha}$ are the antighosts of $\Omega$, $\gamma$, $s_{\alpha}$, $\epsilon_{\alpha}$ and $\theta^{\alpha}$, respectively (hatted and unhatted). The antighosts will be generically represented by $\bar{\Phi}_{a}$ (with antifields $\bar{\Phi}_{*}^{a}$), where the index $a$ denotes the different components of the gauge parameters. In addition, the extended phase space of the model will include the Nakanishi-Lautrup fields, $\Lambda_{a}$, and antifields, $\Lambda_{*}^{a}$.
The gauge fixed action can be determined by evaluating the non-minimal master action, defined as $$S_{nm}=S+\int d\tau d\sigma\left(\bar{\Phi}_{*}^{a}\Lambda_{a}\right),$$ at $$\begin{array}{rclcrclcrcl}
\Phi_{I}^{*} & = & \frac{\delta\Xi}{\delta\Phi^{I}}, & & \bar{\Phi}_{*}^{a} & = & \frac{\delta\Xi}{\delta\bar{\Phi}_{a}}, & & \Lambda_{*}^{a} & = & \frac{\delta\Xi}{\delta\Lambda_{a}}.\end{array}$$ For the particular choice of gauge fixing fermion , the non-vanishing antifields are given by $$\begin{array}{rclcrcl}
X_{m}^{*} & = & -\mathcal{T}\partial_{\sigma}[\tfrac{(\pi\gamma^{m}\Lambda)}{(\Lambda\lambda)}]+\mathcal{T}\partial_{\sigma}[\tfrac{(\hat{\pi}\gamma^{m}\hat{\Lambda})}{(\hat{\Lambda}\hat{\lambda})}], & & A_{*}^{\tau} & = & \bar{\Omega},\\
\delta P_{*}^{m} & = & \mathcal{T}\tfrac{(\pi\gamma^{m}\Lambda)}{(\Lambda\lambda)}+\mathcal{T}\tfrac{(\hat{\pi}\gamma^{m}\hat{\Lambda})}{(\hat{\Lambda}\hat{\lambda})}, & & \hat{A}_{*}^{\tau} & = & \hat{\bar{\Omega}},\\
\lambda_{\alpha}^{*} & = & -\tfrac{(\pi\gamma^{m}\Lambda)}{(\Lambda\lambda)^{2}}(P_{m}+\mathcal{T}\partial_{\sigma}X_{m})\Lambda_{\alpha}, & & \chi_{*}^{\tau} & = & \beta,\\
\hat{\lambda}_{\hat{\alpha}}^{*} & = & -\tfrac{(\hat{\pi}\gamma^{m}\hat{\Lambda})}{(\hat{\Lambda}\hat{\lambda})^{2}}(P_{m}-\mathcal{T}\partial_{\sigma}X_{m})\hat{\Lambda}_{\hat{\alpha}}, & & \hat{\chi}_{*}^{\tau} & = & \hat{\beta},\\
L_{\alpha}^{*} & = & -\pi_{\alpha}, & & \hat{L}_{\hat{\alpha}}^{*} & = & -\hat{\pi}_{\hat{\alpha}},\\
w_{\sigma*}^{\alpha} & = & -r^{\alpha}, & & \hat{w}_{\sigma*}^{\hat{\alpha}} & = & -\hat{r}^{\hat{\alpha}},\\
p_{\sigma*}^{\alpha} & = & \eta^{\alpha}, & & \hat{p}_{\sigma*}^{\hat{\alpha}} & = & \hat{\eta}^{\hat{\alpha}}.
\end{array}$$
Since the master action $S$ was consistently built with the pure spinor constraints , the antighosts are constrained accordingly: $$\begin{array}{rclcrcl}
\xi^{\alpha}\pi_{\alpha} & = & 0, & & \hat{\xi}^{\hat{\alpha}}\hat{\pi}_{\hat{\alpha}} & = & 0,\\
\lambda^{\alpha}\pi_{\alpha} & = & 0, & & \hat{\lambda}^{\hat{\alpha}}\hat{\pi}_{\hat{\alpha}} & = & 0,\\
(\lambda\gamma^{m}r) & = & 0, & & (\hat{\lambda}\gamma^{m}\hat{r}) & = & 0,\\
(\lambda\gamma^{mn}\pi) & = & 0, & & (\hat{\lambda}\gamma^{mn}\hat{\pi}) & = & 0,\\
(\lambda\gamma^{m}\eta)+(r\gamma^{m}\xi) & = & 0, & & (\hat{\lambda}\gamma^{m}\hat{\eta})+(\hat{r}\gamma^{m}\hat{\xi}) & = & 0.
\end{array}\label{eq:PSconstraintsantighosts}$$
After solving for the equations of motion of the Nakanishi-Lautrup fields, the gauge fixed action can be written as $$\begin{aligned}
S_{fixed} & = & \int d\tau d\sigma\{P_{m}\partial_{\tau}X^{m}-\tfrac{1}{2\mathcal{T}}(P_{m}P^{m}+\mathcal{T}^{2}\partial_{\sigma}X_{m}\partial_{\sigma}X^{m})\}\nonumber \\
& & +\int d\tau d\sigma\{w_{\alpha}\partial_{\tau}\lambda^{\alpha}+p_{\alpha}\partial_{\tau}\xi^{\alpha}+\pi_{\alpha}\partial_{\tau}\theta^{\alpha}+r^{\alpha}\partial_{\tau}s_{\alpha}+\eta^{\alpha}\partial_{\tau}\epsilon_{\alpha}\}\nonumber \\
& & +\int d\tau d\sigma\{A\partial_{\tau}B+\Sigma\partial_{\tau}\chi+\bar{\Omega}\partial_{\tau}\Omega+\beta\partial_{\tau}\gamma\}\nonumber \\
& & +\int d\tau d\sigma\{\tfrac{1}{2}\partial_{\sigma}(\lambda\gamma^{m}\theta)\tfrac{(\pi\gamma^{m}\Lambda)}{(\Lambda\lambda)}-\tfrac{1}{8T}\tfrac{(\lambda\gamma^{n}\gamma^{m}\Lambda)}{(\Lambda\lambda)}(P_{m}+\mathcal{T}\partial_{\sigma}X_{m})(\theta\gamma_{n}r)\}\nonumber \\
& & +\int d\tau d\sigma\{\hat{w}_{\hat{\alpha}}\partial_{\tau}\hat{\lambda}^{\hat{\alpha}}+\hat{p}_{\hat{\alpha}}\partial_{\tau}\hat{\xi}^{\hat{\alpha}}+\hat{\pi}_{\hat{\alpha}}\partial_{\tau}\hat{\theta}^{\hat{\alpha}}+\hat{r}^{\hat{\alpha}}\partial_{\tau}\hat{s}_{\hat{\alpha}}+\hat{\eta}^{\hat{\alpha}}\partial_{\tau}\hat{\epsilon}_{\hat{\alpha}}\}\nonumber \\
& & +\int d\tau d\sigma\{\hat{A}\partial_{\tau}\hat{B}+\hat{\Sigma}\partial_{\tau}\hat{\chi}+\hat{\bar{\Omega}}\partial_{\tau}\hat{\Omega}+\hat{\beta}\partial_{\tau}\hat{\gamma}\}\nonumber \\
& & +\int d\tau d\sigma\{-\tfrac{1}{2}\partial_{\sigma}(\hat{\lambda}\gamma^{m}\hat{\theta})\tfrac{(\hat{\pi}\gamma^{m}\hat{\Lambda})}{(\hat{\Lambda}\hat{\lambda})}+\tfrac{1}{8T}\tfrac{(\hat{\lambda}\gamma^{n}\gamma^{m}\hat{\Lambda})}{(\hat{\Lambda}\hat{\lambda})}(P_{m}-\mathcal{T}\partial_{\sigma}X_{m})(\hat{\theta}\gamma_{n}\hat{r})\},\label{eq:PSfixed1}\end{aligned}$$ where some of the fields were renamed in order to simplify the notation, $$\begin{array}{rclcrclcrclcrcl}
w_{\alpha} & \equiv & w_{\alpha}^{\tau}, & & A & \equiv & A_{\sigma}, & & \hat{w}_{\hat{\alpha}} & \equiv & \hat{w}_{\hat{\alpha}}^{\tau}, & & \hat{A} & \equiv & \hat{A}_{\sigma},\\
p_{\alpha} & \equiv & p_{\alpha}^{\tau}, & & \chi & \equiv & \chi_{\sigma}, & & \hat{p}_{\hat{\alpha}} & \equiv & \hat{p}_{\hat{\alpha}}^{\tau}, & & \hat{\chi} & \equiv & \hat{\chi}_{\sigma}.
\end{array}\label{eq:rename}$$
The BV-BRST transformations can be readily determined next. From equation , the transformations of the fields can be written as
\[eq:BV-BRSTfields\] $$\begin{aligned}
\delta X^{m} & = & \tfrac{1}{4\mathcal{T}}[(\lambda\gamma^{m}\theta)+(\hat{\lambda}\gamma^{m}\hat{\theta})],\\
\delta P_{m} & = & \tfrac{1}{4}\partial_{\sigma}[(\lambda\gamma_{m}\theta)-(\hat{\lambda}\gamma_{m}\hat{\theta})],\\
\delta\lambda^{\alpha} & = & \Omega\lambda^{\alpha},\\
\delta w_{\alpha} & = & \gamma p_{\alpha}-\epsilon_{\alpha}\chi-\Omega w_{\alpha}-\nabla_{\sigma}s_{\alpha}-\tfrac{1}{4\mathcal{T}}[P_{m}+\mathcal{T}\partial_{\sigma}X_{m}-\tfrac{1}{4}(r\gamma_{m}\theta)](\theta\gamma^{m})_{\alpha},\\
\delta A & = & -\partial_{\sigma}\Omega,\\
\delta B & = & \gamma\Sigma-\lambda^{\alpha}s_{\alpha},\\
\delta\xi^{\alpha} & = & \gamma\lambda^{\alpha},\\
\delta p_{\alpha} & = & -\partial_{\sigma}\epsilon_{\alpha},\\
\delta\chi & = & \partial_{\sigma}\gamma-A\gamma-\Omega\chi,\\
\delta\Sigma & = & \Omega\Sigma-\lambda^{\alpha}\epsilon_{\alpha},\\
\delta\hat{\lambda}^{\hat{\alpha}} & = & \hat{\Omega}\hat{\lambda}^{\hat{\alpha}},\\
\delta\hat{w}_{\hat{\alpha}} & = & \hat{\gamma}\hat{p}_{\hat{\alpha}}-\hat{\epsilon}_{\hat{\alpha}}\hat{\chi}-\Omega\hat{w}_{\hat{\alpha}}-\hat{\nabla}_{\sigma}\hat{s}_{\hat{\alpha}}-\tfrac{1}{4\mathcal{T}}[P_{m}-\mathcal{T}\partial_{\sigma}X_{m}+\tfrac{1}{4}(\hat{r}\gamma_{m}\hat{\theta})](\gamma^{m}\hat{\theta})_{\hat{\alpha}},\\
\delta\hat{A} & = & -\partial_{\sigma}\hat{\Omega},\\
\delta\hat{B} & = & \hat{\gamma}\hat{\Sigma}-\hat{\lambda}^{\hat{\alpha}}\hat{s}_{\hat{\alpha}},\\
\delta\hat{\xi}^{\hat{\alpha}} & = & \hat{\gamma}\hat{\lambda}^{\hat{\alpha}},\\
\delta\hat{p}_{\hat{\alpha}} & = & -\partial_{\sigma}\hat{\epsilon}_{\hat{\alpha}},\\
\delta\hat{\chi} & = & \partial_{\sigma}\hat{\gamma}-\hat{A}\hat{\gamma}-\hat{\Omega}\hat{\chi},\\
\delta\hat{\Sigma} & = & \hat{\Omega}\hat{\Sigma}-\hat{\lambda}^{\hat{\alpha}}\hat{\epsilon}_{\hat{\alpha}},\end{aligned}$$
and, using equations and , the transformations of the ghosts are given by
\[eq:BV-BRSTghosts\] $$\begin{aligned}
\delta\pi_{\alpha} & = & \Omega\pi_{\alpha}-\tfrac{1}{4\mathcal{T}}(\gamma_{m}\lambda)_{\alpha}(P^{m}+\mathcal{T}\partial_{\sigma}X^{m})-\tfrac{1}{8\mathcal{T}}(\gamma_{m}\lambda)_{\alpha}(\theta\gamma^{m}r),\\
\delta\theta^{\alpha} & = & -\Omega\theta^{\alpha},\\
\delta r^{\alpha} & = & \partial_{\sigma}\lambda^{\alpha}+A\lambda^{\alpha}+\Omega r^{\alpha},\\
\delta s_{\alpha} & = & -\Omega s_{\alpha}-\gamma\epsilon_{\alpha}-\tfrac{1}{16\mathcal{T}}(\lambda\gamma_{m}\theta)(\gamma^{m}\theta)_{\alpha},\\
\delta\bar{\Omega} & = & -\lambda^{\alpha}w_{\alpha}+\partial_{\sigma}B+r^{\alpha}s_{\alpha}+\pi_{\alpha}\theta^{\alpha}+\Sigma\chi+\beta\gamma,\\
\delta\Omega & = & 0,\\
\delta\beta & = & \lambda^{\alpha}p_{\alpha}-\partial_{\sigma}\Sigma-A\Sigma+\Omega\beta-r^{\alpha}\epsilon_{\alpha},\\
\delta\gamma & = & -\Omega\gamma,\\
\delta\eta^{\alpha} & = & \partial_{\sigma}\xi^{\alpha}-\lambda^{\alpha}\chi-\gamma r^{\alpha},\\
\delta\epsilon_{\alpha} & = & 0,\\
\delta\hat{\pi}_{\hat{\alpha}} & = & \hat{\Omega}\hat{\pi}_{\hat{\alpha}}-\tfrac{1}{4\mathcal{T}}(\gamma_{m}\hat{\lambda})_{\hat{\alpha}}(P^{m}-\mathcal{T}\partial_{\sigma}X^{m})+\tfrac{1}{8\mathcal{T}}(\gamma_{m}\hat{\lambda})_{\hat{\alpha}}(\hat{\theta}\gamma^{m}\hat{r}),\\
\delta\hat{\theta}^{\hat{\alpha}} & = & -\hat{\Omega}\hat{\theta}^{\hat{\alpha}},\\
\delta\hat{r}^{\hat{\alpha}} & = & \partial_{\sigma}\hat{\lambda}^{\hat{\alpha}}+\hat{A}\hat{\lambda}^{\hat{\alpha}}+\hat{\Omega}r^{\hat{\alpha}},\\
\delta\hat{s}_{\hat{\alpha}} & = & -\hat{\Omega}\hat{s}_{\hat{\alpha}}-\hat{\gamma}\hat{\epsilon}_{\hat{\alpha}}+\tfrac{1}{16\mathcal{T}}(\hat{\lambda}\gamma_{m}\hat{\theta})(\gamma^{m}\hat{\theta})_{\hat{\alpha}},\\
\delta\hat{\bar{\Omega}} & = & -\hat{\lambda}^{\hat{\alpha}}\hat{w}_{\hat{\alpha}}+\partial_{\sigma}\hat{B}+\hat{r}^{\hat{\alpha}}\hat{s}_{\hat{\alpha}}+\hat{\pi}_{\hat{\alpha}}\hat{\theta}^{\hat{\alpha}}+\hat{\Sigma}\hat{\chi}+\hat{\beta}\hat{\gamma},\\
\delta\hat{\Omega} & = & 0,\\
\delta\hat{\beta} & = & \hat{\lambda}^{\hat{\alpha}}\hat{p}_{\hat{\alpha}}-\partial_{\sigma}\hat{\Sigma}-\hat{A}\hat{\Sigma}+\hat{\Omega}\hat{\beta}-\hat{r}^{\hat{\alpha}}\hat{\epsilon}_{\hat{\alpha}},\\
\delta\hat{\gamma} & = & -\hat{\Omega}\hat{\gamma},\\
\delta\hat{\eta}^{\hat{\alpha}} & = & \partial_{\sigma}\hat{\xi}^{\hat{\alpha}}-\hat{\lambda}^{\hat{\alpha}}\hat{\chi}-\hat{\gamma}\hat{r}^{\hat{\alpha}},\\
\delta\hat{\epsilon}_{\hat{\alpha}} & = & 0.\end{aligned}$$
The action does not look like an ordinary action in the conformal gauge. This can be fixed with the addition of a BRST trivial expression. Consider the operators
$$\begin{aligned}
I & \equiv & -\tfrac{1}{2}(r\gamma^{m}\theta)\tfrac{(\pi\gamma^{m}\Lambda)}{(\Lambda\lambda)}+w_{\alpha}r^{\alpha}-p_{\alpha}\eta^{\alpha}+A\bar{\Omega}+\chi\beta,\\
\hat{I} & \equiv & \tfrac{1}{2}(\hat{r}\gamma^{m}\hat{\theta})\tfrac{(\hat{\pi}\gamma^{m}\hat{\Lambda})}{(\hat{\Lambda}\hat{\lambda})}+\hat{w}_{\hat{\alpha}}\hat{r}^{\hat{\alpha}}-\hat{p}_{\hat{\alpha}}\hat{\eta}^{\hat{\alpha}}+\hat{A}\hat{\bar{\Omega}}+\hat{\chi}\hat{\beta},\end{aligned}$$
and their BRST variation. It is then straightforward to demonstrate that $$\begin{aligned}
\tilde{S} & \equiv & S_{fixed}+\left\{ Q,\int d\tau d\sigma\,(I+\hat{I})\right\} \nonumber \\
& = & \int d\tau d\sigma\{P_{m}\partial_{\tau}X^{m}-\tfrac{1}{2\mathcal{T}}(P_{m}P^{m}+\mathcal{T}^{2}\partial_{\sigma}X_{m}\partial_{\sigma}X^{m})\}\nonumber \\
& & +\int d\tau d\sigma\{w_{\alpha}\partial_{-}\lambda^{\alpha}+p_{\alpha}\partial_{-}\xi^{\alpha}+\pi_{\alpha}\partial_{\tau}\theta^{\alpha}+r^{\alpha}\partial_{-}s_{\alpha}+\eta^{\alpha}\partial_{-}\epsilon_{\alpha}\}\nonumber \\
& & +\int d\tau d\sigma\{A\partial_{-}B+\Sigma\partial_{-}\chi+\bar{\Omega}\partial_{-}\Omega+\beta\partial_{-}\gamma+\tfrac{1}{2}(\lambda\gamma^{m}\partial_{\sigma}\theta)\tfrac{(\pi\gamma^{m}\Lambda)}{(\Lambda\lambda)}\}\nonumber \\
& & +\int d\tau d\sigma\{\hat{w}_{\hat{\alpha}}\partial_{+}\hat{\lambda}^{\hat{\alpha}}+\hat{p}_{\hat{\alpha}}\partial_{+}\hat{\xi}^{\hat{\alpha}}+\hat{\pi}_{\hat{\alpha}}\partial_{\tau}\hat{\theta}^{\hat{\alpha}}+\hat{r}^{\hat{\alpha}}\partial_{+}\hat{s}_{\hat{\alpha}}+\hat{\eta}^{\hat{\alpha}}\partial_{+}\hat{\epsilon}_{\hat{\alpha}}\}\nonumber \\
& & +\int d\tau d\sigma\{\hat{A}\partial_{+}\hat{B}+\hat{\Sigma}\partial_{+}\hat{\chi}+\hat{\bar{\Omega}}\partial_{+}\hat{\Omega}+\hat{\beta}\partial_{+}\hat{\gamma}-\tfrac{1}{2}(\hat{\lambda}\gamma^{m}\partial_{\sigma}\hat{\theta})\tfrac{(\hat{\pi}\gamma^{m}\hat{\Lambda})}{(\hat{\Lambda}\hat{\lambda})}\},\label{eq:PSfixed2}\end{aligned}$$ where $\partial_{\pm}=\partial_{\tau}\pm\partial_{\sigma}$. Now, using the gamma matrix identity $$(\gamma^{mn})_{\alpha}^{\hphantom{\alpha}\beta}(\gamma_{mn})_{\lambda}^{\hphantom{\lambda}\gamma}=4\gamma_{\alpha\lambda}^{m}\gamma_{m}^{\beta\gamma}-2\delta_{\alpha}^{\beta}\delta_{\lambda}^{\gamma}-8\delta_{\lambda}^{\beta}\delta_{\alpha}^{\gamma},\label{eq:gammaproperty}$$ it follows that
$$\begin{aligned}
-\tfrac{1}{2}(\lambda\gamma_{m}\partial_{\sigma}\theta)(\pi\gamma^{m}\Lambda) & = & \tfrac{1}{8}(\pi\gamma^{mn}\lambda)(\Lambda\gamma_{mn}\partial_{\sigma}\theta)+\tfrac{1}{4}(\pi\lambda)(\Lambda\partial_{\sigma}\theta)+(\Lambda\lambda)(\pi\partial_{\sigma}\theta),\\
-\tfrac{1}{2}(\hat{\lambda}\gamma_{m}\partial_{\sigma}\hat{\theta})(\hat{\pi}\gamma^{m}\hat{\Lambda}) & = & \tfrac{1}{8}(\hat{\pi}\gamma^{mn}\hat{\lambda})(\hat{\Lambda}\gamma_{mn}\partial_{\sigma}\hat{\theta})+\tfrac{1}{4}(\hat{\pi}\hat{\lambda})(\hat{\Lambda}\partial_{\sigma}\hat{\theta})+(\hat{\Lambda}\hat{\lambda})(\hat{\pi}\partial_{\sigma}\hat{\theta}).\end{aligned}$$
The first two terms on the right hand side of each equation vanish due to the constraints . Therefore the gauge fixed action can be finally rewritten as $$\begin{aligned}
\tilde{S} & = & \int d\tau d\sigma\{P_{m}\partial_{\tau}X^{m}-\tfrac{1}{2\mathcal{T}}(P_{m}P^{m}+\mathcal{T}^{2}\partial_{\sigma}X_{m}\partial_{\sigma}X^{m})\}\nonumber \\
& & +\int d\tau d\sigma\{w_{\alpha}\partial_{-}\lambda^{\alpha}+p_{\alpha}\partial_{-}\xi^{\alpha}+\pi_{\alpha}\partial_{-}\theta^{\alpha}+r^{\alpha}\partial_{-}s_{\alpha}+\eta^{\alpha}\partial_{-}\epsilon_{\alpha}\}\nonumber \\
& & +\int d\tau d\sigma\{A\partial_{-}B+\Sigma\partial_{-}\chi+\bar{\Omega}\partial_{-}\Omega+\beta\partial_{-}\gamma\}\nonumber \\
& & +\int d\tau d\sigma\{\hat{w}_{\hat{\alpha}}\partial_{+}\hat{\lambda}^{\hat{\alpha}}+\hat{p}_{\hat{\alpha}}\partial_{+}\hat{\xi}^{\hat{\alpha}}+\hat{\pi}_{\hat{\alpha}}\partial_{+}\hat{\theta}^{\hat{\alpha}}+\hat{r}^{\hat{\alpha}}\partial_{+}\hat{s}_{\hat{\alpha}}+\hat{\eta}^{\hat{\alpha}}\partial_{+}\hat{\epsilon}_{\hat{\alpha}}\}\nonumber \\
& & +\int d\tau d\sigma\{\hat{A}\partial_{+}\hat{B}+\hat{\Sigma}\partial_{+}\hat{\chi}+\hat{\bar{\Omega}}\partial_{+}\hat{\Omega}+\hat{\beta}\partial_{+}\hat{\gamma}\},\label{eq:PSfixed3}\end{aligned}$$ in which the separation between right and left-moving sectors is manifest.
The BRST current can be computed using the transformations and . It has two components, one left and one right-moving, given by
\[eq:PS-BRSTcurrent\] $$\begin{aligned}
J & = & -\tfrac{1}{4\mathcal{T}}(\lambda\gamma_{m}\theta)(P^{m}+\mathcal{T}\partial_{\sigma}X^{m})+\tfrac{1}{16\mathcal{T}}(r\gamma^{m}\theta)(\lambda\gamma_{m}\theta)-\lambda^{\alpha}\partial_{\sigma}s_{\alpha}\nonumber \\
& & +A\lambda^{\alpha}s_{\alpha}-\Omega(\lambda^{\alpha}w_{\alpha}-r^{\alpha}s_{\alpha}+\theta^{\alpha}\pi_{\alpha}-\partial_{\sigma}B-\Sigma\chi-\beta\gamma)\nonumber \\
& & +\gamma(\lambda^{\alpha}p_{\alpha}-r^{\alpha}\epsilon_{\alpha}-\partial_{\sigma}\Sigma-A\Sigma)+\epsilon_{\alpha}\partial_{\sigma}\xi^{\alpha}-\chi\lambda^{\alpha}\epsilon_{\alpha},\\
\hat{J} & = & -\tfrac{1}{4\mathcal{T}}(\hat{\lambda}\gamma_{m}\hat{\theta})(P^{m}-\mathcal{T}\partial_{\sigma}X^{m})-\tfrac{1}{16\mathcal{T}}(\hat{r}\gamma^{m}\hat{\theta})(\hat{\lambda}\gamma_{m}\hat{\theta})-\hat{\lambda}^{\hat{\alpha}}\partial_{\sigma}\hat{s}_{\alpha}\nonumber \\
& & +\hat{A}\hat{\lambda}^{\hat{\alpha}}\hat{s}_{\alpha}-\hat{\Omega}(\hat{\lambda}^{\hat{\alpha}}\hat{w}_{\hat{\alpha}}+\hat{\theta}^{\hat{\alpha}}\hat{\pi}_{\hat{\alpha}}-\hat{r}^{\hat{\alpha}}\hat{s}_{\hat{\alpha}}-\partial_{\sigma}\hat{B}-\hat{\Sigma}\hat{\chi}-\hat{\beta}\hat{\gamma})\nonumber \\
& & +\hat{\gamma}(\hat{\lambda}^{\hat{\alpha}}\hat{p}_{\hat{\alpha}}-\hat{r}^{\hat{\alpha}}\hat{\epsilon}_{\hat{\alpha}}-\partial_{\sigma}\hat{\Sigma}-\hat{A}\hat{\Sigma})+\hat{\epsilon}_{\hat{\alpha}}\partial_{\sigma}\hat{\xi}^{\hat{\alpha}}-\hat{\chi}\hat{\lambda}^{\hat{\alpha}}\hat{\epsilon}_{\hat{\alpha}},\end{aligned}$$
such that $\partial_{-}J=\partial_{+}\hat{J}=0$.
As one final consistency check, observe that the generators of reparametrization symmetry are exact. By defining,
$$\begin{aligned}
B_{+} & \equiv & \tfrac{(\Lambda\gamma^{m}\pi)}{(\Lambda\lambda)}[P_{m}+\mathcal{T}\partial_{\sigma}X_{m}-\tfrac{1}{2}(r\gamma_{m}\theta)]-r^{\alpha}w_{\alpha}+\eta^{\alpha}p_{\alpha}-\beta\chi-A\bar{\Omega},\\
B_{-} & \equiv & -\tfrac{(\hat{\Lambda}\gamma^{m}\hat{\pi})}{(\hat{\Lambda}\hat{\lambda})}[P_{m}-\mathcal{T}\partial_{\sigma}X_{m}+\tfrac{1}{2}(\hat{r}\gamma_{m}\hat{\theta})]-\hat{r}^{\hat{\alpha}}\hat{w}_{\hat{\alpha}}+\hat{\eta}^{\hat{\alpha}}\hat{p}_{\hat{\alpha}}-\hat{\beta}\hat{\chi}-\hat{A}\hat{\bar{\Omega}},\end{aligned}$$
it is straightforward to compute their BV-BRST transformation, *c.f.* equations and ,
$$\begin{aligned}
\delta B_{+} & = & -\tfrac{1}{4\mathcal{T}}(P_{m}+\mathcal{T}\partial_{\sigma}X_{m})(P^{m}+\mathcal{T}\partial_{\sigma}X^{m})-w_{\alpha}\partial_{\sigma}\lambda^{\alpha}-p_{\alpha}\partial_{\sigma}\xi^{\alpha}-A\partial_{\sigma}B\nonumber \\
& & -\chi\partial_{\sigma}\Sigma-\pi_{\alpha}\partial_{\sigma}\theta^{\alpha}-r^{\alpha}\partial_{\sigma}s_{\alpha}-\eta^{\alpha}\partial_{\sigma}\epsilon_{\alpha}-\beta\partial_{\sigma}\gamma-\bar{\Omega}\partial_{\sigma}\Omega,\\
\delta B_{-} & = & \tfrac{1}{4\mathcal{T}}(P^{m}-\mathcal{T}\partial_{\sigma}X^{m})(P_{m}-\mathcal{T}\partial_{\sigma}X_{m})-\hat{w}_{\hat{\alpha}}\partial_{\sigma}\hat{\lambda}^{\hat{\alpha}}-\hat{p}_{\hat{\alpha}}\partial_{\sigma}\hat{\xi}^{\hat{\alpha}}-\hat{A}\partial_{\sigma}\hat{B}\nonumber \\
& & -\hat{\chi}\partial_{\sigma}\hat{\Sigma}-\hat{\pi}_{\hat{\alpha}}\partial_{\sigma}\hat{\theta}^{\hat{\alpha}}-\hat{r}^{\hat{\alpha}}\partial_{\sigma}\hat{s}_{\hat{\alpha}}-\hat{\eta}^{\hat{\alpha}}\partial_{\sigma}\hat{\epsilon}_{\hat{\alpha}}-\hat{\beta}\partial_{\sigma}\hat{\gamma}-\hat{\bar{\Omega}}\partial_{\sigma}\hat{\Omega},\end{aligned}$$
which constitute the generalization of $H^{\pm}$ in .
Field redefinition and emergent spacetime supersymmetry\[subsec:Fieldred\]
--------------------------------------------------------------------------
Although not obviously, the BRST structure of the action can be greatly simplified. Consider the field redefinitions $$\begin{array}{rclcrcl}
\lambda^{\alpha} & \to & \gamma^{-1}\lambda^{\alpha}, & & p_{\alpha} & \to & p_{\alpha}+\partial_{\sigma}s_{\alpha},\\
w_{\alpha} & \to & \gamma w_{\alpha}, & & \pi_{\alpha} & \to & \gamma^{-1}\pi_{\alpha},\\
A & \to & A+\frac{\partial_{\sigma}\gamma}{\gamma}, & & r^{\alpha} & \to & \gamma^{-1}(r^{\alpha}+\partial_{\sigma}\xi^{\alpha}),\\
\chi & \to & \gamma\chi, & & s_{\alpha} & \to & \gamma s_{\alpha},\\
\Sigma & \to & \gamma^{-1}\Sigma, & & \beta & \to & \beta+\gamma^{-1}(\lambda^{\alpha}w_{\alpha}+\theta^{\alpha}\pi_{\alpha}-\partial_{\sigma}B)\\
\theta^{\alpha} & \to & \gamma\theta^{\alpha}, & & & & +\gamma^{-1}(s_{\alpha}r^{\alpha}+s_{\alpha}\partial_{\sigma}\xi^{\alpha}+\chi\Sigma),
\end{array}\label{eq:fieldredefinition}$$ and analogous operations in the hatted sector, which leave the action invariant. Note that the pure spinor constraints have to be modified accordingly.
Since the ghosts fields $\gamma$ and $\hat{\gamma}$ transform under scaling, the field redefinitions above leave all the spacetime spinors scale invariant at the price of shifting their ghost number. In particular, the pure spinors $\lambda^{\alpha}$ and $\hat{\lambda}^{\hat{\alpha}}$ now have ghost number $+1$ while the $\theta^{\alpha}$ and $\hat{\theta}^{\hat{\alpha}}$ have ghost number zero. Also, due to the ordering of the operators, the scaling parts of the BRST current (with $\Omega$ and $\hat{\Omega}$) receive quantum corrections which can be cast as
\[eq:U(1)ordering\] $$\begin{aligned}
\Omega(\lambda^{\alpha}w_{\alpha}-r^{\alpha}s_{\alpha}+\theta^{\alpha}\pi_{\alpha}-\partial_{\sigma}B-\Sigma\chi-\beta\gamma) & \to & -\Omega(\beta\gamma+c_{\#}\partial_{\sigma}\ln\gamma),\\
\hat{\Omega}(\hat{\lambda}^{\hat{\alpha}}\hat{w}_{\hat{\alpha}}+\hat{\theta}^{\hat{\alpha}}\hat{\pi}_{\hat{\alpha}}-\hat{r}^{\hat{\alpha}}\hat{s}_{\hat{\alpha}}-\partial_{\sigma}\hat{B}-\hat{\Sigma}\hat{\chi}-\hat{\beta}\hat{\gamma}) & \to & -\hat{\Omega}(\hat{\beta}\hat{\gamma}+\hat{c}_{\#}\partial_{\sigma}\ln\hat{\gamma}),\end{aligned}$$
where $c_{\#}$ and $\hat{c}_{\#}$ are numerical constants which will be fixed later in Appendix \[sec:decoupledsector\].
The BRST currents are then rewritten as
$$\begin{aligned}
J & = & -\tfrac{1}{4\mathcal{T}}(\lambda\gamma_{m}\theta)\Big[P^{m}+\mathcal{T}\partial_{\sigma}X^{m}-\tfrac{1}{4}(\partial_{\sigma}\xi\gamma^{m}\theta)\Big]\nonumber \\
& & +\lambda^{\alpha}p_{\alpha}-\Omega(\beta\gamma+c_{\#}\partial_{\sigma}\ln\gamma)-A\Sigma-r^{\alpha}\epsilon_{\alpha}\nonumber \\
& & +\tfrac{1}{16\mathcal{T}}(r\gamma^{m}\theta)(\lambda\gamma_{m}\theta)+A\lambda^{\alpha}s_{\alpha}-\chi\lambda^{\alpha}\epsilon_{\alpha}-\partial_{\sigma}(\gamma\Sigma),\label{eq:JfieldR}\\
\hat{J} & = & -\tfrac{1}{4\mathcal{T}}(\hat{\lambda}\gamma_{m}\hat{\theta})\Big[P^{m}-\mathcal{T}\partial_{\sigma}X^{m}+\tfrac{1}{4}(\partial_{\sigma}\hat{\xi}\gamma^{m}\hat{\theta})\Big]\nonumber \\
& & +\hat{\lambda}^{\hat{\alpha}}\hat{p}_{\hat{\alpha}}-\hat{\Omega}(\hat{\beta}\hat{\gamma}+\hat{c}_{\#}\partial_{\sigma}\ln\hat{\gamma})-\hat{A}\hat{\Sigma}-\hat{r}^{\hat{\alpha}}\hat{\epsilon}_{\hat{\alpha}}\nonumber \\
& & -\tfrac{1}{16\mathcal{T}}(\hat{r}\gamma^{m}\hat{\theta})(\hat{\lambda}\gamma_{m}\hat{\theta})+\hat{A}\hat{\lambda}^{\hat{\alpha}}\hat{s}_{\alpha}-\hat{\chi}\hat{\lambda}^{\hat{\alpha}}\hat{\epsilon}_{\hat{\alpha}}-\partial_{\sigma}(\hat{\gamma}\hat{\Sigma}).\label{eq:JhatfieldR}\end{aligned}$$
The last term in each current can be disregarded, since they correspond to total derivatives and do not contribute to the BRST charges. Furthermore, the remaining terms in the third lines of and can be removed by similarity transformations of the form $J^{'}\equiv e^{-U}Je^{U}$ and $\hat{J}^{'}\equiv e^{-\hat{U}}\hat{J}e^{\hat{U}}$, where $U$ and $\hat{U}$ are invariant under the pure spinor symmetries and given by
$$\begin{aligned}
U & = & \int d\sigma\Bigg\{\chi\lambda^{\alpha}s_{\alpha}-\tfrac{1}{32\mathcal{T}}(\lambda\gamma^{m}\theta)(\lambda\gamma^{n}\theta)\frac{(\eta\gamma_{mn}\Lambda)}{(\Lambda\lambda)}\Bigg\},\\
\hat{U} & = & \int d\sigma\Bigg\{\hat{\chi}\hat{\lambda}^{\hat{\alpha}}\hat{s}_{\hat{\alpha}}+\tfrac{1}{32\mathcal{T}}(\hat{\lambda}\gamma^{m}\hat{\theta})(\hat{\lambda}\gamma^{n}\hat{\theta})\frac{(\hat{\eta}\gamma_{mn}\hat{\Lambda})}{(\hat{\Lambda}\hat{\lambda})}\Bigg\},\end{aligned}$$
It is then straightforward to show that
$$\begin{aligned}
J^{'} & = & \lambda^{\alpha}p_{\alpha}-\tfrac{1}{4\mathcal{T}}(\lambda\gamma_{m}\theta)\Big[P^{m}+\mathcal{T}\partial_{\sigma}X^{m}-\tfrac{1}{4}(\partial_{\sigma}\xi\gamma^{m}\theta)\Big]\nonumber \\
& & -\Omega(\beta\gamma+c_{\#}\partial_{\sigma}\ln\gamma)-A\Sigma-r^{\alpha}\epsilon_{\alpha},\\
\hat{J}^{'} & = & \hat{\lambda}^{\hat{\alpha}}\hat{p}_{\hat{\alpha}}-\tfrac{1}{4\mathcal{T}}(\hat{\lambda}\gamma_{m}\hat{\theta})\Big[P^{m}-\mathcal{T}\partial_{\sigma}X^{m}+\tfrac{1}{4}(\partial_{\sigma}\hat{\xi}\gamma^{m}\hat{\theta})\Big]\nonumber \\
& & -\hat{\Omega}(\hat{\beta}\hat{\gamma}+\hat{c}_{\#}\partial_{\sigma}\ln\hat{\gamma})-\hat{A}\hat{\Sigma}-\hat{r}^{\hat{\alpha}}\hat{\epsilon}_{\hat{\alpha}}.\end{aligned}$$
It is important to emphasize that the field redefinitions in are well defined only if $\gamma$ and $\hat{\gamma}$ are assumed to be non-vanishing in every point of the worldsheet. This is clear in the definition of the conformal field theory of the decoupled sector, which is singular for $\gamma=0$ and $\hat{\gamma}=0$. In a path integral formulation, this singularity can be avoided by choosing a convenient parametrization for the ghosts, *e.g.* $\gamma=e^{\sigma}$ and $\beta=\rho e^{-\sigma}$, therefore enforcing the non-vanishing condition. Here, $\sigma$ is a chiral worldsheet scalar with conjugate $\rho$. More details can be found in the appendix \[sec:decoupledsector\].
Using the quartet argument, the BRST cohomology can be shown to be independent of $A$, $B$, $\chi$, $\sigma$, $\rho$, $\Sigma$, $r^{\alpha}$, $s_{\alpha}$, $\eta^{\alpha}$ and $\epsilon_{\alpha}$ (hatted and unhatted). Therefore, these fields can be eliminated from the theory and the gauge fixed action is further simplified to $$\begin{aligned}
\tilde{S} & = & \int d\tau d\sigma\{P_{m}\partial_{\tau}X^{m}-\tfrac{1}{2\mathcal{T}}(P_{m}P^{m}+\mathcal{T}^{2}\partial_{\sigma}X_{m}\partial_{\sigma}X^{m})\}\nonumber \\
& & +\int d\tau d\sigma\{w_{\alpha}\partial_{-}\lambda^{\alpha}+p_{\alpha}\partial_{-}\xi^{\alpha}+\pi_{\alpha}\partial_{-}\theta^{\alpha}+\hat{w}_{\hat{\alpha}}\partial_{+}\hat{\lambda}^{\hat{\alpha}}+\hat{p}_{\hat{\alpha}}\partial_{+}\hat{\xi}^{\hat{\alpha}}+\hat{\pi}_{\hat{\alpha}}\partial_{+}\hat{\theta}^{\hat{\alpha}}\}.\label{eq:PSfixed}\end{aligned}$$
The pure spinor constraints are reduced to $$\begin{array}{rclcrcl}
(\lambda\gamma^{m}\lambda) & = & 0, & & (\hat{\lambda}\gamma^{m}\hat{\lambda}) & = & 0,\\
(\lambda\gamma^{m}\xi) & = & 0, & & (\hat{\lambda}\gamma^{m}\hat{\xi}) & = & 0,\\
\xi^{\alpha}\pi_{\alpha} & = & 0, & & \hat{\xi}^{\hat{\alpha}}\hat{\pi}_{\hat{\alpha}} & = & 0,\\
\lambda^{\alpha}\pi_{\alpha} & = & 0, & & \hat{\lambda}^{\hat{\alpha}}\hat{\pi}_{\hat{\alpha}} & = & 0,\\
(\lambda\gamma^{mn}\pi) & = & 0, & & (\hat{\lambda}\gamma^{mn}\hat{\pi}) & = & 0,
\end{array}\label{eq:PSconstraintsBRST}$$ and the action is invariant under the implied pure spinor gauge transformations
\[eq:PSsymmetriesBRST\] $$\begin{aligned}
\delta w_{\alpha} & = & d_{m}(\gamma^{m}\lambda)_{\alpha}+e_{m}(\gamma^{m}\xi)_{\alpha}-\bar{f}\pi_{\alpha}-\bar{f}_{mn}(\gamma^{mn}\pi)_{\alpha},\\
\delta p_{\alpha} & = & e_{m}(\gamma^{m}\lambda)_{\alpha}-\bar{g}\pi_{\alpha},\\
\delta\theta^{\alpha} & = & \bar{f}\lambda^{\alpha}+\bar{f}_{mn}(\gamma^{mn}\lambda)^{\alpha}+\bar{g}\xi^{\alpha},\\
\delta\hat{w}_{\hat{\alpha}} & = & \hat{d}_{m}(\gamma^{m}\hat{\lambda})_{\hat{\alpha}}+\hat{e}_{m}(\gamma^{m}\hat{\xi})_{\hat{\alpha}}-\hat{\bar{f}}\hat{\pi}_{\hat{\alpha}}-\hat{\bar{f}}_{mn}(\gamma^{mn}\hat{\pi})_{\hat{\alpha}},\\
\delta\hat{p}_{\hat{\alpha}} & = & \hat{e}_{m}(\gamma^{m}\hat{\lambda})_{\hat{\alpha}}-\hat{\bar{g}}\hat{\pi}_{\hat{\alpha}},\\
\delta\hat{\theta}^{\hat{\alpha}} & = & \hat{\bar{f}}\hat{\lambda}^{\hat{\alpha}}+\hat{\bar{f}}_{mn}(\gamma^{mn}\hat{\lambda})^{\hat{\alpha}}+\hat{\bar{g}}\hat{\xi}^{\hat{\alpha}}.\end{aligned}$$
These transformations are the key to spacetime supersymmetry. The parameter $e_{m}$ can be tuned in such a way that the gauge dependent components of $p_{\alpha}$ are identified with the independent components of the constrained spinor $\pi_{\alpha}$. Furthermore, the gauge parameters $\bar{f}$ and $\bar{f}_{mn}$ can be similarly chosen such that the gauge dependent components of $\theta^{\alpha}$ are identified with the independent components of the constrained spinor $\xi^{\alpha}$. This is demonstrated in Appendix \[sec:Partial-gauge-fixing\]. An analogous gauge fixing can be performed in the hatted sector. The outcome of the partial gauge fixing of the pure spinor symmetries is the action $$\begin{aligned}
S & = & \int d\tau d\sigma\{P_{m}\partial_{\tau}X^{m}-\tfrac{1}{2\mathcal{T}}(P_{m}P^{m}+\mathcal{T}^{2}\partial_{\sigma}X_{m}\partial_{\sigma}X^{m})\}\nonumber \\
& & +\int d\tau d\sigma\{w_{\alpha}\partial_{-}\lambda^{\alpha}+p_{\alpha}\partial_{-}\theta^{\alpha}+\hat{w}_{\hat{\alpha}}\partial_{+}\hat{\lambda}^{\hat{\alpha}}+\hat{p}_{\hat{\alpha}}\partial_{+}\hat{\theta}^{\hat{\alpha}}\},\label{eq:PSaction}\end{aligned}$$ but now with unconstrained $p_{\alpha}$, $\theta^{\alpha}$, $\hat{p}_{\hat{\alpha}}$ and $\hat{\theta}^{\hat{\alpha}}$. The action $S$ corresponds to the type II-B superstring in the pure spinor formalism. The type II-A is similarly obtained by reverting the spinor chirality of one of the sectors, either hatted or unhatted. The heterotic superstring is obtained when only one of the twistor-like constraints is imposed, but then worldsheet reparametrization has be taken into account in the construction of the master action.
The non-vanishing BRST transformations can be cast as $$\begin{array}{rclcrcl}
\delta X^{m} & = & \tfrac{1}{4\mathcal{T}}(\lambda\gamma^{m}\theta)+\tfrac{1}{4\mathcal{T}}(\hat{\lambda}\gamma^{m}\hat{\theta}), & & \delta P_{m} & = & \tfrac{1}{4}\partial_{\sigma}[(\lambda\gamma_{m}\theta)]-\tfrac{1}{4}\partial_{\sigma}[(\hat{\lambda}\gamma_{m}\hat{\theta})],\\
\delta w_{\alpha} & = & p_{\alpha}-\tfrac{1}{16\mathcal{T}}(\theta\gamma_{m}\partial_{\sigma}\theta)(\gamma^{m}\theta)_{\alpha} & & \delta\hat{w}_{\hat{\alpha}} & = & \hat{p}_{\hat{\alpha}}+\tfrac{1}{16\mathcal{T}}(\hat{\theta}\gamma_{m}\partial_{\sigma}\hat{\theta})(\gamma^{m}\hat{\theta})_{\hat{\alpha}}\\
& & -\tfrac{1}{4\mathcal{T}}(P_{m}+\mathcal{T}\partial_{\sigma}X_{m})(\gamma^{m}\theta)_{\alpha}, & & & & -\tfrac{1}{4\mathcal{T}}(P_{m}-\mathcal{T}\partial_{\sigma}X_{m})(\gamma^{m}\hat{\theta})_{\hat{\alpha}},\\
\delta\theta^{\alpha} & = & \lambda^{\alpha}, & & \delta\hat{\theta}^{\hat{\alpha}} & = & \hat{\lambda}^{\hat{\alpha}},\\
\delta p_{\alpha} & = & -\tfrac{1}{4\mathcal{T}}(P_{m}+\mathcal{T}\partial_{\sigma}X_{m})(\gamma^{m}\lambda)_{\alpha} & & \delta\hat{p}_{\hat{\alpha}} & = & -\tfrac{1}{4\mathcal{T}}(P_{m}-\mathcal{T}\partial_{\sigma}X_{m})(\gamma^{m}\hat{\lambda})_{\hat{\alpha}}\\
& & +\tfrac{1}{16\mathcal{T}}(\partial_{\sigma}\lambda\gamma_{m}\theta)(\gamma^{m}\theta)_{\alpha} & & & & -\tfrac{1}{16\mathcal{T}}(\partial_{\sigma}\hat{\lambda}\gamma_{m}\hat{\theta})(\gamma^{m}\hat{\theta})_{\hat{\alpha}}\\
& & +\tfrac{3}{8\mathcal{T}}(\lambda\gamma_{m}\theta)(\gamma^{m}\partial_{\sigma}\theta)_{\alpha} & & & & -\tfrac{3}{8\mathcal{T}}(\hat{\lambda}\gamma_{m}\hat{\theta})(\gamma^{m}\partial_{\sigma}\hat{\theta})_{\hat{\alpha}}
\end{array}$$ generated by the BRST charges
$$\begin{aligned}
Q & = & \int d\sigma\{\lambda^{\alpha}p_{\alpha}-\tfrac{1}{4\mathcal{T}}(\lambda\gamma^{m}\theta)(P_{m}+\mathcal{T}\partial_{\sigma}X_{m})+\tfrac{1}{16\mathcal{T}}(\lambda\gamma^{m}\theta)(\theta\gamma_{m}\partial_{\sigma}\theta)\},\label{eq:PSBRSTholo}\\
\hat{Q} & = & \int d\sigma\{\hat{\lambda}^{\hat{\alpha}}\hat{p}_{\hat{\alpha}}-\tfrac{1}{4\mathcal{T}}(\hat{\lambda}\gamma^{m}\hat{\theta})(P_{m}-\mathcal{T}\partial_{\sigma}X_{m})-\tfrac{1}{16\mathcal{T}}(\hat{\lambda}\gamma^{m}\hat{\theta})(\hat{\theta}\gamma_{m}\partial_{\sigma}\hat{\theta})\},\label{eq:PSBRSTantiholo}\end{aligned}$$
Finally, note that the BRST charges and the action are spacetime supersymmetric and the supersymmetry generators can be expressed as
\[eq:SUSYcharges\] $$\begin{aligned}
q_{\alpha} & = & \int d\sigma\Big\{ p_{\alpha}+\tfrac{1}{4\mathcal{T}}(P_{m}+\mathcal{T}\partial_{\sigma}X_{m})(\gamma^{m}\theta)_{\alpha}+\tfrac{1}{48\mathcal{T}}(\theta\gamma_{m}\partial_{\sigma}\theta)(\gamma^{m}\theta)_{\alpha}\Big\},\\
\hat{q}_{\hat{\alpha}} & = & \int d\sigma\Big\{\hat{p}_{\hat{\alpha}}+\tfrac{1}{4\mathcal{T}}(P_{m}-\mathcal{T}\partial_{\sigma}X_{m})(\gamma^{m}\hat{\theta})_{\hat{\alpha}}-\tfrac{1}{48\mathcal{T}}(\hat{\theta}\gamma_{m}\partial_{\sigma}\hat{\theta})(\gamma^{m}\hat{\theta})_{\hat{\alpha}}\Big\},\end{aligned}$$
satisfying the algebra
\[eq:SUSYalgebra\] $$\begin{aligned}
\{q_{\alpha},q_{\beta}\} & = & \tfrac{1}{2\mathcal{T}}\gamma_{\alpha\beta}^{m}\int d\sigma\,P_{m},\\
\{\hat{q}_{\hat{\alpha}},\hat{q}_{\hat{\beta}}\} & = & \tfrac{1}{2\mathcal{T}}\gamma_{\hat{\alpha}\hat{\beta}}^{m}\int d\sigma\,P_{m}.\end{aligned}$$
Therefore, spacetime supersymmetry is made manifest with the help of the field redefinitions and the pure spinor gauge symmetries , demonstrating the the action can be seen as the underlying gauge theory of the pure spinor superstring
Final remarks\[sec:Final-remarks\]
==================================
The pure spinor action can be rewritten in a more traditional way by solving the equation of motion for $P_{m}$ and Wick-rotating the worldsheet time coordinate $\tau$. The resulting action is $$S=\int d^{2}z\{\tfrac{1}{2}\partial X^{m}\bar{\partial}X_{m}+w_{\alpha}\bar{\partial}\lambda^{\alpha}+p_{\alpha}\bar{\partial}\theta^{\alpha}+\hat{w}_{\hat{\alpha}}\partial\hat{\lambda}^{\hat{\alpha}}+\hat{p}_{\hat{\alpha}}\partial\hat{\theta}^{\hat{\alpha}}\},\label{eq:PSactiontraditional}$$ with $z$ ($\bar{z}$) denoting the usual (anti)holomorphic coordinate, $\partial\equiv\tfrac{\partial}{\partial z}$, $\bar{\partial}\equiv\tfrac{\partial}{\partial\bar{z}}$ and $\mathcal{T}=1$ (string tension). The BRST charge takes its standard form in the pure spinor formalism as $$Q_{\tiny{PS}}=\ointctrclockwise\,\lambda^{\alpha}d_{\alpha},\label{eq:PSBRSTcharge}$$ where $d_{\alpha}$ denotes the field realization of the supersymmetric derivative and is expressed as $$d_{\alpha}\equiv p_{\alpha}-\tfrac{1}{2}\partial X^{m}(\gamma_{m}\theta)_{\alpha}-\tfrac{1}{8}(\theta\gamma^{m}\partial\theta)(\gamma_{m}\theta)_{\alpha}.$$
While the twistor-like symmetry can be understood as a way to rewrite the generators of worldsheet reparametrization with a linear dependence on $P_{m}$, the extra fermionic pair $\{\xi^{\alpha},p_{\alpha}\}$ and the fermionic symmetry do not have a clear physical interpretation. These ingredients are ultimately responsible for the emergence of spacetime supersymmetry but from the worldsheet point of view their existence lacks a more fundamental understanding. The operator $\lambda^{\alpha}p_{\alpha}$ resembles part of a possible worldsheet supersymmetry generator with “wrong” conformal dimension so it might be possible that the gauge action can be embedded in a bigger model with twisted worldsheet supersymmetry. It would be interesting to investigate potential connections with (1) the superembedding origin of the heterotic pure spinor superstring, presented in [@Matone:2002ft]; and (2) the twisted formulation of the pure spinor superstring introduced in [@Berkovits:2016xnb] and its relation to the spinning string. Also in this direction, it is worth studying alternative gauge choices for the master action and whether they could be related to the Green-Schwarz superstring. This idea was first proposed in [@Berkovits:2015yra] for Berkovits’ action and it would be interesting to develop a similar approach here. In order to do that, it seems that worldsheet reparametrization has to be explicitly included in the construction of the master action.
**Acknowledgments:** I would like to thank Thales Azevedo and Nathan Berkovits for comments of the draft. Also, I would like to thank Dmitri Sorokin for reference suggestions and for pointing out a possible connection between the results presented here and the superembedding approach. This research has been supported by the Czech Science Foundation - GAČR, project 19-06342Y.
Partial gauge fixing of the pure spinor symmetries\[sec:Partial-gauge-fixing\]
==============================================================================
The aim of this appendix is to demonstrate that the action can be rewritten in terms of unconstrained spacetime spinors $p_{\alpha}$, $\theta^{\alpha}$, $\hat{p}_{\hat{\alpha}}$ and $\hat{\theta}^{\hat{\alpha}}$, provided that the pure spinor gauge symmetries are partially fixed in a precise form. In order to do this, the pure spinor constraints will be explicitly solved in a Wick-rotated scenario and the $SO(10)$ spinors will be expressed in terms of $U(5)$ components. To illustrate the procedure, only the unhatted (left-moving) sector will be analyzed, but it can be easily extended to the right-moving sector.
$U(5)$ decomposition {#u5-decomposition .unnumbered}
--------------------
Given an $SO\left(10\right)$ chiral spinor $\xi^{\alpha}$ (antichiral $\chi_{\alpha}$), with $\alpha=1,\ldots,16$, it is possible to determine its $U\left(5\right)$ components using the projectors $P_{A}^{\alpha}$ and $\left(P_{A}^{\alpha}\right)^{-1}\equiv P_{\alpha}^{I}$, where $A=\{+,a,ab\}$ denotes the $U\left(5\right)$ indices, respectively the $U(1)$-charged singlet, the vector and the adjoint representations, with $a=1,\ldots,5$, defined in such a way that $$\begin{array}{rclcrcl}
\xi^{\alpha} & = & P_{+}^{\alpha}\xi^{+}+\frac{1}{2}P_{ab}^{\alpha}\xi^{ab}+P^{\alpha a}\xi_{a}, & & \chi_{\alpha} & = & P_{\alpha}^{+}\chi_{+}+\frac{1}{2}P_{\alpha}^{ab}\chi_{ab}+P_{\alpha a}\chi^{a},\\
\xi^{+} & = & P_{\alpha}^{+}\xi^{\alpha}, & & \chi_{+} & = & P_{+}^{\alpha}\chi_{\alpha},\\
\xi^{ab} & = & P_{\alpha}^{ab}\xi^{\alpha}, & & \chi_{ab} & = & P_{ab}^{\alpha}\chi_{\alpha},\\
\xi_{a} & = & P_{\alpha a}\xi^{\alpha}, & & \chi^{a} & = & P^{\alpha a}\chi_{\alpha}.
\end{array}\label{eq:u5spinordecomposition}$$ These projectors satisfy the orthogonality equations $$\begin{array}{rcl}
P_{+}^{\alpha}P_{\alpha}^{+} & = & 1,\\
P^{\alpha a}P_{\alpha b} & = & \delta_{b}^{a},\\
P_{ab}^{\alpha}P_{\alpha}^{cd} & = & \delta_{a}^{c}\delta_{b}^{d}-\delta_{b}^{c}\delta_{a}^{d}\\
P_{+}^{\alpha}P_{\beta}^{+}+\frac{1}{2}P_{ab}^{\alpha}P_{\beta}^{ab}+P^{\alpha a}P_{\beta a} & = & \delta_{\beta}^{\alpha},
\end{array}\label{eq:orthoprojectorsSO(10)}$$ and more generally $P_{A}^{\alpha}P_{\alpha}^{B}=0$ for $A\neq B$. They can be used as building blocks of the symmetric $g$-matrices, defined as $$\begin{array}{rcl}
(g^{a})^{\alpha\beta} & \equiv & \sqrt{2}(P_{+}^{\alpha}P^{\beta a}+P_{+}^{\beta}P^{\alpha a}+\frac{1}{4}\epsilon^{abcde}P_{bc}^{\alpha}P_{de}^{\beta}),\\
(\bar{g}_{a})^{\alpha\beta} & \equiv & \sqrt{2}(P^{\alpha b}P_{ba}^{\beta}+P^{\beta b}P_{ba}^{\alpha}),\\
(g^{a})_{\alpha\beta} & \equiv & \sqrt{2}(P_{\beta b}P_{\alpha}^{ba}+P_{\alpha b}P_{\beta}^{ba}),\\
(\bar{g}_{a})_{\alpha\beta} & \equiv & \sqrt{2}(P_{\alpha}^{+}P_{\beta a}+P_{\beta}^{+}P_{\alpha a}+\frac{1}{4}\epsilon_{abcde}P_{\alpha}^{bc}P_{\beta}^{de}),
\end{array}\label{eq:gmatricesSO(10)}$$ where $\epsilon^{abcde}$ and $\epsilon_{abcde}$ are the totally antisymmetric $U\left(5\right)$ tensors, with $\epsilon^{12345}=\epsilon_{54321}=1$, satisfying the algebra
$$\begin{aligned}
\{g^{a},g^{b}\}_{\alpha}^{\phantom{\alpha}\beta} & \equiv & g_{\alpha\gamma}^{a}(g^{b})^{\gamma\beta}+g_{\alpha\gamma}^{b}(g^{a})^{\gamma\beta}\nonumber \\
& = & 0,\\
\{\bar{g}_{a},\bar{g}_{b}\}_{\alpha}^{\phantom{\alpha}\beta} & \equiv & (\bar{g}_{a})_{\alpha\gamma}\bar{g}_{b}^{\gamma\beta}+(\bar{g}_{b})_{\alpha\gamma}\bar{g}_{a}^{\gamma\beta}\nonumber \\
& = & 0,\\
\{g^{a},\bar{g}_{b}\}_{\alpha}^{\phantom{\alpha}\beta} & \equiv & g_{\alpha\gamma}^{a}\bar{g}_{b}^{\gamma\beta}+(\bar{g}_{b})_{\alpha\gamma}(g^{a})^{\gamma\beta}\nonumber \\
& = & 2\delta^{\beta}\delta_{b}^{a}.\end{aligned}$$
Therefore, the $g$-matrices represent the $U(5)$ components of the gamma matrices $\gamma^{m}$.
Solving the pure spinor constraints {#solving-the-pure-spinor-constraints .unnumbered}
-----------------------------------
The action is invariant under the pure spinor gauge transformations $$\begin{array}{rcl}
\delta w_{\alpha} & = & d_{m}(\gamma^{m}\lambda)_{\alpha}+e_{m}(\gamma^{m}\xi)_{\alpha}-\bar{f}_{mn}(\gamma^{mn}\pi)_{\alpha},\\
\delta p_{\alpha} & = & e_{m}(\gamma^{m}\lambda)_{\alpha},\\
\delta\theta^{\alpha} & = & \bar{f}_{mn}(\gamma^{mn}\lambda)^{\alpha},
\end{array}\label{eq:PSgaugeAP}$$ where $d_{m},$ $e_{m}$ and $\bar{f}_{mn}$ are the parameters. Note that the parameters $\bar{f}$ and $\bar{g}$ in are redundant, for they can be absorbed by a shift of the other parameters: $$\begin{array}{rcl}
d_{m} & \to & d_{m}-\tfrac{4\bar{f}}{5}\frac{(\Lambda\gamma_{m}\pi)}{(\Lambda\lambda)}+\tfrac{\bar{g}}{5}\frac{(\Lambda\xi)}{(\Lambda\lambda)^{2}}(\Lambda\gamma_{m}\pi),\\
e_{m} & \to & e_{m}+\tfrac{\bar{g}}{2}\frac{(\Lambda\gamma_{m}\pi)}{(\Lambda\lambda)},\\
\bar{f}_{mn} & \to & \bar{f}_{mn}+\tfrac{\bar{f}}{10}\frac{(\Lambda\gamma_{mn}\lambda)}{(\Lambda\lambda)}+\tfrac{\bar{g}}{8}\frac{(\Lambda\gamma_{mn}\xi)}{(\Lambda\lambda)}-\tfrac{\bar{g}}{40}\frac{(\Lambda\xi)(\Lambda\gamma_{mn}\lambda)}{(\Lambda\lambda)^{2}}.
\end{array}$$ This can be easily demonstrated with the help of the gamma matrix property .
In terms of $U(5)$ components, $\lambda^{\alpha}$, $\theta^{\alpha}$ and $\pi_{\alpha}$ can be parametrized as $\lambda^{\alpha}=(\lambda^{+},\lambda^{ab},\lambda_{a})$, $\xi^{\alpha}=(\xi^{+},\xi^{ab},\xi_{a})$, and $\pi_{\alpha}=(\pi_{+},\pi_{ab},\pi^{a})$. Therefore, using the $g$-matrices , the constraints can be rewritten as $$\begin{array}{rclcrcl}
\lambda^{ab}\lambda_{b} & = & 0, & & \lambda^{+}\pi_{ab} & = & \frac{1}{2}\epsilon_{abcde}\pi^{c}\lambda^{de},\\
\lambda^{+}\lambda_{a} & = & -\tfrac{1}{8}\epsilon_{abcde}\lambda^{bc}\lambda^{de}, & & \lambda^{+}\pi_{+}- & = & \tfrac{3}{5}\lambda_{a}\pi^{a}-\tfrac{1}{10}\lambda^{ab}\pi_{ab},\\
\lambda^{ab}\xi_{b} & = & -\xi^{ab}\lambda_{b}, & & \lambda_{a}\pi^{b}-\lambda^{bc}\pi_{ac} & = & \tfrac{1}{5}\delta_{a}^{b}(\lambda_{c}\pi^{c}-\lambda^{cd}\pi_{cd}),\\
\lambda^{+}\xi_{a} & = & -\xi^{+}\lambda_{a}-\tfrac{1}{4}\epsilon_{abcde}\lambda^{bc}\xi^{de}, & & \lambda^{+}\pi_{+}- & = & \lambda_{a}\pi^{a}+\tfrac{1}{2}\lambda^{ab}\pi_{ab},\\
\lambda^{ab}\pi_{+} & = & \frac{1}{2}\epsilon^{abcde}\lambda_{c}\pi_{de}, & & \xi^{+}\pi_{+}- & = & \xi_{a}\pi^{a}+\tfrac{1}{2}\xi^{ab}\pi_{ab}.
\end{array}$$
Assuming $\lambda^{+}\neq0$, these constraints can be explicitly solved by $$\begin{array}{rcl}
\lambda_{a} & = & -\tfrac{1}{8\lambda^{+}}\epsilon_{abcde}\lambda^{bc}\lambda^{de},\\
\xi_{a} & = & -\tfrac{\xi^{+}}{\lambda^{+}}\lambda_{a}-\tfrac{1}{4\lambda^{+}}\epsilon_{abcde}\lambda^{bc}\xi^{de},\\
\pi_{+} & = & \tfrac{1}{\lambda^{+}}\lambda_{a}\pi^{a},\\
\pi_{ab} & = & \tfrac{1}{2\lambda^{+}}\epsilon_{abcde}\pi^{c}\lambda^{de},
\end{array}\label{eq:PSsolutions}$$
Next, the transformations can be used to conveniently choose the gauge $p_{\alpha}=\{p_{+},p_{ab},\pi^{a}\}$ and $\theta^{\alpha}=\{\xi^{+},\xi^{ab},\theta_{a}\}$, where $p_{+}$, $p_{ab}$ and $\theta_{a}$ denote the independent components of $p_{\alpha}$ and $\theta^{\alpha}$. Using this gauge and the solutions , their contribution to the action can be cast as $$\begin{aligned}
S_{sp} & = & \int d\tau d\sigma\{p_{\alpha}\partial_{-}\xi^{\alpha}+\pi_{\alpha}\partial_{-}\theta^{\alpha}\},\nonumber \\
& = & \int d\tau d\sigma\{p_{+}\partial_{-}\xi^{+}+\tfrac{1}{2}p_{ab}\partial_{-}\xi^{ab}+p^{a}\partial_{-}\xi_{a}\}\nonumber \\
& & +\int d\tau d\sigma\{\pi_{+}\partial_{-}\theta^{+}+\tfrac{1}{2}\pi_{ab}\partial_{-}\theta^{ab}+\pi^{a}\partial_{-}\theta_{a}\},\nonumber \\
& = & \int d\tau d\sigma\Big\{ p_{\alpha}\partial_{-}\theta^{\alpha}-\pi^{a}\xi^{+}\partial_{-}\Big(\tfrac{\lambda_{a}}{\lambda^{+}}\Big)-\tfrac{1}{4}\epsilon_{abcde}\pi^{a}\xi^{de}\partial_{-}\Big(\tfrac{\lambda^{bc}}{\lambda^{+}}\Big)\Big\}.\end{aligned}$$ Note that the last two terms can be absorbed by a redefinition of $w_{\alpha}$ and the resulting action depends only on the unconstrained pair $\{p_{\alpha},\theta^{\alpha}\}$.
Furthermore, the BRST current derived from the action is rewritten as well in terms of the pair $\{p_{\alpha},\theta^{\alpha}\}$. The terms $\lambda^{\alpha}p_{\alpha}$ and $(\lambda\gamma^{m}\theta)$ preserve their shape with the gauge choice above and after a simple similarity transformation, the BRST current is given by $$J=\lambda^{\alpha}p_{\alpha}-\tfrac{1}{4\mathcal{T}}(\lambda\gamma_{m}\theta)\Big[P^{m}+\mathcal{T}\partial_{\sigma}X^{m}+\tfrac{1}{4}(\theta\gamma^{m}\partial_{\sigma}\theta)\Big],$$ corresponding to the usual pure spinor BRST current plus a $U(1)$ decoupled sector.
The $U(1)_{R}\times U(1)_{L}$ sector\[sec:decoupledsector\]
===========================================================
This section presents some properties of the $U(1)_{R}\times U(1)_{L}$ sector, which is constituted by the ghosts coming from the scaling symmetry and the fermionic symmetry , but after the field redefinition .
After a Wick-rotation of the worldsheet time $\tau$, their contribution to the action can be written as $$S_{*}=\int d^{2}z\{\bar{\Omega}\bar{\partial}\Omega+\beta\bar{\partial}\gamma+\hat{\bar{\Omega}}\partial\hat{\Omega}+\hat{\beta}\partial\hat{\gamma}\},\label{eq:BU1action}$$ with holomorphic and anti-holomorphic BRST currents given by
\[eq:BU1BRST\] $$\begin{aligned}
J_{*} & = & \Omega\big(\tfrac{1}{2}\partial\ln\gamma-\beta\gamma\big),\\
\hat{J}_{*} & = & \hat{\Omega}\big(\tfrac{1}{2}\bar{\partial}\ln\hat{\gamma}-\hat{\beta}\hat{\gamma}\big).\end{aligned}$$
Note that the constants $c_{\#}$ and $\hat{c}_{\#}$ in were fixed by requiring nilpotency of the BRST charges $Q_{*}\equiv\ointctrclockwise\,J_{*}$ and $\hat{Q}_{*}\equiv\varointclockwise\,\hat{J}_{*}$.
The cohomology of $Q_{*}$ has only two elements, the identity operator $\mathbbm{1}$ and $\Omega$, which is BRST singular, $$\Omega=\lim_{\epsilon\to0}\tfrac{1}{\epsilon}[Q_{*},\gamma^{\epsilon}].$$ Any other BRST-closed operator can be shown to be $Q_{*}$-exact and there is no operator trivializing the cohomology.
Because of the field redefinition , $\beta$ and $\hat{\beta}$ are not conformal primary operators. In fact, the components of the energy-momentum tensor have a non-standard form given by
$$\begin{aligned}
T_{*} & = & -\bar{\Omega}\partial\Omega-\beta\partial\gamma+\tfrac{1}{2\gamma^{2}}[\gamma\partial^{2}\gamma-(\partial\gamma)^{2}],\\
\hat{T}_{*} & = & -\hat{\bar{\Omega}}\bar{\partial}\hat{\Omega}-\hat{\beta}\bar{\partial}\hat{\gamma}+\tfrac{1}{2\hat{\gamma}^{2}}[\hat{\gamma}\bar{\partial}^{2}\hat{\gamma}-(\bar{\partial}\hat{\gamma})^{2}].\end{aligned}$$
Using the fundamental OPE $$\gamma(z)\,\beta(y)\sim\frac{1}{(z-y)},$$ the following results are obtained
$$\begin{aligned}
T_{*}(z)\,\gamma(y) & \sim & \frac{\partial\gamma}{(z-y)},\\
T_{*}(z)\,\beta(y) & \sim & \frac{\gamma^{-1}}{(z-y)^{3}}+\frac{\beta}{(z-y)^{2}}+\frac{\partial\beta}{(z-y)},\\
T_{*}(z)\,\Omega(y) & \sim & \frac{\partial\Omega}{(z-y)},\\
T_{*}(z)\,\bar{\Omega}(y) & \sim & \frac{\bar{\Omega}}{(z-y)^{2}}+\frac{\partial\bar{\Omega}}{(z-y)},\\
T_{*}(z)\,J_{*}(y) & \sim & \frac{J_{*}}{(z-y)^{2}}+\frac{\partial J_{*}}{(z-y)},\\
T_{*}(z)\,T_{*}(y) & \sim & \frac{2T_{*}}{(z-y)^{2}}+\frac{\partial T_{*}}{(z-y)}.\end{aligned}$$
Apart from $\beta$, all the other operators above are primary fields and the central charge of the model vanishes.
As discussed in the main text, the ghost $\gamma$ should not have any zeros on the worldsheet, otherwise the field redefinitions are ill defined. This condition can be enforced by the parametrization
$$\begin{aligned}
\gamma & = & e^{\sigma},\\
\beta & = & \rho e^{-\sigma},\end{aligned}$$
and similarly for the hatted sector, such that the holomorphic part of the action and its associated BRST charge are given by $$\begin{aligned}
S_{*} & = & \int d^{2}z\{\bar{\Omega}\bar{\partial}\Omega+\rho\bar{\partial}\sigma\},\\
Q_{*} & = & \ointctrclockwise\,\Omega\rho.\end{aligned}$$ Using the quartet argument, it is easy to show that the cohomology of $Q_{*}$ contains only one element, the identity operator.
A non-minimal formalism with fundamental $(b,c)$ ghosts\[sec:NMBC\]
===================================================================
Because worldsheet reparametrization is a redundant symmetry of the action , the gauge fixed model does not contain the fundamental $(b,c)$ ghosts which are directly connected to the definition of perturbative string theory as a sum over different worldsheet topologies. In order to implement such structure in the pure spinor formalism, Berkovits proposed the composite $b$ ghost in [@Berkovits:2001us], which was later made super Poincar� covariant with the introduction of the non-minimal formalism in [@Berkovits:2005bt]. In its simplest form [@Oda:2005sd], the pure spinor composite $b$ ghost can be defined classically as $$B\equiv\tfrac{\Lambda_{\alpha}}{4(\Lambda\lambda)}\Big(2\partial X^{m}(\gamma_{m}d)^{\alpha}+(\theta\gamma^{m}\partial\theta)(\gamma_{m}d)^{\alpha}+\tfrac{1}{2}(w\gamma_{mn}\lambda)(\gamma^{mn}\partial\theta)^{\alpha}+(w\lambda)\partial\theta^{\alpha}\Big),$$ and satisfies $\{Q,B\}=T_{\tiny{PS}}$, where $Q$ is the BRST charge and $T_{\tiny{PS}}$ denotes the holomorphic component of the energy-momentum tensor associated to and can be expressed as $$T_{\tiny{PS}}=-\tfrac{1}{2}\partial X^{m}\partial X_{m}-p_{\alpha}\partial\theta^{\alpha}-w_{\alpha}\partial\lambda^{\alpha}.$$
When the master action is extended to include all the gauge and gauge-for-gauge symmetries described in subsection , it is possible to choose a gauge in which the reparametrization ghosts survive in the form of non-minimal variables such that the resulting BRST charge can be expressed as $$Q=e^{-U}\Big(\ointctrclockwise\{\lambda^{\alpha}d_{\alpha}+b\phi\}\Big)e^{U},\label{eq:BerkovitsbcBRST}$$ where $U$ is the generator of a similarity transformation given by $$U=\ointctrclockwise\{-cB+\bar{\phi}c\partial c\}.$$ Here, $b$, $c$, $\bar{\phi}$ and $\phi$ are the (fundamental) Virasoro ghosts and ghost-for-ghosts with vanishing contribution to the central charge of the action (-26+26=0).
The BRST charge has the structure of a generic coupling of the pure spinor superstring to topological two-dimensional gravity, as analyzed in [@Hoogeveen:2007tu], and was already suggested in [@Berkovits:2014aia], but it will be further explored here. First note that it can be rewritten as $$Q=\ointctrclockwise\{\lambda^{\alpha}d_{\alpha}+c(T_{\tiny{PS}}-2\bar{\phi}\partial\phi-\phi\partial\bar{\phi}-b\partial c)+b\phi\},$$ such that the fundamental $b$ ghost satisfies $$\{Q,b\}=T,$$ where $T$ is the energy-momentum tensor of the non-minimal action, $$T=T_{\tiny{PS}}-\bar{\phi}\partial\phi-\partial(\bar{\phi}\phi)-b\partial c-\partial(bc).$$
Therefore, the similarity transformation in helps to expose the familiar construction from gauge fixing worldsheet diffeomorphisms. On the other hand, there does not seem to be any relevant advantage in making such structure manifest.
The whole machinery related to picture changing operators can be immediately built. The bosonic ghosts $\phi$ and $\bar{\phi}$ can be *fermionized* as follows, $$\begin{aligned}
\phi & \cong & e^{\sigma}\eta,\\
\bar{\phi} & \cong & e^{-\sigma}\partial\xi,\end{aligned}$$ such that the BRST cohomology of can be written at picture $-1$ as $$V^{(-1)}=e^{-U}\big(ce^{-\sigma}V\big)e^{U},$$ where $V$ is an element of the pure spinor cohomology, with BRST charge . Integrated vertex operators, $I$, can be built with the action of the fundamental $b$ ghost on the unintegrated vertex operators. In the picture $-1$, they can be built as $$\begin{aligned}
I^{(-1)} & \equiv & \ointctrclockwise\,b\cdot V^{(-1)},\nonumber \\
& = & e^{-U}\Big(\ointctrclockwise\{e^{-\sigma}V+\ldots\}\Big)e^{U}.\end{aligned}$$
Picture changing operators are also defined in a simple way. The picture raising operator is given by $$\begin{aligned}
Y & \equiv & \{Q,\xi\}\nonumber \\
& = & c\partial\xi+(b-B)e^{\sigma},\end{aligned}$$ while the picture lowering is given by $$Z\equiv ce^{-\sigma},$$ such that $$\lim_{z\to y}Z(z)Y(y)=1.$$ Note that because of the manifest spacetime supersymmetry, Neveu-Schwarz and Ramond states are treated on equal footing and there are no half-integer pictures.
At tree level, the integration measure for the reparametrization ghosts and ghost-for-ghosts is simply $$\left\langle c\partial c\partial^{2}ce^{-3\sigma}\right\rangle =1.$$ $N$-point amplitudes can then be computed with $3$ vertices with picture $-1$ to saturate the background charge ($\sigma$) and $(N-3)$ vertices with picture $0$.
In order to have a fully covariant formulation of this model, the composite $B$ ghost introduced by Berkovits in [@Berkovits:2005bt] can be used. Observe that for higher genus, $B$ will enter through the picture changing operator, so it agrees with the usual pure spinor prescription in which the $B$ ghost helps to saturate the number of fermionic fields ($d_{\alpha}$).
The sectorized and ambitwistor pure spinor superstrings\[sec:sectorized\]
=========================================================================
The gauge relies on the connection between the twistor-like constraints and the generators of worldsheet reparametrization in the first order formalism, represented through the identifications displayed in . As mentioned earlier, the gauge $L^{\alpha}=\hat{L}^{\hat{\alpha}}=0$ would imply a degenerate worldsheet metric. To see this, consider a different gauge fixing fermion of the form $$\Xi=\int d\tau d\sigma\{\bar{\Omega}A_{\tau}+\beta\chi_{\tau}-r^{\alpha}w_{\alpha}^{\sigma}+\eta^{\alpha}p_{\alpha}^{\sigma}-\pi_{\alpha}L^{\alpha}+\hat{\bar{\Omega}}\hat{A}_{\tau}-\hat{r}^{\hat{\alpha}}\hat{w}_{\hat{\alpha}}^{\sigma}+\hat{\beta}\hat{\chi}_{\tau}+\hat{\eta}^{\hat{\alpha}}\hat{p}_{\hat{\alpha}}^{\sigma}-\hat{\pi}_{\hat{\alpha}}\hat{L}^{\hat{\alpha}}\},$$ thus implementing the singular gauge. The gauge fixed action can then be shown to be $$\begin{aligned}
\tilde{S} & = & \int d\tau d\sigma\{P_{m}\partial_{\tau}X^{m}+w_{\alpha}\partial_{\tau}\lambda^{\alpha}+p_{\alpha}\partial_{\tau}\xi^{\alpha}+\pi_{\alpha}\partial_{\tau}\theta^{\alpha}+r^{\alpha}\partial_{\tau}s_{\alpha}+\eta^{\alpha}\partial_{\tau}\epsilon_{\alpha}+A_{\sigma}\partial_{\tau}B\}\nonumber \\
& & +\int d\tau d\sigma\{\hat{w}_{\hat{\alpha}}\partial_{\tau}\hat{\lambda}^{\hat{\alpha}}+\hat{p}_{\hat{\alpha}}\partial_{\tau}\hat{\xi}^{\hat{\alpha}}+\hat{\pi}_{\hat{\alpha}}\partial_{\tau}\hat{\theta}^{\hat{\alpha}}+\hat{r}^{\hat{\alpha}}\partial_{\tau}\hat{s}_{\hat{\alpha}}+\hat{\eta}^{\hat{\alpha}}\partial_{\tau}\hat{\epsilon}_{\hat{\alpha}}+\hat{A}_{\sigma}\partial_{\tau}\hat{B}\}\nonumber \\
& & +\int d\tau d\sigma\{\Sigma\partial_{\tau}\chi_{\sigma}+\bar{\Omega}\partial_{\tau}\Omega+\beta\partial_{\tau}\gamma+\hat{\Sigma}\partial_{\tau}\hat{\chi}_{\sigma}+\hat{\bar{\Omega}}\partial_{\tau}\hat{\Omega}+\hat{\beta}\partial_{\tau}\hat{\gamma}\}.\end{aligned}$$
The problem with this action is a residual gauge symmetry (aside from the pure spinor symmetries) of the form $\delta\tau\to f(\tau)$, a reparametrization of the worldsheet time coordinate. Therefore, the singular gauge proposed in [@Berkovits:2015yra] does not completely fix the gauge symmetries of the action .
There is, however, another interesting singular gauge. Consider now the gauge fixing fermion $$\begin{gathered}
\Xi=\int d\tau d\sigma\{\bar{\Omega}A_{\tau}+\beta\chi_{\tau}-r^{\alpha}w_{\alpha}^{\sigma}+\eta^{\alpha}p_{\alpha}^{\sigma}+\hat{\bar{\Omega}}\hat{A}_{\tau}-\hat{r}^{\hat{\alpha}}\hat{w}_{\hat{\alpha}}^{\sigma}+\hat{\beta}\hat{\chi}_{\tau}+\hat{\eta}^{\hat{\alpha}}\hat{p}_{\hat{\alpha}}^{\sigma}\}\\
-\int d\tau d\sigma\{\pi_{\alpha}[L^{\alpha}-\tfrac{(\gamma^{m}\Lambda)^{\alpha}}{(\Lambda\lambda)}(P_{m}+\mathcal{T}\partial_{\sigma}X_{m})]+\hat{\pi}_{\hat{\alpha}}[\hat{L}^{\hat{\alpha}}+\tfrac{(\gamma^{m}\hat{\Lambda})^{\hat{\alpha}}}{(\hat{\Lambda}\hat{\lambda})}(P_{m}-\mathcal{T}\partial_{\sigma}X_{m})]\},\end{gathered}$$ which differs from by a sign in the last term. In an analogy with the first order Polyakov action , this gauge can be interpreted as $e_{+}=-e_{-}=1$, *cf*. [^3]. In terms of the worldsheet metric, , this gauge is equivalent to $g_{\tau\tau}=0$. The gauge fixed action, after the addition of a BRST expression, similarly to what is described in subsection , can be written as $$\begin{aligned}
\tilde{S} & = & \int d\tau d\sigma\{P_{m}\partial_{-}X^{m}+w_{\alpha}\partial_{-}\lambda^{\alpha}+p_{\alpha}\partial_{-}\xi^{\alpha}+A\partial_{-}B+\hat{w}_{\hat{\alpha}}\partial_{-}\hat{\lambda}^{\hat{\alpha}}+\hat{p}_{\hat{\alpha}}\partial_{-}\hat{\xi}^{\hat{\alpha}}+\hat{A}\partial_{-}\hat{B}\}\nonumber \\
& & +\int d\tau d\sigma\{\pi_{\alpha}\partial_{-}\theta^{\alpha}+r^{\alpha}\partial_{-}s_{\alpha}+\eta^{\alpha}\partial_{-}\epsilon_{\alpha}+\hat{\pi}_{\hat{\alpha}}\partial_{-}\hat{\theta}^{\hat{\alpha}}+\hat{r}^{\hat{\alpha}}\partial_{-}\hat{s}_{\hat{\alpha}}+\hat{\eta}^{\hat{\alpha}}\partial_{-}\hat{\epsilon}_{\hat{\alpha}}\}\nonumber \\
& & +\int d\tau d\sigma\{\Sigma\partial_{-}\chi+\bar{\Omega}\partial_{-}\Omega+\beta\partial_{-}\gamma+\hat{\Sigma}\partial_{-}\hat{\chi}+\hat{\bar{\Omega}}\partial_{-}\hat{\Omega}+\hat{\beta}\partial_{-}\hat{\gamma}\},\end{aligned}$$ and all the worldsheet fields satisfy the equation of motion $\partial_{-}=0$, constituting a chiral model. Furthermore, following the procedure described in subsection , the resulting action is given by $$S=\int d\tau d\sigma\{P_{m}\partial_{-}X^{m}+w_{\alpha}\partial_{-}\lambda^{\alpha}+p_{\alpha}\partial_{-}\theta^{\alpha}+\hat{w}_{\hat{\alpha}}\partial_{-}\hat{\lambda}^{\hat{\alpha}}+\hat{p}_{\hat{\alpha}}\partial_{-}\hat{\theta}^{\hat{\alpha}}\},\label{eq:Sectorizedaction}$$ which corresponds to the sectorized string introduced in [@Jusinskas:2016qjd]. The two chiral components of the BRST current can be cast as
\[eq:SectorizedBRST\] $$\begin{aligned}
J & = & \lambda^{\alpha}p_{\alpha}-\tfrac{1}{4\mathcal{T}}(\lambda\gamma_{m}\theta)(P^{m}+\tfrac{\mathcal{T}}{2}\partial_{+}X^{m})+\tfrac{1}{32\mathcal{T}}(\lambda\gamma_{m}\theta)(\theta\gamma^{m}\partial_{+}\theta),\\
\hat{J} & = & \hat{\lambda}^{\hat{\alpha}}\hat{p}_{\hat{\alpha}}-\tfrac{1}{4\mathcal{T}}(\hat{\lambda}\gamma_{m}\hat{\theta})(P^{m}-\tfrac{\mathcal{T}}{2}\partial_{+}X^{m})-\tfrac{1}{32\mathcal{T}}(\hat{\lambda}\gamma_{m}\hat{\theta})(\hat{\theta}\gamma^{m}\partial_{+}\hat{\theta}).\end{aligned}$$
As a consequence of the singular gauge choice, the string tension disappears from the action , although it is still present in the BRST current. Note also that the spacetime spinors can be made dimensionless via a scale transformation of the form $$\begin{array}{rclcrcl}
\lambda^{\alpha} & \to & \mathcal{T}^{1/2}\lambda^{\alpha}, & & \theta^{\alpha} & \to & \mathcal{T}^{1/2}\theta^{\alpha},\\
w_{\alpha} & \to & \mathcal{T}^{-1/2}w_{\alpha}, & & p_{\alpha} & \to & \mathcal{T}^{-1/2}p_{\alpha},
\end{array}$$ and similarly for the hatted fields.
Now, the model has a well defined tensionless limit and the BRST currents are given by
$$\begin{aligned}
J & = & \lambda^{\alpha}p_{\alpha}-\tfrac{1}{4}(\lambda\gamma^{m}\theta)P_{m},\\
\hat{J} & = & \hat{\lambda}^{\hat{\alpha}}\hat{p}_{\hat{\alpha}}-\tfrac{1}{4}(\hat{\lambda}\gamma^{m}\hat{\theta})P_{m},\end{aligned}$$
which correspond to the ambitwistor pure spinor superstring [@Berkovits:2013xba], in agreement with the results of [@Azevedo:2017yjy].
The heterotic sectorized or ambitwistor strings in the pure spinor formalism are obtained in a similar way. Worldsheet reparametrization and only one of the twistor-like constraints has to be taken into account in building the master action . Apart from that, the gauge fixing procedure should be very similar.
[99]{} N. Berkovits, “Super Poincare covariant quantization of the superstring,” JHEP [**0004**]{}, 018 (2000) doi:10.1088/1126-6708/2000/04/018 \[hep-th/0001035\].
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[^1]: [email protected]
[^2]: Twistor-like variables arise naturally using superembedding techniques and the first appearance of pure spinors in this context was in [@Tonin:1991ii].
[^3]: I would like to thank Thales Azevedo for observing this is the gauge choice leading to the bosonic sectorized model.
|
---
abstract: 'The paper deals with Hilbert space valued fields over any locally compact Abelian group $G$, in particular over $G={\mathbb{Z}}^n\times {\mathbb{R}}^m$, which are periodically correlated (PC) with respect to a closed subgroup of $G$. PC fields can be regarded as multi-parameter extensions of PC processes. We study structure, covariance function, and an analogue of the spectrum for such fields. As an example a weakly PC field over ${\mathbb{Z}}^2$ is thoroughly examined.'
author:
- |
Dominique Dehay\
[*Institut de Recherche Mathématique de Rennes, CNRS umr 6625,* ]{}\
[*Université Rennes 2, cs 24307, 35043 Rennes, FRANCE; [email protected]*]{}\
Harry Hurd\
[*Department of Statistics, University of North Carolina,*]{}\
[*Chapel Hill, NC 27599-2630, USA; [email protected]* ]{}\
Andrzej Makagon [^1]\
[*Department of Mathematics, Hampton University*]{}\
[*Hampton, VA 26668, USA; [email protected]*]{}
title: Spectrum of Periodically Correlated Fields
---
Introduction
============
Periodically correlated (PC) processes and sequences have been studied for almost half of the century and at present they are very well understood mainly due to works of Gladyshev [@gladyshev61; @gladyshev63], Hurd [@hurd69; @hurd74; @hurd74b; @hurd89b; @hurdk91] and other authors [@dehay94; @honda82; @makms94; @mak99; @mak00; @miamee90; @miamees79]. A summary of the theory of PC sequences can be found in [@HM]. Surprisingly, there are only several works [@Bose98; @Bose01; @Chen96; @dehayhurd; @DragJav82; @Gardner06; @gaspar04; @hurdkf; @serp05] dealing with PC fields, and each one concentrates on a particular type, namely coordinate-wise strong periodicity. An intention of this paper is to sketch a unified theory of fields over any locally compact Abelian (LCA) group $G$ which are periodically correlated with respect to an arbitrary closed subgroup $K$ of $G$. We emphasize the case of $G={\mathbb{Z}}^n\times{\mathbb{R}}^m$ to illustrate the results. This work includes stationary fields as well the weakly periodically correlated fields, that is the fields whose covariance function exhibits periodicity (or stationarity) in fewer directions than the dimension of the group. In the latter case we assume a certain integrability condition (see Definition \[def:square-integr\]) in order to develop some simple spectral analysis of those fields. A work in progress treats the case where this condition is not satisfied.
The paper is organized as follows. In the remaining part of this section we introduce notation and vocabulary used in the paper, review needed facts from harmonic analysis on LCA groups, and outline the theory of one-parameter PC processes. In the next three sections we study the covariance function, the notion of the spectrum, and the structure of a $K$-periodically correlated field. These sections include the main results of the paper (Theorems \[thm-Corr\], \[thm-spectrum\] and \[PCF-structure\]). The last section contains examples that illustrate the theory developed. In particular Example \[ex-wk\] gives a complete analysis of the weakly periodically correlated fields over ${\mathbb{Z}}^2$, introduced in [@hurdkf].
Background {#background .unnumbered}
----------
To avoid confusion and to set the notations of the paper we recall some features of group theory, Haar measures, Fourier transform, and periodic functions. For more information on these subjects the authors refer to [@HR; @reiter; @rudin].
1.. Let $G$ be an additive locally compact Abelian (LCA) group, ${\widehat}{G}$ be its dual (group of continuous characters), and let ${\left\langle \chi,t \right\rangle}$ denote the value of a character $\chi \in {\widehat}{G}$ at $t\in G$. The dual ${\widehat}{G}$ can be given a topology that makes it an LCA group such that ${\widehat}{({\widehat}{G})} = G$. Let $K$ be a closed subgroup of $G$. The symbol $G/K$ will stand for the quotient group and ${\widehat}{(G/K)}$ for its dual. Let $\imath$ denote the natural homomorphism of $G$ onto $G/K$, $\imath(t) {:=}t+K$, and $\imath^*$ be its dual map $\imath^*:{\widehat}{G/K}\to{\widehat}{G}$, defined as ${\left\langle \imath^*(\eta),t \right\rangle} = {\left\langle \eta , (t+K) \right\rangle}$ for $\eta\in{\widehat}{G/K}$ and $t\in G$. The mapping $\imath^*$ is injective and continuous, and for each $\eta \in {\widehat}{G/K}$, ${\left\langle \imath^*(\eta),\cdot \right\rangle}$ is a $K$-periodic function on $G$ (see below). Consequently ${\widehat}{G/K}$ can be identified with a closed subgroup $\Lambda_K$ of ${\widehat}{G}$ consisting of the elements $\lambda \in
{\widehat}{G}$ such that ${\left\langle \lambda,t \right\rangle} =1$ for any $t\in K$. In the sequel we use the notation ${\left\langle \lambda,t \right\rangle}{:=}{\left\langle \lambda,\imath(t) \right\rangle}$, for all $\lambda\in\Lambda_K$ and $t\in G$. By ${{\mathcal{B}}}(G)$ we denote the $\sigma$-algebra of Borel sets on $G$. A *cross-section* $\xi$ for $G/K$ is a mapping $\xi: G/K\to G$ such that
-- ------- ---------------------------------------------------------------------------------------------------
(i) $\xi$ is Borel,
(ii) $\xi(G/K)$ is a measurable subset of $G$,
(iii) $\xi(0) = 0$ and $\xi\circ\imath(t) \in t+K$ for all $t\in G$, where $t+K{:=}\{t+k\!:\!k\in K\}$.
-- ------- ---------------------------------------------------------------------------------------------------
For existence and other properties of a cross-section please see [@EK:84; @varadarajan]. For each cross-section $\xi$ for $G/K$, the sets $k+\xi(G/K)$, $k\in K$, are disjoint and their union is $G$, and hence each element $t\in G$ has a unique representation $t = k(t)
+ \xi(\imath(t))$, where $k(t)\in K$. Note that the function $\xi$ is not additive, that is $\xi(x+y)$ may be different than $\xi(x)+
\xi(y)$, $x,y \in G/K$.
Any LCA group has a nonnegative translation-invariant measure, unique up to a multiplicative constant, called a Haar measure. The Haar measures on $G$ and ${\widehat}{G}$ can be normalized in such a way that the following implication holds
$$\begin{gathered}
\mbox{if}\quad f\in L^1(G),\quad{\widehat}{f}(\chi) {:=}\int_{{G}} {\left\langle \chi,t \right\rangle} f(t)\,{{\hbar}}_G (dt)\quad \mbox{for}\,\,\chi \in {\widehat}{G},\quad\mbox{and}\quad {\widehat}{f}\in L^1({\widehat}{G}) \quad\\
\mbox{then} \quad f(t) = \int_{{\widehat}{G}} \overline{{\left\langle \chi,t \right\rangle}} {\widehat}{f}(\chi)
\,{{\hbar}}_{{\widehat}{G}} (d\chi) \quad \mbox{for a.e.}\,\, t\in {G}. $$
The function ${\widehat}{f}$ above is called the *Fourier transform* of $f$. Here and what follows $L^1(G)$ stands for the space of complex functions on $G$ which are integrable with respect to ${{\hbar}}_G$, and ${{\hbar}}_G$ denotes the normalized Haar measure on the group indicated in the subscript. Note that the normalization of the Haar measures of $ G$ and ${\widehat}{G}$ is not unique. We follow the usual convention that if $G$ is compact and infinite then the normalization is such that ${{\hbar}}_G(G) = 1$; if $G$ is discrete and infinite then the normalized Haar measure of any single point is 1; if $G$ is both compact and finite then its dual is also and the Haar measure on $G$ is normalized to have a mass 1 while the Haar measure on ${\widehat}{G}$ is counting measure. The normalized Haar measure on ${\mathbb{R}}$ is the Lebesgue measure divided by $\sqrt{2\pi}$. Finally, if $K$ is a closed subgroup of $G$ then the normalized Haar measures satisfy Weil’s formula $$\label{eq:Weil}
\int_{G/K}\left(\int_K f(k+s)\,{{\hbar}}_K(dk)\right){{\hbar}}_{G/K}(d\dot{s})=\int_G f(t)\,{{\hbar}}_G(dt), \quad f\in L^1(G).$$ The inner integral above depends only on the coset $\dot{s}{:=}s+K$. See e.g. [@reiter SectionIII.3.3].
If $f\in L^1(G)$ then ${\widehat}{f}$ is a continuous bounded function on ${\widehat}{G}$ but not necessarily integrable. The Fourier transform, which is customarily denoted by the integral ${\widehat}{f}(\chi) = \int_{{G}} {\left\langle \chi,t \right\rangle} f(t)\,{{\hbar}}_G (dt)$ (even if $f$ is not integrable) extends from $L^1(G)\cap L^2(G)$ to an isometry from $L^2(G)$ onto $L^2({\widehat}{G})$ (Plancherel theorem [@rudin]). If there is a danger of confusion we will recognize the difference by writing $${\widehat}{f}(\chi) \stackrel{L^2}{=} \int_{{G}} {\left\langle \chi,t \right\rangle} f(t)\,{{\hbar}}_G (dt),\quad f\in L^2(G).$$
In the sequel we say that *the inverse formula holds for $f$* if the function $f$ is the inverse Fourier transform of ${\widehat}{f}$. If both $f$ and ${\widehat}{f}$ are integrable then clearly the inverse formula holds for both. Also if $G$ is discrete and $f\in L^2(G)$, then the inverse formula holds for $f$. Indeed, in this case ${\widehat}{f} \in L^1({\widehat}{G})$ because ${\widehat}{G}$ is compact, and hence $f(t) = \int_{{\widehat}{G}} \overline{{\left\langle \chi,t \right\rangle}} {\widehat}{f}(\chi)\,
{{\hbar}}_{{\widehat}{G}} (d\chi)$ for all $t\in {G}$. For a separable Hilbert space ${{\cal H}}$ with inner product $(\cdot,\cdot )_{{\cal H}}$ and norm $\|\cdot\|_{{\cal H}}$, let $L^p(G;{{\cal H}}){:=}L^p(G,{{\hbar}}_G;{{\cal H}})$, $p=1$ or $2$, be the space of ${{\cal H}}$-valued fields on $G$ which are $p$-integrable with respect to Haar measure ${{\hbar}}_G$, that is, $f\in L^p(G;{{\cal H}})$ means that $f:G\to{{\cal H}}$ is ${{\hbar}}_G$-measurable and the real-valued function $t\mapsto \|f(t)\|_{{\cal H}}^p$ is integrable with respect to ${{\hbar}}_G$. It is well known that the space $L^1(G;{{\cal H}})$ is a Banach space with the norm $\|f\|_{L^1}{:=}\int_G\|f(t)\|_{{{\cal H}}}\,{{\hbar}}_G(dt),\, f\in L^1(G;{{\cal H}})$, and the space $L^2(G;{{\cal H}})$ is a separable Hilbert space with the inner product $\big(f,g\big)_{L^2}{:=}\int_G \big(f(t),g(t)\big)_{{{\cal H}}}\,{{\hbar}}_G(dt),\, f,g\in L^2(G;{{\cal H}}).$ See e.g. [@DS Chapter III] (see also [@Dinc2000; @Hille48; @RudinFA]). Whenever $f\in L^1(G;{{\cal H}})$ then $f$ is Bochner integrable (also called strongly integrable) and its Fourier transform exits. Futhermore Plancherel theorem applies and defines an isometry from $L^2(G;{{\cal H}})$ onto $L^2({\widehat}{G};{{\cal H}})$ (one-to-one), so ${\widehat}{f}\in L^2({\widehat}{G};{{\cal H}})$ is also well defined for $f\in L^2(G;{{\cal H}})$.
2.. Given $G$ and a closed subgroup $K$ of $G$, it is natural to call a function $f$ defined on $G$ to be *$K$-periodic* if $$f(t+k) = f(t)\quad\mbox{for all}\,\,t\in G\,\,\mbox{and}\,\,k\in K.$$ In this case, the function $f$ is constant on cosets of $K$. Hence a function $f$ on $G$ is $K$-periodic if and only if $f$ is of the form $f = f_K\circ \imath $, where $f_K$ is a function on $G/K$. The concrete realization $\Lambda_K{:=}\imath^*({\widehat}{G/K})\subset{\widehat}{G}$ of ${\widehat}{G/K}$ as a subgroup of ${\widehat}{G}$ will be in the sequel called *the domain of the spectrum of $f$*. Note that $\Lambda_K$ is not determined uniquely by $f$, for a $K$-periodic function can be at the same time periodic with respect to a larger subgroup $K'\supset K$; in other words we will not be assuming that $K$ is the “smallest” period of $f$.
If $f\in L^1(G/K;{{\cal H}})$, ${{\cal H}}$ being the set of complex numbers ${\mathbb{C}}$ or any separable Hilbert space, we consider the Fourier transform of $f_K$ at $\lambda \in
\Lambda_K$ $$\label{f_K-lambda}
{\widehat}{f_K}(\lambda) {:=}\int_{{G/K}} {\left\langle \lambda , x \right\rangle}
f_K(x)\, {{\hbar}}_{G/K} (dx)$$ that will be referred to as the *spectral coefficient of $f$ at frequency* $\lambda \in\Lambda_K$.
A couple of remarks regarding the above definition and its relation to the standard notions of the spectrum and its domain are certainly due here. The word [*spectrum*]{} comes originally from physics, operator theory, and more recently from signal processing. It is widely used in the theory of second order stochastic processes. Intuitively, the spectrum of a scalar function $f$ is a Fourier transform of $f$ in whatever sense it exists. If $f$ is a locally integrable function on $G={\mathbb{Z}}^n\times{\mathbb{R}}^m$ then the spectrum $F$ of $f$ is a Schwartz distribution on ${\widehat}{G}$, which is a functional on a certain space of functions on ${\widehat}{G}$ determined by the relation ${}F\big({\widehat}{\phi}\big) = \int_G {\phi}(t) f(t)\, {{\hbar}}_{G} (dt)$, where $\phi$ runs over the set of compactly supported functions on $G$ which are infinitely many times differentiable in last $m$ variables. One can show that if $f$ is additionally $K$-periodic, then the support of $F$ (as defined in [@RudinFA]) is a subset of $\Lambda_K$. This rationalizes the name ”*domain of the spectrum*” that we have assigned for $\Lambda_K$, as well the phrase ”*the spectrum sits on* $\Lambda_K$” which we will use sometimes. The first task in understanding the spectrum of a $K$-PC field is thus to identify the domain of its spectrum or its second order spectrum. (See below).
The coefficient ${\widehat}{f_K}(\lambda)$ defined in (\[f\_K-lambda\]) represents an “amplitude” of the harmonic ${\left\langle \lambda, \cdot \right\rangle}$ in a spectral decomposition of $f$. Indeed, if $f_K$ and ${\widehat}{f_K}$ are integrable, then ${}f_K(x) = \int_{\Lambda_K} \overline{{\left\langle \lambda,x \right\rangle}} \, {\widehat}{f_K}(\lambda)
\,{{\hbar}}_{\Lambda_K} (d\lambda)$, $x\in G/K$, and as a consequence of Weil’s formula (\[eq:Weil\]) and the fact that $ {\left\langle \lambda , t \right\rangle}={\left\langle \lambda , \imath(t) \right\rangle} $, $t\in G$, $\lambda \in \Lambda_K$, we conclude that $$\label{FTofPF}
f(t) = \int_{\Lambda_K}
\overline{{\left\langle \lambda, t \right\rangle}} \, {\widehat}{f_K}(\lambda)\, {{\hbar}}_{\Lambda_K}
(d\lambda)
, \quad t\in G.$$ If ${\widehat}{f_K}$ is not integrable, then equality (\[FTofPF\]) holds only for ${{\hbar}}_G$-almost every $t\in G$ or is not valid as stated, but ${a}_{\lambda}$ still retains its interpretation.
For illustration suppose that $f$ is a continuous scalar function on ${\mathbb{R}}$ which is periodic with period $T>0$, that is such that $f(t) = f(t+T)$ for every $t\in {\mathbb{R}}$. In this case $G=
{\mathbb{R}}$, $K=\{kT\!:\! k\in {\mathbb{Z}}\}$, the quotient group $G/K$ can be identified with $[0,T)$ with addition *modulo $T$*, the mapping $\imath$ is defined as $\imath(t) = \big[t\big]_{T}$, the remainder in integer division of $t$ by $T$, and the identity $
\xi(x) = x$, $x\in[0,T)$, is the most natural cross-section for $G/K$. The function $f_K$ is defined as $f_K(x) = f(\xi(x)) = f(x)$, $x\in [0,T)$. The dual of $G/K$ is identified with the subgroup $\Lambda_K=\{2\pi j/T\!:\! j\in {\mathbb{Z}}\}$ of ${\mathbb{R}}$, and with this identification ${\left\langle \lambda , \imath(t) \right\rangle} = {\left\langle \lambda , t \right\rangle} = e^{-i\lambda t}$, $\lambda
\in \Lambda_K$, $t \in {\mathbb{R}}$. The normalized Haar measures on $[0,T)$ and $\Lambda_K$ are the Lebesgue measure divided by $T$ and the counting measure, respectively. The domain of the spectrum of $f$ is therefore the set $\Lambda_K$. The spectral coefficient $\hat{f}_j$ of $f$ at $\lambda = 2\pi j/T$, is given by $${}\hat{f}_{j}=\frac{1}{T}\int_0^T e^{-i2\pi jt/T}f(t)\,dt,\quad j\in{\mathbb{Z}}.$$ Note that the sequence $\{\hat{f}_j\}$ is square-summable and consequently ${}f(t) = \sum_{j= -\infty}^{\infty} e^{i2\pi j t/T} \hat{f}_{j}$, where the series above converges in $L^2[0,T]$, so in $L^2([-A,A])$ for every $0<A<\infty$. The spectrum $F$ of $f$ is defined by the relation ${}F\big({\widehat}{\phi}\big) = \frac{1}{\sqrt{2\pi}}
\int_{-\infty}^{\infty}{\phi}(t) f(t)\,dt
$. If $\phi$ is an infinitely times differentiable with compact support then $$F\big({\widehat}{\phi}\big) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} {\phi}(t) f(t)\,dt
= \int_{{\mathbb{R}}} {\widehat}{\phi}(t) F(dt),$$ where ${}F = \sum_{j= -\infty}^{\infty} \hat{f}_j\, \delta_{\{2\pi j/T\}}$, and $\delta_a$ denotes the measure of mass 1 concentrated at $\{a\}$. The spectrum of $f$ can be therefore identified with a $\sigma$-additive complex measure $F$ on ${\mathbb{R}}$ sitting on $\Lambda_K$ and defined by ${}F = \sum_{j= -\infty}^{\infty} \hat{f}_j\, \delta_{\{2\pi j/T\}}$. If the sequence $\{\hat{f}_j\}$ is summable, then $F$ is a finite measure, but it does not have to be in general.
Periodically Correlated Fields
==============================
Let ${{\cal H}}$ be a separable Hilbert space with the inner product $(\cdot,\cdot )_{{\cal H}}$. In a probabilistic context the space ${{\cal H}}$ represents the space of zero-mean complex random variables with finite variance. A *(stochastic) field* $X=\{X(t)\!:\!t\in G\}$ is a measurable function $X: G\to {{\cal H}}$. Let ${{\cal H}}_X {:=}{\overline{\mbox{span}}\left\{ X(t)\!:\! t\in G \right\} }$ be the smallest closed linear subspace of ${{\cal H}}$ that contains all $X(t)$, $t\in G$. The function ${{\mathbf K}}_X(t,s) {:=}\big(X(t), X(s)\big)_{{{\cal H}}}$, $t,s\in G$, is referred to as the *covariance function* of the field $X$. A field $X$ is called *stationary* if it is continuous and for all $t,s\in G$, the function ${{\mathbf K}}_X(t+u,s+u)$ does not depend on $u\in G$. If $X$ is stationary then ${{\mathbf K}}_X(t+s,s)={{\mathbf K}}_X(t+0,0) =:{{\mathbf R}}_X(t)$, $t,s \in G$, and ${{\mathbf R}}_X$ has the form $${{\mathbf R}}_X(t) = \int_{{\widehat}{G}}\overline{{\left\langle \chi,t \right\rangle}}\,\Gamma(d\chi),$$ where $\Gamma$ is a non-negative Borel measure on ${\widehat}{G}$ (Bochner Theorem [@reiter SectionIV.4.4]). A field $X$ is called *harmonizable* if there is a (complex) measure $\digamma$ on ${\widehat}{G}\times{\widehat}{G}$ such that $$\label{harm}
{{\mathbf K}}_X(t,s) =
\int\!\!\!\int_{{\widehat}{G}^2} \overline{{\left\langle \chi,t \right\rangle}} {\left\langle \beta,s \right\rangle}
\,\digamma (d\chi, d\beta),\quad t,s\in{G},$$ see [@rao; @rao2005]. The measure $\digamma$ above is called the *second order spectral (SO-spectral) measure* of the harmonizable field $X$. Note that every stationary field is harmonizable with the measure $\digamma$ sitting on the diagonal: $\digamma (\Delta) =\Gamma\{\chi\in {\widehat}{G}\!:\!(\chi,\chi) \in \Delta\}$, $\Delta
\in {{\mathcal{B}}}({\widehat}{G}\times{\widehat}{G})$. The SO-spectrum of a harmonizable field $X$ is the spectral measure $\digamma$ associated with function ${{\mathbf K}}_X(t, -s)$, $ s,t \in G$ via relation (\[harm\]). Here we adopt the terminology of ”*second order spectrum (SO-spectrum)*” of the field $X$, instead of the usual term ”spectrum”, in order to avoid confusion with the spectrum of a periodic function (or field). By this way we point at the fact that we are considering not the field $X$ by itself but its covariance function ${{\mathbf K}}_X$. This leads to the following definition.
\[def:spectrum of X\] Let $X$ be a continuous stochastic field over $G
$. The *second order spectrum (SO-spectrum)* of the field $X$ is the spectrum (Fourier transform, in whatever sense it may exist) of the function $ G\times G \ni (t,s) \mapsto {{\mathbf K}}_X(t, -s). $ The domain of the SO-spectrum of the field $X$ is defined as the domain of the spectrum of this function.
Let $K$ be a closed subgroup of $G$. A field $X$ is called *$K$-periodically correlated ($K$-PC)* if $X$ is continuous and the function $G\ni u \mapsto {{\mathbf K}}_X(t+u,s+u)$ is $K$-periodic in $u$ for all $t,s\in G$. The group $K$ will be called the period of the PC process $X$.
If $G={\mathbb{R}}$ (or ${\mathbb{Z}}$) and $K=\{kT\!:\! k\in{\mathbb{Z}}\}$ then we will use the phrase ”*PC process (or PC sequence) with period $T>0$*”, rather than $K$-PC field. Note that every stationary field over $G$ is $K$-PC field with $K=G$. A $K$-PC field is labeled [*strongly PC*]{} if $G/K$ is compact, and [*weakly PC*]{} otherwise.
For example if $X$ is a field over ${\mathbb{R}}^2$ (or ${\mathbb{Z}}^2$) such that for every ${{\mathbf{s}}},{{\mathbf{t}}}\in {\mathbb{R}}^2$ (or ${\mathbb{Z}}^2$), ${{\mathbf K}}_X({{\mathbf{s}}},{{\mathbf{t}}}) = {{\mathbf K}}_X\big({{\mathbf{s}}}+(T_1,0),{{\mathbf{t}}}+(T_1,0)\big)= {{\mathbf K}}_X\big({{\mathbf{s}}}+(0,T_2),{{\mathbf{t}}}+(0,T_2)\big)$, $0<T_1, T_2 < \infty$, then $X$ is strongly $K$-PC with $ K = \{(k_1 T_1,k_2,T_2)\!:\! k_1,k_2 \in {\mathbb{Z}}\}$. Since ${{\mathbf K}}_X$ is invariant under shifts from $K$ this leads to the existence of unitary operators $U_1$, $U_2$ in ${{\cal H}}_X$ such that $U_1 X({{\mathbf{t}}}) = X(t_1+T_1,t_2)$ and $U_2 X({{\mathbf{t}}}) = X(t_1,t_2+T_2)$ for every ${{\mathbf{t}}}=(t_1,t_2)$. If the field $X$ instead satisfies ${{\mathbf K}}_X({{\mathbf{s}}},{{\mathbf{t}}}) = {{\mathbf K}}_X\big({{\mathbf{s}}}+(T_1,T_2),{{\mathbf{t}}}+(T_1,T_2)\big)$, then $X$ is weakly $K$-PC with $ K = \{k(T_1,T_2)\!:\! k \in {\mathbb{Z}}\}$. This leads to a unitary operator $U$ such that $U X({{\mathbf{t}}}) = X\big({{\mathbf{t}}}+ (T_1,T_2)\big)$, ${{\mathbf{t}}}=(t_1,t_2)$.
Examples of PC fields on ${\mathbb{Z}}^2$ can be constructed by a periodic amplitude or time deformation of a stationary field. Suppose $X({{\mathbf{t}}})=f({{\mathbf{t}}}) Y({{\mathbf{t}}})$, ${{\mathbf{t}}}=(t_1,t_2)\in {\mathbb{Z}}^2$, where $Y$ is a stationary field and $f$ is a non-random periodic function such that $f(t_1,t_2)=f(t_1+T_1,t_2)=f(t_1,t_2+T_2)$. Then the field $X$ is strongly $K$-PC with $ K = \{(k_1 T_1,k_2T_2)\!:\! k_1,k_2 \in {\mathbb{Z}}\}$. If $f$ above instead satisfies $f(t_1,t_2)=f(t_1+T_1,t_2+T_2)$, then $X$ is weakly $K$-PC with $K = \{k(T_1,T_2)\!:\! k \in {\mathbb{Z}}\}$. If the function $f$ is two-dimensional integer valued, then the field $X$ defined by $X({{\mathbf{t}}})=Y\big({{\mathbf{t}}}+f({{\mathbf{t}}})\big)$ will be weakly PC.
Remark that generally ${{\mathbf K}}_X(t+u,s+u) = {{\mathbf K}}_X(t-s+s+u,s+u)$, so a continuous field $X$ is $K$-PC if and only if ${{\mathbf K}}_X(t+u,u)$ is a $K$-periodic function of $u$ for every $t\in G$. If $X$ is $K$-PC field then for all $t,s\in G$ there is a unique function $x\mapsto b_X(t,s;x)$ on $G/K$ such that $${{\mathbf K}}_X(t+u,s+u)= b_X(t,s; \imath(u)),\quad t,s,u\in G.$$ The canonical map $\imath:G\to G/K$ is continuous and open [@reiter SectionIII.1.6], so the function $x\mapsto b_X(s,t;x)$ is continuous. Denote $${{\mathbf B}}_X(t;x) {:=}b_X(t,0;x),\qquad t\in G,\, x\in G/K.$$ Note that $b_X(t,s;\imath(u))=b_X\big(t-s,0;\imath(s+u)\big)= {{\mathbf B}}_X\big(t-s; \imath(s+u)\big)$ for all $t,s,u\in G$.
In this work we need the following notion to proceed to the spectral analysis.
\[def:square-integr\]A $K$-PC field $X$ over $G$ is called $G/K$-square integrable if the function ${{\mathbf B}}_X(0; \cdot)$ is integrable with respect to the Haar measure on $G/K$.
If $X$ is a $G/K$-square integrable $K$-PC field, then from translation-invariance of the Haar measure it follows that for every $t\in G$, $b_X(t,t;\cdot)$ is ${{\hbar}}_{G/K}$-integrable, and $$\begin{aligned}
&&\int_{\xi(G/K)} \|X(t+u)\|_{{{\cal H}}}^2\, ({{\hbar}}_{G/K} \circ {\xi}^{-1})(du) =
\int_{G/K} b_X(t,t;x)\, {{\hbar}}_{G/K}(dx)\\
&&\qquad
= \int_{G/K}{{\mathbf B}}_X(0;x)\, {{\hbar}}_{G/K}(dx)
=\int_{\xi(G/K)} \|X(u)\|_{{{\cal H}}}^2\, ({{\hbar}}_{G/K} \circ {\xi}^{-1})(du) < \infty,\end{aligned}$$ where $\xi$ is any cross-section for $G/K$. Also note that if $b_X(t,t;\cdot)$ is ${{\hbar}}_{G/K}$-integrable for any $t\in G$, then by Cauchy-Schwarz inequality $b_X(t,s;\cdot)$ is ${{\hbar}}_{G/K}$-integrable for all $t,s\in G$.
When $X=P$ is a $K$-periodic continuous field, then it is a PC field and we can readilly prove the following equivalence
$P$ is $G/K$-square integrable $\Longleftrightarrow$ $P_K\in L^2(G/K;{{\cal H}})$,
where $P_K$ is the field defined on $G/K$ by $P=P_K\circ\imath$.
When $X$ is a PC process on ${\mathbb{R}}$ with period $T>0$ (i.e. ${{\mathbf K}}_X(t,s) ={{\mathbf K}}_X (t+T,s+T)$ for all $t,s \in {\mathbb{R}}$), it is well known that the SO-spectrum of $X$ can be described as a sequence of complex measures $\gamma_j$, $j\in {\mathbb{Z}}$, on ${\mathbb{R}}$ (cf. [@HM]). If in addition $\sum_j \mathrm{Var}(\gamma_j) < \infty$ then the process $X$ is harmonizable and $$\label{1par-spectrum}
{{\mathbf K}}_X(t,s) = \int\!\!\!\int_{{\mathbb{R}}^2} e^{i(u t-v s)}\,\digamma(du,dv),$$ where $\digamma\!{:=}\sum_j \digamma_{\!\!j}$, and $\digamma_{\!\!j}$ is the image of $\gamma_j$ via the mapping $\ell_{j}(u) {:=}(u, u-2\pi
j/T)$. If $\sum_j \mathrm{Var}(\gamma_j) =\infty$ then $\digamma\!= \sum_j\digamma_{\!\!j}$ can still be viewed as the SO-spectrum of $X$ in the framework of the Schwartz distributions theory (see [@mak00]). For more discussion about PC processes please see Example \[ex-TPC\] in Section \[sect:Examples\]. A corresponding description of the SO-spectrum is available for PC sequences ($G={\mathbb{Z}}$). Let us remark here that a PC sequence is always harmonizable, but there are continuous PC processes which are not, see e.g. [@gladyshev61; @gladyshev63].
The above description of the SO-spectrum of a PC process, which originates from Gladyshev’s papers [@gladyshev61; @gladyshev63], can be easily extended to the case of coordinate-wise strongly periodically correlated fields over ${\mathbb{R}}^n$ or ${\mathbb{Z}}^n$ (see e.g. [@Alekseev91; @DragJav82; @dehayhurd; @gaspar04; @hurdkf]). The purpose of this work is to describe the SO-spectrum of a $K$-periodically correlated field for any closed subgroup $K$ of an LCA group $G$ and as a particular case when $G = {\mathbb{R}}^m \times {\mathbb{Z}}^n$. We also briefly address the question of structure of $K$-PC fields.
Covariance Function of a PC Field
=================================
This section contains an extension of Gladyshev’s description of the covariance function of one-parameter PC processes (see [@gladyshev61; @gladyshev63]) to the case of $K$-PC fields. For any $G/K$-square integrable $K$-PC field $X$, define *the spectral covariance function of the field $X$* (also called cyclic covariance in signal theory, see e.g. [@Gardner06]) by $$\label{alambda}
{a}_{\lambda} (t){:=}\int_{{G/K}} {\left\langle \lambda,x \right\rangle} {{\mathbf B}}_X(t;x)\, {{\hbar}}_{G/K}(dx), \quad
\lambda \in \Lambda_K.$$ Let $\xi$ be a fixed cross-section for $G/K$. For each $\lambda\in \Lambda_K$ and $t\in G$ let us define an ${{\cal H}}_X$-valued function ${{Z}}^{\lambda}(t)$ on $G/K$ by $$\label{Zlambda}
{{Z}}^\lambda(t) (x){:=}{\left\langle \lambda,(\imath(t)+x) \right\rangle} X\big(t+\xi(x)\big),
\quad x \in G/K.$$ Notice that $Z^{\lambda}(t)(x)$ depends on the chosen cross-section $\xi$. From $G/K$-square integrability of $X$ it follows that for all $\lambda \in \Lambda_K$ and $t\in G$, $Z^{\lambda}(t)$ is an element of the Hilbert space $L^2(G/K;{{\cal H}})$.
\[thm-Corr\] Let $X$ be an ${{\cal H}}$-valued $G/K$-square integrable $K$-PC field, and let ${a}_{\lambda}(t)$ and $Z^\lambda(t)(x)$ be as above. Then the cross-covariance function $ {{\mathbf K}}_Z^{\lambda, \mu}(t,s) {:=}\big(Z^\lambda(t), Z^\mu(s)\big)_{{\cal K}}$ of the family $\{Z^\lambda(t)\!:\! \lambda
\in \Lambda_K,t\in G\}$ $$\label{SCorr}
{{\mathbf K}}_Z^{\lambda,\mu}(t,s) ={\left\langle \lambda,(t-s) \right\rangle}\, a_{\lambda-\mu}(t-s) =: {{\mathbf R}}^{\lambda,\mu}(t-s).$$ If additionally
[**\[A\]**]{}
: the function${}G\ni t
\longmapsto a_0(t)$ is continuous at $t=0$,
then $\{Z^\lambda\!:\! \lambda \in \Lambda_K\}$ is a family of jointly stationary fields over $G$ in $L^2(G/K;{{\cal H}}_X)$.
[*Proof*. ]{}Let $\xi$ be a fixed cross-section for $G/K$. Since $X$ is $G/K$-square integrable and $\imath\circ\xi(x) = x$ for any $x\in G/K$, the function $B(0;\cdot)$ is ${{\hbar}}_{G/K}$-integrable, so $Z^{\lambda}(t)\in{{\cal K}}$ and $$\begin{aligned}
\big(Z^\lambda(t), Z^\mu(s)\big)_{{{\cal K}}}
&=& \int_{G/K} {\left\langle \lambda,(\imath(t)+x) \right\rangle}
\overline{{\left\langle \mu,(\imath(s)+x) \right\rangle}}b_X\big(t,s;\imath\circ\xi(x)\big)\, {{\hbar}}_{G/K} (dx) \\
&=& \int_{G/K} {\left\langle \lambda ,(\imath(t)+x) \right\rangle}
\overline{{\left\langle \mu,(\imath(s)+x) \right\rangle}} {{\mathbf B}}_X\big(t-s; \imath(s)+ x\big)\, {{\hbar}}_{G/K} (dx) \nonumber \\
&=& {\left\langle \lambda ,(t-s) \right\rangle} \int_{G/K} {\left\langle (\lambda - \mu), y \right\rangle} {{\mathbf B}}_X(t-s; y)\, {{\hbar}}_{G/K} (dy)\nonumber \\
&=& {\left\langle \lambda,(t-s) \right\rangle}\, a_{\lambda -\mu}(t-s) \nonumber\end{aligned}$$ for all $s,t\in G$. In view of relation (\[SCorr\]), in order to complete the proof it is enough to show that for every $\lambda \in \Lambda_K$, the function $G\ni t \to Z^\lambda(t)\in {{\cal K}}$ is continuous provided condition [**\[A\]**]{} is satisfied, and this is obvious since by equality (\[SCorr\]), $$\begin{aligned}
\big\|Z^\lambda(t) - Z^\lambda(s)\big\|^2_{{\cal K}}&= &{{\mathbf K}}_Z^{\lambda,\lambda}(t,t) - {{\mathbf K}}_Z^{\lambda,\lambda}(t,s) -{{\mathbf K}}_Z^{\lambda,\lambda}(s,t) + {{\mathbf K}}_Z^{\lambda,\lambda}(s,s) \\
&=& 2a_0(0) - {\left\langle \lambda, (t-s) \right\rangle} a_0 (t-s) - {\left\langle \lambda, (t-s) \right\rangle} a_0 (s-t).\mbox{{\hfill $\blacksquare$}}\end{aligned}$$
The condition [**\[A\]**]{} in Theorem \[thm-Corr\] is satisfied if either
------- ---------------------------------------------------------------------------------------
(i) $G$ is discrete, or
(ii) $G/K$ is compact, or
(iii) $X$ is bounded, and ${{\mathbf B}}_X(0;\cdot)^{1/2}$ is ${{\hbar}}_{G/K}$-integrable.
------- ---------------------------------------------------------------------------------------
[*Proof*. ]{}Property [**\[A\]**]{} is evident when the group $G$ is discrete. When $G/K$ is compact then $X$ is clearly bounded because $\|X(t)\|_{{{\cal H}}} = \| X(\xi(x))\|_{{{\cal H}}}$ where $x = \imath(t) \in G/K$, and $x\mapsto\| X(\xi(x))\|_{{{\cal H}}}$ is continuous. Since ${{\hbar}}_{G/K}$ is finite, the continuity of the function ${}t\mapsto a_0(t)=\int_{G/K} {{\mathbf B}}_X(t;x)\, {{\hbar}}_{G/K}(dx)$ follows therefore from Lebesgue dominated convergence theorem.
Suppose now that ${}\int_{G/K}{{\mathbf B}}_X(0;x)^{1/2} \,{{\hbar}}_{G/K}(dx) <\infty$. In this case for all $t, s,x$ $$\begin{aligned}
&&\big|{{\mathbf B}}_X(t;x)-{{\mathbf B}}_X(s;x)\big|
\,=\,\big|{{\mathbf K}}_X\big(t+\xi(x),\xi(x)\big)-{{\mathbf K}}_X\big(s+\xi(x),\xi(x)\big)\big|\\
&&\qquad\qquad\leq\, \big\|X\big(t+\xi(x)\big)-X\big(s+\xi(x)\big)\big\|_{{{\cal H}}} \, \big\|X(\xi(x))\big\|_{{{\cal H}}} \leq 2\sup_t\|X\|_{{{\cal H}}}
{{\mathbf B}}_X(0,x)^{1/ 2},\end{aligned}$$ and ${}\lim_{u\to t}{{\mathbf B}}_X(u;x)={{\mathbf B}}_X(t;x)$. Hence Lebesgue dominated convergence theorem applies and we conclude that $\lim_{s\to t}a_0(s)=a_0(t),$ so condition [**\[A\]**]{} is satisfied. [$\blacksquare$]{}
Relation (\[SCorr\]) in Theorem \[thm-Corr\] can be also obtained using Gladyshev’s technique, that is by showing non-negative definiteness of ${}\left[{{\mathbf R}}^{\lambda, \mu}(t)\right]_{\lambda,\mu \in \Lambda_K}$, i.e. that $$\label{posdef}
\sum_{j=1}^n \sum_{k=1}^n c_j\overline{c_k}\,{{\mathbf R}}^{\lambda_j,\lambda_k}(t_j - t_k) \geq 0,$$ for any finite set of complex numbers $\{c_1, \dots, c_n\}$. Our method, which is an adaptation of the technique used in [@mak99], has the advantage that it gives an explicit construction of an associated stationary family of fields. We want to point out here that even in the case of PC processes on ${\mathbb{R}}$ with period $T$ not every matrix function ${}\left[{{\mathbf R}}^{m, n}(t)\right]_{m,n \in {\mathbb{Z}}} $ with continuous entries, which is non-negative definite in the sense of (\[posdef\]) and such that $ {{\mathbf R}}^{m,n}(t)\,e^{i2\pi mt/T }$ depends only on $
m-n$, is associated with a continuous PC process of period $T$ through the relation (\[SCorr\]). To achieve the one-to-one correspondence one has to consider not necessarily continuous PC processes (see e.g. [@mak99]).
To complete the analysis of the family of fields $\{Z^{\lambda}\!:\!\lambda\in\Lambda_K\}$ defined by (\[Zlambda\]), consider the space ${{\cal H}}_Z{:=}{\overline{\mbox{span}}\left\{ Z^\lambda(t)\!:\!\lambda \in \Lambda_K, t\in G \right\} }$. Clearly ${{\cal H}}_Z$ is a subspace of $L^2(G/K; {{\cal H}}_X)$. In the case where $G={\mathbb{Z}}^n\times{\mathbb{R}}^m$, these Hilbert spaces coincide. More precisely
\[prop:MZ=L2cHX\] Let $X$ be an ${{\cal H}}$-valued $G/K$-square integrable $K$-PC field, and $Z^{\lambda}$ be defined as above by (\[Zlambda\]). Assume that the LCA group $G$ admits a countable dense subset, which is true when $G={\mathbb{Z}}^n\times{\mathbb{R}}^m$. Then ${{\cal H}}_Z=L^2(G/K; {{\cal H}}_X)$.
[*Proof*. ]{}We know that ${{\cal H}}_Z\subset L^2(G/K; {{\cal H}}_X)$. To show the equality, let $f\in L^2(G/K; {{\cal H}}_X)$ be such that $\big(Z^{\lambda}(t), f\big)_{{{\cal K}}} = 0$ for all $\lambda\in \Lambda_K$ and $t\in G$, i.e. $$\int_{G/K} {\left\langle \lambda,(\imath(t)+x) \right\rangle} \big(X\big(t+\xi(x)\big),f(x)\big)_{{{\cal H}}}\, {{\hbar}}_{G/K}(dx)= 0, \quad\lambda\in \Lambda_K,\,t\in G.$$ Since $\Lambda_K\sim{\widehat}{G/K}$ and ${\left\langle \lambda,\imath(t) \right\rangle}\neq 0$, the scalar product $\big(X(t+\xi(x)),f(x)\big)_{{{\cal H}}} = 0$, for ${{\hbar}}_{G/K}$-almost every $x\in G/H$ and for every $t\in G$ [@HR Theorem 23.11]. This implies that for each $t$ there is a negligible Borel subset $\Xi_t$ of $G/K$ such that $X\big(t+\xi(x)\big)\perp_{{{\cal H}}} f(x) $ for every $x\notin \Xi_t$. Note that for every $x$, ${\overline{\mbox{span}}\left\{ X\big(t+\xi(x)\big)\!:\! t\in G \right\} }= {\overline{\mbox{span}}\left\{ X(t)\!:\! t\in G \right\} } = {{\cal H}}_X$. If $G$ admits a countable dense subset $G^*$, which is true in the case when $G={\mathbb{Z}}^m\times{\mathbb{R}}^n$, then from continuity of $X$, it follows that also ${\overline{\mbox{span}}\left\{ X\big(t+\xi(x)\big)\!:\! t\in G^* \right\} } = {{\cal H}}_X$. Therefore $f(x)\perp_{{{\cal H}}} {{\cal H}}_X$ for all $x$ which are not in the negligible set $\bigcup_{t\in G^*}\Xi_t$. Hence $f(x)=0$ for ${{\hbar}}_{G/K}$-almost every $x\in G/K$ and Proposition \[prop:MZ=L2cHX\] is proved. [$\blacksquare$]{}
SO-spectrum of a PC Field {#sec-spectrum}
=========================
Let $X$ be a $G/K$-square integrable $K$-PC over $G$ and let ${{\mathbf K}}_X(t,s)$ be its covariance function. The objective is to describe the domain of the SO-spectrum of $X$, which by definition (see Definition \[def:spectrum of X\]) is the domain of the spectrum of the function $\Phi(t,s) ={{\mathbf K}}_X(t,-s)$.
First we give a description of the domain of the domain of the SO-spectrum of the PC field $X$ in the simplest case where $G/K$ is compact.
\[lem:domain of PC\] Let $X$ be a $G/K$-square integrable $K$-PC over $G$, and let $\Lambda_K = \imath^*({\widehat}{G/K})\subseteq {\widehat}{G}$. Then the domain of the SO-spectrum of $X$ is the subgroup $L$ of ${\widehat}{G}\times {\widehat}{G}$ given by $$L = \{(\gamma, \gamma - \lambda)\!:\! \lambda \in \Lambda_K, \gamma\in
{\widehat}{G}\}.$$
Note that $ L $ can be viewed as the union of [*hyperplanes*]{} $$\label{FT-K}
L = \bigcup_{\lambda \in \Lambda_K} L_\lambda\qquad\mbox{where}\quad L_\lambda {:=}\{(\gamma,
\gamma - \lambda)\!:\! \gamma\in {\widehat}{G}\}.$$ [*Proof*. ]{}Since ${{\mathbf K}}_X(t+u,s+u)$ is $K$-periodic in $u$, the function $\Phi(t,s) ={{\mathbf K}}_X(t,-s)$ is itself a periodic function on ${G} \times {G}$ with the period $D = \{(k,-k)\!:\! k\in K
\}\subseteq G \times G$. The domain of the SO-spectrum of $X$ is therefore the dual group of $(G \times
G)/D$ viewed as a subgroup of ${\widehat}{G}\times {\widehat}{G}$. Note that the subgroup $D$ is the image of the subgroup $\{0\} \times K$ through the isomorphism ${\cal D}\!:\! G \times G \ni (t,s) \mapsto (t+s,-s) \in G \times G$, and this induces an isomorphism from the quotient group $(G\times G)/D$ onto $(G\times G)/(\{0\}\times K)$. Futhermore, since $(G \times G)/(\{0\} \times K) = G \times G/K$ and its dual is ${\widehat}{G} \times \Lambda_K$, we deduce that the dual of $(G \times G)/D$ can be identified with the subgroup $L$ of ${\widehat}{G} \times {\widehat}{G}$ consisting of the elements of the form $(\chi, \chi- \lambda)$, $\chi \in {\widehat}{G}$, $
\lambda \in \Lambda_K$. [$\blacksquare$]{}
We have not used the fact that ${{\mathbf K}}_X$ is a covariance function of a process. It turns out that this additional property of ${{\mathbf K}}_X$ (i.e. the fact that it is nonnegative definite) implies that the “part of the SO-spectrum” that sits on each $L_\lambda$ is a measure. We want to point out that the set $\Lambda_K$ may be uncountable.
\[thm-spectrum\] Suppose that $X$ is a $G/K$-square integrable $K$-PC field that satisfies the condition [**\[A\]**]{} of Theorem \[thm-Corr\]. Then for every $\lambda \in \Lambda_K$ there is a unique Borel complex measure $\gamma_\lambda$ on ${\widehat}{G}$ such that $$\label{inta}
a_\lambda (t) = \int_{{\widehat}{G}}\overline{{\left\langle \chi, t \right\rangle}}\, \gamma_\lambda(d\chi),
\quad t\in G.$$ Furthermore $\sup_\lambda \mathrm{Var}(\gamma_\lambda ) < \infty$ and for each $\lambda\in\Lambda_K$, $\gamma_{\lambda}$ is absolutely continuous with respect to $\gamma_0$.
In this paper by a *representation of $G$ in a Hilbert space ${{\cal K}}$* we mean a weakly continuous group ${\mathcal{U}}{:=}\{U^t\!:\!t\in G\}$ of unitary operators in ${{\cal K}}$ (see [@HR Section22]). In this case there exists a weakly countably additive orthogonally scattered (w.c.a.o.s) Borel operator-valued measure $E$ on ${\widehat}{G}$ such that for every Borel set $\Delta$ the operator $E(\Delta)$ is an orthogonal projection in ${{\cal K}}$, and for every $u,v \in {{\cal K}}$, $$\big(U^tu,v\big)_{{{\cal K}}} = \int_{{\widehat}{G}} \overline{{\left\langle \chi,t \right\rangle}}\,\big(E(d\chi)u,v\big)_{{{\cal K}}}, \quad
t\in G.$$ ”*Orthogonally scattered*” means that $\big(E(\Delta_1)u, E(\Delta_2)v\big)_{{{\cal K}}}= 0$ for all disjoint $\Delta_1, \Delta_2$ and $u,v\in {{\cal K}}$. The measure $E$ will be referred to as the *spectral resolution of the unitary operator group ${\mathcal{U}}$*.
[*Proof*. ]{}*of Theorem \[thm-spectrum\]*. The joint stationarity of the fields $\{Z^\lambda\!:\!\lambda \in \Lambda_K\}$ defined by (\[Zlambda\]), implies that each $Z^\lambda(t) = U^t Z^\lambda(0)$ where ${\mathcal{U}}{:=}\{U^t\!:\!t\in G\}$ is the common shift operators group. Condition [**\[A\]**]{} guarantees the continuity of the representation ${\mathcal{U}}$ of $G$ in $L^2(G/K;{{\cal H}}_X)$, and hence $$Z^\lambda(t) = \int_{{\widehat}{G}} \overline{{\left\langle \chi, t \right\rangle}}\, E(d\chi)Z^\lambda(0),$$ where $E$ is the spectral resolution of ${\mathcal{U}}$. Therefore for every $\lambda, \mu \in \Lambda_K$ there is a complex measure $\Gamma^{\lambda, \mu}$ on ${\widehat}{G}$ such that $ {{\mathbf R}}_Z^{\lambda,\mu}(t) = \int_{{\widehat}{G}} \overline{{\left\langle \chi, t \right\rangle}}\,\Gamma^{\lambda,\mu}(d\chi), $ namely, $\Gamma^{\lambda, \mu}(\Delta) = \big(E(\Delta) Z^\lambda(0), Z^\mu(0)\big)_{{\cal K}}$, where ${{\cal K}}{:=}L^2(G/K;{{\cal H}}_X)$. Consequently, from relations (\[alambda\]) and (\[SCorr\]) we conclude that $${a}_{\lambda} (t)=\int_{G/K} {\left\langle \lambda,x \right\rangle} {{\mathbf B}}_X(t;x)\,{{\hbar}}_{G/K} (dx) = {{\mathbf R}}_Z^{0, -\lambda}(t) = \int_{{\widehat}{G}} \overline{{\left\langle \chi,t \right\rangle}}\, \Gamma^{0,-\lambda}(d\chi).$$ Then equality (\[inta\]) is satisfied with $\gamma_\lambda {:=}\Gamma^{0, -\lambda}$. From Cauchy-Schwarz inequality we have $$\begin{aligned}
&&\left|\Gamma^{0, -\lambda}(\Delta) \right| =
\left| \big(E(\Delta) Z^0(0), E(\Delta)Z^{-\lambda}(0)\big)_{{\cal K}}\right|\\
&&\qquad\qquad\quad\leq \,
\sqrt{\Gamma^{0, 0}(\Delta)}\sqrt{\Gamma^{-\lambda,-\lambda} (\Delta)} = \sqrt{\Gamma^{0,0}(\Delta)} \sqrt{\Gamma^{0, 0}(\Delta -\lambda)},\end{aligned}$$ and we deduce the absolute continuity of $\gamma_{\lambda}$ with respect to $\gamma_0$ for any $\lambda \in \Lambda_K$.
Finally note that the total variations of measures $\Gamma^{\lambda, \lambda}$, $\lambda \in \Lambda_K$, are all equal to $\Gamma^{0,0}({\widehat}{G})$. Indeed, since the measures $\Gamma^{\lambda, \lambda}$, $\lambda \in \Lambda_K$, are non-negative $$\mathrm{Var}\left(\Gamma^{\lambda, \lambda}\right) = \Gamma^{\lambda,\lambda}({\widehat}{G})
={{\mathbf R}}_Z^{\lambda,\lambda}(0)
=\int_{G/K} {{\mathbf B}}_X(0;y)\,{{\hbar}}_{G/K}(dy) = {{\mathbf R}}_Z^{0,0}(0) = \Gamma^{0,0}({\widehat}{G}).$$ Hence all total variations $\mathrm{Var}\left(\Gamma^{\lambda, \mu}\right)
\leq\sqrt{\Gamma^{\lambda, \lambda}({\widehat}{G})} \sqrt{\Gamma^{\mu,\mu}({\widehat}{G})}$, $\lambda,\mu\in{\widehat}{G}$, are bounded by the same constant, and in consequence all measures $\gamma_\lambda$, $\lambda \in \Lambda_K$, have uniformly bounded total variations.[$\blacksquare$]{}
Remark that when the field $X=P$ is $K$-periodic and $G/K$-square integrable, the field $P_K$ is ${{\hbar}}_{G/K}$-square integrable and thanks to Parseval equality, the spectral covariance function of the field $P$ can be expressed as $$\begin{aligned}
a_{\lambda}^P(t)&=&
\int_{G/K}{\left\langle \lambda,x \right\rangle}\big(P_K(\imath(t)+x),P_K(x)\big)_{{{\cal H}}}\,{{\hbar}}_{G/K}(dx)\\
&=&
\int_{\Lambda_K}\overline{{\left\langle \chi,\imath(t) \right\rangle}}\big({\widehat}{P_K}(\chi),{\widehat}{P_K}(\chi-\lambda)\big)_{{{\cal H}}}\,{{\hbar}}_{\Lambda_K}(d\chi)\end{aligned}$$ where ${\widehat}{P_K}$ is the Fourier Plancherel transform of the field $P_K$. Then, we deduce that the function $\chi\mapsto \big({\widehat}{P_K}(\chi),{\widehat}{P_K}(\chi-\lambda)\big)_{{{\cal H}}}$ is the density function of the SO-spectral measure $\gamma_{\lambda}^P$ of the field $P$ with respect to ${{\hbar}}_{\Lambda_K}$, $$\label{gamma-P}
\gamma_{\lambda}^P(\Delta)=\int_{\Delta\cap\Lambda_K}\big({\widehat}{P_K}(\chi),{\widehat}{P_K}(\chi-\lambda)\big)_{{{\cal H}}}\,{{\hbar}}_{\Lambda_K}(d\chi)$$ for any $\Delta\in{{\mathcal{B}}}({\widehat}{G})$. Particulary $$\gamma_{0}^P(\Delta)=\int_{\Delta\cap\Lambda_K}\big\|{\widehat}{P_K}(\chi)\big\|_{{{\cal H}}}^2\,{{\hbar}}_{\Lambda_K}(d\chi).$$ Notice that SO-spectral measure $\gamma_{\lambda}^P$ is concentrated on $\Lambda_K\subset{\widehat}{G}$ : $\gamma_{\lambda}^P(\Delta)=\gamma_{\lambda}^P(\Delta\cap\Lambda_K)$ for any $\Delta\in{{\mathcal{B}}}({\widehat}{G})$. When in addition ${\widehat}{P_k}$ is ${{\hbar}}_{\Lambda_K}$-integrable, then the fields $P_K$ and $P$ are harmonizable with $$\begin{aligned}
{{\mathbf K}}_P(t,s)&=&{{\mathbf K}}_{P_K}(\imath(t),\imath(s))\\
&=&\int\!\!\int_{\Lambda_K\times\Lambda_K}\overline{{\left\langle \lambda,\imath(t) \right\rangle}}{\left\langle \mu,\imath(s) \right\rangle}\big({\widehat}{P_K}(\lambda),{\widehat}{P_K}(\mu)\big)_{{{\cal H}}}\,{{\hbar}}_{\Lambda_K}(d\lambda){{\hbar}}_{\Lambda_K}(d\mu)\\
&=&\int\!\!\int_{{\widehat}{G}\times{\widehat}{G}}
\overline{{\left\langle \chi,t \right\rangle}}{\left\langle \beta,s \right\rangle}\,\digamma^P(d\chi,d\beta)\end{aligned}$$ where $\digamma^{\!P}$ is the measure on ${\widehat}{G}\times{\widehat}{G}$ concentrated on $\Lambda_K\times\Lambda_K$ defined by $$\digamma^{\!P}(\Delta){:=}\int\!\!\int_{\Delta\cap(\Lambda_K\times\Lambda_K)}\big({\widehat}{P_K}(\chi),{\widehat}{P_K}(\beta)\big)_{{{\cal H}}}\,{{\hbar}}_{\Lambda_K}(d\chi){{\hbar}}_{\Lambda_K}(d\beta),\quad\Delta\in{{\mathcal{B}}}({\widehat}{G}\times{\widehat}{G}).$$ From relation (\[gamma-P\]) we find out that the measure $\gamma_{\lambda}^P$ or more precisely its image $\digamma_{\!\!\lambda}^P{:=}\gamma_{\lambda}^P\circ \ell_{\lambda}^{-1}$ through the mapping $\ell_\lambda: {\widehat}{G} \to {\widehat}{G}^2 $ defined by $\ell_\lambda(\chi) = (\chi,
\chi-\lambda)$, $\chi\in {\widehat}{G}$, is the restriction of the measure $\digamma^P$ to the hyperplane $L_{\lambda}{:=}\{(\chi,\chi-\lambda)\!:\!\chi\in{\widehat}{G}\}$.
More generally, when $X$ is a PC field, the family $\{\gamma_\lambda\!:\!\lambda\in\Lambda_K\}$ is commonly referred to as the *SO-spectral family of* $X$. The measure $\gamma_\lambda$ or more precisely its image $\digamma_{\!\!\lambda}=\gamma_\lambda \circ \ell_\lambda^{-1}$ represents the part of the SO-spectrum of $X$ that sits on the hyperplane $L_\lambda = \{(\chi, \chi-\lambda)\!:\! \gamma\in {\widehat}{G}\}$, see relation (\[FT-K\]).
Next, we give a sufficient condition for a PC field to be harmonizable.
\[thm-harmsp\] Let $X$ be a $G/K$-square integrable $K$-PC field that satisfies the condition [**\[A\]**]{} of Theorem \[thm-Corr\], and let $\{\gamma_\lambda\!:\!\lambda\in\Lambda_K\}$ be the SO-spectral family of $X$. Suppose that there is an ${{\hbar}}_{\Lambda_K}$-integrable non-negative function $\omega$ on $\Lambda_K$ such that for every $\lambda \in \Lambda_K$ $$\label{harm-cond}
|\gamma_\lambda(\Delta) | \leq \omega(\lambda)
\quad\mbox{for any Borel }\,\,\Delta \in {{\mathcal{B}}}({\widehat}{G}).$$ Then the field $X$ is harmonizable and the SO-spectral measure of $X$ is given by $$\label{F}
\digamma\!(\Delta) = \int_{\Lambda_K}\digamma_{\!\!\lambda}(\Delta)\, {{\hbar}}_{\Lambda_K}(d\lambda),
\quad\mbox{for any Borel }\,\,\Delta \in {{\mathcal{B}}}({\widehat}{G}\times {\widehat}{G}),$$ where $\digamma_{\!\!\lambda}{:=}\gamma_\lambda \circ \ell_\lambda^{-1}$ and $\ell_\lambda(\chi) {:=}(\chi,
\chi-\lambda)$, $\chi\in {\widehat}{G}$.
Notice that condition (\[harm-cond\]) is satisfied by any $G/K$-square integrable $K$-periodic field $P$ such that ${\widehat}{P_K}$ is ${{\hbar}}_{\Lambda_K}$-integrable. Here we can take $\omega(\lambda)$ equal to the total variation of the SO-spectral measure $\gamma^P_{\lambda}$ of the field $P$ $$\omega(\lambda)=\int_{\Lambda_K}\left|\big({\widehat}{P_K}(\chi),{\widehat}{P_K}(\chi-\lambda)\big)_{{{\cal H}}}\right|\,{{\hbar}}_{\Lambda_K}(d\chi).$$ The integrability condition on ${\widehat}{P_K}:\Lambda_K\to{{\cal H}}$ is always satisfied when $\Lambda_K$ is compact that is when $G/K$ is discrete, and in particular when $G={\mathbb{Z}}^n$.
[*Proof*. ]{}*of Theorem \[thm-harmsp\]*. Let $\{Z^\lambda\!:\!\lambda \in \Lambda_K\}$ be as in Theorem \[thm-Corr\]. From the proof of Theorem \[thm-spectrum\] it follows that $$\gamma_\lambda(\Delta) =
\Gamma^{0, -\lambda}(\Delta) = \big(E(\Delta) Z^0(0), Z^{\-\lambda}(0) \big)_{{\cal K}}$$ Thanks to definition (\[Zlambda\]) and Lebesgue dominated convergence theorem it follows that the field $\Lambda_K \ni \lambda \mapsto
Z^{\-\lambda}(0)\in {{\cal K}}$ is continuous, and hence by assumption (\[harm-cond\]), $\lambda \mapsto \gamma_\lambda(\Delta)$ is integrable over $\Lambda_K$ for every Borel $\Delta$ of ${\widehat}{G}$. For all Borel $D\subseteq \Lambda_K$ and $\Delta\subseteq {\widehat}{G}$ let us define $$\tilde{\digamma}(\Delta \times D) {:=}\int_D\gamma_\lambda(\Delta)\, {{\hbar}}_{\Lambda_K}(d\lambda)
= \int_D \big(E(\Delta) Z^0(0), Z^{\-\lambda}(0)\big)_{{\cal K}}\,{{\hbar}}_{\Lambda_K}(d\lambda).$$
Condition (\[harm-cond\]) and again Lebesgue dominated convergence theorem entail that the function $\tilde{\digamma}(\Delta \times D)$ is countably additive in $\Delta$ and $D$ separately. So to show that the bimeasure $\tilde{\digamma}$ extends to a Borel measure on ${\widehat}{G}\times
\Lambda_K$, it is sufficient to show that its Vitali variation is finite (see [@DS; @rao]), that is $$\sup\left\{\sum_{i=1}^n \sum_{j=1}^n \big|\tilde{\digamma}(\Delta_i \times D_j)\big|\!:\!\Delta_i\cap\Delta_j=\emptyset\,\,\mbox{and}\,\,D_i\cap D_j=\emptyset\,\,\mbox{for}\,\, i\neq j\,\, \mbox{in}\,\, \{1,\dots,n\}\right\}< \infty.$$ Since $\sum_{i=1}^n \big|\gamma_\lambda(\Delta_j)\big| \leq
\mathrm{Var}(\gamma_\lambda) \leq 4 \omega(\lambda)$ and the function $\omega$ is ${{\hbar}}_{\lambda}$-integrable, $$\begin{aligned}
\sum_{i=1}^n \sum_{j=1}^n \big|\tilde{\digamma}(\Delta_i \times D_j)\big|
&\leq &
\sum_{j=1}^n \int_{D_j}\sum_{i=1}^n \big|\gamma_\lambda(\Delta_i)\big|\,{{\hbar}}_{\Lambda_K}(d\lambda) \\
&\leq & \int_{\bigcup_jD_j} 4 \omega(\lambda)\,{{\hbar}}_{\Lambda_K}(d\lambda)
\leq \int_{\Lambda_K} 4 \omega(\lambda)\,{{\hbar}}_{\Lambda_K}(d\lambda) <\infty.\end{aligned}$$ Hence $\tilde{\digamma}$ is a measure and in particular Fubini and Lebesgue dominated convergence theorems hold for $\tilde{\digamma}$. Let $\Delta\subseteq {\widehat}{G}$ be fixed and let $\varphi(\lambda) {:=}\sum_{j=1}^n b_j 1_{D_j}(\lambda)$ be a simple function on $\Lambda_K$. Then $$\int_{\Lambda_K} \varphi(\lambda) \tilde{\digamma}(\Delta, d\lambda)
= \sum_{j=1}^n b_j\int_{D_j} \gamma_\lambda(\Delta) \, {{\hbar}}_{\Lambda_K}(d\lambda)
=\int_{\Lambda_K} \varphi(\lambda) \gamma_\lambda(\Delta) \,{{\hbar}}_{\Lambda_K}(d\lambda).$$ From condition (\[harm-cond\]) we deduce that $ \int_{\Lambda_K} \varphi(\lambda)\, \tilde{\digamma}(\Delta, d\lambda)
=\int_{\Lambda_K} \varphi(\lambda) \gamma_\lambda(\Delta)\, {{\hbar}}_{\Lambda_K}(d\lambda)$ for any bounded Borel function $\varphi$. Consequently, for any simple function $\phi$ on ${\widehat}{G}$ and bounded $\varphi$ on $\Lambda_K$ $$\label{iterated}
\int\!\!\!\int_{{\widehat}{G}\times\Lambda_K}\phi(\chi) \varphi(\lambda)\,\tilde{\digamma}(d\chi, d\lambda)
=\int_{\Lambda_K} \varphi(\lambda)\left[\int_{{\widehat}{G}}\phi(\chi)\,\gamma_\lambda(d\chi)\right]{{\hbar}}_{\Lambda_K}(d\lambda).$$ If $|\phi|$ is bounded by some finite $c>0$ then by condition (\[harm-cond\]), the integral $\int_{{\widehat}{G}} |\phi(\chi)|\,\gamma_\lambda(d\chi)$ is bounded by $4c\,\omega(\lambda)$ which is an ${{\hbar}}_{\lambda}$-integrable function of $\lambda$. Therefore by Lebesgue dominated convergence theorem, relation (\[iterated\]) holds for any two bounded measurable functions $\phi$ on ${\widehat}{G}$ and $\varphi$ on $\Lambda_K$. In particular $$\begin{aligned}
\label{GFT}
\int\!\!\!\int_{{\widehat}{G}\times\Lambda_K} \overline{{\left\langle \chi, t \right\rangle}}\, \overline{{\left\langle \lambda, x \right\rangle}}\,\tilde{\digamma}(d\chi,d\lambda)
&=& \int_{\Lambda_K} \overline{{\left\langle \lambda,x \right\rangle}} \left[\int_{{\widehat}{G}} \overline{{\left\langle \chi,t \right\rangle}}\, \gamma_\lambda(d\chi)\,\right]{{\hbar}}_{\Lambda_K}(d\lambda) \\
&=& \int_{\Lambda_K} \overline{{\left\langle \lambda,x \right\rangle}}a_\lambda(t)\, {{\hbar}}_{\Lambda_K}(d\lambda) = {{\mathbf B}}_X(t; x)= {{\mathbf K}}_X(t+x, x) \nonumber\end{aligned}$$ for all $t\in G$ and $x\in G/K$.
Let $\ell: {\widehat}{G}\times \Lambda_K \to {\widehat}{G}^{\,2}$ be defined by $\ell(\chi,\lambda) {:=}\ell_{\lambda}(\chi)= (\chi, \chi-\lambda)$, and let $\digamma = \tilde{\digamma}\! \circ \ell^{-1}$ be the image of the measure $\tilde{\digamma}\!$ through the mapping $\ell$, that is $ \digamma\!(\Delta) =\tilde{\digamma}\!\{ (\chi,\lambda)\!:\! (\chi, \chi-\lambda) \in \Delta\}$. Then $\digamma\!$ is a Borel measure on ${\widehat}{G}^{\,2}$ and change of variables formula yields that $$\label{chvar}
\int\!\!\!\int_{{\widehat}{G}\times{\widehat}{G}} \psi(\chi_1,\chi_2)\, \digamma\!(d\chi_1,d\chi_2) = \int\!\!\!\int_{{\widehat}{G}\times\Lambda_K} \psi(\chi,\chi-\lambda)\,\tilde{\digamma}\!(d\chi,d\lambda)$$ for any bounded Borel function $\psi:{\widehat}{G}\times{\widehat}{G}\to{\mathbb{C}}$. In particular, in view of relation (\[GFT\]) $$\begin{aligned}
\int\!\!\!\int_{{\widehat}{G}\times\Lambda_K} \overline{{\left\langle \chi,t \right\rangle}} {\left\langle \lambda,s \right\rangle}\,\digamma\!(d\chi, d\lambda)
&=&
\int\!\!\!\int_{{\widehat}{G}\times\Lambda_K}\overline{{\left\langle \chi,t \right\rangle}}{\left\langle (\chi-\lambda),s \right\rangle}\,
\tilde{\digamma}\!(d\chi, d\lambda) \\
&=& {{\mathbf B}}_X\big(t-s; \imath(s)\big) = {{\mathbf K}}_X(t,s).\end{aligned}$$ for all $t,s\in G$ (recall that ${\left\langle \lambda,s \right\rangle}={\left\langle \lambda,\imath(s) \right\rangle}$ for all $\lambda\in\Lambda_K$ and $s\in G$). Thus the field $X$ is harmonizable and $\digamma\!$ is its SO-spectral measure. Note that relation (\[iterated\]) holds true if the product $\phi(\chi) \varphi(\lambda)$ is replaced by any bounded measurable function $\psi(\chi,\lambda)$ of two variables. Thanks to such upgraded relation (\[iterated\]) and to relation (\[chvar\]) with $\psi = 1_\Delta$, we get $$\digamma\!(\Delta)= \int\!\!\!\int_{{\widehat}{G}\times\Lambda_K} 1_\Delta(\chi,\chi-\lambda)\,\tilde{\digamma}\!(d\chi,d\lambda)
=\int_{\Lambda_K} \left[\int_{{\widehat}{G}}1_{\Delta}(\chi,\chi-\lambda)\,\gamma_\lambda(d\chi)\right]{{\hbar}}_{\Lambda_K}(d\lambda)$$ So, by the definition of $\tilde{\digamma}\!$, we deduce relation (\[F\]). [$\blacksquare$]{}
Note that if $G = {\mathbb{Z}}^n$ then condition \[A\] is satisfied, $\Lambda_K$ is compact, and condition (\[harm-cond\]) holds true with $\omega(\lambda) = \mathrm{Var}(\gamma_\lambda) <\infty$. Therefore we generalize the property of harmonizability of the PC sequences proved in [@gladyshev61].
\[cor-harm\] Any $G/K$-square integrable $K$-PC field over $G={\mathbb{Z}}^n$ is harmonizable.
All the results above simplify significantly if $G/K$ is compact, because then every $K$-PC field over $G$ is $G/K$-square integrable and condition \[[**A**]{}\] in Theorem \[thm-Corr\] is always satisfied.
\[Spectrum-compact\] Suppose that $X$ is a $K$-PC field and that $G/K$ is compact. Then for every $\lambda \in \Lambda_K$ there is a Borel complex measure $\gamma_\lambda$ on ${\widehat}{G}$ such that $$a_\lambda (t) =\int_{{\widehat}{G}} \overline{{\left\langle \chi,t \right\rangle}}\, \gamma_\lambda(d\chi),\quad t\in G.$$ Moreover the set $\Lambda_K$ is countable, ${}\sum_{\lambda \in \Lambda_K} |a_\lambda (t)|^2 <\infty$ and for every $t\in G$ $$\label{series}
{{\mathbf B}}_X(t;x) \stackrel{L^2}{=}
\sum_{\lambda \in \Lambda_K} \overline{{\left\langle \lambda,x \right\rangle}} a_\lambda (t)$$ (the series (\[series\]) converges in $L^2(G/K)$ with respect to $x$). Additionally:
1. if ${}\sum_{\lambda\in\Lambda_K} |a_\lambda (t)| <\infty$ for every $t\in G$, then the series (\[series\]) converges also pointwise and uniformly with respect to $x\in G/K$, and for all $t,s \in G$ we have $${}{{\mathbf K}}_X(t+s,s) = \sum_{\lambda\in\Lambda_K} \overline{{\left\langle \lambda,s \right\rangle}} a_\lambda (t);$$
2. if ${}\sum_{\lambda\in\Lambda_K} \mathrm{Var}(\gamma_\lambda) < \infty$, then $X$ is harmonizable, and for all $t,s \in G$ we have $${}{{\mathbf K}}_X(t,s) = \int\!\!\!\int_{{\widehat}{G}\times{\widehat}{G}} \overline{{\left\langle \chi,t \right\rangle}}\, {{\left\langle \varrho,s \right\rangle}}\,{\digamma}\!(d\chi, d\varrho),$$ where the SO-spectral measure $\digamma\!$ is given by ${}\digamma\!(\Delta) = \sum_{\lambda \in \Lambda_K} \digamma_{\!\!\lambda}(\Delta)$, $\digamma_{\!\!\lambda} {:=}\gamma_\lambda \circ \ell_\lambda^{-1}$ and $\ell_\lambda: {\widehat}{G} \to {\widehat}{G} \times {\widehat}{G} $ is defined as $\ell_\lambda(\chi) {:=}(\chi,\chi-\lambda)$.
[*Proof*. ]{}Existence of $\gamma_\lambda$ follows from Theorem \[thm-spectrum\]. Since for each $t\in G$ the function $x \mapsto
{{\mathbf B}}_X(t;x)$ is bounded, it is in $L^2(G/K)$ and hence its Fourier transform $\lambda\mapsto a_\lambda(t)$ is in $L^2(\Lambda_K)$. Formula (\[series\]) is just the inverse formula for ${{\mathbf B}}_X(t;\cdot)$. Item (i) follows from the uniqueness of the Fourier transform, while item (ii) from Theorem \[thm-harmsp\]. [$\blacksquare$]{}
Structure of PC fields {#sect:structure}
======================
When $X$ is a $K$-PC field then for every $k\in K$ the mapping $V^k: X(t) \mapsto X(t+k)$, $t\in G$, is well defined and extends linearly to an isometry from ${{\cal H}}_X={\overline{\mbox{span}}\left\{ X(t)\!:\! t\in G \right\} }$ onto itself. The group ${\mathcal{V}}{:=}\{V^k\!:\!k\in K\}$ is a unitary representation of $K$ in ${{\cal H}}_X$ and is called the *$K$-shift of $X$*.
\[PCF-structure\] A continuous field $X$ over $G$ is $K$-PC if and only if there are a unitary representation ${\mathcal{U}}=\{U^t\!:\!t\in G\}$ of $G$ in ${{\cal H}}_X$, and a continuous $K$-periodic field $P$ over $G$ with values in ${{\cal H}}_X$ such that $X(t) = U^t P(t)$, $t\in G$.
[*Proof*. ]{}The “if” part is obvious. Prove the other part. Let $X$ be a $K$-PC field, ${\mathcal{V}}=\{V^k\!:\!k\in K\}$ be the $K$-shift of $X$, and $E$ be the spectral resolution of ${\mathcal{V}}$. Hence $E$ is a w.c.a.o.s. Borel operator-valued measure defined on ${\widehat}{K}$. Since ${\widehat}{K}$ is isomorphic to ${\widehat}{G}/ \Lambda_K $, the measure $E$ can be seen as a measure on ${\widehat}{G}/\Lambda_K$, see [@rudin Section2.1.2]. Let $\zeta$ be a cross-section for ${\widehat}{G}/\Lambda_K$. For every Borel subset $\Delta$ of ${\widehat}{G}$ let us define $\tilde{E} (\Delta) {:=}E\left( \zeta^{-1}(\Delta)\right)$. Then $\tilde{E}$ is a w.c.a.o.s. Borel operator-valued measure on ${\widehat}{G}$ whose support is contained in a measurable set $\zeta({\widehat}{G}/\Lambda_K)$, and whose values are orthogonal projections in ${{\cal H}}_X$. Since for all $\chi \in {\widehat}{K}$ and $k\in K$, ${\left\langle \chi, k \right\rangle} = {\left\langle \zeta(\chi),k \right\rangle}$, by change of variable we obtain that $$V^k = \int_{{\widehat}{G}} \overline{{\left\langle \chi,k \right\rangle}}\, \tilde{E}(d\chi), \quad k \in K.$$ Following Gladyshev’s idea ([@gladyshev61]) for every $t\in G$ define the operator on ${{\cal H}}_X$, $$U^t {:=}\int_{{\widehat}{G}} \overline{{\left\langle \chi,t \right\rangle}}\, \tilde{E}(d\chi), \quad t\in G.$$ Clearly ${\mathcal{U}}{:=}\{U^t\!:\!t\in G\}$ is a group of unitary operators indexed by $G$. Moreover for every $v\in {{\cal H}}_X$, $$\|(U^t - I)v \|_{{{\cal H}}}^2 = \int_{{\widehat}{G}} \left| \overline{{\left\langle \chi,t \right\rangle}} -1 \right|^2\mu_v(dx)$$ where $\mu_v(dx) = \|E(dx)v\|_{{{\cal H}}}^2$ is a finite non-negative measure on ${\widehat}{G}$. From Lebesgue dominated convergence theorem we therefore conclude that the unitary operator group ${\mathcal{U}}$ is continuous, and hence it is a unitary representation of $G$ in ${{\cal H}}_X$. Note that for $t=k\in K$, we have $U^k = V^k$. Define $P(t) {:=}U^{-t}X(t)$, $t\in G$. Then $P$ is continuous and $$P(t+k) = U^{-t}U^{-k}X(t+k) = U^{-t}V^{-k}X(t+k) = U^{-t}X(t) = P(t), \quad t\in G,\,k\in K.$$ So $P$ is a continuous $K$-periodic field with values ${{\cal H}}_X$ and $ X(t) = U^t P(t)$, for every $t\in G$. [$\blacksquare$]{}
Theorem \[PCF-structure\] gives a good insight on the origin of the measures $\gamma_{\lambda}$, $\lambda\in\Lambda_K$. Indeed let $X$ be a $G/K$-square integrable $K$-PC field, ${\mathcal{U}}$ and $P$ be as defined in Theorem \[PCF-structure\]. Then $\|X(t)\|_{{{\cal H}}}=\|P(t)\|_{{{\cal H}}}$ and $\big(X(t+u),X(u)\big)_{{{\cal H}}}=\big(U^tP(t+u),P(u)\big)_{{{\cal H}}}$ for all $t,u\in G$. The PC field $X$ being $G/K$-square integrable, the field $P_K:G/K\to{{\cal H}}_X$ defined by $P=P_K\circ \imath$ is square integrable as well as the field $x\mapsto U^tP_K(\imath(t)+x)$ defined on $G/K$, for any $t\in G$. Denoting by ${\widehat}{P_K}:\Lambda_K\to{{\cal H}}_X$ the Fourier Plancherel transform of $P_K$, the Fourier Plancherel transform of $U^tP_K(\imath(t)+\cdot)$ coincides with the function $$\mu\mapsto\int_{{\widehat}{G}}\overline{{\left\langle \chi+\mu,t \right\rangle}}\,\tilde{E}(d\chi){\widehat}{P_K}(\mu).$$ where $\tilde{E}$ is the Borel operator-valued measure on ${\widehat}{G}$ defined in the proof of Theorem \[PCF-structure\]. Then thanks to Parseval equality, the spectral covariance function of the PC field $X$ verifies $$\begin{aligned}
a_{\lambda}(t)&=&\int_{G/K}{\left\langle \lambda,x \right\rangle}\big(U^tP_K(\imath(t)+x),P_K(x)\big)_{{{\cal H}}}\,{{\hbar}}_{G/K}(dx)\\
&=&
\int_{\Lambda_K}\left(\int_{{\widehat}{G}}\overline{{\left\langle \chi+\mu,t \right\rangle}}\,\tilde{E}(d\chi){\widehat}{P_K}(\mu),{\widehat}{P_K}(\mu-\lambda)\right)_{{{\cal H}}}\,{{\hbar}}_{\Lambda_K}(d\mu)\\
&=&
\int_{{\widehat}{G}}\overline{{\left\langle \rho,t \right\rangle}}\left(\int_{\Lambda_K}\left(\tilde{E}(d\rho-\mu){\widehat}{P_K}(\mu),{\widehat}{P_K}(\mu-\lambda)\right)_{{{\cal H}}}\,{{\hbar}}_{\Lambda_K}(d\mu)\right)\end{aligned}$$ for all $\lambda\in\Lambda_K$ and $t\in G$. The SO-spectral measure of the field $X$ is $$\gamma_{\lambda}(\Delta)=\int_{\Lambda_K}\left(\tilde{E}(\Delta-\mu){\widehat}{P_K}(\mu),{\widehat}{P_K}(\mu-\lambda)\right)_{{{\cal H}}}\,{{\hbar}}_{\Lambda_K}(d\mu), \quad\Delta\in{{\mathcal{B}}}({\widehat}{G}),\lambda\in\Lambda_K.$$ In comparison with expression (\[gamma-P\]), we see that the spectral resolution $\tilde{E}$ of the unitary operators group ${\mathcal{U}}$, in some sense, “spreads” the SO-spectral measure $\gamma_{\lambda}^P$ over ${\widehat}{G}$ to form $\gamma_\lambda$.
Theorem \[PCF-structure\] also suggests a possibility to decompose a PC field into stationary components. If the $K$-PC field $X$ is $G/K$-square integrable, we can therefore *formally* write $$\label{FTofX}
X(t) \approx\int_{\Lambda_K} \overline{{\left\langle \lambda, t \right\rangle}}X^\lambda(t)\,{{\hbar}}_{\Lambda_K} (d\lambda)$$ where $\{X^\lambda(t){:=}U^t {\widehat}{P_K}(\lambda): t\in G\}$, $\lambda\in\Lambda_K $, is a family of jointly stationary fields over $G$. If in addition ${\widehat}{P_K}$ is integrable, then integral (\[FTofX\]) exists, and we have equality for any $t$. The integrability condition on ${\widehat}{P_K}$ being satisfied if $\Lambda_K$ is compact, that is in particular when $G={\mathbb{Z}}^n$, we deduce the following result.
\[cor-Xrep\] Let $X$ be a $G/K$-square integrable $K$-PC field over $G={\mathbb{Z}}^n$. Then there exists a family $\{X^\lambda\!:\! \lambda \in \Lambda_K\}$ of jointly stationary fields over $G$ in ${{\cal H}}_X$ such that $$X(t) = \int_{\Lambda_K} e^{i\lambda t'} X^\lambda(t)\, {{\hbar}}_{\Lambda_K} (d\lambda),
\quad t\in G,$$
Note that the pair $({\mathcal{U}}, P)$ in Theorem \[PCF-structure\] is highly non-unique since there are many ways to extend ${\mathcal{V}}=\{V^k\!:\!k\in K\}$ into ${\mathcal{U}}=\{U^t\!:\!t\in G\}$. Consequentlty the family $\{X^\lambda\!:\! \lambda \in\Lambda_K\}$ above is likewise not unique.
If $G/K$ is compact, then every $K$-PC field is $G/K$-square integrable, $\Lambda_K$ is countable, and the integrals above become series. If $G = {\mathbb{Z}}^n$ and $G/K$ is compact (and hence finite), then $\Lambda_K$ is finite and Corollary \[cor-Xrep\] yields the following ${\mathbb{Z}}^n$ version of Gladyshev’s representation of PC sequences included in [@gladyshev61].
Suppose that $X$ is a $K$-PC field over ${\mathbb{Z}}^n$ and that ${\mathbb{Z}}^n/K$ is compact. Then $\Lambda_K$ is finite and there is a finite family $\{X^\lambda\!:\! \lambda \in \Lambda_K\}$ of jointly stationary fields over $G$ in ${{\cal H}}_X$ such that for every $t\in G$ $$X(t) = \sum_{\lambda \in \Lambda_K} e^{i\lambda t'}X^\lambda(t).$$
If $\Lambda_K$ is not compact then even in the case of a periodic function, its Fourier transform does not have to converge everywhere.
Examples {#sect:Examples}
========
In this section $G={\mathbb{Z}}^n\times {\mathbb{R}}^m$, the Haar measure on ${\mathbb{Z}}^n$ is the counting measure, the Haar measure on ${\mathbb{R}}^m$ is $dt/(\sqrt{2\pi})^{m}$ where $dt$ is the Lebesgue measure on ${\mathbb{R}}^m$, ${\widehat}{{\mathbb{Z}}^n}$ will be identified with $[0,2\pi)^n$ with addition *mod* $2\pi$, ${\widehat}{{\mathbb{R}}^m}$ will be identified with ${\mathbb{R}}^m$, the Haar measures on $[0,2\pi)^n$ is $dt/(2\pi)^{n}$, ${\widehat}{G} = [0,2\pi)^n \times {\mathbb{R}}^m$, elements of $G$ and ${\widehat}{G}$ are row vectors, and the value of a character $\chi \in {\widehat}{G}$ at $t\in G$ is ${\left\langle \chi,t \right\rangle} = e^{-i \lambda t'}$, where $t'$ is transpose of $t$. Remembering previous sections, in order to describe the domain of the SO-spectrum of a $K$-PC field $X$ over $G$ the only task is to identify $G/K$ and ${\widehat}{G/K}$ as concrete subsets $Q$ and $\Lambda_K$ of $G= {\mathbb{Z}}^n\times {\mathbb{R}}^m$ and ${\widehat}{G} = [0,2\pi)^n\times {\mathbb{R}}^m$, respectively, in the way that the value of character $\lambda \in \Lambda_K$ at $t\in Q$ is still ${\left\langle \lambda,t \right\rangle} = e^{-i \lambda t'}$. This identification, which is obvious when $X$ is coordinate-wise PC, may be less trivial in the case of more complex $K$. It may be helpful, and is worth, to note that any closed nontrivial subgroup $K$ of $G={\mathbb{Z}}^n\times {\mathbb{R}}^m$ is isomorphic to ${\mathbb{Z}}^k\times {\mathbb{R}}^l$ for some $k, l\in{\mathbb{N}}$ such that $l\leq m$ and $1\leq k+l\leq n+m$. This isomorphism, which at least in the case of $G={\mathbb{Z}}^n$ or $G= {\mathbb{R}}^m$ can be found by selecting a proper basis for $G$ (see [@HR Theorem 9.11 and A.26]), provides a description and a parametrization of the sets $Q$ and $\Lambda_K$. As before $\big[a\big]_{b}$ will denote the remained in integer division of $a$ by $b$, $b>0$.
First we briefly revisit a one-parameter case and its slight extension.
\[ex-TPC\]
Suppose that $X$ is a PC process with period $T>0$. Then $K = \big\{kT\!:\! k\in {\mathbb{Z}}\big\}$, ${\mathbb{R}}/K = [0,T)$ with addition modulo $T$ and $\Lambda_K = \big\{ \frac{2\pi k}{T}\!:\! k \in {\mathbb{Z}}\big\}$. Moreover for every $\lambda = \frac{2\pi k}{T}\in \Lambda_K$, $ a_\lambda (t) {:=}a_k(t) = \frac{1}{T} \int_0^{T} e^{-i s \frac{2\pi k}{T}} {{\mathbf K}}_X(t+s,s)\, ds$ and there is a measure $\gamma_k$ on ${\mathbb{R}}$ such that $ a_k(t) = \int_{{\mathbb{R}}} e^{itu}\, \gamma_k(du)$ (Theorem \[thm-spectrum\]). The domain of the SO-spectrum of $X$ is $L = \bigcup_{k\in {\mathbb{Z}}} L_k$, where $L_k {:=}\big\{\big(u, u-\frac{2\pi k}{T}\big)\!:\! u \in {\mathbb{R}}\big\}$. The part of the SO-spectrum that sits on $L_k$ is a measure $\digamma_{\!\!k}$ defined as $\digamma_{\!\!k}=\gamma_k\circ\ell_{k}^{-1}$ where $\ell_{k}: {\mathbb{R}}\to {\mathbb{R}}^2$, $\ell_{k}(u) = \big(u, u-\frac{2\pi k}{T}\big)$. If $\sum_k \mathrm{Var}(\gamma_k) < \infty$ then the process $X$ is harmonizable and $\digamma\! {:=}\sum_k \digamma_{\!\!k}$ is a measure on ${\mathbb{R}}^2$ which satisfies relation (\[1par-spectrum\]).
If $X$ is stationary then $K = {\mathbb{R}}$, ${\mathbb{R}}/K = \{0\}$, $\Lambda_K = \{0\}$, $a_0 (t) = \int_{\{0\}} {{\mathbf K}}_X(t+s,s)\, \delta_0(ds) = {{\mathbf K}}_X(t,0)$. By Theorem \[thm-spectrum\] there is a measure $\gamma_0$ on ${\mathbb{R}}$ such that $a_0 (t) = \int_{{\mathbb{R}}} e^{itu}\, \gamma_0(du)$. Consequently, the SO-spectrum of $X$ is the measure $\digamma\!\! = \digamma_{\!\!0} = \gamma_0\circ \ell_{0}^{-1}$, which sits on the diagonal $L_0 = \big\{(u, u)\!:\! u \in {\mathbb{R}}\big\}$.
To see the need for the square integrability assumption, let us add one parameter to the above process; that is, let us consider a field $X=\{X(s,t)\!:\! (s,t)\in{\mathbb{R}}^2\}$ such that ${{\mathbf K}}_X\big((s,t), (u,v)\big) = {{\mathbf K}}_X\big((s+T,t), (u+T,v)\big)$, $s,t,u,v \in {\mathbb{R}}$ ($T>0$ is fixed). Then $K = \big\{(kT,0)\!:\! k\in {\mathbb{Z}}\big\}$, ${\mathbb{R}}^2/K = [0,T)\times {\mathbb{R}}$ with addition modulo $T$ on the first coordinate, $\Lambda_K = \big\{ \big(\frac{2\pi k}{T},x\big)\!:\! k \in {\mathbb{Z}}, x\in {\mathbb{R}}\big\}$ and $$a_\lambda (s,t) {:=}a_{k,x}(s,t) = \frac{1}{T\sqrt{2\pi}} \int_0^{T}\!\! \int_{{\mathbb{R}}} e^{-i \big(u\frac{2\pi k}{T} + vx\big) } {{\mathbf K}}_X\big((s+u,t+v), (u,v)\big)\, du dv$$ for $\lambda=\big(\frac{2\pi k}{T},x\big)\in \Lambda_K$. The square integrability assumption $\int_0^{T} \left[ \int_{{\mathbb{R}}} \| X(s,t)\|_{{{\cal H}}}^2 dt \right]\,ds < \infty$ assures that the above integral exists. If it does then, in view of Theorem \[thm-spectrum\], for every $k\in {\mathbb{Z}}$ and $x\in {\mathbb{R}}$ there exists a measure $\gamma_{k,x}$ on ${\mathbb{R}}^2$ such that $a_{k,x}(s,t) = \int_{{\mathbb{R}}} e^{i (su + tv) }\, \gamma_{k,x}(du,dv)$. The domain of the SO-spectrum of $X$ is $L = \bigcup_{k\in {\mathbb{Z}}} \bigcup_{x\in {\mathbb{R}}} L_{k,x}$, where $L_{k,x}$ is a two-dimensional plane in ${\mathbb{R}}^4$, $L_{k,x} {:=}\big\{\big(u, v, u-\frac{2\pi k}{T}, v-x\big)\!:\! u,v \in {\mathbb{R}}\big\}$. The “part” of the SO-spectrum that sits on $L_{k,x}$ is a measure $\digamma_{\!\!k,x}$ defined as $\digamma_{\!\!k,x}{:=}\gamma_{k,x}\circ\ell_{k,x}^{-1}$, where $\ell_{k,x}: {\mathbb{R}}^2 \to {\mathbb{R}}^4$ is defined by $\ell_{k,x}(u,v) {:=}\big(u, v, u-\frac{2\pi k}{T}, v-x\big)$. If $\mathrm{Var}(\digamma_{\!\!k,x}) \leq \omega(k,x)$ and $\sum_k \int_{{\mathbb{R}}} \omega(k,x)\, dx < \infty$, then $X$ is harmonizable and the SO-spectral measure of $X$ is $\digamma\! = \frac{1}{\sqrt{2\pi}} \sum_k \int_{{\mathbb{R}}} \digamma_{\!\!k,x}\, dx$ (see Theorem \[thm-harmsp\]). Note that $L$ above is, in fact, the union of countably many three-dimensional hyperplanes $D_k$ in ${\mathbb{R}}^4$, $D_k {:=}\bigcup_{x\in {\mathbb{R}}} L_{k,x} =\big\{(u, v, u-\frac{2\pi}{T}, v-x)\!:\! u,v,x \in {\mathbb{R}}\big\}$, which are parallel to the “diagonal” $D_0$.
If the field $X=\{X(s,t)\!:\!(s,t)\in{\mathbb{R}}^2\}$ is stationary in $s$, then $\Lambda_K = \big\{ (0,x)\!:\! x\in {\mathbb{R}}\big\}$, the condition of the square integrability of $X$ means that $ \int_{{\mathbb{R}}} \|X(0,t)\|^2_{{{\cal H}}}\, dt < \infty$ and, if the latter is satisfied, the SO-spectrum of $X$ sits on the three-dimensional hyperplane in ${\mathbb{R}}^4$, $L {:=}D_0 = \big\{(u, v+x, u, x)\!:\! u,v,x \in {\mathbb{R}}\big\}$. [$\blacksquare$]{}
Next example contains a complete analysis of the SO-spectrum of a weakly PC field.
\[ex-wk\]
Let $T$ and $S$ be two non-zero integers. Suppose that the field $X$ on ${\mathbb{Z}}^2$ is [weakly PC]{} with period $(T,S)$ ([@hurdkf]), that is $${{\mathbf K}}_X\big((m,n),(u,v)\big) = {{\mathbf K}}_X\big((m+T,n+S),(u+T,v+S)\big), \quad\mbox{for all}\,\, n,m,u,v\in{\mathbb{Z}}$$ Here $K=\big\{k(T,S)\!:\! k\in {\mathbb{Z}}\big\}$. We assume that at least one of $T$ or $S$ is positive. Let $d{:=}\gcd(T,S)$ be the greatest common positive integer divisor of $T$ and $S$, so that $(T,S) = d\times (T_1,S_1)$ and $\gcd(T_1,S_1) = 1$. From Bezout’s lemma there are integers $q,p$ such that $T_1 q - S_1 p$ = 1. Let $\phi$ be a mapping of ${\mathbb{Z}}^2$ onto itself, given by $\phi(m,n) = (m,n)\Phi'$, where $\Phi = \left(\begin{array}{cc}
T_1 & p \\ S_1 & q \\
\end{array} \right)$, and $\Phi'$ stands for the transpose matrix of the matrix $\Phi$. Because $\det \Phi = 1$, $\phi$ is an isomorphism. Since $\phi(dk,0) = (kT,kS)$ for $k \in {\mathbb{Z}}$, we have $K = \phi(d{\mathbb{Z}}\times \{0\})$ and we identify $G/K$ to $$Q {:=}\phi\big(\{0,\dots, d-1\} \times {\mathbb{Z}}\big) = \big\{(kT_1+lp, kS_1+lq)\!:\! k=0,\dots,d-1, l\in
{\mathbb{Z}}\big\}.$$ The dual mapping $\psi(s,t) = \big[(s,t) \Phi^{-1}\big]_{2\pi} = \big(\big[qs - S_1t\big]_{2\pi}, \big[- ps + T_1 t\big]_{2\pi}\big)$, $s, t \in [0,2\pi)$, maps $\big\{\frac{2\pi k}{d}\!:\! k=0,\dots,d-1\big\} \times [0,2\pi)$, which is the dual of $\{0,\dots, d-1\} \times {\mathbb{Z}}$, onto the dual $\Lambda_K$ of $Q$. The construction that we use produces a convenient parametrization of $\Lambda_K$ as the union of $d$ lines : $\Lambda_K = \bigcup_{k=0}^{d-1}\Lambda_k$ where $$\Lambda_k {:=}\left\{\left(\left[\frac{2\pi kq}{d} - S_1t\right]_{2\pi},\left[\frac{- 2\pi kp}{d} + T_1 t\right]_{2\pi}\right)\!:\! t \in [0,2\pi)\right\}.$$ Note that the value of a character $(u,v)=\psi(s,t) \in \Lambda_K$ at $(m,n) = \phi(k,l) \in Q$ is equal to $e^{-i (mu+nv)} = e^{-i (s,t) \Phi^{-1} \Phi (k,l)'} = e^{-i (ks + lt)}$, as required. Assume that $X$ is $G/K$-square integrable, for example that $ \sum_{n = -\infty}^{\infty} \|X(m,n)\|^2_{{{\cal H}}} < \infty$, for any $m=1,\dots,T-1$. Then from the previous discussion and the results of Section \[sec-spectrum\] we deduce the following properties.
1. If $(u,v) \in \Lambda_K$, then $u= \big[\frac{2\pi kq}{d} - S_1t\big]_{2\pi}$, $v= \big[\frac{- 2\pi kp}{d} + T_1 t\big]_{2\pi}$ for some unique $t\in[0,2\pi)$ and unique $k=0,\dots,d-1$. Hence the spectral covariance $a_{(u,v)} (m,n) =: a_{k,t} (m,n)$ of $X$ at $(m,n) \in {\mathbb{Z}}^2$ is equal to $$\hspace{-2mm}
a_{k,t} (m,n) = \frac{1}{d} \sum_{j=0}^{d-1} \sum_{l = -\infty}^{\infty} e^{-i\big(j\frac{2\pi k q }{d} + lt\big)} {{\mathbf K}}_X\big((m+jT_1+lp, n+jS_1+lq),(jT_1+lp, jS_1+lq)\big).$$
2. For each $k=0,\dots,d-1$ and $t\in [0,2\pi)$ there exists a measure $\gamma_{k,t}$ on $[0,\pi)^2$ such that $$a_{k,t} (m,n) = \int_0^{2\pi} \int_0^{2\pi} e^{i(mx+ny)}\, \gamma_{k,t} (dx, dy).$$
3. The SO-spectrum of $X$ sits on the set $ L = \bigcup_{k=1}^{d-1} \bigcup_{t\in [0,2\pi)} L_{k,t}$, where $ L_{k,t} $ is a two-dimensional plane in $[0,2\pi)^4$ $$L_{k,t} {:=}\left \{ \left(x,y,\left[x-\frac{2\pi kq}{d} + S_1t\right]_{2\pi},\left[y+\frac{ 2\pi kp}{d} - T_1t\right]_{2\pi}\right)\!:\!
x, y \in [0,2\pi)\right \}.$$ Note that $D_k = \bigcup_{t\in [0,2\pi)} L_{k,t}$ is a three-dimensional hyperplane in $[0,2\pi)^4$, so $L$ is, in fact, the union of $d$ three-dimensional hyperplanes. If $d=\gcd(T,S)=1$, then $L = D_0 = \left \{ \left(x,y,\big[x+St\big]_{2\pi},\big[y-Tt\big]_{2\pi}\right) \!:\! x, y, t \in [0,2\pi) \right\}.$ In this case the field $X$ is a rotation of the field $Y$ defined by $Y(m,n){:=}X\big((m,n) \Phi'\big)$, $m,n\in{\mathbb{Z}}$, which is stationary in $m$. Indeed $X(m,n)= Y\big((m,n)(\Phi')^{-1}\big)$.
4. If there is an integrable function $\omega:[0,2\pi)\to[0,\infty)$ such that $ Var (\gamma_{k,t}) \leq \omega(t)$ for all $k$ and $t$, then $X$ is harmonizable and $${{\mathbf K}}_X\big((m,n),(j,r)\big) = \int_0^{2\pi} \int_0^{2\pi} \int_0^{2\pi} \int_0^{2\pi} e^{i (mu + nv - jx - ry)}\,\digamma\! (du,dv,dx,dy).$$ The SO-spectral measure $\digamma\!$ of $X$ is given by $ \digamma\! (\Delta) = \sum_{k=0}^{d-1} \int_0^{2\pi} \digamma_{\!\!k,t} (\Delta)\, dt $, where $\digamma_{\!\!k,t}$ is the complex measure on $[0,\pi)^4$ whose support is contained in the plane $L_{k,t}$ and defined by $$\hspace{-9mm}\digamma_{\!\!k,t}(\Delta)\!{:=}\gamma_{k,t} \left\{ (x,y)\in [0,2\pi)^2\!:\! \left(x,y,\left[x-\frac{2\pi kq}{d} + S_1t\right]_{2\pi},\left[y+\frac{2\pi kp}{d} - T_1t\right]_{2\pi}\right)\in \Delta \right\}.$$
Figure 1 is the graph of the set $\Lambda_K$ defined previously in the case when $T=12$ and $S = 9$ ($d=3$, $p=q=1$). Then $\Lambda_K=\Lambda_0\cup\Lambda_1\cup\Lambda_2$ consists of three lines, which are shown with different width pattern. If now from each point on the graph we draw the rectangle $[0,2\pi)\times [0,2\pi)$ then the resulting three-dimensional body in $[0,2\pi)^4$ is the domain of the SO-spectrum of the field $X$. [$\blacksquare$]{}
\[fig1\] {width="2.5in" height="2.5in"}\
Fig 1: Graph of $\Lambda_K$
The last example is a particular example of strongly PC fields over ${\mathbb{R}}\times {\mathbb{Z}}^2$. It combines a mixture of continuous and discrete structures.
Suppose that $X$ is a field over $G {:=}{\mathbb{R}}\times{\mathbb{Z}}^2$ such that $X(t,m,n) = X(t+4,m,n) = X(t,m+1,n+3) = X(t,m+2,n)$ for all $t\in {\mathbb{R}}$, $m,n\in {\mathbb{Z}}.$ Then $K = \{k(4,0,0)+j(0,1,3) + l(0,2,0)\!:\! k,l,j \in {\mathbb{Z}}\}$. In order to describe $G/K$ and $\Lambda_K$ we consider a change of basis of $G={\mathbb{R}}\times{\mathbb{Z}}^2$ defined by the mapping $$\phi(t,m,n) {:=}(t,m,n)\Phi',\qquad
\mbox{where}\quad\Phi = \left(
\begin{array}{ccc}
1& 0 & 0 \\
0 &1 & 0\\
0 &3 & 1 \\
\end{array}\right).$$ Then the mapping $\phi$ is an isomorphism of $G$ onto itself and $K = \phi(P)$, where $P= \{(4k, j, 6l)\!:\! k,l,j \in {\mathbb{Z}}\} = 4{\mathbb{Z}}\times {\mathbb{Z}}\times 6{\mathbb{Z}}$. To see this note that $2(0,1,3) - (0,2,0)= (0,0,6)$, so that $K$ is generated by the 3-tuples $(4,0,0)$, $(0,1,3)$ and $(0,0,6)$, which are respectively equal to $\phi(4,0,0)$, $\phi(0,1,0)$, and $\phi(0,0,6)$. The quotient $G/P = [0,4)\times \{0\} \times \{0,\dots,5\}$, so we take $Q{:=}\phi(G/P) = \left\{ (s,0,l)\!:\! s\in [0,4), l=0,\dots 5\right\}$. The dual of $G/P$ is $\Lambda_P = \frac{2\pi}{T} {\mathbb{Z}}\times \{0\} \times \big\{\frac{\pi r}{3}\!:\! r=0,\dots,5\big\} $ and hence the dual of $G/K$ can be represented as $\Lambda_K = \psi(\Lambda_P)$, where $\psi$ is the isomorphism of ${\widehat}{G} = {\mathbb{R}}\times [0,2\pi)^2$ onto itself defined by $\psi(t,u,v) {:=}(t,u,v)\Phi^{-1} = \big(t, \big[u-3v\big]_{2\pi}, v\big)$. Therefore $$\Lambda_K = \left\{\left(\frac{2\pi k}{T}, \big[-\pi r\big]_{2\pi}, \frac{\pi r}{3}\right)\!:\! k\in {\mathbb{Z}}, r=0,\dots,5 \right\},$$ is countable. Note that $\big[-\pi r\big]_{2\pi}$ is either $\pi$ (if $r$ is odd) or 0. For each $\lambda_{k,r} = \big(\frac{2\pi k}{T}, \big[-\pi r\big]_{2\pi}, \frac{\pi r}{3}\big) \in \Lambda_K$, the corresponding spectral covariance is given by $$a_{k,r} (t,m,n) = \frac{1}{24}\sum_{l=0}^5\int_0^4 e^{-i(\frac{2\pi k s}{T} + \frac{\pi r l}{3})} {{\mathbf K}}_X\big((t+s, m, n+ l),(s, 0, l)\big)\, ds,$$ and for each ${k,r}$ there exists a measure $\gamma_{k,r}$ on ${\mathbb{R}}\times [0,2\pi)^2$ such that $$a_{k,r} (t,m,n) = \int_{{\mathbb{R}}} \int_0^{2\pi} \int_0^{2\pi}e^{i(ts+mu+nv)}_, \gamma_{k,r}(ds, du,dv).$$ The SO-spectrum of the field $X$ sits on the union of countably many hyperplanes $$L_{k,r} {:=}\left\{\left(s,u,v,s-\frac{2\pi k}{T},\left[u+\pi r\right]_{2\pi}, \left[v-\frac{\pi r}{3}\right]_{2\pi}\right)\!:\! s\in {\mathbb{R}}, u,v \in [0,2\pi)\right\},$$ $k\in {\mathbb{Z}}$, $r=0,\dots,5$, of ${\mathbb{R}}\times [0,2\pi)^2\times{\mathbb{R}}\times [0,2\pi)^2 $. If the sum of total variations of measures $\gamma_{k,r}$ is finite, then $X$ is harmonizable and its SO-spectral measure $\digamma\! = \sum_{k=-\infty}^{\infty} \sum_{r=0}^{5} \digamma_{\!\!k,r}$, where $\digamma_{\!\!k,r} = \gamma_{k,r}\circ \ell_{k,r}^{-1}$ and $\ell_{k,r}(s,u,v) {:=}\big(s,u,v,s-\frac{2\pi k}{T},\big[u+\pi r\big]_{2\pi}, \big[v-\frac{\pi r}{3}\big]_{2\pi}\big)$.
Note that if we define $Y(t,n,m) {:=}X\big((t,n,m) \Phi'\big)$, then $Y$ is PC in $t$ with period $T=4$, stationary in $n$, and PC in $m$ with period $M=6$. Since $X(t,n,m) = Y\big((t,n,m) (\Phi')^{-1}\big)$, one can therefore say that the field $X$ is periodically correlated in direction $(1,0,0)$ with period $T=4$, stationary in direction of $(0,1,3)$ and periodically correlated in direction $(0,0,1)$ with period $M=6$. [$\blacksquare$]{}
#### [****]{}ACKNOWLEDGEMENTS
The paper was partially written during the author’s stay at Université Rennes 2, Rennes, France, in June 2011.
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[^1]: The paper was partially written during the author’s stay at Université Rennes 2 - Haute Bretagne, Rennes, France, in June 2011.
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abstract: 'While slowly turning the ends of a single molecule of DNA at constant applied force, a discontinuity was recently observed at the supercoiling transition, when a small plectoneme is suddenly formed. This can be understood as an abrupt transition into a state in which stretched and plectonemic DNA coexist. We argue that there should be discontinuities in both the extension and the torque at the transition, and provide experimental evidence for both. To predict the sizes of these discontinuities and how they change with the overall length of DNA, we organize a phenomenological theory for the coexisting plectonemic state in terms of four parameters. We also test supercoiling theories, including our own elastic rod simulation, finding discrepancies with experiment that can be understood in terms of the four coexisting state parameters.'
author:
- 'Bryan C. Daniels'
- Scott Forth
- 'Maxim Y. Sheinin'
- 'Michelle D. Wang'
- 'James P. Sethna'
bibliography:
- 'PlectonemePaper.bib'
title: Discontinuities at the DNA supercoiling transition
---
=1
A DNA molecule, when overtwisted, can form a [*plectoneme*]{} [@StrAllBen96; @CruKosSei07] (inset of [Fig. \[fig:ExtensionAndTorqueVsK\]]{}), a twisted supercoil structure familiar from phone cords and water hoses, which stores added turns (linking number) as ‘writhe.’ The plectoneme is not formed when the twisted DNA goes unstable (as in water hoses [@HeiNeuGos03]), but in equilibrium when the free energies cross — this was vividly illustrated by a recent experiment [@ForDeuShe08] ([Fig. \[fig:Hopping\]]{}), which showed repeated transitions between the straight “stretched state” (SS, described by the worm-like chain model [@MarSig95b]), and a coexisting state (CS) of stretched DNA and plectoneme [@Mar07]. This transition, in addition to being both appealing and biologically important, provides an unusual opportunity for testing continuum theories of coexisting states. Can we use the well-established continuum theories of DNA elasticity to explain the newly discovered [@ForDeuShe08] jumps in behavior at the transition?
The recent experiment measures the extension (end-to-end distance) and torque of a single molecule of DNA held at constant force as it is slowly twisted [@ForDeuShe08]. A straightforward numerical implementation of the elastic rod model [@FaiRudOst96; @MarSig95a; @Neu04] for DNA in these conditions (with fluctuations incorporated via entropic repulsion [@MarSig95a]) leads to two quantitative predictions that are at variance with the experiment. First, the experiment showed a jump $\Delta z$ in the extension as the plectoneme formed ([Fig. \[fig:ExtensionAndTorqueVsK\]]{}) that appeared unchanged for each applied force as the overall DNA length was varied from 2.2 kbp to 4.2 kbp, whereas the simulation showed a significant increase in $\Delta z$ at the longer DNA length. Second, no discontinuity was observed in the (directly measured) filtered torque data ([Fig. \[fig:ExtensionAndTorqueVsK\]]{}), yet the simulation predicted a small jump.
Simulation is not understanding. Here we analyze the system theoretically, focusing on the physical causes of the behavior at the transition. We use as our framework Marko’s two-phase coexistence model [@Mar07; @Mar08], which we generalize to incorporate extra terms that represent the interfacial energy between the plectoneme and straight regions of the DNA. We show that any model of the supercoiling transition in this parameter regime can be summarized by four force-dependent parameters. After extracting these parameters directly from the experiments, we use them to predict the torque jump (which we then measure) and to explain why the extension jump appears length independent. Finally, we use our formulation to test various models of plectonemes, finding discrepancies mainly at small applied force.
The transition occurs at the critical linking number $K^*$ when the two states have the same free energy ${\mathcal{F}}$, where ${\mathcal{F}}$ is defined by the ensemble with constant applied force and linking number. We therefore need models for the free energy ${\mathcal{F}}$ and extension $z$ of the SS and CS.
![\[fig:ExtensionAndTorqueVsK\]Extension and torque as a function of linking number $K$, for $L=2.2$ kbp at $F=2$ pN. Black lines show data from Ref. [@ForDeuShe08], smoothed using a “boxcar” average of nearby points. The green lines show worm-like chain (WLC) predictions below the transition \[in the unsupercoiled “stretched” state (SS)\], and fits to the data after the transition \[in the “coexisting” state (CS)\], linear for the extension and constant for the torque. The size of the torque jump, not visible in the smoothed data, is implied by the coexisting torque $\tau$, the CS fit, and the transition linking number $K^*$ in the extension data. Inset: Simulated DNA showing the CS of a plectoneme and straight DNA, ignoring thermal fluctuations. The ends are held with fixed orientation and pulled with a constant force $F$, here 2 pN. ](ExtensionAndTorqueVsK2pN_smaller_landscape_inset){width="\linewidth"}
![\[fig:Hopping\]Directly measuring the torque jump by observing thermal hopping, for the same conditions as [Fig. \[fig:ExtensionAndTorqueVsK\]]{}. As linking number $K$ is slowly increased near $K^*$, thermal fluctuations induce hopping between states with (CS) and without (SS) a plectoneme. Averaging over these two states gives a direct way of measuring the torque jump: analogously to a lock-in amplifier, we set a threshold in the extension signal to separately average the SS (black) and CS (red) data near the transition. Using multiple traces, we find an average torque jump of $\Delta \tau = 2.9 \pm 0.7$ pN nm for $L=2.2$ kbp at $F=2$ pN. Additionally, this value of $\Delta \tau$ implies (see text) that the transition should happen over a range of linking number $K$ (top) of about 0.9 turns, as it does. ](Hopping){width="\linewidth"}
The properties of stretched, unsupercoiled DNA are well-established. At small enough forces and torques that avoid both melting and supercoiling, DNA acts as a torsional spring with twist elastic constant $C$ [@Mar07][^1]: ${\mathcal{F}}_{\mathrm{SS}}(K,L) = \frac{C}{2} \left( 2\pi \frac{K}{L} \right)^2 L - {F_{\mathrm{eff}}}L$, where $K$ is the added linking number, $L$ is the overall (basepair) length of DNA, the effective force ${F_{\mathrm{eff}}}= F - kT\sqrt{F/B}$ [@Mar07] (see supplemental material), $F$ is the force applied to the ends of the DNA, $B = 43 \pm 3$ nm$\times kT$ is the DNA’s bending elastic constant, $C$ = $89 \pm 3$ nm$\times kT$, and the thermal energy $kT=$ 4.09 pN nm for this experiment (at 23.5[$^\circ$]{}C). Differentiating with respect to $K$ gives the torque: $\tau_{\mathrm{SS}}= \frac{1}{2\pi} \frac{d {\mathcal{F}}_{\mathrm{SS}}}{dK} = 2\pi C \frac{K}{L}$. The extension of unsupercoiled DNA is shortened by thermal fluctuations, and in the relevant force regime is approximately given by $z_{\mathrm{SS}}= \xi(\tau_{\mathrm{SS}}) L$, where [@MorNel98] $$\label{xi}
\xi(\tau) = 1 - \frac{1}{2} \left[ \frac{B F}{(k T)^2}
-\left(\frac{\tau}{2 k T}\right)^2
-\frac{1}{32} \right]^{-1/2}.$$
Since supercoiling theories must include contact forces, they are less amenable to traditional theoretical methods. Even so, many theories have been successful in predicting properties of the CS; such methods have included detailed Monte Carlo simulations [@VolMar97], descriptions of the plectoneme as a simple helix [@MarSig95a; @Neu04; @ClaAudNeu08], and a more phenomenological approach [@Mar07]. However, none of these theories has yet been used to predict discontinuities at the SS–CS transition. Here we connect the free energy and extension predictions from any given model to the corresponding predictions for discontinuities at the transition.
We will use the framework of two-phase coexistence adopted by Marko [@Mar07; @Mar08] to describe the CS as consisting of two phases, each with constant free energy and extension per unit length of DNA [^2]. Since phase coexistence leads to a linear dependence on $K$ of the fraction of plectonemic DNA (keeping the torque fixed), in this model both ${\mathcal{F}}_{\mathrm{CS}}$ and $z_{\mathrm{CS}}$ are linear functions of added linking number $K$ and length $L$ (just as the free energy of an ice-water mixture is linear in the total energy, and the temperature remains fixed, as the ice melts). This linearity, along with the known properties of the SS, allows us to write ${\mathcal{F}}_{\mathrm{CS}}$ and $z_{\mathrm{CS}}$ as (see supplemental material) $$\begin{aligned}
\label{F_coex}
{\mathcal{F}}_{{\mathrm{CS}}}(K,L) &= {\mathcal{F}}_0
+ 2\pi \tau K - \left( \frac{\tau^2}{2C} + {F_{\mathrm{eff}}}\right) L ; \\
\label{z_coex}
z_{{\mathrm{CS}}}(K,L) &= -z_0
- q K
+ \left( \xi(\tau) + \frac{\tau}{2\pi C}q \right)L,\end{aligned}$$ where $q$ is the slope of extension versus linking number and $\tau$ is the CS torque. That is, ${\mathcal{F}}_{\mathrm{CS}}$ and $z_{\mathrm{CS}}$ are specified by four force-dependent values: their slopes with respect to $K$ ($\tau$ and $q$), which describe how the plectonemic phase coexists with the stretched phase; and $K=L=0$ offsets (${\mathcal{F}}_0$ and $z_0$), which describe the extra free energy and extension necessary to form the interface between the phases — the end loop and tails of the plectoneme.
The experimental observables can then be written in terms of these four values. Easiest are $\tau$ and $q$, which are directly measured. Next, the linking number $K^*$ at the transition is found by equating the CS free energy with that of the SS: ${\mathcal{F}}_{{\mathrm{CS}}}(K^*,L) = {\mathcal{F}}_{\mathrm{SS}}(K^*,L)$ implies $$\label{Kstar}
K^* = \frac{L}{2\pi C}(\tau + \Delta \tau),
\mathrm{with}~\Delta \tau = \sqrt{ \frac{2C}{L} {\mathcal{F}}_0 },$$ where $\Delta \tau$ is the jump in the torque at the transition. Lastly, inserting $K^*$ from [Eq. (\[Kstar\])]{} into [Eq. (\[z\_coex\])]{}, we find the change in extension at the transition: $$\begin{aligned}
\Delta z &=
z_0 + q \sqrt{\frac{L{\mathcal{F}}_0}{2\pi^2 C} }
- L \Big( \xi(\tau) - \xi( \tau + \sqrt{ 2C{\mathcal{F}}_0/L }) \Big).
\label{Deltaz}\end{aligned}$$
![\[fig:FourPlots\]The four parameters describing the CS (coexisting torque $\tau$, extension versus linking number slope $q$, and the extra free energy ${\mathcal{F}}_0$ and extension $z_0$ necessary to form the end loop and tails of the plectoneme), as a function of applied force. The circles show values calculated from experimental data taken at two different overall DNA lengths $L$. Model predictions for our simulation [@endnote20] and Marko’s model [@Mar07] are shown as solid and dashed lines, respectively (using $S=0$ for ${\mathcal{F}}_0$ predictions). The circular end-loop model uses average $\tau$ and $q$ values from the experiment to predict ${\mathcal{F}}_0$ and $z_0$, shown as dotted lines. ](FourPlots_smaller){width="\linewidth"}
 Predicted length-dependence of the extension and torque jumps at $F=2$ pN. Using the CS parameters extracted from the experiment at two different lengths, Eqs. (\[Kstar\]) and (\[Deltaz\]) predict the $L$-dependence of $\Delta z$ and $\Delta \tau$. The circles show experimentally-measured values \[with the torque jump here calculated from $K^*$ using [Eq. (\[Kstar\])]{}\]. Without entropic corrections to ${\mathcal{F}}_0$ ($S=0$; dot-dashed lines) $\Delta z$ depends noticeably on $L$, but including an initial estimate of $S$ (solid lines) shows that entropic effects can significantly reduce this length-dependence. (Right) Force-dependence of the extension and torque jumps, and predictions from two models. Disagreements with experimental data can be understood in terms of the four CS parameters in [Fig. \[fig:FourPlots\]]{}. Also plotted as a diamond is $\Delta \tau$ measured using the direct method depicted in [Fig. \[fig:Hopping\]]{}. ](Discontinuities_and_LengthDependence){width="\linewidth"}
To additionally include entropic effects, we can write ${\mathcal{F}}_0 = \mu - TS$, where $\mu$ is the energy cost for the end-loop and tails, and $S$ is the entropy coming from fluctuations in the location, length, and linking number of the plectoneme. Using an initial calculation of $S$ that includes these effects (in preparation; see supplemental material), we find that $S$ varies logarithmically with $L$, and that setting $S=0$ is a good approximation except when $L$ changes by large factors.
Given experimental data ($\tau$, $q$, $K^*$, and $\Delta z$), we can solve for the four CS parameters. The results from Ref. [@ForDeuShe08] are shown as circles in [Fig. \[fig:FourPlots\]]{} for the two overall DNA lengths tested. If we assume that the DNA is homogeneous, we expect the results to be independent of $L$ (except for a logarithmic entropic correction to ${\mathcal{F}}_0$ that would reduce it at the longer $L$ by about $kT \log 2 \approx 5$ pN nm; see supplemental material). We do expect ${\mathcal{F}}_0$ and $z_0$ to be sensitive to the local properties of the DNA in the end-loop of the plectoneme, so we suspect that the difference in $z_0$ between the two measured lengths could be due to sequence dependence. With this data, we can also predict the length-dependence of the discontinuities, as shown in [Fig. \[fig:Discontinuities\]]{} (left). Here we included entropic corrections to ${\mathcal{F}}_0$ (see supplemental material), and we find that entropic effects significantly decrease the length-dependence of the extension jump.
Note that here we are solving for the experimental size of the torque jump using the observed $K^*$ and $\tau$ in [Eq. (\[Kstar\])]{}. We also find direct evidence of $\Delta \tau$ in the data by averaging over the torque separately in the SS and CS near the transition ([Fig. \[fig:Hopping\]]{}). With data taken at $F=2$ pN and $L=2.2$ kbp, we find $\Delta \tau = 2.9 \pm 0.7$ pN nm, in good agreement with the prediction from $K^*$ ($3.9 \pm 2.6$ pN nm; see [Fig. \[fig:Discontinuities\]]{}). We can also predict the width of the range of linking numbers around $K^*$ in which hopping between the two states is likely (where $|\Delta {\mathcal{F}}|<kT$): expanding to first order in $K-K^*$ gives a width of $2 kT / (\pi \Delta \tau$). This predicts a transition region width of about 0.9 turns for the conditions in [Fig. \[fig:Hopping\]]{}, agreeing well with the data.
We can now use various plectoneme models to calculate the four CS parameters, which in turn give predictions for the experimental observables. The results are shown as lines in [Fig. \[fig:FourPlots\]]{} and [Fig. \[fig:Discontinuities\]]{} (right). As we expect entropic corrections to be small (changing ${\mathcal{F}}_0$ by at most about 5 pN nm), we set $S=0$ for these comparisons.
First, we test Marko’s phase coexistence model [@Mar07]. The plectoneme is modeled as a phase with zero extension and an effective twist stiffness $P < C$. Shown as dashed lines in [Fig. \[fig:FourPlots\]]{}, the Marko model predicts the coexisting torque and extension slope well, with $P$ as the only fit parameter (we use $P=26$ nm). However, the Marko model (and any model that includes only terms in the free energy proportional to $L$) produces ${\mathcal{F}}_0=0$ and $z_0=0$.
In order to have a discontinuous transition, we must include the effects of the end loop and tails of the plectoneme. The simplest model assumes that the coexistence of stretched and plectonemic DNA requires one additional circular loop of DNA. Minimizing the total free energy for this circular end-loop model gives $$\begin{aligned}
{\mathcal{F}}_0 &= 2\pi \sqrt{2B{F_{\mathrm{eff}}}} - 2\pi \tau {\mathrm{Wr}}_{\mathrm{loop}}; \label{circularz0} \\
z_0 &= 2\pi \xi(\tau) \sqrt{B/(2{F_{\mathrm{eff}}})} - q{\mathrm{Wr}}_{\mathrm{loop}}, \label{circularF0}\end{aligned}$$ where ${\mathrm{Wr}}_{\mathrm{loop}}$ is the writhe taken up by the loop. For a perfect circle, ${\mathrm{Wr}}_{\mathrm{loop}}=1$, and ${\mathrm{Wr}}_{\mathrm{loop}}<1$ for a loop with two ends not at the same location. We chose ${\mathrm{Wr}}_{\mathrm{loop}}=0.8$ as a reasonable best fit to the data. Using the experimentally measured $\tau$ and $q$, the predictions are shown as solid lines in [Fig. \[fig:FourPlots\]]{} and [Fig. \[fig:Discontinuities\]]{}; ${\mathcal{F}}_0$ is fit fairly well, but $z_0$ is underestimated, especially at small applied forces.
In an attempt to more accurately model the shape of the plectoneme, we use an explicit simulation of an elastic rod, with elastic constants set to the known values for DNA. We must also include repulsion between nearby segments to keep the rod from passing through itself. Physically, this repulsion has two causes: screened Coulomb interaction of the charged strands and the loss of entropy due to limited fluctuations in the plectoneme. We use the repulsion free energy derived for the helical part of a plectoneme in Ref. [@MarSig95a], modified to a pairwise potential form (see supplemental material). We find that the simulation does form plectonemes (inset of [Fig. \[fig:ExtensionAndTorqueVsK\]]{}), and we can extract the four CS parameters, shown as solid lines in [Fig. \[fig:FourPlots\]]{} [^3]\[footnote:RodSimulationChoice\]. Since ${\mathcal{F}}_0$ and $z_0$ are nonzero, we find discontinuities in the extension and torque at the transition; their magnitudes are plotted in [Fig. \[fig:Discontinuities\]]{}.
Both the circular loop model and the simulation produce torque and extension jumps of the correct magnitude, but in both cases $\Delta z$ has an incorrect dependence on force and too much dependence on length. Our approach provides intuition about the causes of the discrepancies by singling out the four values (connected to different physical effects) that combine to produce the observed behavior. Specifically, we can better understand why the models’ predictions are length-dependent: as displayed in [Fig. \[fig:Discontinuities\]]{} (top left), the negligible length-dependence observed in experiment is caused by a subtle cancellation of a positive length-dependence \[smaller than either model, and described by [Eq. (\[Deltaz\])]{}\] combined with a negative contribution coming from entropic effects. One would expect, then, that any plectoneme model (even one that explicitly includes entropic fluctuations) might easily miss this cancellation. In general, without this intuition, it is difficult to know where to start in improving the DNA models. The largest uniform discrepancy happens at small applied forces, where both models underestimate $z_0$ [^4], leading to an underestimate of $\Delta z$. We have examined various effects that could alter $z_0$, but none have caused better agreement (see also supplemental material). Adding to the circular end-loop model softening or kinking [@DuKotVol08] at the plectoneme tip, or entropic terms from DNA cyclization theories [@ShiYam84; @Odi96], uniformly [*decreases*]{} $z_0$. Increasing $B$ in [Eq. (\[circularz0\])]{} by a factor of four (perhaps due to sequence dependence) does raise $z_0$ into the correct range, but it also raises ${\mathcal{F}}_0$ from [Eq. (\[circularF0\])]{} to values well outside the experimental ranges. Finally, $z_0$ would be increased if multiple plectonemes form at the transition, but we find that the measured values of ${\mathcal{F}}_0$ are too large to allow for more than one plectoneme in this experiment.
Support is acknowledged from NSF Grants DMR-0705167 and MCB-0820293, NIH Grant GM059849, and the Cornell Nanobiotechnology Center.
[^1]: As described in Ref. [@MorNel97] (see also supplemental material), $C$ is renormalized to a smaller value by bending fluctuations. We use $C$ calculated from the torque measured in the experiment, which gives its renormalized value.
[^2]: The language of phase coexistence is approximate in that the finite barrier to nucleation in one-dimensional systems precludes a true (sharp) phase transition.
[^3]: We have also explored increasing the entropic repulsion by a constant factor of up to 3. Though this does bring the torques closer to the experiment, the only other significant change is a decrease in ${\mathcal{F}}_0$ (data not shown) — specifically, this does not change the discussed discrepancies between the simulation and experiment.
[^4]: Though the 4.2 kbp $z_0$ data alone would be arguably consistent with the model predictions, the 2.2 kbp data highlights the discrepancy at small applied forces.
|
---
abstract: 'Researchers in the humanities are among the many who are now exploring the world of big data. They have begun to use programming languages like Python or R and their corresponding libraries to manipulate large data sets and discover brand new insights. One of the major hurdles that still exists is incorporating visualizations of this data into their projects. Visualization libraries can be difficult to learn how to use, even for those with formal training. Yet these visualizations are crucial for recognizing themes and communicating results to not only other researchers, but also the general public. This paper focuses on producing meaningful visualizations of data using machine learning. We allow the user to visually specify their code requirements in order to lower the barrier for humanities researchers to learn how to program visualizations. We use a hybrid model, combining a neural network and optical character recognition to generate the code to create the visualization.'
author:
-
title: 'Advancing Visual Specification of Code Requirements for Graphs\'
---
Introduction
============
A new field has been formed within the traditional humanities fields called Digital Humanities (DH), or Computational Humanities. DH scholars are interested in harnessing the power of computers to analyze novels, poetry, and other writing to answer their research questions. Whether or not they have a specific question in mind, they want to manipulate their data and create visualizations to see if they can find surprising patterns. This indirect approach is called exploratory programming, and while it contains a lot of trial and error, it can yield fascinating results.\
Exploratory programming as a concept in the DH field has been developing over the past decade, where researchers use languages like Python or R and their corresponding libraries to provide new insight into their studies, utilizing natural language processing (NLP) or other text analysis techniques. With the ability to now analyze higher volumes of text than was previously possible to do manually (i.e., distant reading vs. close reading), there has been an influx of new results. One of the pioneers of the exploratory programming approach is Nick Montfort; this approach is introduced in the text *Exploratory Programming for the Arts and Humanities* [@b1]. Montfort discusses that when we are dealing with large amounts of data, the best way to both discover conclusions and communicate them is to create visualizations. Montfort points out that visualizations are crucial for exploratory programming in that they allow the researcher to see intermediary results and make decisions about what to look into next. Therefore, in many different regards, data visualizations are important for DH, and other fields.\
Most tools that are currently available for generating visualizations fall into one of two categories: (1) those that require significant programming experience and (2) those that never expose a user to the code. Researchers in non-Computer Science fields, and even some within it, don’t necessarily have the programming background to make good use of the such toolkits in (1). Matplotlib, for example, requires an understanding of Python and object-oriented concepts to some degree. While new users can find code examples online and perhaps cobble something together, they are limited by what search terms they know, and by their knowledge of programming. Furthermore, official documentation for such libraries can be hard to find and confusing to those unaccustomed with how to decipher it.\
For the tools that require no coding (e.g., Tableau, Voyant), while they are made to be easy to learn and get started quickly, they ultimately limit user options as they don’t provide access to the code directly. Users are left to either choose from the capabilities currently available, or request new features. Developers of these tools may take a long time to make and deploy these features, or never even implement them if they are not general enough. We wanted a tool that fits somewhere in between, that has a lower barrier to entry for those that are new to programming, but will allow users a full range of capabilities. This led to the idea that we could allow a user to visually specify what they want their graph to look like. This is what makes our tool novel compared to others. In this way it will also serve as an education aid, helping teach researchers who are simultaneously learning to code.\
Our primary target audience for this tool is DH researchers, who have started programming in Python environments, but still need help creating visualizations. We wanted our tool to generate fully executable example code that the researcher can then manipulate themselves. We trained a Convolutional Neural Network (CNN) to classify an input target image into one of many Matplotlib code templates. We also used optical character recognition (OCR) to scrape important text (e.g., title, labels) off of the example image and fill that into the template so that the researcher can have more insight into how the code works. If a user needs to generate a visualization for a paper or presentation, they would simply search online for a visualization that they would like to recreate. They would upload it to our tool and would get the Matplotlib code they need. The user would now only have to update some variable names and values to fill in their data and create their desired graph. This allows them to quickly have a working visualization but also grants them the ability to make as many modifications as they get more comfortable with programming.\
The tool we created is called *Graph2Library*, or G2L, and was developed in collaboration with a working group of DH and Literature Researchers. Their feedback was instrumental in determining the output of G2L, as well as the types of visualizations G2L should be able to generate. We implemented it as a web-based tool so that it is easy to access and use. Contributions of this paper include:
- A first step towards larger goal of Visual Requirements Engineering. Allowing researchers to specify their needs in terms of a graph visualization.
- Creation of a hybrid model that combines a neural classifier with OCR to produce editable code.
- Novel use of a code-based generator to produce a practically infinite synthetic training set.
- Evaluation that showed users found the system to be effective and performance-enhancing.
We present recent related work and how our approach differs in Section II. In Section III, we discuss the approach and models. Further, Sections IV and V share our results and conclusions and Section VI discusses possible future extensions.
Related Work
============
Related research work can be categorized into one of two areas: program synthesis (code generation), and creating hybrid models.
Program Synthesis
-----------------
Perhaps the most similar and interesting research has been done in the area of program synthesis. One such project generated libmypaint and CAD (Computer-Aided Design) code from a target image to then reconstruct one that cannot be distinguished from the original[@b6]. In this project, they adversarially trained a recurrent neural network (RNN) to generate images and attempt to fool a discriminator CNN that was being trained to distinguish between the original and synthetic. They used Earth Mover’s Distance (Wasserstein metric) to compare images and showed that it was more effective than $L^2$ Distance.\
The generative adversarial network (GAN) approach they took proved successful for generating their code, since they did not use a template based approach. However, we deemed this approach overly complicated for our use case of classifying graphs to their corresponding Matplotlib template. We like the GAN approach and may turn to it if we decide that we need more fine-grained code generation, i.e., lower-level Matplotlib code that is below our template level (see section VI Future Work). They worked with hand-drawn images, simple CAD models and, color portrait photographs. We will be using graphs, which on a complexity scale are somewhere in between the CAD models and color photographs, though because of the specific text on them we had to add another processing step.\
Other recent research in this area includes the use of RNNs for natural language to code, as Lin used [@b5], which will be discussed in further detail in II-B Hybrid Models. Seq2seq is a popular approach for these RNNs, because it generates output vectors of a different size than the input vectors. This is necessary when the input, e.g., an English sentence, differs in size from the output, e.g., a code snippet. However, since we will be *classifying* an image into one of a handful of code templates we do not need to use an RNN to generate variable length code output, but instead, a CNN.\
Another interesting foray into program synthesis is the creation of AutoPandas [@b19]. In this project they endeavor to solve a similar challenge, the complexity of modern libraries being too much for the novice programmer to learn. They take on the Pandas library, a Python based, Microsoft Excel-like data manipulating tool. They created a specification language of sorts by asking the user to provide an I/O example of how they want to transform their data. This example then gets passed to their neurally-backed program generator and code is generated and presented to the user. Their tool covers an impressive 119 transform functions in Pandas.\
There have also been projects that work in the opposite direction, generating visualizations from natural language[@b3] and from raw data[@b2]. We did not explore this avenue, as we liked the idea of simultaneously being able to teach the user how to code, instead of removing that visibility entirely.
Hybrid Models
-------------
While much can be accomplished when using one machine learning model alone, there is a recent rise in using multiple techniques to fit the problem domain. Different than an ensemble model, where many algorithms work to classify one thing, a hybrid model is a conglomeration of two or more algorithms or technologies, to work on different parts on the problem.\
For example, Lin’s project, Tellina, used an RNN and K-Nearest Neighbors (KNN). The RNN took natural language input (i.e., “I want to delete all files from June 5") and generated a Bash code template that would execute this command. To deal with specific values, such as a date like June 5, they removed it from the input to the RNN and used KNN to determine it’s variable type, which was then slotted back into the output code template [@b5]. Tellina is a web based tool that delivers results in real time. To evaluate their model, Lin conducted user studies that contrasted a control group of users that could only use the internet to get help completing some bash tasks with a group using Tellina. They summarized both quantitative and qualitative results from this study. Participants in the study spent an average of 22% less time to complete tasks when using Tellina. They also found that users felt positively about Tellina helping them to complete tasks faster, and that it did not hinder them as much as it assisted them (results from a survey of the participants). We conduct a similar user evaluation, but take a slightly different approach as described in section IV.\
{width="\linewidth"}
Another project explored generating LaTeXcode from hand drawn images which met with some success, they supported primitive drawing commands to create a line, circle, and rectangle. They trained a neural network with attention mechanism to learn to infer specs which in turn can render an image in one or more steps. In addition to this neural architecture, they corrected mistakes by adding a Sequential Monte Carlo (SMC) sampling scheme which helps guide the output. Their results show that the combined model with the neural architecture and SMC significantly outperforms each on their own [@b4].
Graph2Library
=============
This section describes the approach for building G2L, to complete (A) training of the CNN, (B) OCR of the images, (C) putting the templates together, and (D) the final output. Figure 3 is provided for reference with corresponding subsections.
Training the CNN
----------------
One challenge of using a neural network is that they typically require thousands of *labeled* training examples in order to learn effectively. This labeled data can be difficult to acquire and a very manual process, especially when working with a new dataset. We navigated this issue in two ways, by (1) using transfer learning, and (2) by synthesizing hundreds of example graphs with their corresponding labels.
**Graph Type** **Number Generated**
---------------- ----------------------
Bar Graph 300
Scatter Plot 300
Pie Chart 200
Heat Map 200
Color Map 200
: Training Data Generated
\[tab1\]
\
We chose to use Google’s *Inception* model as the pre-trained neural network for the transfer learning portion. This model has demonstrated 78.1% accuracy on the ImageNet dataset [@b8]. We are able to use the trained model (on over a million images) and retrain only the final layer with our specific categories of graphs. We benefit from the earlier layers of the CNN already being able to recognize key image features, such as edges. Also, because we added hundreds of new training images of graphs specific to what we are classifying we achieved over 98% training accuracy rate.\
To generate the specific training data, we wrote python code, that uses the Matplotlib library to generate graphs. We created six different classes of graphs: pie charts, bar graphs, stacked bar graphs, grouped bar graphs, scatter plots, and grouped scatter plots. We start with one code template per class, which is also considered the label of each synthesized graph. Each graph is generated, saved, and then the code template is automatically tweaked slightly (i.e., values, labels), while the bones of the template stay the same, and the process starts over again. See Figure 2 for an example of a synthesized grouped bar graph in which we manipulated the axes, data values, and title to generate many similar ones.
{width="\linewidth"}
{width="\linewidth"}
[**G2L Flow Diagram**]{}
The end result of this part of the process is that we have a CNN that can classify our graphs into their types. Each type has its corresponding code template. Now with a test image we can classify it into a specific type of graph and the code template, once completed in the following step, will generate a similar graph, of the same type (see Figure 1 and section A of Figure 3).
Optical Character Recognition
-----------------------------
The neural net alone can classify an image to the correct graph type and corresponding code template but there are other elements of the graph that are important to preserve in order to generate fully executable code on the other end. Elements such as the number of labels on the x-axis of a graph can indicate how many data points are on the graph, or at the very least how many x-ticks are labeled. We use the Google library *Tesseract* to conduct OCR on each test image. OCR allows us to gather the exact text, and information about its position on the image. We use the graph type obtained from the CNN and the text from our OCR step and can then perform more robust analysis to deliver a more complete code template. See Figure 3, section B for reference.\
The OCR step provides crucial information to G2L, specifically, any text on the image as well as where on the image it is located. This information is provided as the coordinates of the bounding box around each word found. Other helpful information, such as whether a set of words are considered part of a “sentence" (based on position and proximity) is also part of what this step provides to us. During trials of *Tesseract*, we discovered that while almost all horizontally aligned text is easily scraped, it is not able to scrape all text when it is rotated diagonally or vertically, but it can still pick up some. OCR is a tool that specializes in recognizing text, which means that it is not made for picking up lines, shapes, or other graphic features of the image. This is why we rely on the CNN to correctly categorize the image based on those features, and use OCR to get any additional information.
Merging into Executable Code Templates
--------------------------------------
Once we know the type of the graph we are working with from the CNN, and have all the text scraped off of the image, we can conduct additional semantic analysis and customize the code template. This happens in section C of Figure 3. For example, a word or sequence of words on the far left of a bar graph is most likely to be a y-axis label. This differs from a pie chart in that if text is found in the same position on it, it is more likely to be a label of one section of the pie. *Tesseract* helps provide this information by denoting it a sentence or phrase. Since graphs vary widely, these generalizations don’t work in every scenario, but they provide a general guideline. We instead use default values such as “Title" when our semantic analysis returns inconclusive results.\
There is also some analysis that can be conducted regardless of graph type. For example, if we find that our test image has a key or legend we know we will need to add the code for generating a key to the template. A title, while typically at the top and center of an image is also usually more than one word, see Algorithm 1 for how this is determined.\
title = “"
Final Output
------------
The final output of our hybrid model is fully executable Python code that utilizes the Matplotlib library. See the following four images as an example of the process. Image 1 is an example of a graph that a user would find in a research paper that they want to recreate. This would be considered the target image fed into G2L and would be classified as a stacked bar graph. Once classified, OCR is conducted to scrape important features off of the graph and an output code template is produced. See Section D of Figure 3 for how this fits into the full G2L process.\
This code template contains specific data from the original image (e.g., Title, labels) so as to demonstrate to the user what parts of the graph are generated where. Also, variable names correspond with the object they represent, for example, where the data gets loaded in the template we use “x = x\_data" so the user knows this has to do with the actual data.\
The user can then copy and paste the code template into their development environment of choice (e.g., Jupyter Notebook, Google Colab) and edit the code to include their own data. The output code template not only contains fully executable code, but also contains comments for how to update the code in some of the most common ways. This gives them the opportunity to customize in any way they need, but they will have working example code to start. The first thing they will want to change is to fill in their own data. We suggest the use of the Numpy or Pandas library to load data so as to keep solutions consistent and better assist the user. Figure 4 shows an example input image. Figure 5 shows the results of running the code template produced by Figure 4, a new graph that mimics Figure 4 but now has been updated to display the user’s data.\
We heard from our working group that tools that require a lot of overheard for setup and getting started can be especially challenging. So when thinking about usage we decided to package all this functionality inside of a web application. This means it runs in the browser and requires no download or installation to begin using. A user simply uploads the file they want to get the code template for and they are given the output code template.
Evaluation
==========
This section describes (A) the evaluation protocol, which was a two phased user study, (B) how we scored the accuracy of users, and (C) the post-evaluation survey results.
![Example Input or Target Graph from an External Source [@b7]](eval_image.jpg){width="\linewidth"}
{width="\linewidth"}
Evaluation Protocol and Scoring
-------------------------------
To evaluate our model we conducted two phases of user evaluation. The first was with three users that are studying Computer Science, and the second was with two Literature researchers. The first phase helped to work out any bugs in the evaluation process and get initial feedback on usability. The second phase was an evaluation on how G2L works for researchers in the humanities domain. In both phases, users were assigned the same series of tasks that contained milestones (T1-T6). In these tasks the user is provided with a starter Jupyter notebook that contains the necessary imports at the top, and pre-loaded data using the Numpy library.\
Selecting subjects was based on two main criteria: field of study and experience with specific technologies. For phase one we selected users who are undergraduate and graduate students in Computer Science. In phase two we selected graduate students in Literature (from the English Department). All users needed to have some experience with Python and Jupyter notebooks in order to complete the evaluation of G2L. Skills in Matplotlib and Pandas were not required. All self-reported skill levels for the four technologies mentioned are outlined in Table II.
[|c|c|c|]{} **Skill** & ***User (Phase)***& ***Score$^{\mathrm{a}}$***\
& A (1) & 4\
&B (1) & 5\
&C (1) & 3\
&D (2)& 3\
&E (2)& 1\
& A (1) & 4\
&B (1) & 4\
&C (1) & 2\
&D (2)& 2\
&E (2)& 2\
& A (1) & 2\
&B (1) & 4\
&C (1) & 0\
&D (2)& 1\
&E (2)& 2\
& A (1) & 3.5\
&B (1) & 3\
&C (1) & 2\
&D (2)& 2\
&E (2)& 1\
\[tab1\]
To conduct the user evaluations we combined elements from the widely-used Thinking Aloud Protocol [@b9; @b11; @b13] with the measurement system in the Lemoncello et. al. paper [@b17]. Though the original Thinking Aloud Protocol process was developed over 30 years ago, it is still considered a tried and true method today for evaluating software for usability. Other research has shown that after evaluation by five users, this protocol finds 75-85% of the problems in the software, and beyond this brings diminishing returns [@b10]. The process in general, involves a test user using the software in front of an observer. The test user vocalizes their actions and thoughts throughout the time it takes them to complete a task. The observer takes notes and the entire interaction is recorded and later transcribed and analyzed. We evaluated with five total users in two phases, the initial three and the final two. We recorded video and audio for all users and they were able to vocalize thoughts but also questions as they completed the tasks. Scripted answers were given to clarify the tasks if a user was unsure how to proceed. The time limit to complete all tasks in the evaluation was 30 minutes.\
Accuracy measurement of users completion of task milestones was based on the accuracy scale system that Lemoncello et. al. utilized in testing how well users could follow directions. This 0-6 scale adequately captured the different options for how well a subject completed tasks in the G2L evaluation. Audio and video was reviewed for each evaluation and each milestone was given a score on a 6 point scale which is defined as follows: 0 = unable; 1 = required intervention; 2 = asked for assistance; 3 = asked for verification; 4 = self-corrected; and 5 = correct and independent [@b17]. Some examples of this follow. If a user was unable to complete a task within the allotted time for the full evaluation then they received a 0 for that task milestone. If they made (a) an initial erroneous attempt at the task before (b) reaching the milestone successfully and (c) without voicing any questions they then received an accuracy score of 4. These accuracy results are discussed further in the next section and the scores are encapsulated in Table III.
-- ------------- -------------- --------------- ---------------
**User**
**(Phase)** ***Rating*** ***Average*** ***Std Dev***
A (1) 5
B (1) 5
C (1) 5
D (2) 5
E (2) 5
A (1) 5
B (1) 5
C (1) 5
D (2) 5
E (2) 5
A (1) 5
B (1) 5
C (1) 4
D (2) 2
E (2) 3
A (1) 5
B (1) 5
C (1) 5
D (2) 5
E (2) 5
A (1) 5
B (1) 5
C (1) 5
D (2) 2
E (2) 5
A (1) 5
B (1) 5
C (1) 5
D (2) 5
E (2) 0
-- ------------- -------------- --------------- ---------------
: Accuracy Score Per Task Milestone Per User
\[tab1\]
In between phases one and two of the evaluation we made improvements to G2L based on feedback from phase one. The main changes were related to the content of the output code template. For example, all three users in phase one mentioned that they would like to be able to change the size of their new visualization more easily, even though this was not one of the assigned tasks. They felt it would be easier to see and also that it would be an update researchers would want to make to better fit presentations and papers. We then saw a user in phase two take advantage of the newly added functionality while completing their evaluation. Additional changes included: changing the phrasing for some of the comments, and adding more variables to the upper section of the template so that they were easier to update.
Evaluation Results and Analysis
-------------------------------
The task set is outlined as follows. The users were given a Jupyter notebook that included (1) an example heatmap to “copy," (2) starter code (imports and pre-loaded data), and (3) instructions to recreate the graph using the Matplotlib code from G2L and the pre-loaded data. All users were told they could also use internet search at any time if needed. Successful completion of the task set was determined if users completed all six task milestones: (1) use G2L to get code template, (2) paste the code into Jupyter, (3) incorporate pre-loaded data into code template, (4) change the title, (5) remove the overlaid numbers on the heatmap, and (6) change the color schema of the visualization. The first four milestones are considered the “core" milestones because they demonstrate the main function of G2L. Completion of these four milestones results in a visualization with the user’s data. The latter two milestones are minor aesthetic updates.\
Five out of six users were able to complete all six milestones in around 15 minutes or less. One user had trouble with the final task milestone (6) and thus ran out of time, but was able to complete the other five. Total completion times for all milestones are outlined in Table IV, User E shows incomplete and 30 minutes because of task milestone six. As background, Milestone six was to change the color schema. In the template comments there was a link provided to official Matplotlib documentation that contained color options. User E did follow the link. Unfortunately, the documentation is a very verbose page, with the information needed towards the bottom. User E tried many of the code snippets at the beginning of the page, thoroughly read the unrelated information at the top, and never scrolled far enough. While more information could have been provided in the comment to guide the user, this is also a good example of when official documentation is too complicated to decipher. This is a common problem with official documentation; too often the provided examples are irrelevant, too simple, or too complex.\
Results show that there is a correlation between how a user rated themselves skill-wise and the speed at which they completed all tasks. Users A and B rated themselves significantly higher than other users in Python and Jupyter skills, and also slightly higher in Matplotlib and Pandas skills. Perhaps unsurprisingly they were able to complete the tasks the quickest, with both taking under 10 minutes for all six milestones. However, seeing as the target audience of the tool is less experienced users, it is encouraging to see that even those who rated themselves lower were able to complete the tasks in around 15 minutes. We would like to extend these same evaluations to more users to increase the sample size and thus the overall accuracy of our measurements.
[|c|c|c|c|]{} **User**&\
& ***Phase***& ***Completed?***& ***Time (minutes)***\
A & 1& Yes & 9:30\
B & 1& Yes & 7:10\
C & 1& Yes & 15:50\
D & 2& Yes & 13:50\
E & 2& No & 30:00\*\
\[tab1\]
Analysis of the recorded sessions and observer notes show that G2L users were able to get the output code templates from G2L very quickly. Of the “core" milestones, milestone three, manipulating the template to incorporate the pre-loaded data, proved the most challenging. With many hints incorporated into the code template, only two of the five users used internet search to help complete the tasks. Only one of these searches was for a “core" milestone, specifically milestone three. This shows that the core functionality of G2L was quite successful.
Post-Evaluation Survey Results
------------------------------
In addition to measuring time and accuracy scores for task completion, the user survey provided valuable insight. A total of six questions were asked of each user immediately after they completed their evaluation of G2L. These questions and results are outlined in Table V. Results were generally positive, with three out of five users saying they would use G2L a lot in the future. Another important finding from the survey is that all users believed G2L helped limit the time they would spend searching the internet (average score 4.8). All five users felt that G2L helped them a lot overall. G2L helped the least in explaining the Matplotlib code. One remedy to this would be to update the comments and structure of the G2L output code template.
[|p[3cm]{}|c|c|c|c|]{} **Skill** & ***User***& ***Score$^{\mathrm{a}}$*** & Average & Std Dev\
& A & 2 & &\
&B& 3 & &\
&C& 5 & &\
&D& 5 & &\
&E& 5 & &\
& A & 4 & &\
&B& 4 & &\
&C& 5 & &\
&D& 3 & &\
&E& 4 & &\
& A & 5 & &\
&B& 4 & &\
&C& 5 & &\
&D& 5 & &\
&E& 5 & &\
& A & 5 & &\
&B& 4 & &\
&C& 5 & &\
&D& 3 & &\
&E& 4 & &\
& A & 5 & &\
&B& 3 & &\
&C& 5 & &\
&D& 2 & &\
&E& 4 & &\
& A & 5 & &\
&B& 5 & &\
&C& 5 & &\
&D& 5 & &\
&E& 5 & &\
\[tab1\]
Conclusion
==========
This paper describes the G2L web based tool and its demonstrated effectiveness in using a hybrid of machine learning, OCR, and rule-based text placement models to help researchers generate visualizations of their data. Using G2L means they are not required to have a detailed knowledge of Matplotlib, or which search terms they need to use on the internet to create a visualization. G2L requires some Python knowledge, but also provides assistance in learning how to use Python and Matplotlib. We have proven success with both Computer Science and Literature researchers using the tool and incorporating the results into a Jupyter notebook.\
While the evaluation in this project was done using Jupyter notebooks, it could have been any python interpreter. We chose Jupyter as it is one of the more commonly used interpreters, especially by researchers who want their work to be easy to recreate and verify. Visualizations display quite naturally within a Jupyter notebook alongside code and text, and rerunning code is easy, which make it a natural choice. However, there are also many pain points associated with Jupyter notebooks, including setup, kernel crashes, and code management [@b18]. Given these reasons may be enough to keep some away, it would be interesting to conduct evaluations using different interpreters as well. The flexibility of the usage of the G2L output code template is a strength that makes it useful for not just researchers but a wide spectrum of programmers.\
In addition to helping DH researchers we have taken a step towards advancing the field of Visual Requirements Engineering. By allowing the users to specify what they want their graph to look like and generating code based on those specifications we have shown the success and potential of this approach and model. We have also proven that we can effectively synthesize the labelled training data that can often prove challenging to acquire for researchers using machine learning models. We have plans to make the G2L codebase publicly available and open source for people to view and add their own contributions, if desired.
Future Work
===========
One direction for potential future work is to apply this same idea to visualizations using other code libraries (e.g., Plotly, Seaborn). To do so, we would need to abstract up from matplotlib. Fortuitously, there is related research being done on the grammar of graphics [@b16]. “A grammar of graphics is a framework which follows a layered approach to describe and construct visualizations or graphics in a structured manner.” [@b15] In fact, this extension, called a “layered grammar of graphics” was refined and used by the creators of *ggplot2*, an R based visualization library [@b20]. Seven layered components of a graphic can be modeled as a pyramid, as identified by Sarkar in [@b15]. These components are data, aesthetics (i.e. axes), scale, geometric objects (i.e. bar), statistics (i.e. mean), facets (i.e. subplots), and coordinate system. The pyramid showing these seven components and how they build on each other is in Figure 6 below. This is promising in that it speaks to the potential of extending beyond just Matplotlib code to any type of visualization library. If these seven common building blocks can be identified by G2L, then they can be utilized to generalize the tool and generate different code output.
{width="\linewidth"}
In an even broader extension of this work, visualizations could be treated as a new way to express software requirements in general. Visualizations have long been important in the Requirements Engineering field for comprehending traditional requirements. For instance, Magee et al. have built an animator for formal specifications using the executable language FSP [@b12]. We plan to explore whether animation (e.g., a video) can be used not only as an output once a specification has been developed, but also as the input itself to produce a specification. This is exciting since it extends single images into a sequence of images, and could make the programming of animations easier. Images are an extremely effective way to communicate ideas and results, and videos can be even more so, as they provide more dimensions of information.\
[00]{} Exploratory Programming for the Arts and Humanities, Nick Montfort, The MIT Press, 2nd Edition, 2019. Whitepaper: Preparing Data for Natural Language Interaction in Ask Data. https://www.tableau.com/learn/whitepapers/preparing-data-nlp-in-ask-data. Accessed 2019-08-05. Dibia, Victor and Çagatay Demiralp. “Data2Vis: Automatic Generation of Data Visualizations Using Sequence to Sequence Recurrent Neural Networks.” CoRR abs/1804.03126 (2018): n. pag. Ellis, Kevin et al. “Learning to Infer Graphics Programs from Hand-Drawn Images.” NeurIPS (2018). Lin, Xi Victoria. “Program Synthesis from Natural Language Using Recurrent Neural Networks.” (2017). Ganin, Yaroslav et al. “Synthesizing Programs for Images using Reinforced Adversarial Learning.” ICML (2018). Zaric, Drazen. Better Heatmaps and Correlation Matrix Plots in Python. https://towardsdatascience.com/better-heatmaps-and-correlation-matrix-plots-in-python-41445d0f2bec. Accessed 2020-02-15. Google Inception. https://cloud.google.com/tpu/docs/inception-v3-advanced. Someren, Maarten van. “The think aloud method : a practical guide to modelling cognitive processes.” (1994). Nielsen, Jakob. “Estimating the number of subjects needed for a thinking aloud test.” Int. J. Hum.-Comput. Stud. 41 (1994): 385-397. AmosOrley, M.. “Thinking Aloud : Reconciling Theory and Practice.” (2000). Magee, J., Pryce, N., Giannakopoulou, D., Kramer, J. (2000). Graphical animation of behavior models. Proceedings of the 2000 International Conference on Software Engineering. ICSE 2000 the New Millennium, 499-508. Ericsson, Karin and Herbert A. Simon. “Protocol Analysis: Verbal Reports as Data.” (1984). Lin, Xi Victoria, Chenglong Wang, Luke S. Zettlemoyer and Michael D. Ernst. “NL2Bash: A Corpus and Semantic Parser for Natural Language Interface to the Linux Operating System.” ArXiv abs/1802.08979 (2018) Dipanjan Sarkar. A Comprehensive Guide to the Grammar of Graphics for Effective Visualization. https://towardsdatascience.com/a-comprehensive-guide-to-the-grammar-of-graphics-for-effective-visualization-of-multi-dimensional-1f92b4ed4149, 2018. Accessed 2019-08-01. The Grammar of Graphics, Leland Wilkinson, Springer, 2nd Edition, 1999. Lemoncello, Rik, McKay Moore Sohlberg and Stephen Fickas. “When directions fail: Investigation of getting lost behaviour in adults with acquired brain injury.” Brain injury 24 3 (2010): 550-9 . Chattopadhyay, Souti et al. “What’s Wrong with Computational Notebooks? Pain Points, Needs, and Design Opportunities.” (2020). Bavishi, Rohan et al. “AutoPandas: neural-backed generators for program synthesis.” Proceedings of the ACM on Programming Languages 3 (2019): 1 - 27. Wickham, Hadley. “A layered grammar of graphics.” Journal of Computational and Graphical Statistics, vol. 19, no. 1, pp. 3–28, 2010.
|
---
abstract: |
We present [[KADABRA]{}]{}, a new algorithm to approximate betweenness centrality in directed and undirected graphs, which significantly outperforms all previous approaches on real-world complex networks. The efficiency of the new algorithm relies on two new theoretical contributions, of independent interest.
The first contribution focuses on sampling shortest paths, a subroutine used by most algorithms that approximate betweenness centrality. We show that, on realistic random graph models, we can perform this task in time $|E|^{\frac{1}{2}+o(1)}$ with high probability, obtaining a significant speedup with respect to the $\Theta(|E|)$ worst-case performance. We experimentally show that this new technique achieves similar speedups on real-world complex networks, as well.
The second contribution is a new rigorous application of the adaptive sampling technique. This approach decreases the total number of shortest paths that need to be sampled to compute all betweenness centralities with a given absolute error, and it also handles more general problems, such as computing the $k$ most central nodes. Furthermore, our analysis is general, and it might be extended to other settings, as well.
author:
- Michele Borassi
- Emanuele Natale
bibliography:
- 'full-version.bib'
title: 'KADABRA is an ADaptive Algorithm for Betweenness via Random Approximation[^1]'
---
Introduction
============
In this work we focus on estimating the *betweenness centrality*, which is one of the most famous measures of *centrality* for nodes and edges of real-world complex networks [@Easley2010NetworksCA; @newman2010networks]. The rigorous definition of betweenness centrality has its roots in sociology, dating back to the Seventies, when Freeman formalized the informal concept discussed in the previous decades in different scientific communities [@bavelas1948mathematical; @shimbel1953structural; @shaw1954group; @cohn1958networks; @Borgatti2006AGP], although the definition already appeared in [@anthonisse1971rush]. Since then, this notion has been very successful in network science [@wasserman1994social; @newman2001scientific; @Geisberger2008ContractionHF; @newman2010networks].
A probabilistic way to define the betweenness centrality[^2] ${{\operatorname{bc}}(v)}$ of a node $v$ in a graph $G=(V,E)$ is the following. We choose two nodes $s$ and $t$, and we go from $s$ to $t$ through a shortest path $\pi$; if the choices of $s$, $t$ and $\pi$ are made uniformly at random, the betweenness centrality of a node $v$ is the probability that we pass through $v$.
In a seminal paper [@Brandes2001], Brandes showed that it is possible to exactly compute the betweenness centrality of all the nodes in a graph in time ${\mathcal{O}}(mn)$, where $n$ is the number of nodes and $m$ is the number of edges. A corresponding lower bound was proved in [@Borassi2015]: if we are able to compute the betweenness centrality of a single node in time ${\mathcal{O}}(mn^{1-\epsilon})$ for some $\epsilon>0$, then the Strong Exponential Time Hypothesis [@Impagliazzo2001] is false.
This result further motivates the rich line of research on computing approximations of betweenness centrality, with the goal of trading precision with efficiency. The main idea is to define a probability distribution over the set of all paths, by choosing two uniformly random nodes $s,t$, and then a uniformly distributed $st$-path ${\boldsymbol{\pi}}$, so that $\Pr(v \in {\boldsymbol{\pi}})={{\operatorname{bc}}(v)}$. As a consequence, we can approximate ${{\operatorname{bc}}(v)}$ by sampling paths ${\boldsymbol{\pi}}_1,\dots,{\boldsymbol{\pi}}_\tau$ according to this distribution, and estimating ${{\tilde{\boldsymbol{b}}}(v)}:=\frac{1}{{\tau}}\sum_{i=1}^{\tau}{\boldsymbol{X}}_i(v)$, where ${\boldsymbol{X}}_i(v)=1$ if $v \in {\boldsymbol{\pi}}_i$ (and $v \neq s,t$), $0$ otherwise.
The tricky part of this approach is to provide probabilistic guarantees on the quality of this approximation: the goal is to obtain a $1-\delta$ confidence interval $\boldsymbol{I}(v)=[{{\tilde{\boldsymbol{b}}}(v)}-{\lambda_{L}},{{\tilde{\boldsymbol{b}}}(v)}+{\lambda_{U}}]$ for ${{\operatorname{bc}}(v)}$, which means that $\Pr(\forall v \in V,{{\operatorname{bc}}(v)} \in \boldsymbol{I}(v))\geq 1-\delta$. Thus, the research for approximating betweenness centrality has been focusing on obtaining, as fast as possible, the smallest possible $\boldsymbol{I}$.
### Our Contribution {#sec:contribshort .unnumbered}
In this work, we propose a new and faster algorithm to approximate betweenness centrality in directed and undirected graphs, named [[KADABRA]{}]{}. In the standard task of approximating betweenness centralities with absolute error at most $\lambda$, we show that, on average, the new algorithm is more than $100$ times faster than the previous ones, on graphs with approximately $10\,000$ nodes. Moreover, differently from previous approaches, our algorithm can perform more general tasks, since it does not need all confidence intervals to be equal. As an example, we consider the computation of the $k$ most central nodes: all previous approaches compute all centralities with an error $\lambda$, and use this approximation to obtain the ranking. Conversely, our approach allows us to use small confidence interval only when they are needed, and allows bigger confidence intervals for nodes whose centrality values are “well separated”. This way, we can compute for the first time an approximation of the $k$ most central nodes in networks with millions of nodes and hundreds of millions of edges, like the Wikipedia citation network and the IMDB actor collaboration network.
Our results rely on two main theoretical contributions, which are interesting in their own right, since their generality naturally extends to other applications.
##### Balanced bidirectional breadth-first search.
By leveraging on recent advanced results, we prove that, on many realistic random models of real-world complex networks, it is possible to sample a random path between two nodes $s$ and $t$ in time $m^{\frac{1}{2}+o(1)}$ if the degree distribution has finite second moment, or $m^{\frac{4-\beta}{2}+o(1)}$ if the degree distribution is power law with exponent $2<\beta<3$. The models considered are the Configuration Model [@Bollobas1980], and all Rank-1 Inhomogeneous Random Graph models [@Hofstad2014 Chapter 3], such as the Chung-Lu model [@Chung2006], the Norros-Reittu model [@Norros2006], and the Generalized Random Graph [@Hofstad2014 Chapter 3]. Our proof techniques have the merit of adopting a unified approach that simultaneously works in all models considered. These models well represent metric properties of real-world networks [@Borassi2016]: indeed, our results are confirmed by practical experiments.
The algorithm used is simply a balanced bidirectional BFS ([bb-BFS]{}): we perform a BFS from each of the two endpoints $s$ and $t$, in such a way that the two BFSs are likely to explore about the same number of edges, and we stop as soon as the two BFSs “touch each other”. Rather surprisingly, this technique was never implemented to approximate betweenness centrality, and it is rarely used in the experimental algorithm community. Our theoretical analysis provides a clear explanation of the reason why this technique improves over the standard BFS: this means that many state-of-the-art algorithm for real-world complex networks can be improved by the [bb-BFS]{}.
##### Adaptive sampling made rigorous.
To speed up the estimation of the betweenness centrality, previous work make use of the technique of adaptive sampling, which consists in testing during the execution of the algorithm whether some condition on the sample obtained so far has been met, and terminating the execution of the algorithm as soon as this happens. However, this technique introduces a subtle stochastic dependence between the time in which the algorithm terminates and the correctness of the given output, which previous papers claiming a formal analysis of the technique did not realize (see Section \[sec:adapshort\] for details). With an argument based on martingale theory, we provide a general analysis of such useful technique. Through this result, we do not only improve previous estimators, but we also make it possible to define more general stopping conditions, that can be decided “on the fly”: this way, with little modifications, we can adapt our algorithm to perform more general tasks than previous ones.
To better illustrate the power of our techniques, we focus on the unweighted, static graphs, and to the centrality of nodes. However, our algorithm can be easily adapted to compute the centrality of edges, to handle weighted graphs and, since its core part consists merely in sampling paths, we conjecture that it may be coupled with the existing techniques in [@Bergamini2015FullyDynamicAO] to handle dynamic graphs.
### Related Work {#sec:relatedshort .unnumbered}
##### Computing Betweenness Centrality.
With the recent event of big data, the major shortcoming of betweenness centrality has been the lack of efficient methods to compute it [@Brandes2001]. In the worst case, the best exact algorithm to compute the centrality of all the nodes is due to Brandes [@Brandes2001], and its time complexity is ${\mathcal{O}}(mn)$: the basic idea of the algorithm is to define the dependency ${\delta_{s}(v)}=\sum_{t \in V} \frac{{\sigma_{st}(v)}}{{\sigma_{st}}}$, which can be computed in time ${\mathcal{O}}(m)$, for each $v \in V$ (we denote by ${\sigma_{st}(v)}$ the number of shortest paths from $s$ to $t$ passing through $v$, and by ${\sigma_{st}}$ the number of $st$-shortest paths). In [@Borassi2015], it is also shown that Brandes algorithm is almost optimal on sparse graphs: an algorithm that computes the betweenness centrality of a single vertex in time ${\mathcal{O}}(mn^{1-\epsilon})$ falsifies widely believed complexity assumptions, such as the Strong Exponential Time Hypothesis [@Impagliazzo2001], the Orthogonal Vector conjecture [@Abboud2016], or the Hitting Set conjecture [@Williams2014]. Corresponding results in the dense, weighted case are available in [@grand2015]: computing the betweenness centrality exactly is as hard as computing the All Pairs Shortest Path, and computing an approximation with a given relative error is as hard as computing the diameter. For both these problems, there is no algorithm with running-time ${\mathcal{O}}(n^{3-\epsilon})$, for any $\epsilon>0$. This shows that, for dense graphs, having an additive approximation rather than a multiplicative one is essential for a provably fast algorithm to exist. These negative results further motivate the already rich line of research on approaches that overcome this barrier. A first possibility is to use heuristics, that do not provide analytical guarantees on their performance [@atalyrek2013ShatteringAC; @Erds2015ADA; @Vella2016AlgorithmsAH]. Another line of research has defined variants of betweenness centrality, that might be easier to compute [@Brandes2008OnVO; @pfeffer2012k; @Dolev2010RoutingBC]. Finally, a third line of research has investigated approximation algorithms, which trade accuracy for speed [@Jacob2004AlgorithmsFC; @Brandes2007; @Geisberger2008ContractionHF; @lim2011online]. Our work follows the latter approach. The first approximation algorithm proposed in the literature [@Jacob2004AlgorithmsFC] adapts Eppstein and Wang’s approach for computing closeness centrality [@Eppstein2001FastAO], using Hoeffding’s inequality and the union bound technique. This way, it is possible to obtain an estimate of the betweenness centrality of every node that is correct up to an additive error $\lambda$ with probability $\delta$, by sampling ${\mathcal{O}}(\frac{D^2}{\lambda^2}\log\frac{n}{\delta})$ nodes, where $D$ is the diameter of the graph. In [@Geisberger2008ContractionHF], it is shown that this can lead to an overestimation. Riondato and Kornaropoulos improve this sampling-based approach by sampling single shortest paths instead of the whole dependency of a node [@Riondato2015], introducing the use of the VC-dimension. As a result, the number of samples is decreased to $\frac{c}{\lambda^2}(\lfloor\log_2(\operatorname{VD}-2)\rfloor+1+\log(\frac{1}{\delta}))$, where $\operatorname{VD}$ is the vertex diameter, that is, the minimum number of nodes in a shortest path in $G$ (it can be different from $D+1$ if the graph is weighted). This use of the VC-dimension is further developed and generalized in [@Riondato2016]. Finally, many of these results were adapted to handle dynamic networks [@Bergamini2015FullyDynamicAO; @Riondato2016].
##### Approximating the top-k betweenness centrality set.
Let us order the nodes $v_1,...,v_n$ such that $\operatorname{bc}(v_1)\geq ...\geq \operatorname{bc}(v_n)$ and define $TOP(k) = \{ (v_i,\operatorname{bc}(v_i)): i \leq k\}$. In [@Riondato2015] and [@Riondato2016], the authors provide an algorithm that, for any given $\delta,\epsilon$, with probability $1-\delta$ outputs a set ${\widetilde{TOP}}(k) = \{(v_i,{{\tilde{\boldsymbol{b}}}(v_i)})\}$ such that: i) If $v \in TOP(k)$ then $v \in {\widetilde{TOP}}(k)$ and $| \operatorname{bc}(v) - {{\tilde{\boldsymbol{b}}}(v)} | \leq \epsilon \operatorname{bc}(v)$; ii) If $v \in {\widetilde{TOP}}(k)$ but $v \not\in TOP(k)$ then ${{\tilde{\boldsymbol{b}}}(v)}\leq (\mathbf{b}_k-\epsilon) (1+\epsilon)$ where $\mathbf{b}_k$ is the $k$-th largest betweenness given by a preliminary phase of the algorithm.
##### Adaptive sampling.
In [@Bader2007; @Riondato2016], the number of samples required is substantially reduced using the adaptive sampling technique introduced by Lipton and Naughton in [@lipton_estimating_1989; @lipton_query_1995]. Let us clarify that, by adaptive sampling, we mean that the termination of the sampling process depends on the sample observed so far (in other cases, the same expression refers to the fact that the distribution of the new samples is a function of the previous ones [@aggarwal_adaptive_2009], while the sample size is fixed in advance). Except for [@pietracaprina_mining_2010], previous approaches tacitly assume that there is little dependency between the stopping time and the correctness of the output: indeed, they prove that, for each *fixed* ${\tau}$, the probability that the estimate is wrong at time ${\tau}$ is below $\delta$. However, the stopping time ${\boldsymbol{\tau}}$ is a random variable, and in principle there might be dependency between the event ${\boldsymbol{\tau}}=\tau$ and the event that the estimate is correct at time $\tau$. As for [@pietracaprina_mining_2010], they consider a specific stopping condition and their proof technique does not seem to extend to other settings. For a more thorough discussion of this issue, we defer the reader to .
##### Bidirectional BFS.
The possibility of speeding up a breadth-first search for the shortest-path problem by performing, at the same time, a BFS from the final end-point, has been considered since the Seventies [@pohl1969bi]. Unfortunately, because of the lack of theoretical results dealing with its efficiency, the bidirectional BFS has apparently not been considered a fundamental heuristic improvement [@Kaindl1997BidirectionalHS]. However, in [@Riondato2015] (and in some public talks by M. Riondato), the bidirectional BFS was proposed as a possible way to improve the performance of betweenness centrality approximation algorithms.
### Structure of the Paper {#structure-of-the-paper .unnumbered}
In , we describe our algorithm, and in we discuss the main difficulty of the adaptive sampling, and the reasons why our techniques are not affected. In , we define the balanced bidirectional BFS, and we sketch the proof of its efficiency on random graphs. In , we show that our algorithm can be adapted to compute the $k$ most central nodes. In we experimentally show the effectiveness of our new algorithm. Finally, all our proofs are in the appendix.
Algorithm Overview {#sec:algoshort}
==================
To simplify notation, we always consider the *normalized* betweenness centrality of a node $v$, which is defined by: $$\operatorname{bc}(v)=\frac{1}{n(n-1)}\sum_{s \neq v \neq t} \frac{{\sigma_{st}(v)}}{{\sigma_{st}}}$$ where ${\sigma_{st}}$ is the number of shortest paths between $s$ and $t$, and ${\sigma_{st}(v)}$ is the number of shortest paths between $s$ and $t$ that pass through $v$. Furthermore, to simplify the exposition, we use bold symbols to denote random variables, and light symbols to denote deterministic quantities. On the same line of previous works, our algorithm samples random paths ${\boldsymbol{\pi}}_1,\dots,{\boldsymbol{\pi}}_{\tau}$, where ${\boldsymbol{\pi}}_i$ is chosen by selecting uniformly at random two nodes $s,t$, and then selecting uniformly at random one of the shortest paths from $s$ to $t$. Then, it estimates ${{\operatorname{bc}}(v)}$ with ${{\tilde{\boldsymbol{b}}}(v)}:=\frac{1}{{\tau}}\sum_{i=1}^{\tau}{\boldsymbol{X}}_i(v)$, where ${\boldsymbol{X}}_i(v)=1$ if $v \in {\boldsymbol{\pi}}_i$, $0$ otherwise. By definition of ${\boldsymbol{\pi}}_i$, ${\mathbb{E}}\left[{{\tilde{\boldsymbol{b}}}(v)}\right]={{\operatorname{bc}}(v)}$.
The tricky part is to bound the distance between ${{\tilde{\boldsymbol{b}}}(v)}$ and its expected value. With a straightforward application of Hoeffding’s inequality (Lemma \[lem:hoeff\] in the appendix), it is possible to prove that $\Pr\left(\left|{{\tilde{\boldsymbol{b}}}(v)}-{{\operatorname{bc}}(v)}\right|\geq \lambda\right) \leq 2e^{-2{\tau}\lambda^2}$. A direct application of this inequality considers a union bound on all possible nodes $v$, obtaining $\Pr(\exists v \in V, |{{\tilde{\boldsymbol{b}}}(v)}-{{\operatorname{bc}}(v)}|\geq \lambda) \leq 2ne^{-2{\tau}\lambda^2}$. This means that the algorithm can safely stop as soon as $2ne^{-2{\tau}\lambda^2} \leq \delta$, that is, after ${\tau}= \frac{1}{2\lambda^2} \log(\frac{2n}{\delta})$ steps.
In order to improve this idea, we can start from Lemma \[lem:cb\] in the appendix, instead of Hoeffding inequality, obtaining that $\Pr\left(\left|{{\tilde{\boldsymbol{b}}}(v)}-{{\operatorname{bc}}(v)}\right|\geq \lambda\right) \leq 2\exp(-\frac{{\tau}\lambda^2}{2({{\operatorname{bc}}(v)}+\lambda/3)})$.
If we assume the error $\lambda$ to be small, this inequality is stronger than the previous one for all values of ${{\operatorname{bc}}(v)}<\frac{1}{4}$ (a condition which holds for almost all nodes, in almost all graphs considered). However, in order to apply this inequality, we have to deal with the fact that we do not know ${{\operatorname{bc}}(v)}$ in advance, and hence we do not know when to stop. Intuitively, to solve this problem, we make a “change of variable”, and we rewrite the previous inequality as $$\label{eq:chernoffmodshort}
\begin{aligned}
\Pr\left({{\operatorname{bc}}(v)} \leq {{\tilde{\boldsymbol{b}}}(v)}-f \right)\leq {\delta_{L}^{(v)}} \quad \text{and} \quad
\Pr\left({{\operatorname{bc}}(v)} \geq {{\tilde{\boldsymbol{b}}}(v)}+g\right) \leq {\delta_{U}^{(v)}},
\end{aligned}$$ for some functions $f=f({{\tilde{\boldsymbol{b}}}(v)},{\delta_{L}^{(v)}}, {\tau}),g=g({{\tilde{\boldsymbol{b}}}(v)},{\delta_{U}^{(v)}}, {\tau})$. Our algorithm fixes at the beginning the values ${\delta_{L}^{(v)}}, {\delta_{U}^{(v)}}$ for each node $v$, and, at each step, it tests if $f({{\tilde{\boldsymbol{b}}}(v)},{\delta_{L}^{(v)}},\tau)$ and $g({{\tilde{\boldsymbol{b}}}(v)},{\delta_{U}^{(v)}},\tau)$ are small enough. If this condition is satisfied, the algorithm stops. Note that this approach lets us define very general stopping conditions, that might depend on the centralities computed until now, on the single nodes, and so on.
Instead of fixing the values ${\delta_{L}^{(v)}},{\delta_{U}^{(v)}}$ at the beginning, one might want to decide them during the algorithm, depending on the outcome. However, this is not formally correct, because of dependency issues (for example, does not even make sense, if ${\delta_{L}^{(v)}},{\delta_{U}^{(v)}}$ are random). Finding a way to overcome this issue is left as a challenging open problem (more details are provided in ).
In order to implement this idea, we still need to solve an issue: holds for each *fixed* time $\tau$, but the stopping time of our algorithm is a random variable ${\boldsymbol{\tau}}$, and there might be dependency between the value of ${\boldsymbol{\tau}}$ and the probability in . To this purpose, we use a stronger inequality ( in the appendix), that holds even if ${\boldsymbol{\tau}}$ is a random variable. However, to use this inequality, we need to assume that ${\boldsymbol{\tau}}< \omega$ for some deterministic $\omega$: in our algorithm, we choose $\omega=\frac{c}{\lambda^2}\left(\lfloor\log_2(\operatorname{VD}-2)\rfloor+1+\log\left(\frac{2}{\delta}\right)\right)$, because, by the results in [@Riondato2015], after $\omega$ samples, the maximum error is at most $\lambda$, with probability $1-\frac{\delta}{2}$. Furthermore, also $f$ and $g$ should be modified, since they now depend on the value of $\omega$. The pseudocode of the algorithm obtained is available in Algorithm \[alg:mainshort\] (as was done in previous approaches, we can easily parallelize the while loop in Line \[line:stopcondshort\]).
$\omega \gets \frac{c}{\lambda^2}\left(\lfloor\log_2(\operatorname{VD}-2)\rfloor+1+\log\left(\frac{2}{\delta}\right)\right)$
$({\delta_{L}^{(v)}},{\delta_{U}^{(v)}}) \gets \computeDeltaV()$ \[line:computedeltavshort\] $\tau \gets 0$
The correctness of the algorithm follows from the following theorem, which is the base of our adaptive sampling, and which we prove in (where we also define the functions $f$ and $g$).
\[thm:mainshort\] Let ${{\tilde{\boldsymbol{b}}}(v)}$ be the output of Algorithm \[alg:mainshort\], and let ${\boldsymbol{\tau}}$ be the number of samples at the end of the algorithm. Then, with probability $1-\delta$, the following conditions hold:
- if ${\boldsymbol{\tau}}=\omega$, $|{{\tilde{\boldsymbol{b}}}(v)}-{{\operatorname{bc}}(v)}| < \lambda$ for all $v$;
- if ${\boldsymbol{\tau}}<\omega$, $ -f({\boldsymbol{\tau}},{{\tilde{\boldsymbol{b}}}(v)},{\delta_{L}^{(v)}},\omega) \leq {{\operatorname{bc}}(v)} - {{\tilde{\boldsymbol{b}}}(v)} \leq g({\boldsymbol{\tau}},{{\tilde{\boldsymbol{b}}}(v)},{\delta_{U}^{(v)}},\omega)$ for all $v$.
This theorem says that, at the beginning of the algorithm, we know that, with probability $1-\delta$, one of the two conditions will hold when the algorithm stops, independently of the final value of ${\boldsymbol{\tau}}$. This is essential to avoid the stochastic dependence that we discuss in .
In order to apply this theorem, we choose $\lambda$ such that our goal is reached if all centralities are known with error at most $\lambda$. Then, we choose the function in a way that our goal is reached if the stopping condition is satisfied. This way, our algorithm is correct, both if ${\boldsymbol{\tau}}=\omega$ and if ${\boldsymbol{\tau}}<\omega$. For example, if we want to compute all centralities with bounded absolute error, we simply choose $\lambda$ as the bound we want to achieve, and we plug the stopping condition $f,g\leq \lambda$ in the function . Instead, if we want to compute an approximation of the $k$ most central nodes, we need a different definition of $f$ and $g$, which is provided in .
To complete the description of this algorithm, we need to specify the following functions.
: The algorithm works for any choice of the ${\delta_{L}^{(v)}},{\delta_{U}^{(v)}}$s, but a good choice yields better running times. We propose a heuristic way to choose them in .
: In order to sample a path between two random nodes $s$ and $t$, we use a balanced bidirectional BFS, which is defined in .
Adaptive Sampling {#sec:adapshort}
=================
In this section, we highlight the main technical difficulty in the formalization of adaptive sampling, which previous works claiming analogous results did not address. Furthermore, we sketch the way we overcome this difficulty: our argument is quite general, and it could be easily adapted to formalize these claims.
As already said, the problem is the stochastic dependence between the time ${\boldsymbol{\tau}}$ in which the algorithm terminates and the event ${\boldsymbol{A}}_{\tau}=$ “at time ${\tau}$, the estimate is within the required distance from the true value”, since both ${\boldsymbol{\tau}}$ and ${\boldsymbol{A}}_{\tau}$ are functions of the same random sample. Since it is typically possible to prove that $\Pr(\neg {\boldsymbol{A}}_{\tau}) \leq \delta$ for every fixed ${\tau}$, one may be tempted to argue that also $\Pr(\neg {\boldsymbol{A}}_{{\boldsymbol{\tau}}}) \leq \delta$, by applying these inequalities at time ${\boldsymbol{\tau}}$. However, this is not correct: indeed, if we have no assumptions on ${\boldsymbol{\tau}}$, ${\boldsymbol{\tau}}$ could even be defined as the smallest ${\tau}$ such that ${\boldsymbol{A}}_{\tau}$ does not hold!
More formally, if we want to link $\Pr(\neg {\boldsymbol{A}}_{{\boldsymbol{\tau}}})$ to $\Pr(\neg {\boldsymbol{A}}_{\tau})$, we have to use the law of total probability, that says that: $$\begin{aligned}
\Pr(\neg {\boldsymbol{A}}_{{\boldsymbol{\tau}}})
&= \sum_{\tau=1}^{\infty} \Pr(\neg {\boldsymbol{A}}_{{\boldsymbol{\tau}}}\,|\, {\boldsymbol{\tau}}={\tau})\Pr({\boldsymbol{\tau}}={\tau})
\label{eq:first_tryshort}\\
&= \Pr(\neg {\boldsymbol{A}}_{{\boldsymbol{\tau}}}\,|\, {\boldsymbol{\tau}}< {\tau})\Pr({\boldsymbol{\tau}}< {\tau}) + \Pr(\neg {\boldsymbol{A}}_{{\boldsymbol{\tau}}}\,|\, {\boldsymbol{\tau}}\geq {\tau})\Pr({\boldsymbol{\tau}}\geq {\tau}).
\label{eq:second_tryshort}\end{aligned}$$ Then, if we want to bound $\Pr(\neg {\boldsymbol{A}}_{{\boldsymbol{\tau}}})$, we need to assume that $$\Pr(\neg A_{{\boldsymbol{\tau}}}\,|\, {\boldsymbol{\tau}}={\tau}) \leq \Pr(\neg A_{\tau}) \quad\text{or that} \quad
\Pr(\neg A_{\tau}\,|\, {\boldsymbol{\tau}}\geq {\tau}) \leq \Pr(\neg A_{\tau}),
\label{eq:first_try_failshort}$$ which would allow to bound or from above. The equations in are implicitly assumed to be true in previous works adopting adaptive sampling techniques. Unfortunately, because of the stochastic dependence, it is quite difficult to prove such inequalities, even if some approaches managed to overcome these difficulties [@pietracaprina_mining_2010].
For this reason, our proofs avoid dealing with such relations: in the proof of , we fix a deterministic time $\omega$, we impose that ${\boldsymbol{\tau}}\leq\omega$, and we apply the inequalities with ${\tau}=\omega$. Then, using martingale theory, we convert results that hold at time $\omega$ to results that hold at the stopping time ${\boldsymbol{\tau}}$ (see ).
Balanced Bidirectional BFS {#sec:bfsshort}
==========================
A major improvement of our algorithm, with respect to previous counterparts, is that we sample shortest paths through a balanced bidirectional BFS, instead of a standard BFS. In this section, we describe this technique, and we bound its running time on realistic models of random graphs, with high probability. The idea behind this technique is very simple: if we need to sample a uniformly random shortest path from $s$ to $t$, instead of performing a full BFS from $s$ until we reach $t$, we perform at the same time a BFS from $s$ and a BFS from $t$, until the two BFSs touch each other (if the graph is directed, we perform a “forward” BFS from $s$ and a “backward” BFS from $t$).
More formally, assume that we have visited up to level $l_s$ from $s$ and to level $l_t$ from $t$, let $\Gamma^{l_s}(s)$ be the set of nodes at distance $l_s$ from $s$, and similarly let $\Gamma^{l_t}(t)$ be the set of nodes at distance $l_t$ from $t$. If $\sum_{v \in \Gamma^{l_s}(s)} \deg(v) \leq \sum_{v \in \Gamma^{l_t}(t)} \deg(v)$, we process all nodes in $\Gamma^{l_s}(s)$, otherwise we process all nodes in $\Gamma^{l_t}(t)$ (since the time needed to process level $l_s$ is proportional to $\sum_{v \in \Gamma^{l_s}(s)} \deg(v)$, this choice minimizes the time needed to visit the next level). Assume that we are processing the node $v \in \Gamma^{l_s}(s)$ (the other case is analogous). For each neighbor $w$ of $v$ we do the following:
- if $w$ was never visited, we add $w$ to $\Gamma^{l_s+1}(s)$;
- if $w$ was already visited in the BFS from $s$, we do not do anything;
- if $w$ was visited in the BFS from $t$, we add the edge $(v,w)$ to the set $\Pi$ of candidate edges in the shortest path.
After we have processed a level, we stop if $\Gamma^{l_s}(s)$ or $\Gamma^{l_t}(t)$ is empty (in this case, $s$ and $t$ are not connected), or if $\Pi$ is not empty. In the latter case, we select an edge from $\Pi$, so that the probability of choosing the edge $(v,w)$ is proportional to $\sigma_{sv}\sigma_{wt}$ (we recall that $\sigma_{xy}$ is the number of shortest paths from $x$ to $y$, and it can be computed during the BFS as in [@Brandes2007]). Then, the path is selected by considering the concatenation of a random path from $s$ to $v$, the edge $(v,w)$, and a random path from $w$ to $t$. These random paths can be easily chosen by backtracking, as shown in [@Riondato2015] (since the number of paths might be exponential in the input size, in order to avoid pathological cases, we assume that we can perform arithmetic operations in ${\mathcal{O}}(1)$ time).
Analysis on Random Graph
------------------------
In order to show the effectiveness of the balanced bidirectional BFS, we bound its running time in several models of random graphs: the Configuration Model (CM, [@Bollobas1980]), and Rank-1 Inhomogeneous Random Graph models (IRG, [@Hofstad2014 Chapter 3]), such as the Chung-Lu model [@Chung2006], the Norros-Reittu model [@Norros2006], and the Generalized Random Graph [@Hofstad2014 Chapter 3]. In these models, we fix the number $n$ of nodes, and we give a weight ${\rho_{u}}$ to each node. In the CM, we create edges by giving ${\rho_{u}}$ half-edges to each node $u$, and pairing these half-edges uniformly at random; in IRG we connect each pair of nodes $(u,v)$ independently with probability close to ${{\rho_{u}} {\rho_{v}}}/{\sum_{w \in V} {\rho_{w}}}$. With some technical assumptions discussed in , we prove the following theorem.
\[thm:bidirectionalshort\] Let $G$ be a graph generated through the aforementioned models. Then, for each fixed $\epsilon>0$, and for each pair of nodes $s,t$, [w.h.p.]{}, the time needed to compute an $st$-shortest path through a bidirectional BFS is ${\mathcal{O}}(n^{\frac{1}{2}+\epsilon})$ if the degree distribution $\lambda$ has finite second moment, ${\mathcal{O}}(n^{\frac{4-\beta}{2}+\epsilon})$ if $\lambda$ is a power law distribution with $2<\beta<3$.
The idea of the proof is that the time needed by a bidirectional BFS is proportional to the number of visited edges, which is close to the sum of the degrees of the visited nodes, which are very close to their weights. Hence, we have to analyze the weights of the visited edges: for this reason, if $V'$ is a subset of $V$, we define the volume of $V'$ as ${\rho_{V'}}=\sum_{v \in V'} {\rho_{v}}$.
Our visit proceeds by “levels” in the BFS trees from $s$ and $t$: if we never process a level with total weight at least $n^{\frac{1}{2}+\epsilon}$, since the diameter is ${\mathcal{O}}(\log n)$, the volume of the set of processed vertices is ${\mathcal{O}}(n^{\frac{1}{2}+\epsilon}\log n)$, and the number of visited edges cannot be much bigger (for example, this happens if $s$ and $t$ are not connected). Otherwise, assume that, at some point, we process a level $l_s$ in the BFS from $s$ with total weight $n^{\frac{1}{2}+\epsilon}$: then, the corresponding level $l_t$ in the BFS from $t$ has also weight $n^{\frac{1}{2}+\epsilon}$ (otherwise, we would have expanded from $t$, because weights and degrees are strongly correlated). We use the “birthday paradox”: levels $l_s+1$ in the BFS from $s$, and level $l_t+1$ in the BFS from $t$ are random sets of nodes with size close to $n^{\frac{1}{2}+\epsilon}$, and hence there is a node that is common to both, [w.h.p.]{}. This means that the time needed by the bidirectional BFS is proportional to the volume of all levels in the BFS tree from $s$, until $l_s$, plus the volume of all levels in the BFS tree from $t$, until $l_t$ (note that we do not expand levels $l_s+1$ and $l_t+1$). All levels except the last have volume at most $n^{\frac{1}{2}+\epsilon}$, and there are ${\mathcal{O}}(\log n)$ such levels because the diameter is ${\mathcal{O}}(\log n)$: it only remains to estimate the volume of the last level.
By definition of the models, the probability that a node $v$ with weight ${\rho_{v}}$ belongs to the last level is about $\frac{{\rho_{v}} {\rho_{{\boldsymbol{\Gamma}^{l_s-1}(s)}}}}{M} \leq {\rho_{v}} n^{-\frac{1}{2}+\epsilon}$: hence, the expected volume of ${\boldsymbol{\Gamma}^{l_s}(s)}$ is at most $\sum_{v \in V} {\rho_{v}} \Pr(v \in {\boldsymbol{\Gamma}^{l_s-1}(s)}) \leq \sum_{v \in V} {\rho_{v}}^2 n^{-\frac{1}{2}+\epsilon}$. Through standard concentration inequalities, we prove that this random variable is concentrated: hence, we only need to compute this expected value. If the degree distribution has finite second moment, then $\sum_{v \in V} {\rho_{v}}^2={\mathcal{O}}(n)$, concluding the proof. If the degree distribution is power law with $2<\beta<3$, then we have to consider separately nodes $v$ such that ${\rho_{v}}<n^{\frac{1}{2}}$ and such that ${\rho_{v}}>n^{\frac{1}{2}}$. In the first case, $\sum_{{\rho_{v}}<n^{\frac{1}{2}}} {\rho_{v}}^2 \approx \sum_{d=0}^{n^{\frac{1}{2}}} nd^2\lambda(d) \approx \sum_{d=0}^{n^{\frac{1}{2}}} nd^{2-\beta}\approx n^{1+\frac{3-\beta}{2}}$. In the second case, we prove that the volume of the set of nodes with weight bigger than $n^{\frac{1}{2}}$ is at most $n^{\frac{4-\beta}{2}}$. Hence, the total volume of ${\boldsymbol{\Gamma}^{l_s}(s)}$ is at most $n^{-\frac{1}{2}+\epsilon}n^{1+\frac{3-\beta}{2}}+n^{\frac{4-\beta}{2}} \approx n^{\frac{4-\beta}{2}}$.
Computing the k Most Central Nodes {#sec:topkshort}
==================================
Differently from previous works, our algorithm is more flexible, making it possible to compute the betweenness centrality of different nodes with different precision. This feature can be exploited if we only want to rank the nodes: for instance, if $v$ is much more central than all the other nodes, we do not need a very precise estimation on the centrality of $v$ to say that it is the top node. Following this idea, in this section we adapt our approach to the approximation of the ranking of the $k$ most central nodes: as far as we know, this is the first approach which computes the ranking without computing a $\lambda$-approximation of all betweenness centralities, allowing significant speedups. Clearly, we cannot expect our ranking to be always correct, otherwise the algorithm does not terminate if two of the $k$ most central nodes have the same centrality. For this reason, the user fixes a parameter $\lambda$, and, for each node $v$, the algorithm does one of the following:
- it provides the exact position of $v$ in the ranking;
- it guarantees that $v$ is not in the top-$k$;
- it provides a value ${{\tilde{\boldsymbol{b}}}(v)}$ such that $|{{\operatorname{bc}}(v)}-{{\tilde{\boldsymbol{b}}}(v)}|\leq\lambda$.
In other words, similarly to what is done in [@Riondato2015], the algorithm provides a set of $k'\geq k$ nodes containing the top-$k$ nodes, and for each pair of nodes $v,w$ in this subset, either we can rank correctly $v$ and $w$, or $v$ and $w$ are almost even, that is, $|{{\operatorname{bc}}(v)}-{{\operatorname{bc}}(w)}|\leq 2\lambda$. In order to obtain this result, we plug into Algorithm \[alg:mainshort\] the aforementioned conditions in the function (see Algorithm \[alg:stopcondtopk\] in the appendix).
Then, we have to adapt the function to optimize the ${\delta_{L}^{(v)}}$s and the ${\delta_{U}^{(v)}}$s to the new stopping condition: in other words, we have to choose the values of ${\lambda_{L}^{(v)}}$ and ${\lambda_{U}^{(v)}}$ that should be plugged into the function (we recall that the heuristic chooses the ${\delta_{L}^{(v)}}$s so that we can guarantee as fast as possible that ${{\tilde{\boldsymbol{b}}}(v)}-{\lambda_{L}^{(v)}} \leq {\operatorname{bc}}(v) \leq {{\tilde{\boldsymbol{b}}}(v)}+{\lambda_{U}^{(v)}}$). To this purpose, we estimate the betweenness of all nodes with few samples and we sort all nodes according to these approximate values $\tilde{b}(v)$, obtaining $v_1,\dots,v_n$. The basic idea is that, for the first $k$ nodes, we set ${\lambda_{U}^{(v_i)}}=\frac{\tilde{b}(v_{i-1})-\tilde{b}(v_i)}{2}$, and ${\lambda_{L}^{(v_i)}}=\frac{\tilde{b}(v_{i})-\tilde{b}(v_{i+1})}{2}$ (the goal is to find confidence intervals that separate the betweenness of $v_i$ from the betweenness of $v_{i+1}$ and $v_{i-1}$). For nodes that are not in the top-$k$, we choose ${\lambda_{L}^{(v)}}=1$ and ${\lambda_{U}^{(v)}}=\tilde{b}(v_{k})-{\lambda_{L}^{(v_k)}}-\tilde{b}(v_{i})$ (the goal is to prove that $v_i$ is not in the top-$k$). Finally, if $\tilde{b}(v_{i})-\tilde{b}(v_{i+1})$ is small, we simply set ${\lambda_{L}^{(v_i)}}={\lambda_{U}^{(v_i)}}={\lambda_{L}^{(v_{i+1})}}={\lambda_{U}^{(v_{i+1})}}=\lambda$, because we do not know if ${{\operatorname{bc}}(v_{i+1})}>{{\operatorname{bc}}(v_i)}$, or viceversa.
Experimental Results {#sec:experimentsshort}
====================
In this section, we test the four variations of our algorithm on several real-world networks, in order to evaluate their performances. The platform for our tests is a server with 1515 GB RAM and 48 Intel(R) Xeon(R) CPU E7-8857 v2 cores at 3.00GHz, running Debian GNU Linux 8. The algorithms are implemented in C++, and they are compiled using gcc 5.3.1. The source code of our algorithm is available at <https://sites.google.com/a/imtlucca.it/borassi/publications>.
### Comparison with the State of the Art {#comparison-with-the-state-of-the-art .unnumbered}
The first experiment compares the performances of our algorithm [[KADABRA]{}]{} with the state of the art. The first competitor is the [[RK]{}]{} algorithm [@Riondato2015], available in the open-source *NetworKit* framework [@Staudt2014]. This algorithm uses the same estimator as our algorithm, but the stopping condition is different: it simply stops after sampling $k=\frac{c}{\epsilon^2}\left(\left\lfloor\log_2(\operatorname{VD}-2)\right\rfloor+1+\log\left(\frac{1}{\delta}\right)\right)$, and it uses a heuristic to upper bound the vertex diameter. Following suggestions by the author of the *NetworKit* implementation, we set to $20$ the number of samples used in the latter heuristic [@elisabetta_personal].
The second competitor is the [[ABRA]{}]{} algorithm [@Riondato2016], available at [http://matteo.rionda.to/software/ABRA-radebetw.
tbz2](http://matteo.rionda.to/software/ABRA-radebetw.
tbz2). This algorithm samples pairs of nodes $(s,t)$, and it adds the fraction of $st$-paths passing from $v$ to the approximation of the betweenness of $v$, for each node $v$. The stopping condition is based on a key result in statistical learning theory, and there is a scheduler that decides when it should be tested. Following the suggestions by the authors, we use both the automatic scheduler [[ABRA-Aut]{}]{}, which uses a heuristic approach to decide when the stopping condition should be tested, and the geometric scheduler [[ABRA-1.2]{}]{}, which tests the stopping condition after $(1.2)^ik$ iterations, for each integer $i$.
The test is performed on a dataset made by $15$ undirected and $15$ directed real-world networks, taken from the datasets SNAP ([snap.stanford.edu/](snap.stanford.edu/)), LASAGNE ([piluc.dsi.unifi.it/lasagne](piluc.dsi.unifi.it/lasagne)), and KONECT (<http://konect.uni-koblenz.de/networks/>). As in [@Riondato2016], we have considered all values of $\lambda \in\{0.03, 0.025, 0.02, 0.015, 0.01, 0.005\}$, and $\delta=0.1$. All the algorithms have to provide an approximation $\tilde{\boldsymbol{b}}(v)$ of ${{\operatorname{bc}}(v)}$ for each $v$ such that $\Pr\left(\forall v, \left|\tilde{\boldsymbol{b}}(v)-{{\operatorname{bc}}(v)}\right| \leq \lambda\right) \geq 1-\delta$. In , we report the time needed by the different algorithms on every graph for $\lambda=0.005$ (the behavior with different values of $\lambda$ is very similar). More detailed results are reported in .
![The time needed by the different algorithms, on all the graphs of our dataset.[]{data-label="fig:running_timeshort"}](Figures/running_time){width="\textwidth"}
From the figure, we see that [[KADABRA]{}]{} is much faster than all the other algorithms, on all graphs: on average, our algorithm is about $100$ times faster than [[RK]{}]{} in undirected graphs, and about $70$ times faster in directed graphs; it is also more than $1\,000$ times faster than [[ABRA]{}]{}. The latter value is due to the fact that the [[ABRA]{}]{} algorithm has large running times on few networks: in some cases, it did not even conclude its computation within one hour. The authors confirmed that this behavior might be due to some bugs in the code, which seems to affect it only on specific graphs: indeed, in most networks, the performances of [[ABRA]{}]{} are better than those of the [[RK]{}]{} algorithm (but, still, not better than [[KADABRA]{}]{}).
![The exponent $\alpha$ such that the average number of edges visited during a bidirectional BFS is $n^\alpha$.[]{data-label="fig:bidbfsshort"}](Figures/bidirectional_experiments){width="\textwidth"}
In order to explain these data, we take a closer look at the improvements obtained through the bidirectional BFS, by considering the average number of edges $m_{\operatorname{avg}}$ that the algorithm visits in order to sample a shortest path (for all our competitors, $m_{\operatorname{avg}}=m$, since they perform a full BFS). In , for each graph in our dataset, we plot $\alpha=\frac{\log(m_{\operatorname{avg}})}{\log(m)}$ (intuitively, this means that the average number of edges visited is $m^{\alpha}$).
The figure shows that, apart from few cases, the number of edges visited is close to $n^{\frac{1}{2}}$, confirming the results in . This means that, since many of our networks have approximately $10\,000$ edges, the bidirectional BFS is about $100$ times faster than the standard BFS. Finally, for each value of $\lambda$, we report in the number of samples needed by all the algorithms, averaged over all the graphs in the dataset.
![The average number of samples needed by the different algorithms.[]{data-label="fig:nsamplesshort"}](Figures/comparison_plot){width="\textwidth"}
From the figure, [[KADABRA]{}]{} needs to sample the smallest amount of shortest paths, and the average improvement over [[RK]{}]{} grows when $\lambda$ tends to $0$, from a factor $1.14$ (resp., $1.14$) if $\lambda=0.03$, to a factor $1.79$ (resp., $2.05$) if $\lambda=0.005$ in the case of undirected (resp., directed) networks. Again, the behavior of [[ABRA]{}]{} is highly influenced by the behavior on few networks, and as a consequence the average number of samples is higher. In any case, also in the graphs where [[ABRA]{}]{} has good performances, [[KADABRA]{}]{} still needs a smaller number of samples.
### Computing Top-k Centralities {#computing-top-k-centralities .unnumbered}
In the second experiment, we let [[KADABRA]{}]{}compute the top-$k$ betweenness centralities of large graphs, which were unfeasible to handle with the previous algorithms.
The first set of graph is a series of temporal snapshots of the IMDB actor collaboration network, in which two actors are connected if they played together in a movie. The snapshots are taken every 5 years from 1940 to 2010, including a last snapshot in 2014, with $1\,797\,446$ nodes and $145\,760\,312$ edges. The graphs are extracted from the IMDB website (<http://www.imdb.com>), and they do not consider TV-series, awards-shows, documentaries, game-shows, news, realities and talk-shows, in accordance to what was done in <http://oracleofbacon.org>.
The other graph considered is the Wikipedia citation network, whose nodes are Wikipedia pages, and which contains an edge from page $p_1$ to page $p_2$ if the text of page $p_1$ contains a link to page $p_2$. The graph is extracted from DBPedia 3.7 (<http://wiki.dbpedia.org/>), and it consists of $4\,229\,697$ nodes and $102\,165\,832$ edges.
![The total time of computation of [[KADABRA]{}]{}on increasing snapshots of the IMDB graph.[]{data-label="fig:imdb_asymshort"}](Figures/actors_running_time){width="\textwidth"}
We have run our algorithm with $\lambda=0.0002$ and $\delta=0.1$: as discussed in , this means that either two nodes are ranked correctly, or their centrality is known with precision at most $\lambda$. As a consequence, if two nodes are not ranked correctly, the difference between their real betweenness is at most $2\lambda$. The full results are available in .
All the graphs were processed in less than one hour, apart from the Wikipedia graph, which was processed in approximately $1$ hour and $38$ minutes. In , we plot the running times for the actor graphs: from the figure, it seems that the time needed by our algorithm scales slightly sublinearly with respect to the size of the graph. This result respects the results in , because the degrees in the actor collaboration network are power law distributed with exponent $\beta \approx 2.13$ (<http://konect.uni-koblenz.de/networks/actor-collaboration>). Finally, we observe that the ranking is quite precise: indeed, most of the times, there are very few nodes in the top-$5$ with the same ranking, and the ranking rarely contains significantly more than $10$ nodes.
##### Acknowledgements.
The authors would like to thank Matteo Riondato for several constructive comments on an earlier version of this work. We also thank Elisabetta Bergamini, Richard Lipton, and Sebastiano Vigna for helpful discussions and Holger Dell for his help with the experiments.
Pseudocode
==========
$\alpha \gets \frac{\omega}{100}$ $\epsilon \gets 0.0001$
Binary search to find $C$ such that $\sum_{v \in V} \exp\left(-\frac{C}{c_L(v)}\right)+\exp\left(-\frac{C}{c_U(v)}\right)=\frac{\delta}{2}-\epsilon\delta$
Sort nodes in decreasing order of ${{\tilde{\boldsymbol{b}}}(v)}$, obtaining $v_1,\dots,v_n$
Concentration Inequalities {#app:concentration}
==========================
\[lem:hoeff\] Let ${\boldsymbol{X}}_1,\dots,{\boldsymbol{X}}_k$ be independent random variables such that $a_i<{\boldsymbol{X}}_i<b_i$, and let ${\boldsymbol{X}}=\sum_{i=1}^k {\boldsymbol{X}}_i$. Then, $$\Pr\left(|{\boldsymbol{X}}-{\mathbb{E}}[X]| \geq \lambda\right) \leq \exp\left\{-\frac{2\lambda^2}{\sum_{i=1}^k(b_i-a_i)^2}\right\}.$$
If we apply Hoeffding’s inequality with ${\boldsymbol{X}}_i={X_{v}^{{\boldsymbol{\pi}}}}$, ${\boldsymbol{X}}=k{\boldsymbol{b}(v)} = \sum_{i=1}^k {X_{v}^{{\boldsymbol{\pi}}}}$, $a_i=0$, $b_i=1$, we obtain that ${\Pr}\left(\left|{\boldsymbol{b}(v)}- {{\operatorname{bc}}(v)}\right|>\lambda\right)<2e^{-2k\lambda^2}$. Then, if we fix $\delta=2e^{-2k\lambda^2}$, the error is $\lambda=\sqrt{\frac{\log(2/\delta)}{2k}}$, and the minimum $k$ needed to obtain an error $\lambda$ on the betweenness of a single node is $\frac{1}{2\lambda^2}\log(2/\delta)$.
\[lem:cb\] Let ${\boldsymbol{X}}_1,\dots,{\boldsymbol{X}}_k$ be independent random variables such that ${\boldsymbol{X}}_i\leq M$ for each $1\leq i\leq n$, and let ${\boldsymbol{X}}=\sum_{i=1}^k {\boldsymbol{X}}_i$. Then, $$\Pr\left({\boldsymbol{X}}\geq {\mathbb{E}}[{\boldsymbol{X}}] + \lambda \right) \leq \exp\left\{- \frac{\lambda^2}{2(\sum_{i=1}^n {\mathbb{E}}[{\boldsymbol{X}}_i^2] + M \lambda /3)}\right\}.$$
\[thm:mcdiarmid\] Let $X$ be a martingale associated with a filter $\mathcal{F}$, satisfying
- $\mathrm{Var}\left(X_{i}\middle|\ \mathcal{F}_{i}\right)\leq\sigma_{i}$ for $1\leq i\leq\ell$,
- $\left|X_{i}-X_{i-1}\right|\leq M$, for $1\leq i\leq\ell$.
Then, we have $$\Pr\left(X-\mathbb{E}\left(X\right)\geq\lambda\right)\leq\exp\left(-\frac{\lambda^{2}}{2\left(\sum_{i=1}^{\ell}\sigma_{i}^{2}+M\lambda/3\right)}\right).$$
Proof of Theorem \[thm:mainshort\] {#sec:adaptive}
==================================
In our algorithm, we sample ${\boldsymbol{\tau}}$ shortest paths ${\boldsymbol{\pi}}_i$, where ${\boldsymbol{\tau}}$ is a random variable such that ${\boldsymbol{\tau}}={\tau}$ can be decided by looking at the first ${\tau}$ paths sampled (see Algorithm \[alg:mainshort\]). Furthermore, thanks to Eq. (3) in [@Riondato2015], we assume that ${\boldsymbol{\tau}}\leq \omega$ for some fixed $\omega \in \mathbb{R}^+$ such that, after $\omega$ steps, $\Pr(\forall v,|{{\tilde{\boldsymbol{b}}}(v)}-{{\operatorname{bc}}(v)}|\leq\lambda) \geq 1-\frac{\delta}{2}$. When the algorithm stops, our estimate of the betweenness is ${\tilde{\boldsymbol{b}}}(v):=\frac{1}{{\boldsymbol{\tau}}}\sum_{i=1}^{{\boldsymbol{\tau}}} {\boldsymbol{X}}_i(v)$, where ${\boldsymbol{X}}_i(v)$ is $1$ if $v$ belongs to ${\boldsymbol{\pi}}_i$, $0$ otherwise.
To estimate the error, we use the following theorem.
\[thm:azuma-borassi\] For each node $v$ and for every fixed real numbers ${\delta_{L}}$, ${\delta_{U}}$, it holds $$\begin{aligned}
\Pr\left({{\operatorname{bc}}(v)}\leq{{\tilde{\boldsymbol{b}}}(v)}-f\left({\tilde{\boldsymbol{b}}}(v),{\delta_{L}}, \omega, {\boldsymbol{\tau}}\right)\right) & \leq{\delta_{L}}\quad\mbox{and}\\
\Pr\left({{\operatorname{bc}}(v)}\geq{{\tilde{\boldsymbol{b}}}(v)}+g\left({{\tilde{\boldsymbol{b}}}(v)},{\delta_{U}}, \omega, {\boldsymbol{\tau}}\right)\right) & \leq{\delta_{U}},\end{aligned}$$ where $$\begin{aligned}
f\left({{\tilde{\boldsymbol{b}}}(v)},{\delta_{L}}, \omega, {\boldsymbol{\tau}}\right)
&=\frac{1}{{\boldsymbol{\tau}}} {\log{\frac 1{{\delta_{L}}}}}\left(\frac{1}{3}-\frac{\omega}{{\boldsymbol{\tau}}}+\sqrt{\left(\frac{1}{3}-\frac{\omega}{{\boldsymbol{\tau}}}\right)^{2}+\frac{2{{\tilde{\boldsymbol{b}}}(v)}\omega}{{\log{\frac 1{{\delta_{L}}}}}}}\right) \quad \text{and}
\label{eq:thm_first}\\
g\left({{\tilde{\boldsymbol{b}}}(v)},{\delta_{U}}, \omega, {\boldsymbol{\tau}}\right)
&=\frac{1}{{\boldsymbol{\tau}}} {\log{\frac 1{{\delta_{U}}}}}\left(\frac{1}{3}+\frac{\omega}{{\boldsymbol{\tau}}}
+\sqrt{\left(\frac{1}{3}+\frac{\omega}{{\boldsymbol{\tau}}}\right)^{2}+\frac{2{{\tilde{\boldsymbol{b}}}(v)}\omega}{{\log{\frac 1{{\delta_{U}}}}}}}\right).
\label{eq:thm_second}\end{aligned}$$
We prove Theorem \[thm:azuma-borassi\] in Section \[sec:azuma-borassi\]. In the rest of this section, we show how this theorem implies . To simplify notation, we often omit the arguments of the function $f$ and $g$.
Let $\boldsymbol{E}_1$ be the event $({\boldsymbol{\tau}}=\omega \wedge \exists v \in V, |{{\tilde{\boldsymbol{b}}}(v)}-{{\operatorname{bc}}(v)}| > \lambda)$, and let $\boldsymbol{E}_2$ be the event $({\boldsymbol{\tau}}<\omega \wedge (\exists v \in V, -f \geq {{\operatorname{bc}}(v)} - {{\tilde{\boldsymbol{b}}}(v)} \vee {{\operatorname{bc}}(v)} - {{\tilde{\boldsymbol{b}}}(v)} \geq g))$. Let us also denote ${{\tilde{\boldsymbol{b}}}_{{\tau}}(v)}=\frac{1}{{\tau}}\sum_{i=1}^{{\tau}} {\boldsymbol{X}}_i(v)$ (note that ${{\tilde{\boldsymbol{b}}}_{{\boldsymbol{\tau}}}(v)}={{\tilde{\boldsymbol{b}}}(v)}$).
By our choice of $\omega$ and Eq. (3) in [@Riondato2015], $$\Pr(\boldsymbol{E}_1) \leq \Pr(\exists v \in V,|{{\tilde{\boldsymbol{b}}}_{\omega}(v)}-{{\operatorname{bc}}(v)}| > \lambda)\leq \frac{\delta}{2}$$ where ${{\tilde{\boldsymbol{b}}}_{\omega}(v)}$ is the approximate betweenness of $v$ after $\omega$ samples. Furthermore, by , $$\begin{aligned}
\Pr(\boldsymbol{E}_2) &\leq \sum_{v \in V}\Pr({\boldsymbol{\tau}}<\omega \wedge -f \geq {{\operatorname{bc}}(v)} - {{\tilde{\boldsymbol{b}}}(v)})+\Pr({\boldsymbol{\tau}}<\omega \wedge {{\operatorname{bc}}(v)} - {{\tilde{\boldsymbol{b}}}(v)} \leq g)\\
&\leq \sum_{v \in V} {\delta_{L}^{(v)}}+{\delta_{U}^{(v)}} \leq \frac{\delta}{2}.\end{aligned}$$ By a union bound, $\Pr(\boldsymbol{E}_1 \vee \boldsymbol{E}_2) \leq \Pr(\boldsymbol{E}_1)+\Pr(\boldsymbol{E}_1) \leq \delta$, concluding the proof of .
Proof of Theorem \[thm:azuma-borassi\] {#sec:azuma-borassi}
--------------------------------------
Since this theorem deals with a single node $v$, let us simply write ${\operatorname{bc}}= {{\operatorname{bc}}(v)}, {\tilde{\boldsymbol{b}}}= {{\tilde{\boldsymbol{b}}}(v)}, {\boldsymbol{X}}_i={\boldsymbol{X}}_i(v)$. Let us consider ${\boldsymbol{Y}}^\tau=\sum_{i=1}^\tau\left({\boldsymbol{X}}_i-{\operatorname{bc}}\right)$ (we recall that ${\boldsymbol{X}}_i=1$ if $v$ is in the $i$-th path sampled, ${\boldsymbol{X}}_i=0$ otherwise). Clearly, ${\boldsymbol{Y}}^\tau$ is a martingale, and ${\boldsymbol{\tau}}$ is a stopping time for ${\boldsymbol{Y}}^\tau$: this means that also ${\boldsymbol{Z}}^\tau={\boldsymbol{Y}}^{\min({\boldsymbol{\tau}},\tau)}$ is a martingale.
Let us apply Theorem \[thm:mcdiarmid\] to the martingales ${\boldsymbol{Z}}$ and $-{\boldsymbol{Z}}$: for each fixed ${\lambda_{L}},{\lambda_{U}}>0$ we have $$\begin{aligned}
\Pr\left({\boldsymbol{Z}}^{\omega}\geq{\lambda_{L}}\right)
& =\Pr\left({\boldsymbol{\tau}}{\tilde{\boldsymbol{b}}}-{\boldsymbol{\tau}}{\operatorname{bc}}\geq{\lambda_{L}}\right)\leq\exp\left(-\frac{{\lambda_{L}}^{2}}{2\left(\omega {\operatorname{bc}}+{\lambda_{L}}/3\right)}\right)={\delta_{L}}\quad\mbox{and}
\label{eq:azuma_apply}\\
\Pr\left(-{\boldsymbol{Z}}^{\omega}\geq{\lambda_{U}}\right)
& =\Pr\left({\boldsymbol{\tau}}{\tilde{\boldsymbol{b}}}-{\boldsymbol{\tau}}{\operatorname{bc}}\leq-{\lambda_{U}}\right)\leq\exp\left(-\frac{{\lambda_{U}}^{2}}{2\left(\omega {\operatorname{bc}}+{\lambda_{U}}/3\right)}\right)={\delta_{U}}. \label{eq:azuma_apply_second}\end{aligned}$$ We now show how to prove from . The way to derive from is analogous.
If we express ${\lambda_{L}}$ as a function of ${\delta_{L}}$ we get $${\lambda_{L}}^{2}=2{\log{\frac 1{{\delta_{L}}}}}\left(\omega {\operatorname{bc}}+\frac{{\lambda_{L}}}{3}\right)\iff{\lambda_{L}}^{2}-\frac{2}{3}{\lambda_{L}}{\log{\frac 1{{\delta_{L}}}}}-2\omega {\operatorname{bc}}{\log{\frac 1{{\delta_{L}}}}}=0,$$ which implies that $${\lambda_{L}}=\frac{1}{3}{\log{\frac 1{{\delta_{L}}}}}\pm\sqrt{\frac{1}{9}\left({\log{\frac 1{{\delta_{L}}}}}\right)^{2}+2\omega {\operatorname{bc}}{\log{\frac 1{{\delta_{L}}}}}}.$$ Since holds for any positive value ${\lambda_{L}}$, it also holds for the value corresponding to the positive solution of this equation, that is, $${\lambda_{L}}=\frac{1}{3}{\log{\frac 1{{\delta_{L}}}}}+\sqrt{\frac{1}{9}\left({\log{\frac 1{{\delta_{L}}}}}\right)^{2}+2\omega {\operatorname{bc}}{\log{\frac 1{{\delta_{L}}}}}}.$$ Plugging this value into , we obtain $$\Pr\left({\boldsymbol{\tau}}{\tilde{\boldsymbol{b}}}-{\boldsymbol{\tau}}{\operatorname{bc}}\geq\frac{1}{3}{\log{\frac 1{{\delta_{L}}}}}+\sqrt{\frac{1}{9}\left({\log{\frac 1{{\delta_{L}}}}}\right)^{2}+2\omega {\operatorname{bc}}{\log{\frac 1{{\delta_{L}}}}}}\right)\leq{\delta_{L}}.
\label{eq:first_azume}$$ By assuming ${\tilde{\boldsymbol{b}}}-{\operatorname{bc}}\geq \frac 1{3{\boldsymbol{\tau}}} \log (\frac{1}{{\delta_{L}}})$, the event in can be rewritten as $$\left({\boldsymbol{\tau}}{\operatorname{bc}}\right)^{2}-2{\operatorname{bc}}\left({\boldsymbol{\tau}}^{2}{\tilde{\boldsymbol{b}}}+\omega{\log{\frac 1{{\delta_{L}}}}}-\frac{1}{3}{\boldsymbol{\tau}}{\log{\frac 1{{\delta_{L}}}}}\right)-\frac{2}{3}{\log{\frac 1{{\delta_{L}}}}}{\boldsymbol{\tau}}{\tilde{\boldsymbol{b}}}+\left({\boldsymbol{\tau}}{\tilde{\boldsymbol{b}}}\right)^{2}\geq0.$$ By solving the previous quadratic equation w.r.t. ${\operatorname{bc}}$ we get $${\operatorname{bc}}\leq
{\tilde{\boldsymbol{b}}}+{\log{\frac 1{{\delta_{L}}}}}\left(
\frac{\omega}{{\boldsymbol{\tau}}^{2}}
-\frac{1}{3{\boldsymbol{\tau}}}
-\sqrt{
\left(\frac{{\tilde{\boldsymbol{b}}}}{{\log{\frac 1{{\delta_{L}}}}}}+\frac{\omega}{{\boldsymbol{\tau}}^{2}}-\frac{1}{3{\boldsymbol{\tau}}}\right)^{2}
-\left(\frac{{\tilde{\boldsymbol{b}}}}{{\log{\frac 1{{\delta_{L}}}}}}\right)^{2}+\frac{2}{3{\boldsymbol{\tau}}}\frac{{\tilde{\boldsymbol{b}}}}{{\log{\frac 1{{\delta_{L}}}}}}}
\right),$$ where we only considered the solution which upper bounds ${\operatorname{bc}}$, since we assumed ${\tilde{\boldsymbol{b}}}-{\operatorname{bc}}\geq \frac 1{3\tau} \log (\frac{1}{{\delta_{L}}})$. After simplifying the terms under the square root in the previous expression, we get $${\operatorname{bc}}\leq
{\tilde{\boldsymbol{b}}}+{\log{\frac 1{{\delta_{L}}}}}\left(
\frac{\omega}{{\boldsymbol{\tau}}^{2}}-\frac{1}{3{\boldsymbol{\tau}}}-\sqrt{\left(\frac{\omega}{{\boldsymbol{\tau}}^{2}}-\frac{1}{3{\boldsymbol{\tau}}}\right)^{2}+\frac{2{\tilde{\boldsymbol{b}}}\omega}{{\boldsymbol{\tau}}^2{\log{\frac 1{{\delta_{L}}}}}}}
\right),$$ which means that $$\Pr\left({\operatorname{bc}}\leq{\tilde{\boldsymbol{b}}}-f\left({\tilde{\boldsymbol{b}}}, {\delta_{L}}, \omega, {\boldsymbol{\tau}}\right)\right)\leq{\delta_{L}},$$ concluding the proof.
How to Choose delta(v) {#sec:deltav}
======================
In , we proved that our algorithm works for any choice of the values ${\delta_{L}^{(v)}}, {\delta_{U}^{(v)}}$. In this section, we show how we can heuristically compute such values, in order to obtain the best performances.
For each node $v$, let ${\lambda_{L}^{(v)}}, {\lambda_{U}^{(v)}}$ be the lower and the upper maximum error that we want to obtain on the betweenness of $v$: if we simply want all errors to be smaller than $\lambda$, we choose ${\lambda_{L}^{(v)}}, {\lambda_{U}^{(v)}} = \lambda$, but for other purposes different values might be needed. We want to minimize the time $\tau$ such that the approximation of the betweenness at time $\tau$ is in the confidence interval required. In formula, we want to minimize $$\label{eq:conddeltav}
\min\left\{\tau \in \mathbb{N}:\forall v \in V, \left(f\left({{\tilde{\boldsymbol{b}}}_{\tau}(v)},{\delta_{L}^{(v)}},\omega,{\tau}\right) \leq {\lambda_{L}^{(v)}} \wedge g\left({{\tilde{\boldsymbol{b}}}_{\tau}(v)},{\delta_{U}^{(v)}},\omega,{\tau}\right) \leq {\lambda_{U}^{(v)}}\right)\right\}$$
where ${{\tilde{\boldsymbol{b}}}_{\tau}(v)}$ is the approximation of ${{\operatorname{bc}}(v)}$ obtained at time $\tau$, and $$\begin{aligned}
f\left(\tau,{\tilde{\boldsymbol{b}}}_\tau,{\delta_{L}}, \omega\right)&=\frac{1}{\tau} {\log{\frac 1{{\delta_{L}}}}}\left(\frac{1}{3}-\frac{\omega}{\tau}+\sqrt{\left(\frac{1}{3}-\frac{\omega}{\tau}\right)^{2}+\frac{2{\tilde{\boldsymbol{b}}}_\tau\omega}{{\log{\frac 1{{\delta_{L}}}}}}}\right) \quad \text{and}\\
g\left(\tau,{\tilde{\boldsymbol{b}}}_\tau,{\delta_{U}}, \omega\right)&=\frac{1}{\tau} {\log{\frac 1{{\delta_{U}}}}}\left(\frac{1}{3}+\frac{\omega}{\tau}+\sqrt{\left(\frac{1}{3}+\frac{\omega}{\tau}\right)^{2}+\frac{2{\tilde{\boldsymbol{b}}}_\tau\omega}{{\log{\frac 1{{\delta_{U}}}}}}}\right).\end{aligned}$$
The goal of this section is to provide deterministic values of ${\delta_{L}^{(v)}},{\delta_{U}^{(v)}}$ that minimize the value in , and such that $\sum_{v \in V} {\delta_{L}^{(v)}}+ {\delta_{U}^{(v)}}<\frac{\delta}{2}$. To obtain our estimate, we replace ${{\tilde{\boldsymbol{b}}}_{\tau}(v)}$ with an approximation $\tilde{b}(v)$, that we compute by sampling $\alpha$ paths, before starting the algorithm (in our code, $\alpha=\frac{\omega}{100}$). Furthermore, we consider a simplified version of : in most cases, ${\lambda_{L}}$ is much smaller than all other quantities in play, and since $\omega$ is proportional to $\frac{1}{{\lambda_{L}}^2}$, we can safely assume $f(\tau,\tilde{b}(v),{\delta_{L}^{(v)}}, \omega) \approx \sqrt{\frac{2\tilde{b}(v)\omega}{\tau^2}{\log{\frac 1{{\delta_{L}}}}}}$ and $g(\tau,\tilde{b}(v),{\delta_{U}^{(v)}}, \omega) \approx \sqrt{\frac{2\tilde{b}(v)\omega}{\tau^2}{\log{\frac 1{{\delta_{U}}}}}}$. Hence, in place of the value in , our heuristic tries to minimize $$\min\left\{\tau \in \mathbb{N}:\forall v \in V, \sqrt{\frac{2\tilde{b}(v)\omega}{\tau^2}{\log{\frac 1{{\delta_{L}^{(v)}}}}}} \leq {\lambda_{L}^{(v)}} \wedge \sqrt{\frac{2\tilde{b}(v)\omega}{\tau^2}{\log{\frac 1{{\delta_{U}^{(v)}}}}}} \leq {\lambda_{U}^{(v)}}\right\}.$$ Solving with respect to $\tau$, we are trying to minimize $$\max_{v \in V} \left(\max\left(\sqrt{\frac{2\tilde{b}(v)\omega}{\left({\lambda_{L}^{(v)}}\right)^2}{\log{\frac 1{{\delta_{L}^{(v)}}}}}}, \sqrt{\frac{2\tilde{b}(v)\omega}{\left({\lambda_{U}^{(v)}}\right)^2}{\log{\frac 1{{\delta_{U}^{(v)}}}}}}\right)\right).$$ which is the same as minimizing $\max_{v \in V} \max \left(c_L(v){\log{\frac 1{{\delta_{L}^{(v)}}}}},c_U(v){\log{\frac 1{{\delta_{U}^{(v)}}}}}\right)$ for some constants $c_L(v), c_U(v)$, conditioned on $\sum_{v \in V} {\delta_{L}^{(v)}}+ {\delta_{U}^{(v)}}<\frac{\delta}{2}$. We claim that, among the possible choices of ${\delta_{L}^{(v)}},{\delta_{U}^{(v)}}$, the best choice makes all the terms in the maximum equal: otherwise, if two terms were different, we would be able to slightly increase and decrease the corresponding values, in order to decrease the maximum. This means that, for some constant $C$, for each $v$, $c_L(v){\log{\frac 1{{\delta_{L}^{(v)}}}}} = c_U(v){\log{\frac 1{{\delta_{L}^{(v)}}}}}=C$, that is, ${\delta_{L}^{(v)}}=\exp(-\frac{C}{c_L(v)})$, ${\delta_{U}^{(v)}}=\exp(-\frac{C}{c_U(v)})$. In order to find the largest constant $C$ such that $\sum_{v \in V} {\delta_{L}^{(v)}}+ {\delta_{U}^{(v)}}\leq \frac{\delta}{2}$, we use a binary search procedure on all possible constants $C$.
Finally, if $c_L(v)=0$ or $c_U(v)=0$, this procedure chooses ${\delta_{L}^{(v)}}=0$: to avoid this problem, we impose $\sum_{v \in V} {\delta_{L}^{(v)}}+ {\delta_{U}^{(v)}}\leq \frac{\delta}{2}-\epsilon\delta$, and we add $\frac{\epsilon\delta}{2n}$ to all the ${\delta_{L}^{(v)}}$s and all the ${\delta_{U}^{(v)}}$s (in our code, we choose $\epsilon=0.001$). The pseudocode of the algorithm is available in Algorithm \[alg:choosedeltav\].
Balanced Bidirectional BFS on Random Graphs {#sec:bbbfs}
===========================================
In this appendix, we formally prove that the bidirectional BFS is efficient in several models of random graphs: the Configuration Model (CM, [@Bollobas1980]), and Rank-1 Inhomogeneous Random Graph models (IRG, [@Hofstad2014 Chapter 3]), such as the Chung-Lu model [@Chung2006], the Norros-Reittu model [@Norros2006], and the Generalized Random Graph [@Hofstad2014 Chapter 3]. All these models are defined by fixing the number $n$ of nodes and $n$ weights ${\rho_{v}}$, and by creating edges at random, in a way that node $v$ gets degree close to ${\rho_{v}}$.
More formally, the edges are generated as follows:
- in the CM, each node is associated to ${\rho_{v}}$ half-edges, or stubs; edges are created by randomly pairing these $M=\sum_{v \in V} {\rho_{v}}$ stubs (we assume the number of stubs to be even, by adding a stub to a random node if necessary).
- in IRG, an edge between a node $v$ and a node $w$ exists with probability $f(\frac{{\rho_{v}}{\rho_{w}}}{M})$, where $M=\sum_{v \in V} {\rho_{v}}$, and the existence of different edges is independent. Different choices of the function $f$ create different models.
- In general, we assume that $f$ satisfies the following conditions:
- $f$ is derivable at least twice in $0$;
- $f$ is increasing;
- $f'(0)=1$;
- in the Chung-Lu model, $f(x)=\min(x,1)$;
- in the Norros-Reittu model, $f(x)=1-e^{-x}$;
- in the Generalized Random Graph model, $f(x)=\frac{x}{1+x}$.
It remains to define how we choose the weights ${\rho_{v}}$, when the number of nodes $n$ tends to infinity. In the line of previous works [@Norros2006; @Fernholz2007; @Hofstad2014], we consider a sequence of graphs $G_i$, whose number of nodes $n_i$ tends to infinity, and whose degree distribution $\lambda_i$ satisfy the following:
1. there is a probability distribution $\lambda$ such that the $\lambda_i$s tend to $\lambda$ in distribution;
2. $M_1(\lambda_i)$ tends to $M_1(\lambda)<\infty$, where $M_1(\lambda)$ is the first moment of $\lambda$;
3. one of the following two conditions hold:
1. $M_2(\lambda_i)$ tends to $M_2(\lambda)<\infty$, where $M_2(\lambda)$ is the second moment of $\lambda$;
2. $\lambda$ is a power law distribution with $2<\beta<3$, and there is a global constant $C$ such that, for each $d$, $\Pr(\lambda_i\geq d)\leq \frac{C}{d^{\beta-1}}$. \[cond:powerlaw\]
For example, these assumptions are satisfied with probability $1$ if we choose the degrees independently, according to a distribution $\lambda$ with finite mean [@Hofstad2014 Section 6.1,7.2].
Note that an aspect often neglected in previous work when it comes to computing shortest paths is the fact that the number of shortest paths between a pair of nodes may be exponential, thus requiring to work with a linear number of bits. While real-world complex networks are typically sparse with logarithmic diameter, in order to avoid such issue it is sufficient to assume that addition and comparison require constant time.
These assumptions cover the Erdös-Renyi random graph with constant average degree, and all power law distributions with $\beta>2$ (because, if $\beta>3$, then $M_2(\lambda)$ is finite).
Assumption \[cond:powerlaw\] seems less natural than the other assumptions. However, it is necessary to exclude pathological cases: for example, assume that $G_i$ has $n-2$ nodes chosen according to a power law distribution, and $2$ nodes $u,v$ with weight $n^{1-\epsilon}$. All assumption except \[cond:powerlaw\] are satisfied, but the bidirectional BFS is not efficient, because if $s$ is a neighbor of $u$ with degree $1$, and $t$ is a neighbor of $v$ with degree $1$, then a bidirectional BFS from $s$ and $t$ needs to visit all neighbors of $u$ or all neighbors of $v$, and the time needed is $\Omega(n^{1-\epsilon})$.
We say that a random graph has a property $\pi$ asymptotically almost surely ([a.a.s.]{}) if $\Pr(\pi(G_i))$ tends to $1$ when $n$ tends to infinity. We say that a random graph has a property $\pi$ with high probability ([w.h.p.]{}) if $\frac{\Pr(\pi(G_i))}{n_i^k}$ tends to $0$ for each $k>0$.
Before proving the main theorem, we need two more definitions and a technical assumption.
In the CM, let ${\rho_{\text{\upshape res}}}=\frac{M_2(\lambda)}{M_1(\lambda)}-1$. In IRG, let ${\rho_{\text{\upshape res}}}=\frac{M_2(\lambda)}{M_1(\lambda)}$ (if $\lambda$ is a power law distribution with $2<\beta<3$, we simply define ${\rho_{\text{\upshape res}}}=+\infty$).
Given a set $V' \subseteq V$, the volume of $V'$ is ${\rho_{V'}}=\sum_{v \in V'} {\rho_{v}}$. Furthermore, if $V'={\boldsymbol{\Gamma}^{d}(s)}$, we abbreviate ${\rho_{{\boldsymbol{\Gamma}^{d}(s)}}}$ with ${\boldsymbol{r}^{l}(s)}$.
The value ${\rho_{\text{\upshape res}}}$ is closely related to $\frac{{\boldsymbol{r}^{l+1}(s)}}{{\boldsymbol{r}^{l}(s)}}$: informally, the expected value of this fraction is ${\rho_{\text{\upshape res}}}$. For this reason, if ${\rho_{\text{\upshape res}}}<1$, then the size of neighbors tends to decrease, and all connected components have ${\mathcal{O}}(\log n)$ nodes. Conversely, if ${\rho_{\text{\upshape res}}}>1$, then the size of neighbors tends to increase, and there is a *giant component* of size $\Theta(n)$ (for a proof of these facts, see [@Hofstad2014 Section 2.3 and Chapter 4]). Our last assumption is that ${\rho_{\text{\upshape res}}}>1$, in order to ensure the existence of the giant component.
Under these assumptions, we prove Theorem \[thm:bidirectionalshort\], following the sketch in . We start by linking the degrees and the weights of nodes.
\[lem:degsweight\] For each node $v$, ${\rho_{v}} n^{-\epsilon} \leq \deg(v) \leq {\rho_{v}} n^{\epsilon}$ [w.h.p.]{}.
We use [@Borassi2016 Lemmas 32 and 37][^3]: these lemmas imply that, for each $\epsilon>0$, if ${\rho_{v}}>n^\epsilon$, $(1-\epsilon) {\rho_{v}} \leq \deg(v) \leq (1+\epsilon) {\rho_{v}}$ [w.h.p.]{}. We have to handle the case where ${\rho_{v}}<n^\epsilon$: one of the two inequalities is empty, while for the other inequality we observe that, if we decrease the weight of $v$, the degree of $v$ can only decrease. Hence, if ${\rho_{v}}<n^\epsilon$, $\deg(v)<(1+\epsilon)n^{\epsilon}$, and the result follows by changing the value of $\epsilon$.
Following the intuitive proof, we have linked the number of visited edges with their weights. Let us define an abbreviation for the volume of the nodes at distance $l$ from $s$.
We denote by ${\boldsymbol{r}^{l}(s)}$ the volume of nodes at distance exactly $l$ from $s$. In the CM, we denote by ${\boldsymbol{R}^{l}(s)}$ the set of stubs at distance $l$ from $s$.
Now, we need to show that, if ${\boldsymbol{r}^{l_s}(s)},{\boldsymbol{r}^{l_t}(t)}>n^{\frac{1}{2}+\epsilon}$, then $d(s,t)\leq l_s+l_t+2$ [w.h.p.]{}.
\[lem:touchbid\] Assume that ${\boldsymbol{r}^{l_s}(s)}>n^{\frac{1}{2}+\epsilon}$, ${\boldsymbol{r}^{l_t}(t)}>n^{\frac{1}{2}+\epsilon}$, and ${\boldsymbol{r}^{l_s-1}(s)},{\boldsymbol{r}^{l_t-1}(t)}<(1-\epsilon)n^{\frac{1}{2}+\epsilon}$. Then, $d(s,t) \leq l_s+l_t+2$.
Let us assume that we know the structure of ${\boldsymbol{N}^{l_s}(s)}$ and ${\boldsymbol{N}^{l_t}(t)}$, that is, for each possible structure $S$ of the subgraph induced by all nodes at distance $l_s$ from $s$ and distance $l_t$ from $t$, let $E_S$ be the event that ${\boldsymbol{N}^{l_s}(s)}$ and ${\boldsymbol{N}^{l_t}(t)}$ are exactly $S$. If we prove that ${\Pr}(d(s,t) \leq l+l'+2 | E_{S})<\epsilon$, then ${\Pr}({\boldsymbol{r}^{l+1}(s)}>{\boldsymbol{r}^{l}(s)})=\sum_{S} {\Pr}({\boldsymbol{r}^{l+1}(s)}>{\boldsymbol{r}^{l}(s)}| E_{S}){\Pr}(E_{S})<\sum_{S} \epsilon {\Pr}(E_{S})=\epsilon$. First of all, if $S$ is such that the two neighborhoods touch each other, ${\Pr}(d(s,t) \leq l+l'+2 | E_{S})=0<\epsilon$. Otherwise, we consider separately the CM and IRG.
In the CM, conditioned on $E_S$, the stubs that are paired with stubs in ${\boldsymbol{R}^{l_s}(s)}$ are a random subset of the set of stubs that are not paired in $S$. This random subset has size at least $\epsilon n^{\frac{1}{2}+\epsilon} \geq n^{\frac{1+\epsilon}{2}}$ (because $\epsilon$ is a fixed constant, and $n$ tends to infinity). Since the total number of stubs is ${\mathcal{O}}(n)$, and since the number of stubs in ${\boldsymbol{R}^{l_t}(t)}$ is at least $\epsilon n^{\frac{1+\epsilon}{2}}$, one of the stubs in ${\boldsymbol{R}^{l_t}(t)}$ is paired with a stub in ${\boldsymbol{r}^{l_s}(s)}$ [w.h.p.]{}, and $d(s,t) \leq l_s+l_t+1$.
In IRG, the probability that a node $v$ is not connected to any node in ${\boldsymbol{\Gamma}^{l_s}(s)}$ is at most $\prod_{w \in {\boldsymbol{\Gamma}^{l_s}(s)}} (1-f(\frac{{\rho_{v}} {\rho_{w}}}{M})) =\prod_{w \in {\boldsymbol{\Gamma}^{l_s}(s)}}(1-\Omega(\frac{{\rho_{w}}}{M}))=\exp({-\sum_{w \in {\boldsymbol{\Gamma}^{l_s}(s)}}\Omega(\frac{{\rho_{w}}}{M})})=\exp({-\Omega(\frac{{\boldsymbol{r}^{l_s}(s)}}{M})})=1-\Omega(\frac{{\boldsymbol{r}^{l_s}(s)}}{M})=1-\Omega(n^{-\frac{1}{2}+\epsilon})$. This means that $v$ belongs to ${\boldsymbol{\Gamma}^{l_s+1}(s)}$ with probability $\Omega(n^{-\frac{1}{2}+\epsilon})$, and similarly it belongs to ${\boldsymbol{\Gamma}^{{l_t}+1}(t)}$ with probability $\Omega(n^{-\frac{1}{2}+\epsilon})$. Since the two events are independent, the probability that $v$ belongs to both is $\Omega(n^{-1+2\epsilon})$. Since, for each node $v$, the events that $v$ belongs to ${\boldsymbol{\Gamma}^{l_s+1}(s)} \cap {\boldsymbol{\Gamma}^{l_t+1}(t)}$ are independent, by a straightforward application of Hoeffding’s inequality, [w.h.p.]{}, there is a node $v$ that belongs to ${\boldsymbol{\Gamma}^{l_s+1}(s)} \cap {\boldsymbol{\Gamma}^{l_t+1}(t)}$, and $d(s,t) \leq l_s+l_t+2$ [w.h.p.]{}, concluding the proof.
The next ingredient is used to bound the first integers $l_s,l_t$ such that ${\boldsymbol{r}^{l_s}(s)},{\boldsymbol{r}^{l_t}(t)}>n^{\frac{1}{2}+\epsilon}$.
\[thm:diameter\] The diameter of a graph generated through the aforementioned models is ${\mathcal{O}}(\log n)$.
The last ingredient of our proof is an upper bound on the size of ${\boldsymbol{r}^{l_s}(s)}$ and ${\boldsymbol{r}^{l_t}(t)}$.
\[lem:lastlevel\] With high probability, for each $s \in V$ and for each $l$ such that $\sum_{i=0}^l {\boldsymbol{r}^{l}(s)}<n^{\frac{1}{2}+\epsilon}$, ${\boldsymbol{r}^{l+1}(s)}<n^{\frac{1}{2}+3\epsilon}$ if $\lambda$ has finite second moment, ${\boldsymbol{r}^{l+1}(s)}<n^{\frac{4-\beta}{2}+3\epsilon}$ if $\lambda$ is power law with $2<\beta<3$.
We consider separately nodes with weight at most $n^{\frac{1}{2}-2\epsilon}$ from nodes with bigger weights: in the former case, we bound the number of such nodes that are in ${\boldsymbol{R}^{l+1}(s)}$, while in the latter case we bound the total number of nodes with weight at least $n^{\frac{1}{2}-2\epsilon}$. Let us start with nodes with the latter case.
**Claim:** for each $\epsilon$, $\sum_{{\rho_{v}} \geq n^{\frac{1}{2}-\epsilon}} {\rho_{v}}$ is smaller than $n^{\frac{1}{2}+3\epsilon}$ if $\lambda$ has finite second moment, and it is smaller than $n^{\frac{4-\beta}{2}+3\epsilon}$ if $\lambda$ is power law with $2<\beta<3$.
If $\lambda$ has finite second moment, by Chebyshev inequality, for each $\alpha$, $${\Pr}\left(\lambda_i>n^{\frac{1}{2}+\alpha}\right)\leq \frac{\operatorname{Var}(\lambda_i)}{n^{1+2\alpha}} \leq \frac{M_2(\lambda_i)}{n^{1+2\alpha}} = {\mathcal{O}}\left(\frac{M_2(\lambda)}{n^{1+2\alpha}}\right) = {\mathcal{O}}\left(n^{-1-2\alpha}\right).$$
For $\alpha=\epsilon$, this means that no node has weight bigger than $n^{\frac{1}{2}+\epsilon}$, and for $\alpha=-\epsilon$, this means that the number of nodes with weight bigger than $n^{\frac{1}{2}-\epsilon}$ is at most $n^{2\epsilon}$. We conclude that $\sum_{{\rho_{v}} \geq n^{\frac{1}{2}-\epsilon}} {\rho_{v}} \leq \sum_{{\rho_{v}} \geq n^{\frac{1}{2}-\epsilon}} n^{\frac{1}{2}+\epsilon} \leq n^{\frac{1}{2}+3\epsilon}$.
If $\lambda$ is power law with $2<\beta<3$, by Assumption \[cond:powerlaw\] the number of nodes with weight at least $d$ is at most $Cnd^{-\beta+1}$. Consequently, using Abel’s summation technique, $$\begin{aligned}
\sum_{{\rho_{v}} \geq n^{\frac{1}{2}-\epsilon}} {\rho_{v}} &= \sum_{d={\rho_{v}}}^{+\infty} d|\{v:{\rho_{v}} = d\}| \\
&= \sum_{d=n^{\frac{1}{2}-\epsilon}}^{+\infty} d(|\{v:{\rho_{v}} \geq d\}|-|\{v:{\rho_{v}} \geq d+1\}|) \\
&= \sum_{d=n^{\frac{1}{2}-\epsilon}}^{+\infty} d|\{v:{\rho_{v}} \geq d\}|-\sum_{d=n^{\frac{1}{2}-\epsilon}+1}^{+\infty}(d-1)|\{v:{\rho_{v}} \geq d\}| \\
&= n^{\frac{1}{2}-\epsilon}|\{v:{\rho_{v}} \geq n^{\frac{1}{2}-\epsilon}\}|+\sum_{d=n^{\frac{1}{2}-\epsilon}+1}^{+\infty} |\{v:{\rho_{v}} \geq d\}| \\
&\leq Cn^{\frac{1}{2}-\epsilon}n^{1-\left(\frac{1}{2}-\epsilon\right)(\beta-1)}+\sum_{d=n^{\frac{1}{2}-\epsilon}+1}^{+\infty}Cnd^{-\beta+1} \\
&={\mathcal{O}}\left(n^{\frac{4-\beta}{2}+\epsilon\beta}+n^{1-\left(\frac{1}{2}-\epsilon\right)(\beta-2)}\right)={\mathcal{O}}\left(n^{\frac{4-\beta}{2}+\epsilon\beta}\right).\end{aligned}$$
By this claim, $\sum_{v \in {\boldsymbol{\Gamma}^{l+1}(s)},{\rho_{v}} \geq n^{\frac{1}{2}-2\epsilon}} {\rho_{v}}$ is smaller than $n^{\frac{1}{2}+6\epsilon}$ if $\lambda$ has finite second moment, and it is smaller than $n^{\frac{4-\beta}{2}+6\epsilon}$ if $\lambda$ is power law with $2<\beta<3$. To conclude the proof, we only have to bound $\sum_{v \in {\boldsymbol{\Gamma}^{l+1}(s)},{\rho_{v}} < n^{\frac{1}{2}-2\epsilon}} {\rho_{v}}$.
**Claim:** with high probability, $\sum_{v \in {\boldsymbol{\Gamma}^{l+1}(s)},{\rho_{v}}<n^{\frac{1}{2}-2\epsilon}} {\rho_{v}}<n^{\frac{1}{2}+\epsilon}$ if $\lambda$ has finite second moment, $\sum_{v \in {\boldsymbol{\Gamma}^{l+1}(s)},{\rho_{v}}<n^{\frac{1}{2}-2\epsilon}} {\rho_{v}}<n^{\frac{4-\beta}{2}+\epsilon}$ if $\lambda$ is power law with $2<\beta<3$.
As in the proof of , we can safely assume that we know the structure $S$ of ${\boldsymbol{N}^{l}(s)}$. Let us sort the stubs in ${\boldsymbol{R}^{l}(s)}$, not paired by $S$, obtaining $a_1,\dots,a_k$, and let $\boldsymbol{a}_i$ be the stub paired with $a_i$. Let $\operatorname{res}(a)$ be the number of stubs of the node $a$, minus $a$, and let ${\boldsymbol{X}}_i=\operatorname{res}(\boldsymbol{a}_i)$ if $\operatorname{res}(\boldsymbol{a}_i)<n^{\frac{1}{2}-2\epsilon}$, $0$ otherwise: clearly, $\sum_{v \in {\boldsymbol{\Gamma}^{l+1}(s)},{\rho_{v}} \leq n^{\frac{1}{2}-2\epsilon}} {\rho_{v}} \leq \sum_{i=1}^{k} {\boldsymbol{X}}_i$ (with equality if there are no horizontal or diagonal edges in the BFS tree). After the first $i-1$ stubs are paired, since $i<n^{\frac{1}{2}+\epsilon}$ and since the number of stubs paired in $S$ is ${\mathcal{O}}\left(n^{\frac{1}{2}+\epsilon}\log n\right)$, for each $k<n^{\frac{1}{2}-2\epsilon}$, $$\begin{aligned}
{\Pr}\left({\boldsymbol{X}}_i=k\right)&={\Pr}\left(\operatorname{res}\left(\boldsymbol{a}_i\right)=k\right) \\
&=\frac{\left|\left\{a \in A: a \text{ unpaired after $i$ rounds}, \operatorname{res}(a)=k\right\}\right|}{\left|\left\{a \in A: a \text{ unpaired after $i$ rounds}\right\}\right|} \\
&=\frac{\left|\left\{a \in A: \operatorname{res}(a)=k\right\}\right|+{\mathcal{O}}\left(n^{\frac{1}{2}+\epsilon}\right)}{\left|A\right|+{\mathcal{O}}\left(n^{\frac{1}{2}+\epsilon}\right)} \\
&=\frac{(k+1)\lambda(k+1)}{M_1(\lambda)}+{\mathcal{O}}\left(n^{-\frac{1}{2}+\epsilon}\right).\end{aligned}$$
Consequently, conditioned on all pairings of $a_j$ for $j<i$, ${\mathbb{E}}\left[{\boldsymbol{X}}_i\right]=\sum_{k=0}^{n^{\frac{1}{2}-2\epsilon}} k\frac{(k+1)\lambda(k+1)}{M_1(\lambda)}+{\mathcal{O}}(n^{-\frac{1}{2}+\epsilon}\log n)=\alpha(n)$, where $\alpha(n)={\mathcal{O}}(1)$ if $\lambda$ has finite second moment, and $\alpha(n)={\mathcal{O}}(n^{\frac{3-\beta}{2}})$ if $\lambda$ is power law with $2<\beta<3$. Hence, for each $\epsilon$, $\sum_{i=1}^k {\boldsymbol{X}}_i-i(M_1(\lambda)+\epsilon)$ is a supermartingale, and by Azuma’s inequality $${\Pr}\left(\sum_{i=1}^k {\boldsymbol{X}}_i-k\alpha(n)\geq \alpha(n)\right) \leq \exp\left({-\frac{\alpha(n)^2}{2\sum_{i=1}^k n^{\frac{1}{2}-2\epsilon}}}\right) \leq \exp(-n^\epsilon).$$ Then, [w.h.p.]{}, $\sum_{i=1}^k {\boldsymbol{X}}_i \leq n^{\frac{1}{2}+\epsilon}(\alpha(n)+2)$, concluding the proof of the claim.
The number of nodes $w$ with weight at most $n^{\frac{1}{2}-2\epsilon}$ that belong to ${\boldsymbol{\Gamma}^{l+1}(s)}$ is at most $\sum_{v \in {\boldsymbol{\Gamma}^{l}(s)},{\rho_{v}}<n^{\frac{1}{2}-2\epsilon}}\sum_{w \in V} {\rho_{w}}{\boldsymbol{X}}_{v,w}$, where ${\boldsymbol{X}}_{v,w}=1$ with probability $f\left(\frac{{\rho_{v}} {\rho_{w}}}{M}\right)={\mathcal{O}}\left(\frac{{\rho_{v}} {\rho_{w}}}{M}\right)$ because ${\rho_{v}}{\rho_{w}} < n^{1-\epsilon}$. Moreover, $${\mathbb{E}}\left[\sum_{v \in {\boldsymbol{\Gamma}^{l}(s)},{\rho_{v}}<n^{\frac{1}{2}-2\epsilon}}\sum_{w \in V} {\rho_{w}}{\boldsymbol{X}}_{v,w}\right] = {\mathcal{O}}\left({\boldsymbol{r}^{l}(s)} \frac{\sum_{v \in V} {\rho_{v}}^2}{n}\right)={\boldsymbol{r}^{l}(s)}\alpha(n)$$
where $\alpha(n)={\mathcal{O}}(1)$ if $\lambda$ has finite second moment, and $\alpha(n)={\mathcal{O}}\left(n^{\frac{3-\beta}{2}}\right)$ if $\lambda$ is power law with $2<\beta<3$.
By Hoeffding inequality, $${\Pr}\left(\sum_{v \in {\boldsymbol{\Gamma}^{l}(s)},{\rho_{v}}<n^{\frac{1}{2}-2\epsilon}}\sum_{w \in V} {\rho_{w}}{\boldsymbol{X}}_{v,w} -{\boldsymbol{r}^{l}(s)} \alpha(n)\geq {\boldsymbol{r}^{l}(s)} \alpha(n)\right) \leq n^{\frac{{\boldsymbol{r}^{l}(s)} \alpha(n)}{{\boldsymbol{r}^{l}(s)} n^{\frac{1}{2}-2\epsilon}}} \leq n^{-\epsilon}.$$ This concludes the proof.
This claim lets us conclude the proof of the lemma.
Let $D_s^i=\sum_{v \in {\boldsymbol{\Gamma}^{i}(s)}} \deg(v)$, $D_t^j=\sum_{w \in {\boldsymbol{\Gamma}^{j}(t)}} \deg(w)$, and let us suppose that we have visited until level $l_s$ from $s$, until level $l_t$ from $t$, and that $D_s^{l_s},D_t^{l_t}>n^{\frac{1}{2}+2\epsilon}$. If this situation never occurs, by , the total number of visited edges is at most ${\mathcal{O}}(\log n)n^{\frac{1}{2}+2\epsilon} = {\mathcal{O}}(n^{\frac{1}{2}+3\epsilon})$, and the conclusion follows. Otherwise, again by , the number of edges visited in the two BFS trees before levels $l_s$ and $l_t$ is ${\mathcal{O}}(n^{\frac{1}{2}+3\epsilon})$. Furthermore, by , ${\boldsymbol{r}^{l_s}(s)},{\boldsymbol{r}^{l_t}(t)}>n^{\frac{1}{2}+2\epsilon}$. We claim that, without loss of generality, we can assume ${\boldsymbol{r}^{l_s-1}(s)}<\epsilon{\boldsymbol{r}^{l_s}(s)}$, to apply . Indeed, if ${\boldsymbol{r}^{l_s-1}(s)}$ is too big, we iteratively decrease $l_s$ until we find a neighbor verifying ${\boldsymbol{r}^{l_s}(s)}>(1-\epsilon'){\boldsymbol{r}^{l_s-1}(s)}$. This process can last at most ${\mathcal{O}}(\log n)$ steps, and hence it is stopped at a point $l_s$ such that ${\boldsymbol{r}^{l_s}(s)}>n^{\frac{1}{2}+2\epsilon}(1-\epsilon')^{{\mathcal{O}}(\log n)} \geq n^{\frac{1}{2}+\epsilon'}$ if $\epsilon'$ is small enough. Similarly, we can suppose without loss of generality that ${\boldsymbol{r}^{l_t}(t)}>(1-\epsilon'){\boldsymbol{r}^{l_t-1}(t)}$. By , $d(s,t) \leq l_s+l_t+2$, and the number of nodes needed to conclude the BFS is at most $D_s^{l_s}+D_t^{l_t}$ (note that, if we extend twice the visit from $s$, it means that $D_s^{l_s+1}<D_t^{l_t}$). By , $D_s^{l_s} \leq n^{\epsilon}{\boldsymbol{r}^{l_s}(s)}$, and by this value is at most $n^{\frac{1}{2}+3\epsilon}$ if $\lambda$ has finite second moment, and $n^{\frac{4-\beta}{2}+3\epsilon}$ if $\lambda$ is power law with $2<\beta<3$. We conclude that the total number of visited nodes is at most $n^{\frac{1}{2}+3\epsilon}+D_s^{l_s}+D_t^{l_t} \leq n^{\frac{1}{2}+3\epsilon}+{\boldsymbol{r}^{l_s}(s)}+{\boldsymbol{r}^{l_t}(t)} \leq n^{\frac{1}{2}+4\epsilon}$ (resp., $n^{\frac{4-\beta}{2}+4\epsilon}$) if $\lambda$ has finite second moment (resp., if $\lambda$ is power law with $2<\beta<3$). The theorem follows by changing the value of $\epsilon$.
Detailed Experimental Results {#app:detailedresults}
=============================
Wikipedia and IMDB Results {#sec:wikipediaimdb}
==========================
In this section, we report our results on the Wikipedia citation network, and on all snapshots of the IMDB actors collaboration network. In the ranking column, we report one number if the position in the ranking is guaranteed with probability $0.9$, otherwise we report a lower and an upper bound, which hold with the same probability.
We remark that, as for the IMDB database, the top-$k$ betweenness centralities of a single snapshot of a similar graph (`hollywood-2009` in [@sebagraph]) have been previously computed exactly, with one week of computation on a $40$-core machine [@seba].
The Results on the IMDB Graph {#sec:imdb_results}
-----------------------------
In 2014, the most central actor is Ron Jeremy, who is listed in the Guinness Book of World Records for “Most Appearances in Adult Films”, with more than 2000 appearances. Among his non-adult ones, we mention *The Godfather Part III*, *Ghostbusters*, *Crank: High Voltage* and *Family Guy*[^4]. His topmost centrality in the actor collaboration network has been previously observed by similar experiments on betweenness centrality [@seba]. Indeed, around 3 actors out of 100 in the IMDB database played in adult movies, which explains why the high number of appearances of Ron Jeremy both in the adult and non-adult film industry rises his betweenness to the top.
The second most-central actor is Lloyd Kaufman, which is best known as a co-founder of *Troma Entertainment Film Studio* and as the director of many of their feature films, including the cult movie *The Toxic Avenger*. His high betweenness score is likely due to his central role in the low-budget independent film industry.
The third “actor” is the historical German dictator Adolf Hitler, since his appearances in several historical footages, that were re-used in several movies (e.g. in *The Imitation Game*), are credited by IMDB as cameo role. Indeed, he appears among the topmost actors since the 1984 snapshot, being the first one in the 1989 and 1994 ones, and during those years many movies about the World War II were produced.
Observe that the betweenness centrality measure on our graph does not discriminate between important and marginal roles. For example, the actress Bess Flowers, who appears among the top actors in the snapshots from 1959 to 1979, rarely played major roles, but she appeared in over 700 movies in her 41 years career.
The Results on the Wikipedia Graph {#sec:wiki_results}
----------------------------------
All topmost pages in the betweenness centrality ranking, except for the World War II, are countries. This is not surprising if we consider that, for most topics (such as important people or events), the corresponding Wikipedia page refers to their geographical context (since it mentions the country of origin of the given person or where a given event took place). It is also worth noting the correlation between the high centrality of the *World War II* Wikipedia page and that of Adolf Hitler in the IMDB graph.
Interestingly, a similar ranking is obtained by considering the closeness centrality measure in the inverse graph, where a link from page $p_1$ to page $p_2$ exists if a link to page $p_1$ appears in page $p_2$ [@Bergamini2016ComputingTC]. However, in contrast with the results in [@Bergamini2016ComputingTC] when edges are oriented in the usual way, the pages about specific years do not appear in the top ranking. We note that the betweenness centrality of a node in a directed graph does not change if the orientation of all edges is flipped.
Finally, the most important pages is the United States, confirming a common conjecture. Indeed, in <http://wikirank.di.unimi.it/>, it is shown that the United States are the center according to harmonic centrality, and many other measures. Further evidence for this conjecture comes from the Six Degree of Wikipedia game (<http://thewikigame.com/6-degrees-of-wikipedia>), where a player is asked to go from one page to the other following the smallest possible number of links: a hard variant of this game forces the player not to pass from the *United States* page, which is considered to be central. Our results thus confirm that the conjecture is indeed true for the betweenness centrality measure.
Ranking Wikipedia page Lower bound Estimated betweenness Upper bound
--------- ---------------- ------------- ----------------------- -------------
1) United States 0.046278 0.047173 0.048084
2) France 0.019522 0.020103 0.020701
3) United Kingdom 0.017983 0.018540 0.019115
4) England 0.016348 0.016879 0.017428
5-6) Poland 0.012092 0.012287 0.012486
5-6) Germany 0.011930 0.012124 0.012321
7) India 0.009683 0.010092 0.010518
8-12) World War II 0.008870 0.009065 0.009265
8-12) Russia 0.008660 0.008854 0.009053
8-12) Italy 0.008650 0.008845 0.009045
8-12) Canada 0.008624 0.008819 0.009018
8-12) Australia 0.008620 0.008814 0.009013
: The top-$k$ betweenness centralities of the Wikipedia graph computed by [[KADABRA]{}]{}with $\delta=0.1$ and $\lambda = 0.0002$.[]{data-label="tab:wikipedia"}
Ranking Actor Lower bound Estimated betweenness Upper bound
--------- ------------------- ------------- ----------------------- -------------
1) Meyer, Torben 0.022331 0.022702 0.023049
2) Roulien, Raul 0.021361 0.021703 0.022071
3) Myzet, Rudolf 0.014229 0.014525 0.014747
4) Sten, Anna 0.013245 0.013460 0.013723
5) Negri, Pola 0.012509 0.012768 0.012943
6-7) Jung, Shia 0.012250 0.012379 0.012509
6-7) Ho, Tai-Hau 0.012195 0.012324 0.012454
8) Goetzke, Bernhard 0.010721 0.010978 0.011201
9-10) Yamamoto, Togo 0.010095 0.010224 0.010354
9-10) Kamiyama, Sōjin 0.010087 0.010215 0.010344
: The top-$k$ betweenness centralities of a snapshot of the IMDB collaboration network taken at the end of 1939 (69011 nodes), computed by [[KADABRA]{}]{}with $\delta=0.1$ and $\lambda = 0.0002$.
Ranking Actor Lower bound Estimated betweenness Upper bound
--------- ------------------- ------------- ----------------------- -------------
1) Meyer, Torben 0.018320 0.018724 0.019136
2) Kamiyama, Sōjin 0.012629 0.012964 0.013308
3-4) Jung, Shia 0.010751 0.010901 0.011053
3-4) Ho, Tai-Hau 0.010704 0.010854 0.011005
5) Myzet, Rudolf 0.010365 0.010514 0.010666
6-7) Sten, Anna 0.009778 0.009928 0.010080
6-7) Goetzke, Bernhard 0.009766 0.009915 0.010066
8) Yamamoto, Togo 0.009108 0.009327 0.009539
9) Parìs, Manuel 0.008649 0.008859 0.009108
10) Hayakawa, Sessue 0.007916 0.008158 0.008369
: The top-$k$ betweenness centralities of a snapshot of the IMDB collaboration network taken at the end of 1944 (83068 nodes), computed by [[KADABRA]{}]{}with $\delta=0.1$ and $\lambda = 0.0002$.
Ranking Actor Lower bound Estimated betweenness Upper bound
--------- -------------------- ------------- ----------------------- -------------
1) Meyer, Torben 0.016139 0.016679 0.017236
2) Kamiyama, Sōjin 0.012351 0.012822 0.013312
3) Parìs, Manuel 0.011104 0.011552 0.011861
4) Yamamoto, Togo 0.010342 0.010639 0.011086
5-6) Jung, Shia 0.008926 0.009120 0.009318
5-6) Goetzke, Bernhard 0.008567 0.008762 0.008962
7-9) Paananen, Tuulikki 0.008147 0.008341 0.008539
7-9) Sten, Anna 0.007969 0.008164 0.008363
7-9) Mayer, Ruby 0.007967 0.008162 0.008362
10-12) Ho, Tai-Hau 0.007538 0.007732 0.007930
10-12) Hayakawa, Sessue 0.007399 0.007593 0.007792
10-12) Haas, Hugo (I) 0.007158 0.007352 0.007552
: The top-$k$ betweenness centralities of a snapshot of the IMDB collaboration network taken at the end of 1949 (97824 nodes), computed by [[KADABRA]{}]{}with $\delta=0.1$ and $\lambda = 0.0002$.
Ranking Actor Lower bound Estimated betweenness Upper bound
--------- -------------------- ------------- ----------------------- -------------
1) Meyer, Torben 0.013418 0.013868 0.014334
2) Kamiyama, Sōjin 0.010331 0.010726 0.011089
3-4) Ertugrul, Muhsin 0.009956 0.010141 0.010331
3-4) Jung, Shia 0.009643 0.009826 0.010013
5-6) Singh, Ram (I) 0.008657 0.008841 0.009030
5-6) Paananen, Tuulikki 0.008383 0.008567 0.008755
7-9) Parìs, Manuel 0.007886 0.008070 0.008257
7-10) Goetzke, Bernhard 0.007802 0.007987 0.008176
7-10) Yamaguchi, Shirley 0.007531 0.007716 0.007905
8-10) Hayakawa, Sessue 0.007473 0.007657 0.007845
: The top-$k$ betweenness centralities of a snapshot of the IMDB collaboration network taken at the end of 1954 (120430 nodes), computed by [[KADABRA]{}]{}with $\delta=0.1$ and $\lambda = 0.0002$.
Ranking Actor Lower bound Estimated betweenness Upper bound
--------- -------------------- ------------- ----------------------- -------------
1-2) Singh, Ram (I) 0.010683 0.010877 0.011075
1-2) Frees, Paul 0.010372 0.010566 0.010763
3) Meyer, Torben 0.009478 0.009821 0.010235
4-5) Jung, Shia 0.008623 0.008816 0.009013
4-5) Ghosh, Sachin 0.008459 0.008651 0.008847
6-7) Myzet, Rudolf 0.007085 0.007278 0.007476
6-7) Yamaguchi, Shirley 0.006908 0.007101 0.007299
8) de Còrdova, Arturo 0.006391 0.006582 0.006778
9-11) Kamiyama, Sōjin 0.005861 0.006054 0.006254
9-12) Paananen, Tuulikki 0.005810 0.006003 0.006202
9-12) Flowers, Bess 0.005620 0.005813 0.006012
10-12) Parìs, Manuel 0.005442 0.005635 0.005835
: The top-$k$ betweenness centralities of a snapshot of the IMDB collaboration network taken at the end of 1959 (146253 nodes), computed by [[KADABRA]{}]{}with $\delta=0.1$ and $\lambda = 0.0002$.
Ranking Actor Lower bound Estimated betweenness Upper bound
--------- -------------------- ------------- ----------------------- -------------
1) Frees, Paul 0.013140 0.013596 0.014067
2) Meyer, Torben 0.007279 0.007617 0.007856
3-4) Harris, Sam (II) 0.006813 0.006967 0.007124
3-5) Myzet, Rudolf 0.006696 0.006849 0.007005
4-5) Flowers, Bess 0.006422 0.006572 0.006726
6) Kong, King (I) 0.005909 0.006104 0.006422
7) Yuen, Siu Tin 0.005114 0.005264 0.005420
8) Miller, Marvin (I) 0.004708 0.004859 0.005015
9-12) de Còrdova, Arturo 0.004147 0.004299 0.004457
9-18) Haas, Hugo (I) 0.003888 0.004039 0.004197
9-18) Singh, Ram (I) 0.003854 0.004004 0.004160
9-18) Kamiyama, Sōjin 0.003848 0.003999 0.004155
10-18) Sauli, Anneli 0.003827 0.003978 0.004135
10-18) King, Walter Woolf 0.003774 0.003923 0.004078
10-18) Vanel, Charles 0.003716 0.003867 0.004024
10-18) Kowall, Mitchell 0.003684 0.003834 0.003990
10-18) Holmes, Stuart 0.003603 0.003752 0.003907
10-18) Sten, Anna 0.003582 0.003733 0.003890
: The top-$k$ betweenness centralities of a snapshot of the IMDB collaboration network taken at the end of 1964 (174826 nodes), computed by [[KADABRA]{}]{}with $\delta=0.1$ and $\lambda = 0.0002$.
Ranking Actor Lower bound Estimated betweenness Upper bound
--------- ------------------- ------------- ----------------------- -------------
1) Frees, Paul 0.010913 0.011446 0.012005
2-3) Yuen, Siu Tin 0.006157 0.006349 0.006547
2-3) Tamiroff, Akim 0.006097 0.006291 0.006490
4-6) Meyer, Torben 0.005675 0.005869 0.006069
4-7) Harris, Sam (II) 0.005639 0.005830 0.006027
4-8) Rubener, Sujata 0.005427 0.005618 0.005815
5-8) Myzet, Rudolf 0.005253 0.005444 0.005641
6-8) Flowers, Bess 0.005136 0.005328 0.005526
9-10) Kong, King (I) 0.004354 0.004544 0.004741
9-10) Sullivan, Elliott 0.004208 0.004398 0.004596
: The top-$k$ betweenness centralities of a snapshot of the IMDB collaboration network taken at the end of 1969 (210527 nodes), computed by [[KADABRA]{}]{}with $\delta=0.1$ and $\lambda = 0.0002$.
Ranking Actor Lower bound Estimated betweenness Upper bound
--------- ------------------ ------------- ----------------------- -------------
1) Frees, Paul 0.008507 0.008958 0.009295
2) Chen, Sing 0.007734 0.008056 0.008507
3) Welles, Orson 0.006115 0.006497 0.006903
4-5) Loren, Sophia 0.005056 0.005221 0.005392
4-7) Rubener, Sujata 0.004767 0.004933 0.005106
5-8) Harris, Sam (II) 0.004628 0.004795 0.004967
5-8) Tamiroff, Akim 0.004625 0.004790 0.004962
6-10) Meyer, Torben 0.004382 0.004548 0.004720
8-12) Flowers, Bess 0.004259 0.004425 0.004598
8-12) Yuen, Siu Tin 0.004229 0.004397 0.004571
9-12) Carradine, John 0.004026 0.004192 0.004364
9-12) Myzet, Rudolf 0.003984 0.004151 0.004325
: The top-$k$ betweenness centralities of a snapshot of the IMDB collaboration network taken at the end of 1974 (257896 nodes), computed by [[KADABRA]{}]{}with $\delta=0.1$ and $\lambda = 0.0002$.
Ranking Actor Lower bound Estimated betweenness Upper bound
--------- ---------------------- ------------- ----------------------- -------------
1) Chen, Sing 0.007737 0.008220 0.008647
2) Frees, Paul 0.006852 0.007255 0.007737
3-5) Welles, Orson 0.004894 0.005075 0.005263
3-6) Carradine, John 0.004623 0.004803 0.004989
3-6) Loren, Sophia 0.004614 0.004796 0.004985
4-6) Rubener, Sujata 0.004284 0.004464 0.004651
7-17) Tamiroff, Akim 0.003516 0.003696 0.003885
7-17) Meyer, Torben 0.003479 0.003657 0.003844
7-17) Quinn, Anthony (I) 0.003447 0.003626 0.003815
7-17) Flowers, Bess 0.003446 0.003625 0.003815
7-17) Mitchell, Gordon (I) 0.003417 0.003596 0.003785
7-17) Sullivan, Elliott 0.003371 0.003551 0.003740
7-17) Rietty, Robert 0.003368 0.003547 0.003735
7-17) Tanba, Tetsurō 0.003360 0.003537 0.003724
7-17) Harris, Sam (II) 0.003331 0.003510 0.003699
7-17) Lewgoy, Josè 0.003223 0.003402 0.003590
7-17) Dalio, Marcel 0.003185 0.003364 0.003553
: The top-$k$ betweenness centralities of a snapshot of the IMDB collaboration network taken at the end of 1979 (310278 nodes), computed by [[KADABRA]{}]{}with $\delta=0.1$ and $\lambda = 0.0002$.
Ranking Actor Lower bound Estimated betweenness Upper bound
--------- ------------------------ ------------- ----------------------- -------------
1) Chen, Sing 0.007245 0.007716 0.008218
2-4) Welles, Orson 0.005202 0.005391 0.005587
2-4) Frees, Paul 0.005174 0.005363 0.005559
2-5) Hitler, Adolf 0.004906 0.005094 0.005290
4-6) Carradine, John 0.004744 0.004932 0.005127
5-7) Mitchell, Gordon (I) 0.004418 0.004606 0.004802
6-8) Jürgens, Curd 0.004169 0.004356 0.004551
7-8) Kinski, Klaus 0.003938 0.004123 0.004318
9-12) Rubener, Sujata 0.003396 0.003585 0.003785
9-12) Lee, Christopher (I) 0.003391 0.003576 0.003771
9-12) Loren, Sophia 0.003357 0.003542 0.003738
9-12) Harrison, Richard (II) 0.003230 0.003417 0.003614
: The top-$k$ betweenness centralities of a snapshot of the IMDB collaboration network taken at the end of 1984 (375322 nodes), computed by [[KADABRA]{}]{}with $\delta=0.1$ and $\lambda = 0.0002$.
Ranking Actor Lower bound Estimated betweenness Upper bound
--------- ------------------------ ------------- ----------------------- -------------
1-2) Hitler, Adolf 0.005282 0.005467 0.005658
1-3) Chen, Sing 0.005008 0.005192 0.005382
2-4) Carradine, John 0.004648 0.004834 0.005027
3-4) Harrison, Richard (II) 0.004515 0.004697 0.004887
5-6) Welles, Orson 0.004088 0.004271 0.004462
5-9) Mitchell, Gordon (I) 0.003766 0.003948 0.004139
6-9) Kinski, Klaus 0.003691 0.003874 0.004065
6-11) Lee, Christopher (I) 0.003610 0.003793 0.003984
6-11) Frees, Paul 0.003582 0.003766 0.003960
8-13) Jürgens, Curd 0.003306 0.003486 0.003676
8-13) Pleasence, Donald 0.003299 0.003479 0.003670
10-13) Mitchell, Cameron (I) 0.003105 0.003285 0.003476
10-13) von Sydow, Max (I) 0.002982 0.003161 0.003350
: The top-$k$ betweenness centralities of a snapshot of the IMDB collaboration network taken at the end of 1989 (463078 nodes), computed by [[KADABRA]{}]{}with $\delta=0.1$ and $\lambda = 0.0002$.
Ranking Actor Lower bound Estimated betweenness Upper bound
--------- ------------------------ ------------- ----------------------- -------------
1) Hitler, Adolf 0.005227 0.005676 0.006164
2-6) Harrison, Richard (II) 0.003978 0.004165 0.004362
2-6) von Sydow, Max (I) 0.003884 0.004069 0.004264
2-7) Lee, Christopher (I) 0.003718 0.003907 0.004106
2-7) Carradine, John 0.003696 0.003883 0.004079
2-7) Chen, Sing 0.003683 0.003871 0.004068
4-10) Jeremy, Ron 0.003336 0.003524 0.003722
7-11) Pleasence, Donald 0.003253 0.003439 0.003637
7-11) Rey, Fernando (I) 0.003234 0.003420 0.003617
7-15) Smith, William (I) 0.003012 0.003199 0.003397
8-15) Welles, Orson 0.002885 0.003072 0.003271
10-15) Mitchell, Gordon (I) 0.002851 0.003036 0.003232
10-15) Kinski, Klaus 0.002705 0.002890 0.003087
10-15) Mitchell, Cameron (I) 0.002671 0.002858 0.003058
10-15) Quinn, Anthony (I) 0.002640 0.002826 0.003026
: The top-$k$ betweenness centralities of a snapshot of the IMDB collaboration network taken at the end of 1994 (557373 nodes), computed by [[KADABRA]{}]{}with $\delta=0.1$ and $\lambda = 0.0002$.
Ranking Actor Lower bound Estimated betweenness Upper bound
--------- ------------------------ ------------- ----------------------- -------------
1) Jeremy, Ron 0.007380 0.007913 0.008484
2) Hitler, Adolf 0.004601 0.005021 0.005480
3-4) Lee, Christopher (I) 0.003679 0.003849 0.004028
3-4) von Sydow, Max (I) 0.003604 0.003775 0.003953
5-6) Harrison, Richard (II) 0.003041 0.003211 0.003390
5-7) Carradine, John 0.002943 0.003114 0.003296
6-11) Chen, Sing 0.002662 0.002834 0.003018
7-14) Rey, Fernando (I) 0.002569 0.002740 0.002922
7-14) Smith, William (I) 0.002559 0.002729 0.002910
7-14) Pleasence, Donald 0.002556 0.002725 0.002906
7-14) Sutherland, Donald (I) 0.002449 0.002617 0.002796
8-14) Quinn, Anthony (I) 0.002307 0.002476 0.002658
8-14) Mastroianni, Marcello 0.002271 0.002440 0.002621
8-14) Saxon, John 0.002251 0.002420 0.002602
: The top-$k$ betweenness centralities of a snapshot of the IMDB collaboration network taken at the end of 1999 (681358 nodes), computed by [[KADABRA]{}]{}with $\delta=0.1$ and $\lambda = 0.0002$.
Ranking Actor Lower bound Estimated betweenness Upper bound
--------- ------------------------ ------------- ----------------------- -------------
1) Jeremy, Ron 0.010653 0.011370 0.012136
2) Hitler, Adolf 0.005333 0.005840 0.006396
3-4) von Sydow, Max (I) 0.003424 0.003608 0.003802
3-4) Lee, Christopher (I) 0.003403 0.003587 0.003781
5-6) Kier, Udo 0.002898 0.003081 0.003275
5-8) Keitel, Harvey (I) 0.002646 0.002828 0.003023
6-12) Hopper, Dennis 0.002424 0.002607 0.002804
6-16) Smith, William (I) 0.002322 0.002504 0.002700
7-17) Sutherland, Donald (I) 0.002241 0.002422 0.002617
7-23) Carradine, David 0.002149 0.002329 0.002526
7-23) Carradine, John 0.002147 0.002328 0.002524
7-23) Harrison, Richard (II) 0.002054 0.002234 0.002430
8-23) Sharif, Omar 0.002043 0.002222 0.002418
8-23) Steiger, Rod 0.001988 0.002165 0.002358
8-23) Quinn, Anthony (I) 0.001974 0.002151 0.002344
8-23) Depardieu, Gèrard 0.001966 0.002148 0.002346
9-23) Sheen, Martin 0.001913 0.002093 0.002291
10-23) Rey, Fernando (I) 0.001866 0.002044 0.002238
10-23) Kane, Sharon 0.001857 0.002038 0.002237
10-23) Pleasence, Donald 0.001859 0.002037 0.002232
10-23) Skarsgard, Stellan 0.001848 0.002026 0.002221
10-23) Mueller-Stahl, Armin 0.001789 0.001969 0.002166
10-23) Hong, James (I) 0.001780 0.001957 0.002152
: The top-$k$ betweenness centralities of a snapshot of the IMDB collaboration network taken at the end of 2004 (880032 nodes), computed by [[KADABRA]{}]{}with $\delta=0.1$ and $\lambda = 0.0002$.
Ranking Actor Lower bound Estimated betweenness Upper bound
--------- ------------------------ ------------- ----------------------- -------------
1) Jeremy, Ron 0.010531 0.011237 0.011991
2) Hitler, Adolf 0.005500 0.006011 0.006568
3-4) Kaufman, Lloyd 0.003620 0.003804 0.003997
3-4) Kier, Udo 0.003472 0.003654 0.003845
5-6) Lee, Christopher (I) 0.003056 0.003240 0.003435
5-8) Carradine, David 0.002866 0.003050 0.003245
6-8) Keitel, Harvey (I) 0.002659 0.002840 0.003034
6-9) von Sydow, Max (I) 0.002532 0.002713 0.002907
8-13) Hopper, Dennis 0.002237 0.002419 0.002616
9-15) Skarsgard, Stellan 0.002153 0.002333 0.002529
9-15) Depardieu, Gèrard 0.002001 0.002181 0.002377
9-15) Hauer, Rutger 0.001894 0.002074 0.002271
9-15) Sutherland, Donald (I) 0.001875 0.002054 0.002250
10-15) Smith, William (I) 0.001811 0.001990 0.002186
10-15) Dafoe, Willem 0.001805 0.001986 0.002186
: The top-$k$ betweenness centralities of a snapshot of the IMDB collaboration network taken at the end of 2009 (1237879 nodes), computed by [[KADABRA]{}]{}with $\delta=0.1$ and $\lambda = 0.0002$.
Ranking Actor Lower bound Estimated betweenness Upper bound
--------- ---------------------- ------------- ----------------------- -------------
1) Jeremy, Ron 0.009360 0.010058 0.010808
2) Kaufman, Lloyd 0.005936 0.006492 0.007100
3) Hitler, Adolf 0.004368 0.004844 0.005373
4-6) Kier, Udo 0.003250 0.003435 0.003631
4-6) Roberts, Eric (I) 0.003178 0.003362 0.003557
4-6) Madsen, Michael (I) 0.003120 0.003305 0.003501
7-9) Trejo, Danny 0.002652 0.002835 0.003030
7-9) Lee, Christopher (I) 0.002551 0.002734 0.002931
7-12) Estevez, Joe 0.002350 0.002534 0.002732
9-17) Carradine, David 0.002116 0.002296 0.002492
9-17) von Sydow, Max (I) 0.002023 0.002206 0.002405
9-17) Keitel, Harvey (I) 0.001974 0.002154 0.002352
10-17) Skarsgard, Stellan 0.001945 0.002125 0.002323
10-17) Dafoe, Willem 0.001899 0.002080 0.002279
10-17) Hauer, Rutger 0.001891 0.002071 0.002269
10-17) Depardieu, Gèrard 0.001763 0.001943 0.002142
10-17) Rochon, Debbie 0.001745 0.001926 0.002126
: The top-$k$ betweenness centralities of a snapshot of the IMDB collaboration network taken in 2014 (1797446 nodes), computed by [[KADABRA]{}]{}with $\delta=0.1$ and $\lambda = 0.0002$.
[^1]: This work was done while the authors were visiting the Simons Institute for the Theory of Computing.
[^2]: As explained in see Section \[sec:algoshort\], to simplify notation we consider the *normalized* betweenness centrality.
[^3]: This paper uses a further assumption on IRG, but the proofs of Lemmas 32 and 39 do not rely on this assumption.
[^4]: The latter is a TV-series, which are not taken into account in our data.
|
---
abstract: 'The first-order, in terms of electron-interaction in the perturbation theory, of the proper linear response function $\Pi ({\bf k}, \omega )$ gives rise to the exchange-contribution to the dielectric function $\epsilon ({\bf k} , \omega)$ in the electron liquid. Its imaginary part, $Im \Pi_1 ({\bf k}, \omega)$, is calculated exactly. An analytical expression for $Im \Pi_1 ({\bf k}, \omega)$ is derived which after refinement has a quite simple form.'
author:
- Zhixin Qian
title: 'Dielectric function with exact exchange contribution in the electron liquid. II. Analytical expression'
---
Introduction with concluding remarks
====================================
Electronic excitations are one of major subjects in solid state physics [@Pines]; the dielectric function $\epsilon ({\bf k} , \omega)$ of the homogeneous electron liquid [@Fetter; @Mahan; @Pines1] has been playing a central role in the description of these excitations. In the preceding paper [@Qian], referred to as I hereafter, the static dielectric function $\epsilon ({\bf k} , 0)$ with exchange contribution was studied. A very simple expression for $\Pi_1 ({\bf k}, 0)$ the first order, in terms of electron-interaction in the perturbation theory, of the static proper linear response function $\Pi ({\bf k}, 0)$ in the electron liquid, was derived. In this paper we set as our task to make like development for $\Pi_1 ({\bf k}, \omega)$, its dynamical counterpart. An analytical expression is obtained for $Im \Pi_1 ({\bf k}, \omega)$, the imaginary part of $\Pi_1 ({\bf k}, \omega)$.
The conceptual importance of $\epsilon ({\bf k} , \omega)$ \[and $\Pi ({\bf k}, \omega)$\] and previous progress made in the study of them have been briefly introduced in I, with emphasis on their static aspect. In general previous works in both of experimental and theoretical respects are enormous. We here limit ourselves to mentioning several of them which bear most close theoretical relation to the present paper [@Bohm; @Lindhard; @Hubbard; @DuBois; @Nozieres; @Osaka; @Glick; @Glick2; @Ninham; @DuBois1; @Kleinman; @Langreth; @Hasegawa; @Toigo; @Rasolt; @Rajagopal; @Niklasson; @Holas1; @Holas2; @Brosens; @Tripathy; @Dharma; @Awa; @Holas3; @Gasser; @Richardson; @Nifosi; @Vignale]. Particularly noteworthy is the work by Holas et al in Ref. [@Holas1] in which an analytical expression for $Im \Pi_1 ({\bf k}, \omega)$ had been reported. Equation (2.18) in Ref. [@Holas1] deserves fully appreciation, for it is the first analytical expression obtained for $Im \Pi_1 ({\bf k}, \omega)$ in terms of one-fold integral. Our expression, given as Eq. (\[Pi-final\]) in Sec. IV, agrees numerically with Eq. (2.18) of Ref. [@Holas1]. The correctness of both of them thus should be beyond doubt. It can be hardly denied that the method invented to obtain Eq. (2.18) in Ref. [@Holas1] is ingenious. Our expression also in terms of one-fold integral has in contrast the character of simplicity. It is also the belief of the present author that this expression has been obtained in optimal way and the derivation is more or less straightforward. Overall, the exchange contribution included in the dielectric function makes a significant improvement over the random-phase approximation (RPA), as had already been shown in Ref. [@Holas1] in several important respects. This will get full confirmation in this series of papers. We must further mention that the singular behavior of $\Pi_1 ({\bf k}, \omega)$ near the characteristic frequencies $\omega_s = (\hbar /2m)|\pm k_Fk +k^2/2| $, which had been elucidated in Ref. [@Holas1] and apparently had made some negative impression of the many-body perturbation theory on those authors [@Holas2], is also confirmed. Indeed explicit expressions of both of the discontinuity jump of $Im \Pi_1 ({\bf k}, \omega)$ at $\omega= \omega_s$ and the corresponding logarithmic divergence there of its real counterpart are obtained in this paper, which are presented in Sec. V.
In a series of papers, Brosens et al [@Brosens] investigated the local field correction to the RPA. They calculated the property $$\begin{aligned}
G({\bf k}, \omega)=
-v^{-1}(k) \Pi_1 ({\bf k}, \omega)/\Pi_0^2 ({\bf k}, \omega)\end{aligned}$$ as an approximation to the local field factor [@Hubbard]. This property is surely not the local field factor including the exact exchange contribution, a fact evidently appreciated by those authors. The latter is instead \[according to Eq. (2) in I\] $$G({\bf k}, \omega) =v(k)^{-1} \biggl [
\frac{1}{\Pi_0 ({\bf k}, \omega)+\Pi_1 ({\bf k}, \omega)}
-\frac{1}{\Pi_0 ({\bf k}, \omega)} \biggr ].$$ They apparently had never elucidated however, for the benifit of readers, that their approximation, obtained by them from the dynamic-exchange decoupling in the equation of motion for the Wigner distribution function, could be also obtained as an (sub-exchange in the sense explained above) approximation in the perturbation theory. (See also the comments made in Ref. [@Holas1] on the earlier ones of the series papers by Brosens et al.) They did point out definitely that several forms obtained before and after them [@Rajagopal; @Tripathy] were very close to or virtually identical to theirs. The relation between the theory of Rajagopal [@Rajagopal] and that by Tripathy and Mandal [@Tripathy] was also pointed out in Ref. [@Tripathy]. Tripathy and Mandal further elucidated the relation between their theory and that proposed in Ref. [@Toigo]. A critical analysis of the relation of the latter (in the static case) to the first order theory was given earlier in Ref. [@Rasolt]. Finally we wish to mention that Richardson and Ashcroft [@Richardson] also had obtained an analytical expression for $\Pi_1 ({\bf k}, \omega)$ but with $\omega$ to be imaginary. Investigations beyond the first order had also been attempted in general, in Refs. [@Holas2; @Gasser; @Richardson] for instance, but mainly in limiting cases, in Refs. [@DuBois1; @Glick2; @Ninham; @Hasegawa; @Holas1; @Holas3; @Nifosi] again for instance.
Expression (\[Pi-final\]) together with (\[H\]) for $Im \Pi_1 (k, \omega)$ is the main result of this paper. \[We remind the reader that $Im \Pi ({\bf k}, \omega)$ determines fully $\Pi ({\bf k}, \omega)$, for its real conjugation can be determined from it via the dispersion relation.\] The aim of this series of papers is to achieve a (relatively speaking) complete and final understanding of the role of the exchange contribution in the dielectric function, taking advantage of the explicit form of expression (\[Pi-final\]) and that for $\Pi_1 ({\bf k}, 0)$ (Eq. (3) in I [@Qian; @Engel]). As an example, we mention that it has been traditionally believed that $Im \Pi_1 ({\bf k}, \omega)$ has the limiting form of $\sim \omega$ for small $\omega$ [@Hasegawa; @Nifosi; @Vignale]. In fact, it was claimed by Mahan [@Mahan] and has been commonly accepted that this must be true also for $Im \epsilon ({\bf k}, \omega)$, the imaginary part of $\epsilon ({\bf k}, \omega)$, in general. We find that this is not the case and $Im \Pi_1 ({\bf k}, \omega)$ actually has the limiting form of $\sim \omega \ln \omega$, (details of which will be presented in a subsequent paper.) The deep subtlety of many-body effects often reveals itself against our intuitive understanding, and does so most definitely and convincingly in the perturbation theory indeed. We end the introduction by further remarking that calculations in the many-body perturbation theory are conventionally known to be notoriously complicated. In this sense, our expression appears quite simple. The derivation to obtain it has also been carried out in a quite manageable manner. Perhaps this is an enlightening revelation about the many-body perturbation theory.
We give our derivation in Sec. III, after presenting the starting formalism in Sec. II.
Starting formalism
==================
The Feynman-diagrammatically obtained expression for $\Pi_1 ({\bf k}, \omega)$ has been shown as Eq. (4) in I. It is, as is well known, the sum of two contributions: $$\label{SE+Ex}
\Pi_1 ({\bf k}, \omega) =\Pi_1^{SE} ({\bf k}, \omega)
+\Pi_1^{Ex} ({\bf k}, \omega);$$ $\Pi_1^{SE} ({\bf k}, \omega)$ and $\Pi_1^{Ex} ({\bf k}, \omega)$ arise, respectively, from the self-energy diagrams and the exchange diagram. We put down below the explicit expressions for them: $$\Pi_1^{SE} ({\bf k}, \omega) =\frac{2}{\hbar^2}
\int \frac{d{\bf p}}{(2 \pi)^3} \frac{d{\bf p}'}{(2 \pi)^3}
v({\bf p}- {\bf p}')
\frac{(n_{\bf p} -n_{{\bf p}+{\bf k}})(n_{{\bf p}'}
-n_{{\bf p}'+{\bf k}})}
{[\omega +\omega_{\bf p} -\omega_{{\bf p}+{\bf k}} +i0^+]^2} ,$$ and $$\Pi_1^{Ex} ({\bf k}, \omega) =-\frac{2}{\hbar^2}
\int \frac{d{\bf p}}{(2 \pi)^3} \frac{d{\bf p}'}{(2 \pi)^3}
v({\bf p}- {\bf p}')
\frac{(n_{\bf p} -n_{{\bf p}+{\bf k}})(n_{{\bf p}'}
-n_{{\bf p}'+{\bf k}})}
{[\omega +\omega_{\bf p} -\omega_{{\bf p}+{\bf k}} +i0^+]
[\omega +\omega_{{\bf p}'}-\omega_{{\bf p}'+{\bf k}} +i0^+]}.$$ (See also Refs. [@DuBois; @Holas1; @Geldart].) The notations in this paper all follow I, and here we have explicitly written $\hbar$. With some manipulation, $\Pi_1^{SE} ({\bf k}, \omega)$ can be cast in the following form: $$\begin{aligned}
\label{sect2a}
\Pi_1^{SE} ({\bf k}, \omega) = \frac{2m^2}{(2 \pi)^6 \hbar^2} \int d {\bf p}
\int d {\bf p}'
n_{{\bf p}-{\bf k}/2} n_{{\bf p}'-{\bf k}/2}
[v({\bf p}- {\bf p}')-v({\bf p}+ {\bf p}')] \nonumber \\
\biggl [ \frac{1}{(m \omega - \hbar {\bf p} \cdot {\bf k}+i0^+)^2}
+\frac{1}{(m \omega + \hbar {\bf p} \cdot {\bf k}+i0^+)^2} \biggr ],\end{aligned}$$ and $\Pi_1^{Ex} ({\bf k}, \omega)$: $$\begin{aligned}
\label{sect2b}
\Pi_1^{Ex} ({\bf k}, \omega) =
&& - \frac{2m^2}{(2 \pi)^6 \hbar^2} \int d {\bf p}
\int d {\bf p}'
n_{{\bf p}-{\bf k}/2} n_{{\bf p}'-{\bf k}/2} \nonumber \\
&& \biggl [ v({\bf p} - {\bf p}')
\biggl ( \frac{1}{(m \omega - \hbar {\bf p} \cdot {\bf k}+i0^+)
(m \omega - \hbar {\bf p}' \cdot {\bf k}+i0^+)} \nonumber \\
&&~~~~~~~~~~~~~~+\frac{1}{(m \omega + \hbar {\bf p} \cdot {\bf k}+i0^+)
(m \omega + \hbar {\bf p}' \cdot {\bf k}+i0^+)} \biggr ) \nonumber \\
&& -v({\bf p} + {\bf p}')
\biggl ( \frac{1}{(m \omega - \hbar {\bf p} \cdot {\bf k}+i0^+)
(m \omega + \hbar {\bf p}' \cdot {\bf k}+i0^+)} \nonumber \\
&&~~~~~~~~~~~~~~+\frac{1}{(m \omega + \hbar {\bf p} \cdot {\bf k}+i0^+)
(m \omega - \hbar {\bf p}' \cdot {\bf k}+i0^+)} \biggr ) \biggr ] . \end{aligned}$$ The imaginary parts of them can be obtained , respectively, as $$\begin{aligned}
\label{Def-SE}
Im \Pi_1^{SE} ({\bf k}, \omega) =&&
\frac{m}{(2 \pi)^5 \hbar^2 } \frac{\partial}{\partial \omega}
\int d {\bf p} \int d {\bf p}'
n_{{\bf p}-{\bf k}/2} n_{{\bf p}'-{\bf k}/2} \nonumber \\
&& [v({\bf p}- {\bf p}')-v({\bf p}+ {\bf p}')]
[\delta (m \omega - \hbar {\bf p} \cdot {\bf k})
+\delta (m \omega + \hbar {\bf p} \cdot {\bf k})] , \end{aligned}$$ and $$\begin{aligned}
\label{Def-Ex}
Im \Pi_1^{Ex} ({\bf k}, \omega) &=& \frac{2 m^2}{(2 \pi)^5 \hbar^2}
\int d {\bf p} \int d {\bf p}'
n_{{\bf p}-{\bf k}/2} n_{{\bf p}'-{\bf k}/2} \nonumber \\
&~& \biggl [ v({\bf p}- {\bf p}')
\biggl ( \frac{1}{m \omega - \hbar {\bf p}' \cdot {\bf k}}
\delta (m \omega - \hbar {\bf p} \cdot {\bf k})
+\frac{1}{m \omega + \hbar {\bf p}' \cdot {\bf k}}
\delta (m \omega + \hbar {\bf p} \cdot {\bf k}) \biggr ) \nonumber \\
&-& v({\bf p}+ {\bf p}')
\biggl ( \frac{1}{m \omega - \hbar {\bf p}' \cdot {\bf k}}
\delta (m \omega + \hbar {\bf p} \cdot {\bf k})
+\frac{1}{m \omega + \hbar {\bf p}' \cdot {\bf k}}
\delta (m \omega - \hbar {\bf p} \cdot {\bf k}) \biggr ) \biggr ] . \nonumber \\
~~\end{aligned}$$ These forms serve our purpose best.
Derivation
===========
$Im \Pi_1^{SE} ({\bf k}, \omega)$
-----------------------------------
The property $\Pi_1 ({\bf k}, \omega)$ depends only on the magnitude of ${\bf k}$ in a uniform system, so it may be written as $\Pi_1 ( k, \omega)$. We first define a dimensionless quantity: $\Omega=m \omega/\hbar k_F^2$. From now on throughout the paper we put $k$ in units of $k_F$, i.e., $k$ will always be dimensionless.
The computation for $Im \Pi_1^{SE} (k, \omega)$ can be made very simple. The integral over the variable ${\bf p}'$ in Eq. (\[Def-SE\]) can be carried out first, which leads to $$\begin{aligned}
\label{SE-1}
Im \Pi_1^{SE} (k, \omega) = \frac{m^2e^2}{2 \pi^2 \hbar^4}
\frac{\partial}{\partial \Omega}
&~& \int_{-a}^b dz \int_0^\lambda dx
[\delta(\Omega -kz) +\delta(\Omega +kz) ] \nonumber \\
&~& [F(\sqrt{z^2 +x -kz +k^2/4})
-F(\sqrt{z^2 +x +kz +k^2/4}) ] ,\end{aligned}$$ where $$\begin{aligned}
F (q) = \frac{1}{4 \pi} \int d {\bf p}
\frac{n_{\bf p}}{|{\bf p} -{\bf q}|^2} . \end{aligned}$$ Explicitly, $$\begin{aligned}
F (q) = \frac{1}{2} + \frac{1 -q^2}{4q}
\ln \biggl |\frac{1+q}{1-q} \biggr |. \end{aligned}$$ We mention once again that the notations here follow I. The integration over $z$ in Eq. (\[SE-1\]) is trivial. After performing it, one gets $$\begin{aligned}
\label{SE-2}
Im \Pi_1^{SE} ( k, \omega) = \frac{m^2 e^2}{2 \pi^2 \hbar^4}\frac{1}{k^2}
&\bigg [& \theta \{(b-\Omega /k)(a+\Omega/k) \}
H^{SE} (k, \Omega/k) \nonumber \\
&~& - \theta \{(b+\Omega /k)(a-\Omega/k) \}
H^{SE} (k, -\Omega/k) \biggr ] ,\end{aligned}$$ with $$\begin{aligned}
H^{SE} (k, z) = \frac{\partial}{\partial z}
\int_0^\lambda dx [F(\sqrt{x -\lambda +1 })-F(\sqrt{x -\lambda +1 + 2kz}) ] .\end{aligned}$$ The $H^{SE} (k, z)$ in the preceding equation can be readily refined into $$\begin{aligned}
H^{SE} (k, z) = (k-2z)F(\sqrt{-\lambda +1}) +(k+2z) F(\sqrt{-\lambda +1+2kz})
-2k F(\sqrt{1+2kz}) . \end{aligned}$$ Explicitly, $$\begin{aligned}
\label{SE-final}
H^{SE} (k, z) = \frac{1}{2}
\biggl [ 2 k^2 z \frac{1}{\sqrt{C_0}} Y(z)
- \lambda W_1 (z) - {\tilde \lambda} W_2 (z) \biggr ].\end{aligned}$$ In Eq. (\[SE-final\]) we have introduced (newly) the symbol ${\tilde \lambda}=(b+z)(a-z)$.
$Im \Pi_1^{Ex} ({\bf k}, \omega)$
-----------------------------------
Our labor lies mainly in the evaluation of $Im \Pi_1^{Ex} (k, \omega)$ expressed in (\[Def-Ex\]). Following paper I, we first carry out the integrals over the azimuthal angular variables of ${\bf p}$ and ${\bf p}'$. After that, we obtain $$\begin{aligned}
Im \Pi_1^{Ex} ( k, \omega)
=&& \frac{m^2 e^2}{4 \pi^2 \hbar^4}
\int_{-a} \int^b dz dz' \biggl [
\biggl \{\frac{1}{\Omega -k z'} \delta (\Omega - k z)
+\frac{1}{\Omega +k z'} \delta (\Omega + k z) \biggr \}
L(\beta^2) \nonumber \\
&&~~~~~~~~~~~~~~~~~~
-\biggl \{ \frac{1}{\Omega -k z'} \delta (\Omega + k z)
+\frac{1}{\Omega +k z'} \delta (\Omega - k z) \biggr \}
L(\alpha^2) \biggr ] .\end{aligned}$$ We then, taking advantage of the presence of the $\delta- $ function, reduce the two-fold integral to one-fold. The $Im \Pi_1^{Ex} ( k, \omega)$ becomes thus $$\begin{aligned}
\label{Ex}
Im \Pi_1^{Ex} ( k, \omega)= - \frac{m^2e^2}{4 \pi^2 \hbar^4} \frac{1}{k^2}
&\bigg [&\theta \{(b-\Omega /k)(a+\Omega/k) \}
H^{Ex} (k, \Omega/k) \nonumber \\
&~& - \theta \{(b+\Omega /k)(a-\Omega/k) \}
H^{Ex} (k, -\Omega/k) ] ,\end{aligned}$$ with the function $H^{Ex} (k, z)$ defined as $$\begin{aligned}
\label{Def-HEx}
H^{Ex} (k, z)= \int_{-a}^b dz'
\biggl [ \frac{1}{\alpha} L(\alpha^2) -\frac{1}{\beta} L(\beta^2) \biggr ] .\end{aligned}$$ The function $L$ has been given in Eq. (9) in I and in Ref. [@Glasser]. There are several components in it, and we separate them in the evaluation of the integral in Eq. (\[Def-HEx\]). Accordingly we write $H^{Ex} (k, z)$ in the following manner: $$\begin{aligned}
\label{HEx-2}
H^{Ex} (k, z)= H_0^{Ex} (k, z) + H_1^{Ex} (k, z)+H_{23}^{Ex} (k, z) ,\end{aligned}$$ with $$\begin{aligned}
H_0^{Ex} (k, z)= \int_{-a}^b dz' z' \bigg [ (\lambda + \lambda')
(2 \ln 2 +1) \frac{1}{\alpha \beta} - 1 \biggr ] ,\end{aligned}$$ $$\begin{aligned}
H_1^{Ex} (k, z)= \int_{-a}^b dz' \bigg [ \frac{1}{\alpha}
\sqrt{R(z, z')} - \frac{1}{\beta} |\beta |\biggr ] ,\end{aligned}$$ and $$\begin{aligned}
\label{Def-H23}
H_{23}^{Ex} (k, z) &=& \int_{-a}^b dz'
\biggl [ \biggl ( \frac{1}{\alpha}-\frac{1}{\beta} \biggr )
\lambda \ln |4 \lambda|
- 2\lambda' \biggl( \frac{1}{\alpha} \ln |\alpha |
- \frac{1}{\beta} \ln |\beta| \biggr ) \nonumber \\
&-& \frac{1}{\alpha} \biggl ( \lambda \ln | \alpha^2 + \lambda'
-\lambda -2 \sqrt{R(z, z')} |
- \lambda' \ln | \alpha^2 - \lambda'
+\lambda +2 \sqrt{R(z, z')} | \biggr ) \nonumber \\
&+& \frac{1}{\beta} \biggl ( \lambda \ln | \beta (k-2z) + 2|\beta| |
- \lambda' \ln | \beta(k-2z') + 2|\beta| | \biggr ) \biggr ] . \end{aligned}$$ The two terms of $J_2$ and $J_3$ [@Glasser] in Eq. (11) of I were combined, for the simplicity of the computation, into one term \[denoted as $J_{23}$ in Eq. (43) there\]. The $H_{23}^{Ex} (k, z)$ here follows suit.
The evaluation for $H_0^{Ex} (k, z)$ and $H_1^{Ex} (k, z)$ is a routine job. It is quite straightforward to get the following result: $$\begin{aligned}
\label{H0}
H_0^{Ex} (k, z)=
(2 \ln 2 +1) [ \lambda W_1 (z) + (ab-z^2) W_2(z) - k ] - k ,\end{aligned}$$ and $$\begin{aligned}
\label{H1}
H_1^{Ex} (k, z)=
k [2 - z W_2(z) - z (2z +k) C_0^{-1/2} Y(z) ] .\end{aligned}$$
We next attack $H_{23}^{Ex} (k, z)$. With a little algebra, we rewrite Eq. (\[Def-H23\]) in the following form: $$\begin{aligned}
\label{H23}
H_{23}^{Ex} (k, z)= - [ W_1(z) + W_2(z) ] \lambda \ln |4 \lambda |
-2 \zeta_1 (z) - \zeta_1 (-z) + \zeta_2 (z) -\zeta_3(z) , \end{aligned}$$ with $$\begin{aligned}
\zeta_1 (z) = \int_{-a}^b dz' \frac{\lambda'}{\alpha}
\ln | \alpha | ,\end{aligned}$$ $$\begin{aligned}
\zeta_2 (z) = \int_{-a}^b dz' \frac{1}{\beta}
\bigg [ \lambda \ln | \beta(k-2z) + 2|\beta| |
- \lambda' \ln | k-2z' + 2 \beta /|\beta| | \biggr ] ,\end{aligned}$$ and $$\begin{aligned}
\label{zeta-3-Def}
\zeta_3 (z) = \int_{-a}^b dz' \frac{1}{\alpha}
\bigg [ \lambda \ln | \alpha^2 + \lambda' -\lambda -2 \sqrt{R(z, z')} |
- \lambda' \ln | \alpha^2 - \lambda' +\lambda + 2 \sqrt{R(z, z')} | \biggr ] .\end{aligned}$$ The reader should not confuse the functions $\zeta_n (z)$ here with the Riemann’s function $\zeta (n)$ that appeared in I. The evaluation for $\zeta_1(z)$ and $\zeta_2(z)$ is a little tedious but clearly straightforward. We thus present only the results: $$\begin{aligned}
\label{zeta-1}
\zeta_1 (z) =
\frac{1}{2} [ (2z+k)(\ln |{\tilde \lambda}| -3 )
+ \{ {\tilde \lambda} (3 -\ln |{\tilde \lambda}|) -2 \} W_2 (z) ] ,\end{aligned}$$ and $$\begin{aligned}
\label{zeta-2}
\zeta_2 (z) =
\frac{1}{2} [\lambda (\ln \lambda -2 \ln2 -3) +2 ] W_1 (z)
+\frac{1}{2}(2z -k) (\ln \lambda -6 \ln2 +3) - \lambda v_1 (z) ,\end{aligned}$$ where $$\begin{aligned}
v_1 (z) = \int_{-a}^z dz' \frac{1}{z' -b } \ln \beta
+\int_{z}^b dz' \frac{1}{z' +a } \ln \beta .\end{aligned}$$
We now turn to $\zeta_3 (z)$. We first rewrite Eq. (\[zeta-3-Def\]) as $$\begin{aligned}
\label{zeta-3}
\zeta_3 (z) = \lambda \zeta_{3a} (z) - \zeta_{3b} (z) ,\end{aligned}$$ with $$\begin{aligned}
\label{zeta-3a}
\zeta_{3a} (z) = \int_{-a}^b dz' \frac{1}{\alpha}
\ln | \alpha^2 + \lambda' -\lambda -2 \sqrt{R(z, z')} | ,\end{aligned}$$ and $$\begin{aligned}
\label{zeta-3b}
\zeta_{3b} (z) = \int_{-a}^b dz' \frac{\lambda '}{\alpha}
\ln | \alpha^2 - \lambda' +\lambda + 2 \sqrt{R(z, z')} | .\end{aligned}$$ The evaluation of $\zeta_{3a} (z)$ is also a routine job. It can be effected with partial integration. One gets in this manner $$\begin{aligned}
\label{3a}
\zeta_{3a} (z) =
- W_2 (z) \ln |2 \lambda | - \int_{-a}^b dz' D_1 (z, z') \ln |\alpha | ,\end{aligned}$$ where $$\begin{aligned}
D_1 (z, z') = \frac{\partial }{\partial z'}
\ln | \alpha^2 + \lambda' - \lambda - 2 \sqrt{R(z, z')} | .\end{aligned}$$ Explicitly, $$\begin{aligned}
D_1 (z, z') =
\frac{1}{\alpha} \biggl [1 - \frac{kz}{\sqrt{R(z, z')}} \biggr ] .\end{aligned}$$ The equation (\[3a\]) can now be readily refined into $$\begin{aligned}
\label{zeta-3a-final}
\zeta_{3a} (z) =
\frac{1}{2} [ \ln |{\tilde \lambda}|
- 2 \ln |2 \lambda | ] W_2 (z) + kz v_2 (z) ,\end{aligned}$$ with $$\begin{aligned}
\label{v2}
v_2 (z) = \int_{-a}^b dz' \frac{1}{\alpha \sqrt{R(z, z')} }
\ln |\alpha | .\end{aligned}$$ The integral on the right hand side of Eq. (\[zeta-3b\]) can also be effected with partial integration. To this end, we employ the following identity: $$\begin{aligned}
\frac{\lambda '}{\alpha} dz' = \frac{1}{2}
d [2 {\tilde \lambda} \ln |\alpha| + \lambda ' + (2z+k) \alpha ], \end{aligned}$$ with the symbol $d$ here denoting the differential operating only on the variable $z'$. We perform in this manner the partial integration and get for $\zeta_{3b} (z)$: $$\begin{aligned}
\label{zeta-3b-1}
\zeta_{3b} (z) &=& \frac{1}{2} (2z +k)
[ 4 \ln 2 +(b+z) \ln r_1 +(a-z) \ln r_2 ]
+ {\tilde \lambda} [ -2 W_2(z) \ln 2 + v_0 (z)] \nonumber \\
&~& - \frac{1}{2} \int_{-a}^bd z'
[2 {\tilde \lambda} \ln |\alpha| + \lambda' + (2z+k) \alpha ] D_2 (z, z') ,\end{aligned}$$ where $$\begin{aligned}
\label{v0}
v_0 (z) = \ln |b+z| \ln r_1 - \ln |a-z| \ln r_2 ,\end{aligned}$$ and $$\begin{aligned}
D_2(z, z') = \frac{\partial }{\partial z'}
\ln | \alpha^2 - \lambda' +\lambda + 2 \sqrt{R(z, z')} | .\end{aligned}$$ We have introduced the following symbols: $$\begin{aligned}
\label{r1-r2}
r_1 = \sqrt{R(z, b)} , ~~~~~~ r_2 =\sqrt{R(z, -a)} \end{aligned}$$ in Eqs. (\[zeta-3b-1\]) and (\[v0\]). Explicitly $$\begin{aligned}
\label{D2}
D_2(z, z') =
\frac{1}{\alpha} \biggl [1 - \frac{kz}{\sqrt{R(z, z')}} \biggr ]
- \frac{1}{2(b-z')} \biggl [ 1- \frac{r_1}{\sqrt{R(z, z')}} \biggr ]
+ \frac{1}{2(a+z')} \biggl [1 - \frac{r_2}{\sqrt{R(z, z')}} \biggr ] .
\nonumber \\ \end{aligned}$$ Making the use of Eq. (\[D2\]) in Eq. (\[zeta-3b-1\]), the expression for $\zeta_{3b} (z)$ can be organized in the form: $$\begin{aligned}
\label{zeta-3b-final}
\zeta_{3b} (z)=&& [ (2z +k)
\{ 4 \ln 2 +(b+z) \ln r_1 +(a-z) \ln r_2 \}
+ {\tilde \lambda} \{ -4 W_2(z) \ln 2 + 2v_0 (z) \nonumber \\
&& +W_2(z) \ln |{\tilde \lambda}| + 2kz v_2 (z) + v_3 (z) \}
- {\bar \zeta}_{3b} (z) ]/2 ,\end{aligned}$$ where $v_3(z)$ and ${\bar \zeta}_{3b} (z)$ are defined as $$\begin{aligned}
v_3 (z) =
\int_{-a}^b d z' \ln |\alpha|
\biggl [\frac{1}{b-z'} \biggl ( 1- \frac{r_1}{\sqrt{R(z, z')}} \biggr )
+\frac{1}{a+z'} \biggl ( -1+ \frac{r_2}{\sqrt{R(z, z')}} \biggr ) \biggr ] ,\end{aligned}$$ and $$\begin{aligned}
\label{bar0}
{\bar \zeta}_{3b} (z)
=\int_{-a}^b dz'[\lambda' +(2z +k) \alpha ] D_2 (z, z') ,\end{aligned}$$ respectively. The integral on the right hand side of Eq. (\[bar0\]) \[with $D_2 (z, z')$ given explicitly in Eq. (\[D2\])\] is basic, although it looks somewhat tedious. With some algebra, we can put it in the following form: $$\begin{aligned}
\label{bar}
{\bar \zeta}_{3b} (z)=&&
-{\tilde \lambda}W_2 (z) - (z/2) [ 3 (k^2 -1 ) +C_0 ] V_0 (z)
+C_0 V_1 (z) - kz {\tilde \lambda} V_{-1} (z, -z) \nonumber \\
&& -\frac{1}{2} (2z +k) \biggl [ - 10 + 4C_0 V_0 (z)
+ (z +b) \biggl ( \int_{-a}^b dz' \frac{1}{b-z'}
+ r_1 V_{-1} (z, b) \biggr ) \nonumber \\
&& - (z -a) \biggl ( \int_{-a}^b dz' \frac{1}{a+z'}
- r_2 V_{-1} (z, -a) \biggr ) \biggr ] ,\end{aligned}$$ where $$\begin{aligned}
V_n (z)=\int_{-a}^b dz' \frac{z'^n}{\sqrt{R(z, z')}} \end{aligned}$$ for $ n=0, 1$, and $$\begin{aligned}
V_{-1} (z, x)=
\int_{-a}^b dz' \frac{1}{z'-x} \frac{1}{\sqrt{R(z, z')}} .\end{aligned}$$ Explicitly [@Qian], $$\begin{aligned}
V_0 (z)=2 C_0^{-1/2} Y(z) ,
~~~ ~~~ ~~~~
V_1 (z)= [2z+k - z(kz + 1 -k^2/2) V_0 (k, z)] C_0^{-1} ,\end{aligned}$$ and $$\begin{aligned}
V_{-1} (z, -z)= - W_2 (z) /kz .\end{aligned}$$ We note that the sum in each of the big curve bracket in Eq. (\[bar\]) is well defined, though their respective components are not. The reader excuses us for the sake of a compact presentation. Indeed, $$\begin{aligned}
\int_{-a}^b dz' \frac{1}{b-z'}
+ r_1 V_{-1} (z, b) = 2 \ln \biggl | \frac{kz}{r_1} \biggr | ,\end{aligned}$$ and $$\begin{aligned}
\int_{-a}^b dz' \frac{1}{a+ z'}
- r_2 V_{-1} (z, -a) = 2 \ln \biggl | \frac{kz}{r_2} \biggr | . \end{aligned}$$ In virtue of the foregoing results, the integral on the right hand side of Eq. (\[bar\]) has now been fully carried out. The final result for ${\bar \zeta}_{3b} (z)$ can after refinement be written as $$\begin{aligned}
\label{bar-final}
{\bar \zeta}_{3b} (z) =
(2z+k) [ 6 - 2(3kz+1)C_0^{-1/2} Y(z) +(z+b) \ln r_1
- (z-a) \ln r_2 - 2 \ln |kz| ] .\end{aligned}$$ The substitution of Eq. (\[bar-final\]) into Eq. (\[zeta-3b-final\]) will give the result for $\zeta_{3b} (z)$. Further substitution of thus obtained result for $\zeta_{3b} (z)$ and the previously obtained one for $\zeta_{3a} (z)$ in Eq. (\[zeta-3a-final\]) into Eq. (\[zeta-3\]) then yields the final result for $\zeta_3 (z)$, which turns out to be $$\begin{aligned}
\label{zeta-3-final}
\zeta_3 (z) =&~& [kz \ln |{\tilde \lambda}| - \lambda \ln | 2 \lambda |
+ 2 {\tilde \lambda} \ln 2 ] W_2 (z) +2 k^2 z^2 v_2 (z)
- ({\tilde \lambda}/2) [2 v_0 (z) + v_3 (z)] \nonumber \\
&-& (2z +k) [ 2 \ln 2 -3 + (3kz +1) C_0^{-1/2} Y(z) + \ln |kz| ] .\end{aligned}$$
One then substitutes $\zeta_1 (z)$ expressed in Eq. (\[zeta-1\]), $\zeta_2 (z)$ in Eq. (\[zeta-2\]), and $\zeta_3 (z)$ in the above equation into Eq. (\[H23\]) to get $$\begin{aligned}
\label{H23-final}
H_{23}^{Ex} (k , z)=&& (-2z +5k) \ln 2 + \mu_1 (k, z) + \mu_1 (k, -z)
- 2 (1+ \ln 2) [\lambda W_1 (z) + {\tilde \lambda} W_2 (z)] \nonumber \\
&&- [2kz \ln 2 +(kz -{\tilde \lambda} ) \ln |{\tilde \lambda}|] W_2 (z)
- \lambda v_1 (z)
+ ( {\tilde \lambda} /2) [2 v_0 (z) +v_3 (z) ] \nonumber \\
&& +(2z +k) [(3kz +1) C_0^{-1/2} Y(z) + \ln |kz|] - 2k^2z^2 v_2 (z) ,\end{aligned}$$ where $$\begin{aligned}
\mu_1 (k, z) = (2z -k) \ln \lambda + [2- (1+\ln 2 ) \lambda ] W_1 (z) .\end{aligned}$$ One then advances further to add \[according to Eq. (\[HEx-2\])\] $H_0^{Ex} (k , z)$ of Eq. (\[H0\]), $H_1^{Ex} (k , z)$ of Eq. (\[H1\]), and $H_{23}^{Ex} (k , z)$ of Eq. (\[H23-final\]) to get the result for $H^{Ex} (k , z)$ which can be in the final form written as $$\begin{aligned}
\label{Ex-final}
H^{Ex} (k , z)=&& (-2z +3k) \ln 2 + \mu_1 (k, z) + \mu_1 (k, -z)
+ ({\tilde \lambda} -kz) W_2 (z) \ln |{\tilde \lambda}| \nonumber \\
&& - \lambda W_1 (z) - {\tilde \lambda} W_2 (z)
- \lambda v_1 (z) + ({\tilde \lambda /2)} [2 v_0 (z) +v_3 (z) ]
\nonumber \\
&& + (2z +k) [\sqrt{C_0} Y(z) + \ln |kz|] - 2k^2z^2 v_2 (z) .\end{aligned}$$
Analytical result
=================
In virtue of Eq. (\[SE+Ex\]), $Im \Pi_1 (k, \omega)$ can be obtained from Eq. (\[SE-2\]) and Eq. (\[Ex\]) as $$\begin{aligned}
\label{Pi-final}
Im \Pi_1 (k, \omega) = \frac{m^2e^2}{(2 \pi)^2 \hbar^4} \frac{1}{k^2}
[ \theta (1-\nu_+^2) H (k, \Omega/k)
- \theta (1-\nu_-^2) H (k, -\Omega/k) ] , \end{aligned}$$ with $\nu_+= \Omega/k - k/2$ , $\nu_-= -\Omega/k - k/2$ , and $$\begin{aligned}
H (k, z)= 2 H^{SE} (k, z) - H^{Ex} (k, z) .\end{aligned}$$ The substitution from Eqs. (\[SE-final\]) and (\[Ex-final\]) will give the result for $H (k, z)$, which we then further refine into the following form: $$\begin{aligned}
\label{H}
H (k, z)= && - (2zC_0 +k) C_0^{-1/2} Y(z) + (2z - 3k) \ln 2
- (k +2z) \ln |kz| \nonumber \\
&& + (kz - {\tilde \lambda} ) \ln |{\tilde \lambda}| W_2 (z)
- \mu_1 (k, z) - \mu_1 (k, -z) - \mu_2 (k, z) +\mu_2 (-k , -z) \nonumber \\
&&+ 2 k^2 z^2 \int_{-a}^b d z' \frac{1}{\alpha \sqrt{R(z, z')}} \ln |\alpha| , \end{aligned}$$ where $$\begin{aligned}
\label{mu2}
\mu_2 (k, z) = && {\tilde \lambda} \ln |z+b| \ln |(k+1)z +b|
-\lambda \int_z^b dz' \frac{1}{z' +a} \ln |\beta| \nonumber \\
&& +\frac{1}{2} {\tilde \lambda} \int_{-a}^b dz' \frac{1}{b-z'}
\biggl [ 1 - \frac{(k+1)z +b}{\sqrt{R(z, z')}} \biggr ] \ln |\alpha| .\end{aligned}$$
Singularity of $\Pi_1 (k, \omega)$ at $\omega = \omega_s$
==========================================================
One can immediately see that $Im \Pi_1 ( k, \omega)$ has the same nonvanishing region as $Im \Pi_0 ( k, \omega)$, the Lindhard function [@Lindhard; @Fetter; @Mahan]. In other words, the region of the single particle-hole continuum remains unchanged with the inclusion of the exchange contribution. The long-wavelength plasmon which has zero linewidth in RPA up to wavevector $k_c$, at which the damping sets in, accordingly remains up to $k_c$ infinitely robust against exchange effect. This truth has been recognized before [@DuBois1; @Glick2; @Ninham; @Hasegawa; @Holas1; @Holas3; @Gasser; @Nifosi]. Such a distinctly drawn conclusion, if understood in an appropriate manner, must also be appreciated as one of the merits of the perturbation theory.
While $Im \Pi_0 ( k, \omega)$ approaches to zero on the edge of the single particle-hole continuum, $Im \Pi_1 (k, \omega)$ shows a discontinuity jump there. In other words, $\Pi_1 (k, \omega)$ exhibits singular behavior at $\omega = \omega_{s}$ with $\omega_s = (\hbar k_F^2/2m)|\pm k +k^2/2| $. This singularity was noticed by Glick [@Glick] before Holas et al [@Holas1], and also by Awa et al [@Awa] after them, and Holas et al had made the most elaborate investigation of it. In fact, all of the three groups of authors had adopted a similar approach in order to remove it. The jump discontinuity, defined as $\bigtriangleup _s (k)
= Im \Pi_1 ( k, \omega_{s}+ 0^+) - Im \Pi_1 ( k, \omega_{s}- 0^+)$ can be explicitly calculated by the use of Eq. (\[Pi-final\]). For $\omega_{s} = (\hbar k_F^2/2m)(k+k^2/2) $, $$\begin{aligned}
\bigtriangleup _s (k) =
\frac{m^2 e^2 }{2 \pi^2 \hbar^4}
\frac{b}{k(1+k)} \ln \biggl |\frac{2b}{k} \biggr | ; \nonumber \\\end{aligned}$$ and, for $\omega_{s} = (\hbar k_F^2/2m) |-k+k^2/2| $, $$\begin{aligned}
\bigtriangleup _s (k) = -
\frac{m^2 e^2} {2 \pi^2 \hbar^4}
\frac{a}{k(1-k)} \ln \biggl |\frac{2a}{k} \biggr | .\end{aligned}$$ The discontinuity in $Im \Pi_1 ( k, \omega)$ gives rise to a logarithmic divergence in $Re \Pi_1 ( k, \omega)$, which has the following form (to the accuracy of the leading logarithmic order): $$\begin{aligned}
Re \Pi_1 ( k, \omega) = \frac{1}{\pi} \bigtriangleup _s (k)
\ln |2m (\omega - \omega_s)/\hbar k_F^2 | \end{aligned}$$ for $\omega \to \omega_s$.
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---
abstract: 'We study the generalization of QED synchrotron radiation to the QCD case with a chromomagnetic field using the Schwinger [*et al*]{} source method. It is shown that the QED case can be obtained as a special limit. The comparison with the path integral approach of Zakharov has shown consistent results.'
author:
- 'Alaa Dbeyssi$^{(1)}$, Dima Al Dirani$^{(2)}$, H. Zaraket$^{(2)}$'
title: '**[Synchrotron radiation in a chromo-magnetic field]{}**'
---
Introduction
============
It is believed that during and after relativistic heavy ion collisions strong chromomagnetic field will form that can be treated as a classical background. Numerical solutions [@FriesK; @LappiM] indicate that ”just“ after collision transverse color electric and color magnetic fields change suddenly from being transverse in the initial state, in the so-called Color-Glass-Condensate state, to being longitudinal. The latter are called glasma flux tubes. Transverse fields then rise, and at some stage after the collision the transverse and longitudinal components of color electric and color magnetic fields reach a ”steady” comparable values. In this rich environment a fast parton escaping from the collision would feel the effect of such a field. Synchrotron and Čerenkov effects are important physical phenomena to be studied, either as an energy loss mechanism or as a coherent gluon radiation process. In this context the Čerenkov radiation was considered in [@Dremin]. Syncrotron radiation was analyzed for a longitudinal field in [@ShuryaZ], and a transverse field in [@Zak1]. We focus in this paper on the synchrotron radiation by a moving quark in a longitudinal chromomagnetic field; this should be distinguished from the case studied in [@Tuchin]. The author of [@Tuchin] has considered the motion of fast fermion in an “electromagnetic” magnetic field not a chromomagnetic field.
The synchrotron radiation of a photon from a relativistic electron was well studied in the last century [@Soko; @Schwinger]. Generalizing the QED study to the QCD case has been done in [@ShuryaZ] for a chromomagnetic field. It would have been the “end of a new chapter” but in [@Zak1] the path integral approach was used to derive the synchrotron radiation of gluons by a parton in a transverse chromomagnetic field. The results found in [@Zak1] seem to be radically different from [@ShuryaZ]; it was also argued that the QED case cannot be obtained from the QCD generalization of [@ShuryaZ]. This motivates reconsidering the QCD generalization of the QED operator/source method used in [@SchwinET]. We show that the QED case is simply obtained from our QCD generalization. After clarifying the differences in the field configurations considered in [@ShuryaZ] and [@Zak1], it is possible to select the closest possible configuration to [@Zak1] and make a comparison between our results and that of [@Zak1]. We show that very similar expressions are found. We analyze some of the possible inconsistencies that led to the erroneous results of [@ShuryaZ].
The radiation in external magnetic fields is applicable in all Lorentz frames where $H^2 - {\cal E}^2 > 0$, where ${\cal E}$ is the electric field. Hence we consider a weak field approximation to avoid the ”Klein catastrophe”, that is, spontaneous pair creation by an electric field. Hence for $|F_{\mu\nu}F^{\mu\nu}|^{1/2}\ll m^2/g$, pair creation effects are negligible. Besides, for the semiclassical approximation to hold we have to have $g^3(F/m^2)(E/m)\ll 1$; having large quark energy the only configuration to have a semiclassical approximation is in the weak field regime, [*i.e.*]{}, $H\ll m^2/g$. Therefore we consider the case where $H\ll m^2/g$.
The paper is organized as follows. The coupling of quarks and gluons to the background field is first represented in terms of color charges. The Schwinger [*et al*]{} method is then presented, the gluon synchrotron rate is calculated, and finally we compare the obtained results with different results in literature.
Charge “representation”
=======================
The main difference between the QED and QCD synchrotron is the interaction between the emitted gluon and the background chromomagnetic field. The non-Abelian generalization of the QED synchrotron radiation can be easily obtained if we characterize the coupling between gluons and the chromomagnetic background field through a fictive color charge. The fictive charge can be explicitly obtained by starting from the QCD Lagrangian, decomposing the gauge field into quantum fluctuations and a classical background field, then regrouping terms that couple to a given background color index. The coupling constant $g$ is then found to be multiplied by a number which can then be interpreted as a color charge. The more formal approach is to use the Cartan subgroup of SU(3), then quarks and gluons can be characterized by two color charges. SU(3) is a unimodular group of 3$\times$3 Hermitian linearly independent matrices of determinant equal to 1. SU(3) has $3^2-1=8$ generators $\lambda_i$ ($i=1\cdots 8$). The maximum number of commuting generators for SU(3) is $2$. In the Gell-Mann fundamental representation $\lambda_3$ and $\lambda_8$ commute hence ${\cal H}_1=\lambda_3/2$ and ${\cal H}_2=\lambda_8/2$ form the Cartan subgroup (they can be seen as the analogue of $J_3$ for SU(2)) and are already diagonal. Define the following combinations of the remaining six generators: $$E_{\pm \alpha}=\lambda_1\pm i\lambda_2 \quad E_{\pm \beta}=\lambda_6\pm i\lambda_7\;\; {\rm and}\; E_{\pm \gamma}=\lambda_4 \pm i\lambda_5$$ these new matrices will play the role of raising and lowering operators, they satisfy $$[{\cal H}_i,E_{\xi}]=\xi({\cal H}_i)E_{\xi}\; .$$ Hence the $E_{\xi}$ are eigenvectors of the $ad({\cal H}_1)$ and $ad({\cal H}_2)$. The weight $\xi$ of the adjoint representation are called the roots. They are the color charge of the gluon. A simple calculation, for the Gell-Mann representation, gives the gluon charges ($\xi({\cal H}_1),\xi({\cal H}_2)$ or $(Q_3,Q_8)$) $$(1,0)\;,\;(-1,0)\;,\;(-\frac{1}{2},\frac{\sqrt{3}}{2})\; ,\; (\frac{1}{2},-\frac{\sqrt{3}}{2})\; ,$$ $$(\frac{1}{2},\frac{\sqrt{3}}{2})\; ,\;(\frac{1}{2},\frac{\sqrt{3}}{2})\; ,\;(-\frac{1}{2},-\frac{\sqrt{3}}{2})\;,(0,0)\; .$$ Quarks are found in the fundamental representation, so the eigenvalues of ${\cal H}_i$ give the color charges of quarks. So quark “states” will be common eigenstates of ${\cal H}_1,{\cal H}_2 $ with eigenvalues $(q_3,q_8)$): $$(\frac{1}{2},\frac{\sqrt{3}}{6})\; ,\; (-\frac{1}{2},\frac{\sqrt{3}}{6})\; ,\; (0,-\frac{\sqrt{3}}{3})\; .$$ The color charge will always be multiplied by the strong interaction coupling constant $g$, so it is possible to absorb the “charge” in the coupling and define a new coupling $g^i_A=gQ_i$ for gluons and $g^i_q=gq_i$. To simplify our calculation we will consider a chromomagnetic field with a given color index, either $3$ or $8$ but not combination of both color indices.
The mass operator
=================
In the Schwinger [*et al*]{} method, the emission process can be seen through a modification of the Dirac equation of a spin $1/2$ particle of mass $m$ in the presence of an external field $H$. Besides the usual modification of the momentum operator (a tree level modification) to include the external field effect ($i\partial_\mu\rightarrow i\partial -qA_\mu=\Pi_\mu$), the emission process can be seen as a modification of the mass by an operator $M$ called the mass operator: $$\left(\gamma_\mu\Pi^\mu+m+M\right)\Psi=0\; .$$ So the mass operator is always acting on a physical spin $1/2$ real particle state. As we will see later, special attention should be paid in some cases to its correct dispersion relation in the presence of an external field.
The total decay rate $\Gamma$ of a particle of mass $m$ and energy $E$ can be obtained from the mass operator $M$ by an “optical theorem”-like relation $$\Gamma=-2\frac{m}{E}{\rm Im}\; M
\label{eq:mass}$$ where $M$ is obtained from the one-loop diagram with full quark and gluon propagators in the presence of the field $H$. So the final expression will be valid to all orders in $gH$ but not to all orders in $\alpha_s$. The decay rate in eq. (\[eq:mass\]) is a number. $M$ is understood as an average of the mass operator over different spin states: $$M\rightarrow \frac{\bar{u}\hat{M} u}{\bar{u}u}\; .$$
However we are usually interested in finding the probability $P$ of emitting a gluon with a given energy $\omega$. In such a case the radiation power for a given energy is related to ${\rm Im}\; M$ by the integral equation $$-\frac{1}{E}{\rm Im}\; M=\int\frac{d\omega}{\omega}P(\omega)\; .$$ Hence to get $P(\omega)$ we have to write $M$ in an integral form as we will see later.
Effective propagators
=====================
Having a strong magnetic field, it is necessary to evaluate the mass operator to all orders in the field $H$. We consider an “external” (classical) chromomagnetic field along the longitudinal direction (the third axis or the $z$ axis): $$H_\mu^a=\delta_{\mu 3}\delta^{ai} h$$ where the color index $i$ is either $3$ or $8$. This field will lead to gauge field $A$ that is linearly dependent on position.
It is necessary to use effective quark and gluon propagators in the presence of $H$.
The quark propagator in a field $H$ can be written in a Fock-Schwinger proper time form as $$S(x,x^\prime)=\phi_q(x,x^\prime) \int \frac{d^4p}{(2\pi)^4}\exp[ip(x-x^\prime)]S(p)$$ where, the non-translation invariant, gauge dependent factor $\phi$ is a Bohm-Aharonov-like phase $$\phi_q(x,x^\prime)=\exp[igq\int_x^{x^\prime} d\chi_\mu A^\mu(\chi)]$$ The “momentum” space quark propagator is $$\begin{aligned}
S(p)=i\int_0^\infty \frac{ds_2}{\cos z} \exp\{-is_2[m^2-p_\parallel^2+\frac{\tan z}{z}p_\perp^2 ]\}
\nonumber\\
\times\{m e^{iz\Sigma_3}+e^{iz\Sigma_3}(\vec{\gamma}\cdot p)_\parallel-\frac{1}{\cos z}(\gamma\cdot p)_\parallel\}\end{aligned}$$ We have explicitly used the longitudinal and transverse decomposition appropriate for the choice of the field: $(\vec{a}\cdot \vec{b})_\perp=a_1b_1+a_2b_2$, $(a\cdot b)_\parallel=a_ob_o-a_3b_3$. The parameter $z$ depends on the color charge of the considered quark $z=gqhs_2$. The matrix $\Sigma_3$ is the usual ($z$-)spin-projection. Graphically, in Feynman diagrams, the effective quark will have a blob to distinguish it from bare propagators.
The nonzero coupling between the gluon and the background leads to a modification of the bare propagator. The gluon effective propagator can be written in a form similar to the quark propagator $$G_{\mu\nu}(x,x^\prime)=\phi_{_{Q}}(x,x^\prime) \int \frac{d^4p}{(2\pi)^4}\exp[ip(x-x^\prime)]G_{\mu\nu}(p)$$ where the gluon phase factor is again $$\phi_{_{Q}}(x,x^\prime)=\exp[igQ\int_x^{x^\prime} d\chi_\mu A^\mu(\chi)]$$ and the “momentum” representation gluon propagator is $$G_{\mu\nu}(p)=i\int_0^\infty \frac{ds_1}{\cos y} \exp\{-is_1[-p_\parallel^2+\frac{\tan y}{y}p_\perp^2 ]\}E_{\mu\nu}$$ where $y=gQhs_1$, and the tensor $E_{\mu\nu}$ can be written (formally) in a compact form in terms of the tensor $F_{\mu\nu}$ as $$E_{\mu\nu}=[\exp(2gQFs_1)]_{\mu\nu}$$ A more practical form is $$E_{\mu\nu}=g_{\mu\nu}^\parallel+g_{\mu\nu}^\perp\cos(2y)-A_{\mu\nu}\sin(2y)\; .$$ The metric tensors $g_{\mu\nu}^{\parallel,\perp}$ stand for the longitudinal and transverse spaces metric. The antisymmetric tensor $A$ is $$A_{\mu\nu}=\delta_{\mu 1}\delta_{\nu 2}-\delta_{\mu 2}\delta_{\nu 1}\; .$$ It is easy to verify that in the limit of the vanishing magnetic field $E_{\mu\nu}\rightarrow g_{\mu\nu}$.
One-loop contribution
=====================
The order $\alpha_s$ contribution to the ”mass operator” is given by the diagram shown in Fig. \[fig:1loop\]. The coordinate space $M$ is given then by $$M(x,x^\prime)=\phi_q(x,x^\prime)\int\frac{d^4p}{(2\pi)^4}\exp[ip(x-x^\prime)]M(p)$$ where we have used the relation between the external quark charge $q$ and the the charges of the internal quark $q^\prime$ and the gluon $Q$: $q=q^\prime+Q$ to write $$\phi_q(x,x^\prime)=\phi_{q^\prime}(x,x^\prime)\phi_{_{Q}}(x,x^\prime)\; .$$ The “momentum” representation mass operator is $$M(p)=-g^2(T^A)_{ik}(T^A)_{ki}\int \frac{d^4k}{(2\pi)^4} G_{\mu\nu}(k)\gamma^\mu S(p-k)\gamma^\nu\; .$$ To get the un-integrated emission probability $P(\omega)$, we use the insertion used in [@SchwinET]: $$1=\int\limits_{-\infty}^\infty d\omega\int\limits_{-\infty}^\infty\frac{d\lambda}{2\pi}\exp\{-i\lambda(\omega-k_o)\}\; .$$ From now on our procedure will follow the steps of [@SchwinET]. To avoid hindering the main result of this paper by technical details we present the main steps and most of the derivation Appendix \[app: a\].
The exact expression of the one-loop momentum representation $M(p)$, after $k$ and $\lambda$ integration, is $$\begin{aligned}
M(p)=-2\frac{g^2}{(4\pi)^2}(T^A)_{ik}(T^A)_{ki}\int\limits_{-\infty}^{\infty} d\omega\int\limits_0^\infty ds \left(\frac{i}{\pi s}\right)^{\frac{1}{2}}\nonumber\\
\int\limits_0^1\frac{du}{\delta^\prime\cos y\cos z}\exp(is\Phi)\left\{ me^{iz\Sigma_3}+\frac{(1-u)\tan y}{y\delta^\prime \cos z} (\vec{\gamma}\cdot\vec{p})_\perp\right.\nonumber\\
\left.\hspace{-2mm}+\left[m-(p_o-\omega)\gamma_o+p_z(1-u)\gamma_3\right]e^{-i(z-2y)\Sigma_3}\right\}
\label{eq: M}\end{aligned}$$ Few definitions are made: $y=(1-u)sgQh$; $z=usgq^\prime h$ $$\delta^\prime = (1-u)\frac{\tan y}{y}+u\frac{\tan z}{z}$$ and the phase factor $$\Phi=-m^2+u(1-u)p_\parallel^2 -u(1-u)\frac{\tan z}{\delta^\prime z}\frac{\tan y}{y}+(up_o-\omega)^2\; .$$ Note that $M(p)$ at this stage is still in a matrix form.
Fourier transform: dispersion relation
======================================
The expression of $M(p)$ found in the previous section \[Eq. (\[eq: M\])\] has to be Fourier transformed and then multiplied by the gauge field depending phase $\phi_q(x,x^\prime)$, with the external quark charge $q$, to give $M(x,x^\prime)$. This can be done using the relations derived by Tsai in [@Tsai]. For instance $$\begin{aligned}
\phi(x,x^\prime)\int\frac{d^4 p}{(2\pi)^4} e^{ip(x-x^\prime)}e^{iA_op_o^2}e^{iA_3p_3^2}e^{-iBp_\perp^2}\nonumber\\
=\cos Z\langle x|e^{iA_o\Pi_o^2}e^{iA_3\Pi_3^2}e^{i{\cal S}\Pi_\perp^2}|x^\prime\rangle
\label{eq: tsai}\end{aligned}$$ The $\Pi$’s are the generalized momentum operators in the presence of a magnetic field. This relation can be easily derived, as shown in Appendix \[app: b\], if one notices that the right-hand side can be decomposed into three quantum non-relativistic propagators: a free particle of some fictive mass related to $A_o$ living in the “$o$” one-dimensional space, a free particle living in the “$3$” one dimensional space, and finally a particle living in the transverse plane and under the action of an external magnetic field $H$ perpendicular to the plane. The three propagators are exactly known which give the above relation with $Z=gq{\cal S}$ and $${\cal S}=\frac{1}{gqh}\tan^{-1}(gqhB)$$ Similar relations are used for the product of exponential factors and $p_\mu$ to the left of Eq. (\[eq: tsai\]). The mass operator can then be obtained. This mass operator will be acting on physical quark states satisfying $(\Pi_\mu\gamma^\mu+m)\psi=0$. Hence it is possible to do the following replacements $$\hat{\Pi}_\perp^2 \rightarrow \hat{\Pi}_\parallel^2+gqh\Sigma_3-m^2 \; ;\quad \hat{\Pi}_\parallel\rightarrow p_\parallel;,$$ where $p_\parallel$ now represents the eigenvalues of $\hat{\Pi}_\parallel$ which is not affected by the longitudinal chromomagnetic field.
Applying Tsai’s relations and the external quark equation of motion to Eq. (\[eq: M\]) the mass operator will be $$\begin{aligned}
\hat{M}=-2\frac{g^2}{(4\pi)^2}(T^A)_{ik}(T^A)_{ki}\int d\omega\int ds \left(\frac{i}{\pi s}\right)^{\frac{1}{2}}\nonumber\\
\int du\frac{\cos Z}{\delta^\prime\cos y\cos z}\exp(is\Phi^\prime)
\left\{ me^{i(z-Z)\Sigma_3} \right.\nonumber\\
\left. +\cos Z\frac{(1-u)\tan y}{y\delta^\prime \cos z} [(\gamma\cdot p)_\parallel-m] \right. \nonumber \\
\left.\!\!\!\!\!\!\!\!+\left[m-(p_o-\omega)\gamma_o+p_z(1-u)\gamma_3\right]e^{-i(z+Z-2y)\Sigma_3}\right\}
\label{eq: Mx}\end{aligned}$$ where $Z=gq{\cal S}$, and $${\cal S}=\frac{1}{gqh}\tan^{-1}\left[su(1-u)\frac{\tan z}{\delta^\prime}\frac{\tan y}{y}\right]$$ defined as mentioned before from the Fourier transform. The new phase factor, which is a scalar now, is $$\Phi^\prime=u(p_o-\omega)^2-(1-u)(up_3-\omega^2)-\frac{\cal S}{s}(p_o^2-p_3^2-m^2)\; .$$
Gamma substitutions
===================
The mass operator has a spinorial/matrix structure. It should be mentioned that we have used the exact dispersion relation for the replacement of $\Pi_\perp$ in the phase factor, without solving the equation of motion. However, the Dirac structure in front of the exponential factor in (\[eq: Mx\]) depends on the gamma matrices in a nontrivial way. We use the same approximation done in the literature, which is the weak point of the method: the spinor structure of the incident quark is the same as that of the free particle, [*i.e.*]{}, the Dirac spinor $u_s(p)$. So sandwiching the mass operator between $u$ and $\bar{u}$, the gamma matrices are replaced by scalar quantities. Simple matrix calculation leads to the following replacement rules: $\gamma_o\rightarrow p_o/m$, $\gamma_3\rightarrow p_3/m$, $\Sigma_3\rightarrow \eta \frac{1}{m}\left(p_o -\frac{p_z^2}{p_o+m}\right)$; $\gamma_o\Sigma_3\rightarrow\eta(1+\frac{p_3^2}{m(p_o+m)})$; $\gamma_3\Sigma_3\rightarrow\eta\frac{p_3}{m}$.
The $\eta=\pm$ are for the different helicities of the quark. The gamma replacement leads to a scalar mass operator that can be used to obtain $P(\omega)$, $$\begin{aligned}
M=-2\frac{g^2}{(4\pi)^2}(T^A)_{ik}(T^A)_{ki}\int d\omega\int ds \left(\frac{i}{\pi s}\right)^{\frac{1}{2}}\int du\nonumber\\
\frac{\cos Z}{\delta^\prime\cos y\cos z}\exp(is\Phi^\prime)\left\{ m(\cos(z-Z)+\cos(Z+z-2y))\right.\nonumber \\
-i\eta\left(p_o-\frac{p_z^2}{p_o+m}\right)(\sin(Z-z)+\sin(Z+z-2y)) \nonumber\\
+\cos Z\frac{(1-u)\tan y}{y\delta^\prime \cos z}\left(\frac{p_o^2-p_3^2-m^2}{m}\right)
\nonumber\\
-(p_o-\omega)\frac{p_o}{m}\cos(Z+z-2y)+\frac{p_z^2}{m}(1-u)\cos(Z+z-2y)\nonumber\\
+i\eta(p_o-\omega)\left(1+\frac{p_3^2}{m(p_o+m)}\right)\sin(Z+z-2y)
\nonumber\\
\left.-i\eta (1-u)\frac{p_z^2}{m^2}\left(p_o-\frac{p_z^2}{p_o+m}\right)\sin(Z+z-2y)\right\}
\label{eq: Mxgamma}\end{aligned}$$ Note that this expression has helicity information that can be explored in a way similar to [@Tuchin].
The Abelian limit
-----------------
The results found in [@SchwinET] can be easily obtained from our expression in Eq. (\[eq: Mxgamma\]). The non-Abelian character can be waived if we set the gluon charge $Q$ to zero hence $y=0$; remove the color factor $T^AT^A$ in $M$; and consider the motion of the incident quark in the transverse plane, [*i.e.*]{}, take $p_3=0$. Besides, we have to set the index of refraction $n=1$ in [@SchwinET]; hence the parameters $\delta$, $E$, $\widetilde\beta$, $\delta^\prime$ of [@SchwinET] will be in our notation (for $y=0=p_3$) $\delta=1$, $E\rightarrow p_o$, $\delta^\prime\rightarrow \delta^\prime $, $\widetilde\beta\rightarrow Z$.
The above simplifications lead to $$\begin{aligned}
M_{abelian}=2\frac{g^2}{(4\pi)^2}\int d\omega\int ds \left(\frac{i}{\pi s}\right)^{\frac{1}{2}}\int du\exp(is\Phi^\prime)\frac{m}{\Delta^{1/2}}\nonumber\\
\left\{ \cos(Z-z) \right.
-i\eta\frac{p_o}{m}\sin(Z-z)\nonumber\\
+\frac{\omega}{m}\frac{p_o}{m}\cos(Z+z) -i\eta\frac{\omega}{m}\sin(Z+z)\nonumber\\
\left. +\frac{p_o^2-m^2}{m^2}\left[\frac{1-u}{\Delta^{1/2}}-\cos(Z+z)\right]\!\!\right\}
\label{eq: Schwin}\end{aligned}$$ whereas in [@SchwinET] $\Delta^{1/2}=(\delta^\prime\cos z)/\cos Z$. Eq. (\[eq: Schwin\]) coincides with equation (3.39 a) of [@SchwinET].
Small $s$ limit
---------------
By analyzing Eq. (\[eq: Mxgamma\]) it is clear that the $s$-integration is dominated by small a $s$ region. The small $s$ expansion of the phase factor $\Phi^\prime$ gives $$\begin{aligned}
\Phi^\prime \approx (up_o-\omega)^2 - m^2u^2 \nonumber\\
-\left[gq^\prime u h
-gQh(1-u)\right]^2\frac{s^2}{3}\left[u^2(1-u)^2(p_o^2-p_3^2)-m^2\right]\end{aligned}$$ The $u$-integration can then be approximated by a Gaussian integration: $$\int_0^1 du f(u)e^{is(up_o-u)^2}\approx f(u\approx \frac{\omega}{p_o})\frac{1}{2ip_o}\sqrt{\frac{\pi}{is}}$$ In this approximation the variable $u$ is replaced by the gluon energy fraction $x=\omega/p_o$. After the $u$-integration the phase factor will be $$s\Phi^\prime \rightarrow -m^2sx^2-\frac{\mathbf{f}^2}{3}s^3 \left[x^2(1-x)^2(p_o^2-p_3^2)-m^2\right]
\label{eq: phase}$$ where we have defined the vector $$\mathbf{f}=gq^\prime x\mathbf{H}-gQ (1-x)\mathbf{H}$$ which represents a fictive magnetic force on an intermediate gluon-quark system.
Zakharov method
---------------
Before proceeding to an explicit comparison of our result with the one found in [@Zak1]; we present Zakharov’s method in its simplest “display.”
Consider first a fast scalar particle whose first quantized wave function satisfying the Klein-Gordon equation $$(\frac{\partial^2}{\partial t^2}-\frac{\partial^2}{\partial\mathbf{r}^2})\psi=m^2\psi\; .$$ For a fast particle moving along the $z$-axis with large energy $E$, the wave function can be written as $$\psi(t,\mathbf{r})=\exp\left(-iE\xi\right)\phi(z,\xi,\mathbf{\rho})$$ where $\xi=t-z$, $\mathbf{r}=(z,\mathbf{\rho})$. For fixed $\xi$ the “transverse” wave function $\phi$ satisfies the equation $$i\frac{\partial\phi}{\partial z}=\left(\frac{\mathbf{p}_\perp^2+m^2}{2E}\right)\phi+\frac{\partial^2\phi}{\partial z^2}$$
Besides, in the low mass limit or for $m\ll E$ the above equation simplifies to $$i\frac{\partial\phi}{\partial z}=\left(\frac{\mathbf{p}_\perp^2+m^2}{2E}\right)\phi\; .$$ Which is a two-dimensional Schrödinger equation with a mass $E$. The solution to the above equation can be easily obtained, $$\phi_{p_z,p_\perp}(z,{\bf \rho})=e^{i{\mathbf{p}_\perp}\cdot{\bf \rho}}e^{-i\int\limits_0^z\left(\frac{\mathbf{p}_\perp^2+m^2}{2p_z}\right)dz}\; ,$$ which can be interpreted as a plane wave. The $\xi$ dependence emerges only via boundary conditions for the transverse wave function. So we have an evolution equation along each line $\xi={\rm const}$.
If we consider now a parton of color charge $Q$ in an external field $H$, represented by a gauge field $A_\mu$, the equation of motion becomes $$i\frac{\partial\phi}{\partial z}=\left\{\frac{(\mathbf{p}-gQ\mathbf{A})_\perp^2+m^2}{2E}+gQ(A^o-A^3)\right\}\phi\; .$$ The field configuration considered in [@Zak1] is such that $\vec{A}=0=A^o$ and $A_3=[\mathbf{H}\times \mathbf{\rho}]_3$; hence the magnetic field is taken to be in the transverse plane. The wave function is then simply found to be $$\phi_{p_z,p_\perp}(z,{\bf \rho})=e^{i{\vec{p}_\perp}\cdot{\bf \rho}}e^{-i\int\limits_0^z\left(\frac{\mathbf{p}_\perp^2+m^2}{2p_z}\right)dz}\; ,$$ But now the transverse momentum is a solution of the “classical equation of motion” $$\frac{d\mathbf{p}_\perp}{dz}=\mathbf{F}\; .$$ This is applied to the incident quark as well as to the outgoing gluon and quark [^1].
The final result of [@Zak1] can be written as $$\begin{aligned}
\frac{dP}{dLdx}=\frac{i\mu}{2\pi}\int\limits_\infty^{-\infty}\frac{d\tau}{\tau}\left[\frac{g_1}{\mu^2}\left(\epsilon^2+\frac{\mathbf{f^2\tau^2}}{2}\right)-g_2\right]\nonumber\\
\exp\left\{-i\left[\frac{\epsilon^2\tau}{2\mu}+\frac{\mathbf{f}^2\tau^3}{24\mu}\right]\right\}
\label{eq: Zak}\end{aligned}$$ where $\mu=Ex(1-x)$ (an effective mass), $g_1=C\alpha_s(1-x+x^2)/x$ (non-spin-flip vertex factor), $g_2=C\alpha_sm_q^2x^3/2\mu^2$ (spin-flip vertex factor), $\epsilon=m_q^2x^2+m_g^2(1-x)^2$, and as in our approach the fictive force of a quark-gluon system is $\mathbf{f}=x\mathbf{F}_{q^\prime}-(1-x)\mathbf{F}_g$.
Comparing a particle propagation in a longitudinal field to that of a transverse field configuration is not intuitive. However, in an infinite (nonrealistic) medium, and for a static (non-propagating) magnetic field, and with a vanishing electric field, it is the propagation direction of the incident parton which gives the meaning of the longitudinal/transverse directions. So if in our approach we take $p_z=0$ we will be the closest to the configuration of [@Zak1], where the transverse direction is mapped into the longitudinal direction in the following ways:
- Present configuration: a particle moving in the transverse plane under the action of a longitudinal field.
- Zakharov’s configuration: a fast particle moving along the longitudinal direction and a transverse magnetic field.
It is closest but not the same, since in [@Zak1], transverse momenta are not set to zero.
So setting $p_z=0$, $p_o=E$ and $ s=\tau/(2\mu)$ in Eq. (\[eq: phase\]) we obtain the same phase given in Eq. (\[eq: Zak\]) for a zero gluon mass. However, not surprisingly, the prefactor of our approximated expression does not match that of Eq. (\[eq: Zak\]).
Shuryak et al
-------------
It is now clear that our agreement with [@Zak1] leads to the conclusion that the results of [@ShuryaZ] are ”defective". As the author of [@ShuryaZ] did not give enough details about their approximation method it is hard to trace the exact source of inconsistency in [@ShuryaZ]. However we expect that their asymmetric treatment of the quark and gluon propagators leads to a nonsystematic approximation/expansion. For instance, the important Bohm-Aharonov phase, which plays an important role in our derivation, was absorbed in an unjustified way. Consequently the fictive force $\mathbf{f}$ was badly approximated.
Conclusions
===========
The generalization of QED synchrotron radiation to the QCD case was done with the minimal possible complications. It is shown that the QED case can be obtained as a special limit. The comparison with the path integral approach of Zakharov has shown consistent results. This will hopefully “solve” the debate about gluon synchrotron radiation.
It is possible now to extend our method to include nonzero gluon mass (thermal mass) and combinations of longitudinal and transverse chromoelectric and chromomagnetic fields with an arbitrary color index. This is under current investigation.
Acknowledgments {#acknowledgments .unnumbered}
===============
The work of H.Z. was supported by the Lebanese University. He would also like to thank his colleagues at the Laboratoire d’Annecy le Vieux de Physique Théorique for their hospitality. This paper was initially planned and discussed with P. Aurenche, we would like to thank him sincerely. We would also like to thank B.G. Zakharov for useful discussions.
Schwinger et al procedure {#app: a}
=========================
Our derivation follows the recipe of [@SchwinET]. We present in what follows the main steps needed to get equation (\[eq: M\]).
The first step to follow is the change of variables: $s_1=(1-u)s$, $s_2=us$, where $0\le u\le 1$, and $s$ is positive. The $\lambda$ and $k$ integration is based on Gaussian integration and saddle point approximation. Hence the exponential factor is factored out in the expression of $M$; this can be written as $$M(p)=\int\limits_0^1 du \int d^4k\int\limits_{-\infty}^\infty d\lambda d\omega \int\limits_0^\infty ds M_{_{\rm Dirac}} \exp \left(is(\chi -\frac{\lambda \omega}{s})\right)$$ where $M_{_{\rm Dirac}}$ contains the Dirac structure of the self-energy diagram and all the non-exponential terms. The function $\chi$ can be split into a momentum dependent part $\chi_1$ and a $\lambda$ dependent function $\chi_o$, $\chi=\chi_o+\chi_1$, such that $$\chi_1=\left(k_o-(up_o-\frac{\lambda}{2s})\right)^2-(k_3-up_3)^2-\delta^\prime\left(\vec{k}_{\perp}-\frac{u\tan z}{z \delta^\prime}\vec{p}_{\perp}\right)^2$$ while $$\chi_o=-\frac{1}{s^2}\left(\frac{\lambda^2}{4}-up_o\lambda s\right)+\Phi$$ where $\delta^\prime$ and $\Phi$ are as defined before.
We perform now the $k$ integration using the Gaussian form(s) $$\begin{aligned}
&&\int \frac{d^4k}{(2\pi)^4}\exp(is\chi_1)\left[1,k_o,k_3,\vec{k}_\perp\right]=\nonumber\\
&&\frac{i}{\delta^\prime(4\pi)^2s^2}\left[1,up_o-\frac{\lambda}{2s},up_3,\frac{u\tan z}{\delta^\prime z}\vec{p}_{\perp}\right]\end{aligned}$$ For the $\lambda$ integration $$\begin{aligned}
&&\int \frac{d\lambda}{2\pi} \exp(is\chi_o-i\lambda \omega)[1,\lambda]=\nonumber\\
&&\left(\frac{s}{i\pi}\right)^{1/2}e^{is(\Phi+(up_o-\omega)^2)}[1,2s(up_o-\omega)]\end{aligned}$$ The remaining, lengthy but straightforward, step needed to get equation (\[eq: M\]) is the Dirac structure simplification/contractions.
Tsai transformation to coordinate space {#app: b}
=======================================
We give a simple derivation for equation (\[eq: tsai\]) which is different from the initial method of Tsai [@Tsai].
Consider a non-relativistic particle of mass $m$ and charge $g$ in a uniform magnetic field $H$. The general expression of the non-relativistic propagator in coordinate space is $$\begin{aligned}
K(\vec{r},\tau,\vec{r}_o,0)=\langle \vec{r}|\exp\left(-i{\cal H}\frac{\tau}{\hbar}\right)|\vec{r}_o\rangle\nonumber\\
=\left(\frac{m}{2i\pi\hbar s}\right)^{3/2}\frac{gH\tau/2m}{\sin(gH\tau)/2m}\exp\left(\frac{iS_{\rm cl}}{\hbar}\right)\end{aligned}$$ where ${\cal H}$ is the particle Hamiltonian $${\cal H}= \frac{\Pi_\perp^2}{2m}-\frac{\Pi_\parallel^2}{2m}$$ with $\Pi$ the generalized momentum. The classical action $S_{\rm cl}$ in the presence of a magnetic field is $$\begin{aligned}
S_{\rm cl}=\frac{gH}{2}\{\frac{1}{2}\left[(x-x_o)^2+(y-y_o)^2\right]\cot(gH\tau/2m)\nonumber\\
+(x_oy-y_ox)\}+\frac{m}{2\tau}(z-z_o)^2\end{aligned}$$ The non-translational invariant term $(x_oy-y_ox)$ in the action gives the Bohm-Aharonov phase, if we assume a straight line trajectory. Hence the four-dimensional generalization will be $$\begin{aligned}
K(x^\mu,\tau,x^{\prime\mu},0)=\Phi(x,x^\prime)\left(\frac{1}{4\pi\tau}\right)^2\frac{z}{\sin z}\nonumber\\
\exp\left(igH\cot z\frac{(\vec{x}_\perp-\vec{x}^\prime_\perp)^2}{4}\right) \exp\left(-i\frac{(x_\parallel-x_\parallel^\prime)^2}{4\tau}\right)\end{aligned}$$ where $z=gH\tau$. It is now possible to write the propagator $K$ in momentum space. We use the Fourier transform $$\begin{aligned}
\frac{z\pi}{i\tau\sin z}\exp\left(igH\cot z\frac{(\vec{x}_\perp-\vec{x}^\prime_\perp)^2}{4}\right)=\nonumber\\
\int\frac{d^2p_\perp}{\cos z}\exp\left(-i\vec{p}_\perp\cdot(\vec{x}_\perp-\vec{x}_\perp^\prime)\right)\exp\left(-i\tau \frac{\tan z}{z}p_\perp^2\right)\end{aligned}$$ This is similarly done for the longitudinal part ($\parallel$) of the propagator. Hence $$\begin{aligned}
K(x^\mu,\tau,x^{\prime\mu},0)=\langle x|\exp\left(i\tau \Pi^2\right)|x^\prime\rangle\nonumber\\
=\Phi(x,x^\prime)\int\frac{d^4P}{(2\pi)^4}e^{iP\cdot(x-x^\prime)}\frac{e^{i\tau p_\parallel^2}}{\cos z}e^{-i\tau\frac{\tan z}{z}p_\perp^2}\end{aligned}$$ where we have set $m=1/2$ and $\hbar=1$ in the nonrelativistic formula. This Formula is the same as equation (\[eq: tsai\]) with appropriate relabeling of field and time parameters.
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[^1]: If we wish to study the longitudinal magnetic field we have to consider keeping $\mathbf{A}_\perp$ which will introduce quadratic terms in the two-dimensional Hamiltonian which will complicate the procedure.
|
---
abstract: |
The accelerated growth of mobile trajectories in location-based services brings valuable data resources to understand users’ moving behaviors. Apart from recording the trajectory data, another major characteristic of these location-based services is that they also allow the users to connect whomever they like or are interested in. A combination of social networking and location-based services is called as location-based social networks (LBSN). As shown in [@cho2013socially], locations that are frequently visited by socially-related persons tend to be correlated, which indicates the close association between social connections and trajectory behaviors of users in LBSNs. In order to better analyze and mine LBSN data, we need to have a comprehensive view to analyze and mine the information from the two aspects, [*i.e.,*]{}the social network and mobile trajectory data.
Specifically, we present a novel neural network model which can jointly model both social networks and mobile trajectories. Our model consists of two components: the construction of social networks and the generation of mobile trajectories. First we adopt a network embedding method for the construction of social networks: a networking representation can be derived for a user. The key of our model lies in the component of generating mobile trajectories. Secondly, we consider four factors that influence the generation process of mobile trajectories, namely user visit preference, influence of friends, short-term sequential contexts and long-term sequential contexts. To characterize the last two contexts, we employ the RNN and GRU models to capture the sequential relatedness in mobile trajectories at different levels, i.e., short term or long term. Finally, the two components are tied by sharing the user network representations. Experimental results on two important applications demonstrate the effectiveness of our model. Especially, the improvement over baselines is more significant when either network structure or trajectory data is sparse.
author:
- 'Cheng Yang$^\dag$ Maosong Sun Wayne Xin Zhao$^{*}$ Zhiyuan Liu$^{*}$ Edward Y.Chang'
bibliography:
- 'main.bib'
title: A Neural Network Approach to Joint Modeling Social Networks and Mobile Trajectories
---
<ccs2012> <concept> <concept\_id>10002951.10003227.10003236.10003101</concept\_id> <concept\_desc>Information systems Location based services</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10002951.10003260.10003261.10003270</concept\_id> <concept\_desc>Information systems Social recommendation</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10003120.10003130.10003131.10003292</concept\_id> <concept\_desc>Human-centered computing Social networks</concept\_desc> <concept\_significance>300</concept\_significance> </concept> <concept> <concept\_id>10010147.10010257.10010293.10010294</concept\_id> <concept\_desc>Computing methodologies Neural networks</concept\_desc> <concept\_significance>100</concept\_significance> </concept> </ccs2012>
$^\dag$ This work was done during the first author’s internship at HTC Beijing Research.
$^{*}$ Corresponding authors
This work was supported by the 973 Program (No. 2014CB340501), the Major Project of the National Social Science Foundation of China (13&ZD190), the National Natural Science Foundation of China (NSFC No.61502502, 61572273 and 61532010), Tsinghua University Initiative Scientific Research Program (20151080406) and Beijing Natural Science Foundation under the grant number 4162032.
Authors¡¯ addresses: Cheng Yang, Maosong Sun and Zhiyuan Liu, Department of Computer Science and Technology, Tsinghua University, Beijing 100084; emails: [email protected], [email protected], [email protected]; W.X.Zhao, School of Information & Beijing Key Laboratory of Big Data Management and Analysis Methods, Renmin University of China, Beijing 100872; email: [email protected]; E.Y.Chang, HTC Research & Innovation, Palo Alto, CA 94306; email: [email protected].
Introduction
============
In recent years, mobile devices ([*e.g.,*]{}smartphones and tablets) are widely used almost everywhere. With the innovation and development on Internet technology, mobile devices have become an essential connection to the broader world of online information for users. In daily life, a user can utilize her smartphone for conducting many life activities, including researching a travel plan, accessing online education, and looking for a job. The accelerated growth of mobile usage brings a unique opportunity to data mining research communities. Among these rich mobile data, an important kind of data resource is the huge amount of mobile trajectory data obtained from GPS sensors on mobile devices. These sensor footprints provide a valuable information resource to discover users’ trajectory patterns and understand their moving behaviors. Several location-based sharing services have emerged and received much attention, such as *Gowalla*[^1] and *Brightkite*[^2].
Apart from recording user trajectory data, another major feature of these location-based services is that they also allow the users to connect whomever they like or are interested in. For example, with *Brightkite* you can track on your friends or any other Brightkite users nearby using the phone’s built in GPS. A combination of social networking and location-based services has lead to a specific style of social networks, termed as *location-based social networks (LBSN)* [@cho2011friendship; @bao2012location; @zheng2015trajectory]. We present an illustrative example for LBSNs in Fig. \[fig:exp\], and it can been seen that LBSNs usually include both the social network and mobile trajectory data. Recent literature has shown that social link information is useful to improve existing recommendation tasks [@machanavajjhala2011personalized; @yuan2014generative; @ma2014measuring]. Intuitively, users that often visit the same or similar locations are likely to be social friends[^3] and social friends are likely to visit same or similar locations. Specially, several studies have found that there exists close association between social connections and trajectory behaviors of users in LBSNs. On one hand, as shown in [@cho2013socially], locations that are frequently visited by socially-related persons tend to be correlated. On the other hand, trajectory similarity can be utilized to infer social strength between users [@pham2013ebm; @zheng2010geolife; @zheng2011recommending]. Therefore we need to develop a comprehensive view to analyze and mine the information from the two aspects. In this paper, our focus is to develop a joint approach to model LBSN data by characterizing both the social network and mobile trajectory data.
*In the first aspect,* social network analysis has attracted increasing attention during the past decade. It characterizes network structures in terms of nodes (individual actors, people, or things within the network) and the ties or edges (relationships or interactions) that connect them. A variety of applications have been developed on social networks, including network classification [@sen2008collective], link prediction [@liben2007link], anomaly detection [@chandola2009anomaly] and community detection [@fortunato2010community]. A fundamental issue is how to represent network nodes. Recently, networking embedding models [@Perozzi:2014:DOL:2623330.2623732] have been proposed in order to solve the data sparsity in networks. *In the second aspect*, location-based services provide a convenient way for users to record their trajectory information, usually called *check-in*. Independent of social networking analysis, many studies have been constructed to improve the location-based services. A typical application task is the location recommendation, which aims to infer users’ visit preference and make meaningful recommendations for users to visit. It can be divided into three different settings: general location recommendation [@zheng2009mining; @cheng2012fused; @ye2011exploiting], time-aware location recommendation [@yuan2013time; @yuan2014graph; @liu2016predicting] and next-location recommendation[@cheng2013you; @ye2013s; @zhang2014lore]. General location recommendation will generate an overall recommendation list of locations for a users to visit; while time-aware or next location recommendation further imposes the temporal constraint on the recommendation task by either specifying the time period or producing sequential predictions.
These two aspects capture different data characteristics on LBSNs and tend to be correlated with each other [@cheng2012fused; @levandoski2012lars]. To conduct better and more effective data analysis and mining studies, there is a need to develop a joint model by capturing both network structure and trajectory behaviors on LBSNs. However, such a task is challenging. Social networks and mobile trajectories are heterogeneous data types. A social network is typically characterized by a graph, while a trajectory is usually modelled as a sequence of check-in records. A commonly used way to incorporate social connections into an application system ([*e.g.,*]{}recommender systems) is to adopt the regularization techniques by assuming that the links convey the user similarity. In this way, the social connections are exploited as the side information but not characterized by a joint data model, and the model performance highly rely on the “homophily principle" of *like associates with like*. In this paper, we take the initiative to jointly model social networks and mobile trajectories using a neural network approach. Our approach is inspired by the recent progress on deep learning. Compared with other methods, neural network models can serve as an effective and general function approximation mechanism that is able to capture complicated data characteristics [@mittal2016survey]. In specific, recent studies have shown the superiority of neural network models on network and sequential data. First, several pioneering studies try to embed vertices of a network into low-dimensional vector spaces [@tang2011leveraging; @Perozzi:2014:DOL:2623330.2623732; @tang2015line], called *networking embedding*. With such a low-dimensional dense vector, it can alleviate the data sparsity that a sparse network representation suffers from. Second, neural network models are powerful computational data models that are able to capture and represent complex input/output relationships. Especially, several neural network models for processing sequential data have been proposed, such as recurrent neural networks (RNN) [@mikolov2010recurrent]. RNN and its variants including LSTM and GRU have shown good performance in many applications. By combining the merits from both network embedding and sequential modelling from deep learning, we present a novel neural network model which can jointly model both social networks and mobile trajectories. In specific, our model consists of two components: the construction of social networks and the generation of mobile trajectories. We first adopt a network embedding method for the construction of social networks: a networking representation can be derived for a user. The key of our model lies in the component generating mobile trajectories. We have considered four factors that influence the generation process of mobile trajectories, namely user visit preference, influence of friends, short-term sequential contexts and long-term sequential contexts. The first two factors are mainly related to the users themselves, while the last factors mainly reflect the sequential characteristics of historical trajectories. We set two different user representations to model the first two factors: a visit interest representation and a network representation. To characterize the last two contexts, we employ the RNN and GRU models to capture the sequential relatedness in mobile trajectories at different levels, [*i.e.,*]{}short term or long term. Finally, the two components are tied by sharing the user network representations: the information from the network structure is encoded in the user networking representation, which is subsequently utilized in the generation process of mobile trajectories.
To demonstrate the effectiveness of the proposed model, we evaluate our model using real-world datasets on two important LBSN applications, namely *next-location recommendation* and *friend recommendation*. For the first task, the trajectory data is the major information signal while network structure serves as auxiliary data. Our method consistently outperforms several competitive baselines. Interestingly, we have found that for users with little check-in data, the auxiliary data ([*i.e.,*]{}network structure) becomes more important to consider. For the second task, the network data is the major information signal while trajectory data serves as auxiliary data. The finding is similar to that in the first task: our method still performs best, especially for those users with few friend links. Experimental results on the two important applications demonstrate the effectiveness of our model. In our approach, network structure and trajectory information complement each other. Hence, the improvement over baselines is more significant when either network structure or trajectory data is sparse.
Our contributions are three-fold summarized below:
- We proposed a novel neural network model to jointly characterize social network structure and users’ trajectory behaviors. In our approach, network structure and trajectory information complement each other. It provides a promising way to characterize heterogeneous data types in LBSNs.
- Our model considered four factors in the generation of mobile trajectories, including user visit preference, influence of friends, short-term sequential contexts and long-term sequential contexts. The first two factors are modelled by two different embedding representations for users. The model further employed both RNN and GRU models to capture both short-term and long-term sequential contexts.
- Experimental results on two important applications demonstrated the effectiveness of our model. Interestingly, the improvement over baselines was more significant when either network structure or trajectory information was sparse.
The remainder of this paper is organized as follows. Section 2 reviews the related work and Section 3 presents the problem formulation. The proposed model together with the learning algorithm is given in Section 4. We present the experimental evaluation in Section 5. Section 6 concludes the paper and presents the future work.
Related Work
============
Our work is mainly related to distributed representation learning, social link prediction and location recommendation.
Distributed Representation Learning and Neural Network Models
-------------------------------------------------------------
Machine learning algorithms based on data representation learning make a great success in the past few years. Representations learning of the data can extract useful information for learning classifiers and other predictors. Distributed representation learning has been widely used in many machine learning tasks [@bengio2013representation], such as computer vision [@krizhevsky2012imagenet] and natural language processing [@mikolov2013distributed].
During the last decade, many works have also been proposed for network embedding learning [@chen2007directed; @tang2009relational; @tang2011leveraging; @Perozzi:2014:DOL:2623330.2623732]. Traditional network embedding learning algorithms learn vertex representations by computing eigenvectors of affinity matrices [@belkin2001laplacian; @yan2007graph; @tang2011leveraging]. For example, DGE [@chen2007directed] solves generalized eigenvector computation problem on combinational Laplacian matrix; SocioDim [@tang2011leveraging] computes $k$ smallest eigenvectors of normalized graph Laplacian matrix as $k$-dimensional vertex representations.
DeepWalk [@Perozzi:2014:DOL:2623330.2623732] adapts Skip-Gram [@mikolov2013distributed], a widely used language model in natural language processing area, for NRL on truncated random walks. DeepWalk which leverages deep learning technique for network analysis is much more efficient than traditional NRL algorithms and makes large-scale NRL possible. Following this line, LINE [@tang2015line] is a scalable network embedding algorithm which models the first-order and second-order proximities between vertices and GraRep [@cao2015grarep] characterizes local and global structural information for network embedding by computing SVD decomposition on $k$-step transition probability matrix. MMDW [@tumax] takes label information into account and learn semi-supervised network embeddings.
TADW [@yang2015network] and PTE [@tang2015pte] extend DeepWalk and LINE by incorporating text information into NRL respectively. TADW embeds text information into vertex representation by matrix factorization framework and PTE learns semi-supervised embeddings from heterogeneous text networks. However both TADW and PTE conduct experiments on document networks and fail to take sequential information between words into consideration.
Neural network models have achieved great success during the last decade. Two well-know neural network architectures are Convolutional Neural Network (CNN) and Recurrent Neural Network (RNN). CNN is used for extracting fix length representation from various size of data [@krizhevsky2012imagenet]. RNN and its variant GRU which aim at sequential modeling have been successfully applied in sentence modeling [@mikolov2010recurrent], speech signal modeling [@chung2014empirical] and sequential click prediction [@zhang2014sequential].
Social Link Prediction
----------------------
Social link prediction has been widely studied in various social networks by mining graph structure patterns such as triadic closure process [@romero2010directed] and user demographics [@huang2014mining]. In this paper, we mainly focus on the applications on trajectory data.
Researchers used to measure user similarity by evaluating sequential patterns. For example, they used a sequence of stay points to represent a user trajectory and evaluated user similarity by a sequence matching algorithm [@li2008mining]. In order to improve these methods, people also took pre-defined tags and weights into consideration to better characterize stay points [@xiao2010finding]. As LBSN becomes increasingly popular, trajectory similarity mining has attracted much more attention. A number of factors was considered to better characterize the similarity. As a result, physical distance [@cranshaw2010bridging], location category [@lee2011user], spatial or temporal co-location rate [@wang2011human] and co-occurrence with time and distance constraints [@pham2011towards] were proposed for social link prediction. The diversity of co-occurrence and popularity of locations [@pham2013ebm] were proved to be important features among all the factors. Using associated social ties to cluster locations, the social strength can be inferred in turn by extracted clusters shared by users [@cho2013socially; @zheng2011recommending].
Location Recommendation
-----------------------
Sequential check-in trajectory modeling originates from the sequential pattern modeling of transactional data [@agrawal1995mining; @han2001prefixspan; @bahar2003clospan; @chiu2004efficient]. Researchers take spatial and temporal pattern into consideration to cluster the locations [@giannotti2007trajectory]. Close locations which are visited by people in the same group are clustered into same regions [@gudmundsson2006computing; @jensen2007continuous; @jeung2008discovery].
One of the most important tasks on trajectory modeling is location recommendation. For general location recommendation, several kinds of side information are considered, such as geographical [@cheng2012fused; @ye2011exploiting], temporal [@zhao2016probabilistic] and social network information [@levandoski2012lars]. To address the data sparsity issue, content information including location category labels is also concerned [@yin2013lcars; @zhougeneral]. The location labels and tags can also be used in probabilistic model such as aggregate LDA [@gao2015content]. Textual information which includes text descriptions [@gao2015content; @li2010contextual; @zhao2015sar] are applied for location recommendation as well. $W^4$ employs tensor factorization on multi-dimensional collaborative recommendation for Who (user), What (location category), When(time) and Where (location) [@zheng2010collaborative; @bhargava2015and]. However, these methods which are mainly based on collaborate filtering, matrix factorization or LDA do not model the sequential information in the trajectory.
For time-aware location recommendation task which recommends locations at a specific time, it is also worth modeling the temporal effect. Collaborate filtering based method [@yuan2013time] unifies temporal and geographical information with linear combination. Geographical-temporal graph was proposed for time-aware location recommendation by doing preference propagation on the graph [@yuan2014graph]. In addition, temporal effect is also studied via nonnegative matrix factorization [@gao2013exploring] and RNN [@liu2016predicting].
Different from general location recommendation, next-location recommendation also need to take current state into account. Therefore, the sequential information is more important to consider in next location recommendation. Most previous works model sequential behaviors, [*i.e.,*]{}trajectories of check-in locations, based on Markov chain assumption which assumes the next location is determined only by current location and independent of previous ones [@rendle2010factorizing; @cheng2013you; @ye2013s; @zhang2014lore]. For example, Factorized Personalized Markov Chain (FPMC) algorithm [@rendle2010factorizing] factorizes the tensor of transition cube which includes transition probability matrices of all users. Personalized Ranking Metric Embedding (PRME) [@feng2015personalized] further extends FPMC by modeling user-location distance and location-location distance in two different vector spaces. Hierarchical Representation Model (HRM) [@wang2015learning], which is originally designed for user purchase behavior modeling, can be easily adapted for modeling user trajectories. HRM builds a two-layer structure to predict items in next transaction with user features and items in last transaction. These methods are applied for next-location recommendation which aims at predicting the next location that a user will visit, given check-in history and current location of the user. Note that Markov chain property is a strong assumption that assumes next location is determined only by current location. In practice, next location may also be influenced by the entire check-in history.
Problem Formalization
=====================
We use $L$ to denote the set of locations (a.k.a. *check-in* points or POIs) . When a user $v$ checks in at a location $l$ at the timestamp $s$, the information can be modeled as a triplet $\langle v, l, s \rangle$. Given a user $v$, her trajectory $T_v$ is a sequence of triplets related to $v$: $\langle v, l_1, s_1 \rangle, ..., \langle v, l_i, s_i \rangle, ..., \langle v, l_{N}, s_{N} \rangle$, where $N$ is the sequence length and the triplets are ordered by timestamps ascendingly. For brevity, we rewrite the above formulation of $T_v$ as a sequence of locations $T_v=\{l_1^{(v)},l_2^{(v)},\dots,l_N^{(v)}\}$ in chronological order. Furthermore, we can split a trajectory into multiple consecutive subtrajectories: the trajectory $T_v$ is split into $m_v$ subtrajectories $T_v^1,\dots,T_v^{m_v}$. Each subtrajectory is essentially a subsequence of the original trajectory sequence. In order to split the trajectory, we compute the time interval between two check-in points in the original trajectory sequence, we follow [@cheng2013you] to make a splitting when the time interval is larger than six hours. To this end, each user corresponds to a trajectory sequence $T_v$ consisting of several consecutive subtrajectories $T_v^1,\dots,T_v^{m_v}$. Let $T$ denote the set of trajectories for all the users.
Besides trajectory data, location-based services provide social connection links among users, too. Formally, we model the social network as a graph $G=(V,E)$, where each vertex $v\in V$ represents a user, each edge $e\in E$ represents the friendship between two users. In real applications, the edges can be either undirected or directed. As we will see, our model is flexible to deal with both types of social networks. Note that these links mainly reflect online friendship, which do not necessarily indicate that two users are friends in actual life.
Given the social network information $G=(V,E)$ and the mobile trajectory information $T$, we aim to develop a joint model which can characterize and utilize both kinds of data resources. Such a joint model should be more effective those built with a single data resource alone. In order to test the model performance, we set up two application tasks in LBSNs.
[**Task I.**]{} For the task of next-location recommendation, our goal is to recommend a ranked list of locations that a user $v$ is likely to visit next at each step.
[**Task II.**]{} For the task of friend recommendation, our goal is to recommend a ranked list of users that are likely to be the friends of a user $v$.
We select these tasks because they are widely studied in LBSNs, respectively representing two aspects for mobile trajectory mining and social networking analysis. Other tasks related to LBSN data can be equally solved by our model, which are not our focus in this paper.
The Proposed Model
==================
In this section, we present a novel neural network model for generating both social network and mobile trajectory data. In what follows, we first study how to characterize each individual component. Then, we present the joint model followed by the parameter learning algorithm. Before introducing the model details, we first summarize the used notations in this paper in Table \[tab:notations\].
Modeling the Construction of the Social Network
-----------------------------------------------
Recently, networking representation learning is widely studied [@chen2007directed; @tang2009relational; @tang2011leveraging; @Perozzi:2014:DOL:2623330.2623732], and it provides a way to explore the networking structure patterns using low-dimensional embedding vectors. Not limited to discover structure patterns, network representations have been shown to be effective to serve as important features in many network-independent tasks, such as demographic prediction [@huang2014mining] and text classification [@yang2015network]. In our task, we characterize the networking representations based on two considerations. First, a user is likely to have similar visit behaviors with their friends, and user links can be leveraged to share common visit patterns. Second, the networking structure is utilized as auxiliary information to enhance the trajectory modelling.
Formally, we use a $d$-dimensional embedding vector of use $F_{v}\in \mathbb{R}^d$ to denote the network representation of user $v$ and matrix $F\in \mathbb{R}^{|V|\times d}$ to denote the network representations for all the users. The network representation is learned with the user links on the social network, and encodes the information for the structure patterns of a user.
The social network is constructed based on users’ networking representations $F$. We first study how to model the generative probability for a edge of $v_i \rightarrow v_j$, formally as $\Pr[(v_i,v_j)\in E]$. The main intuition is that if two users $v_i$ and $v_j$ form a friendship link on the network, their networking representations should be similar. In other words, the inner product $F_{v_i}^\top\cdot F_{v_j}$ between the corresponding two networking representations will yield a large similarity value for two linked users. A potential problem will be such a formulation can only deal with undirected networks. In order to characterize both undirected and directed networks, we propose to incorporate a context representation for a user $v_j$, i.e., $F'_{v_j}$. Given a directed link $v_i \rightarrow v_j$, we model the representation similarity as $F_{v_i}^\top\cdot F'_{v_j}$ instead of $F_{v_i}^\top\cdot F_{v_j}$. The context representations are only used in the network construction. We define the probability of a link $v_i \rightarrow v_j$ by using a sigmoid function as follows
$$\Pr[(v_i,v_j)\in E] = \sigma(-F_{v_i}^\top\cdot F'_{v_j})=\frac{1}{1+\exp(-F_{v_i}^\top\cdot F'_{v_j})}.
\label{edge1}$$
When dealing with undirected networks, a friend pair $(v_i, v_j)$ will be split into two directed links namely $v_i \rightarrow v_j$ and $v_j \rightarrow v_i$. For edges not existing in $E$, we propose to use the following formulation
$$\Pr[(v_i,v_j)\not\in E] = 1-\sigma(-F_{v_i}^\top\cdot F'_{v_j})=\frac{\exp(-F_{v_i}^\top\cdot F'_{v_j})}{1+\exp(-F_{v_i}^\top\cdot F'_{v_j})}.
\label{edge2}$$
Combining Eq. \[edge1\] and \[edge2\], we essentially adopt a Bernouli distribution for modelling networking links. Following studies on networking representation learning [@Perozzi:2014:DOL:2623330.2623732], we assume that each user pair is independent in the generation process. That is to say the probabilities $\Pr[(v_i,v_j)\in E |F]$ are independent for different pairs of $(v_i,v_j)$. With this assumption, we can factorize the generative probabilities by user pairs
$$\begin{aligned}
\mathcal{L}(G)&=\sum_{(v_i,v_j)\in E}\log\Pr[(v_i,v_j)\in E]+\sum_{(v_i,v_j)\not\in E}\log\Pr[(v_i,v_j)\not\in E]\\
&=-\sum_{v_i,v_j}\log(1+\exp(-F_{v_i}^\top\cdot F'_{v_j}))-\sum_{(v_i,v_j)\not\in E}F_{v_i}^\top\cdot F'_{v_j}.
\end{aligned}
\label{network}$$
Modeling the Generation of the Mobile Trajectories
--------------------------------------------------
In Section 3, a user trajectory is formatted as an ordered check-in sequences. Therefore, we model the trajectory generation process with a sequential neural network method. To generate a trajectory sequence, we generate the locations in it one by one conditioned on four important factors. We first summarize the four factors as below
- *General visit preference*: A user’s preference or habits directly determine her own visit behaviors.
- *Influence of Friends*: The visit behavior of a user is likely to be influenced by her friends. Previous studies [@cheng2012fused; @levandoski2012lars] indeed showed that socially correlated users tend to visit common locations.
- *Short-term sequential contexts*: The next location is closely related to the last few locations visited by a user. The idea is intuitive in that the visit behaviors of a user is usually related to a single activity or a series of related activities in a short time window, making that the visited locations have strong correlations.
- *Long-term sequential contexts*: It is likely that there exists long-term dependency for the visited locations by a user in a long time period. A specific case for long-term dependency will be periodical visit behaviors. For example, a user regularly has a travel for vocation in every summer vocation.
The first two factors are mainly related to the two-way interactions between users and locations. While the last two factors mainly reflect the sequential relatedness among the visited locations by a user.
### Characterization of General Visit Preference
We first characterize the general visit preference by the interest representations. We use a $d$-dimensional embedding vector of $P_{v}\in \mathbb{R}^d$ to denote the visit interest representation of user $v$ and matrix $P\in \mathbb{R}^{|V|\times d}$ to denote the visit preference representations for all the users. The visit interest representation encodes the information for the general preference of a user over the set of locations in terms of visit behaviors.
We assume that one’s general visit interests are relatively stable and does not vary too much in a given period. Such an assumption is reasonable in that a user typically has a fixed lifestyle ([*e.g.,*]{}with a relatively fixed residence area) and her visiting behaviors are likely to show some overall patterns. The visit interest representation aims to capture and encode such visit patterns by using a $d$-dimensional embedding vector. For convenience, we call $P_{v}$ as the *interest representation* for user $v$.
### Characterization of Influence of Friends
For characterizing influence of friends, a straightforward approach is to model the correlation between interest representations from two linked users with some regularization terms. However, such a method usually has high computational complexity. In this paper, we adopt a more flexible method: we incorporate the network representation in the trajectory generation process. Because the network representations are learned through the network links, the information from their friends are implicitly encoded and used. We still use the formulation of networking representation $F_v$ introduced in Section 4.1.
### Characterization of Short-Term Sequential Contexts
Usually, the visited locations by a user in a short time window are closely correlated. A short sequence of the visited locations tend to be related to some activity. For example, a sequence “Home $\rightarrow$ Traffic $\rightarrow$ Office" refers to one’s transportation activity from home to office. In addition, the geographical or traffic limits play an important role in trajectory generation process. For example, a user is more likely to visit a nearby location. Therefore, when a user decides what location to visit next, the last few locations visited by herself should be of importance for next-location prediction.
Based on the above considerations, we treat the last few visited locations in a short time window as the sequential history and predict the next location based on them. To capture the short-term visit dependency, we use the Recurrent Neural Network (RNN), a convenient way for modelling sequential data, to develop our model. Formally, given the $j$-th subsequence $T_v^j=\{l_1^{(v,j)},l_2^{(v,j)}\dots l_{m_{v,j}}^{(v,j)}\}$ from the trajectory of user $v$, we recursively define the short-term sequential relatedness as follows: $$S_i =\text{tanh}(U_{l_{i-1}}+W\cdot S_{i-1})
\label{state}$$ where $S_i\in \mathbb{R}^d$ is the embedding representation for the state after visiting location $l_{i-1}$, $U_{l_i}\in \mathbb{R}^d$ is the representation of location $l_i^{(v,j)}$ and $W\in \mathbb{R}^{d\times d}$ is a transition matrix. Here we call $S_i$ *states* which are similar to those in Hidden Markov Models. RNN resembles Hidden Markov Models in that the sequential relatedness is also reflected through the transitions between two consecutive states. A major difference is that in RNN each hidden state is characterized by a $d$-dimensional embedding vector. As shown in Fig. \[fig:str\], we derive the state representation $S_{i}$ by forwarding $S_{i-1}$ with a transformation matrix $W$ and adding the embedding representation for the current location $U_{l_{i-1}}$. The initial representation $S_0$ is invariant among all users because short-term correlation is supposed to be irrelevant to user preference in our model. Our formulation in Eq. \[state\] is essentially a RNN model without outputs. The embedding vector corresponding to each state can be understood as an information summary till the corresponding location in the sequence. Especially, the state corresponding to the last location can be considered the embedding representation for the entire sequence.
![An illustrative example of recurrent neural networks for modelling short-term sequential contexts. []{data-label="fig:str"}](newSTR.pdf){width="0.7\columnwidth"}
### Characterization of Long-Term Sequential Contexts
In the above, short-term sequential contexts (five locations on average for our dataset) aim to capture the sequential relatedness in a short time window. The long-term sequential contexts are also important to consider when modelling trajectory sequences. For example, a user is likely to show some periodical or long-range visit patterns. To capture the long-term dependency, a straightforward approach will be to use another RNN model for the entire trajectory sequence. However, the entire trajectory sequence generated by a user in a long time period tends to contain a large number of locations, [*e.g.,*]{}several hundred locations or more. A RNN model over long sequences usually suffers from the problem of “vanishing gradient".
To address the problem, we employ the Gated Recurrent Unit (GRU) for capturing long-term dependency in the trajectory sequence. Compared with traditional RNN, GRU incorporates several extra gates to control the input and output. Specifically, we use two gates in our model: input gate and forget gates. With the help of input and forget gates, the memory of GRU, [*i.e.,*]{}the state $C_t$ can remember the “important stuff" even when the sequence is very long and forget less important information if necessary. We present an illustrative figure for the architecture for recurrent neural networks with GRUs in Fig. \[fig:gru\].
![An illustrative architecture of recurrent neural networks with GRUs. Let $\widetilde{C_t}$ denote a candidate state. The current state $C_t$ is a mixture of the last state $C_{t-1}$ and the current candidate state $\widetilde{C_t}$. $I_t$ and $F_t$ are input and forget gate respectively, which can control this mixture.[]{data-label="fig:gru"}](GRU.pdf){width="0.8\columnwidth"}
Formally, consider the following location sequence $\{l_1,l_2,\dots,l_m\}$, we denote the initial state by $C_0\in \mathbb{R}^d$ and initial representation by $h_0=\text{tanh}(C_0)\in \mathbb{R}^d$. At a timestep of $t$, the new candidate state is updated as follows
$$\widetilde{C_t} =\text{tanh}(W_{c_1}U_{l_t}+W_{c_2}h_{t-1}+b_c)
\label{candgate}$$
where $W_{c_1}\in \mathbb{R}^{d\times d}$ and $W_{c_2}\in \mathbb{R}^{d\times d}$ are the model parameters, $U_{l_t}$ is the embedding representation of location $l_t$ which is the same representation used in short-term sequential relatedness, $h_{t-1}$ is the embedding representation in the last step and $b_c\in \mathbb{R}^d$ is the bias vector. Note that the computation of $\widetilde{C_t}$ remains the same as that in RNN.
GRU does not directly replace the state with $\widetilde{C_t}$ as RNN does. Instead, GRU tries to find a balance between the last state $C_{t-1}$ and a new candidate state $\widetilde{C_t}$: $$C_t=i_t*\widetilde{C_t}+f_t*C_{t-1}
\label{newstate}$$ where $*$ is entrywise product and $i_t,f_t\in \mathbb{R}^d$ are input and forget gate respectively.
And the input and forget gates $i_t,f_t\in \mathbb{R}^d$ are defined as $$i_t =\sigma(W_{i_1}U_{l_t}+W_{i_2}h_{t-1}+b_i)
\label{inputgate}$$ and $$f_t =\sigma(W_{f_1}U_{l_t}+W_{f_2}h_{t-1}+b_f)
\label{forgetgate}$$ where $\sigma(\cdot)$ is the sigmoid function, $W_{i_1},W_{i_2}\in \mathbb{R}^{d\times d}$ and $W_{f_1},W_{f_2}\in \mathbb{R}^{d\times d}$ are input and forget gate parameters, and $b_i,b_f\in \mathbb{R}^d$ are the bias vectors.
Finally, the representation of long-term interest variation at the timestep of $t$ is derived as follows $$h_t =\text{tanh}(C_t).
\label{ht}$$
Similar to Eq. \[state\], $h_t$ provides a summary which encodes the information till the $t$-th location in a trajectory sequence. We can recursively learn the representations after each visit of a location.
### The Final Objective Function for Generating Trajectory Data
Given the above discussions, we are now ready to present the objective function for generating trajectory data. Given the trajectory sequence $T_v=\{l_1^{(v)},l_2^{(v)},\dots,l_m^{(v)}\}$ of user $v$, we factorize the log likelihood according to the chain rule as follows
$$\begin{aligned}
\mathcal{L}(T_v)&=\log \Pr[l_1^{(v)},l_2^{(v)},\dots,l_m^{(v)} | v, \Phi]\\
&=\sum_{i=1}^m \log\Pr[l_i^{(v)}|l_1^{(v)},\dots,l_{i-1}^{(v)} ,v, \Phi],
\end{aligned}
\label{chain}$$
where $\Phi$ denotes all the related parameters. As we can see, $\mathcal{L}(T_v)$ is characterized as a sum of log probabilities conditioned on the user $v$ and related parameters $\Phi$. Recall that the trajectory $T_v$ is split into $m_v$ subtrajectories $T_v^1,\dots,T_v^{m_v}$. Let $l_i^{(v,j)}$ denote the $i$-th location in the $j$-th subtrajectory. The contextual locations for $l_i^{(v,j)}$ contain the preceding $(i-1)$ locations ([*i.e.,*]{}$l_1^{(v,j)}\dots l_{i-1}^{(v,j)}$) in the same subtrajectory, denoted by $l_1^{(v,j)}:l_{i-1}^{(v,j)}$, and all the locations in previous $(j-1)$ subtrajectories ([*i.e.,*]{}$T_v^1,\dots,T_v^{j-1}$ ), denoted by $T_v^1 : T_v^{j-1}$. With these notions, we can rewrite Eq. \[chain\] as follows
$$\begin{aligned}
\mathcal{L}(T_v) &=\sum_{i=1}^m \log\Pr[l_i^{(v,j)} | \underbrace{l_1^{(v,j)}:l_{i-1}^{(v,j)}}_{\text{short-term contexts}}, \underbrace{T_v^1 : T_v^{j-1}}_{\text{long-term contexts}} , v, \Phi].
\end{aligned}
\label{chain2}$$
Given the target location $l_i^{(v,j)}$, the term of $l_1^{(v,j)}:l_{i-1}^{(v,j)}$ corresponds to the short-term contexts, the term of $T_v^1 : T_v^{j-1}$ corresponds to the long-term contexts, and $v$ corresponds to the user context. The key problem becomes how to model the conditional probability $\Pr[l_i^{(v,j)} | l_1^{(v,j)}:l_{i-1}^{(v,j)}, T_v^1 : T_v^{j-1} , v, \Phi]$.
For short-term contexts, we adopt the RNN model described in Eq. \[state\] to characterize the the location sequence of $l_1^{(v,j)}:l_{i-1}^{(v,j)}$. We use $S_{i}^j$ to denote the derived short-term representation after visiting the $i$-th location in the $j$-th subtrajectory; For long-term contexts, the locations in the preceding subtrajectories $T_v^1\dots T_v^{j-1}$ are characterized using the GRU model in Eq. \[newstate\] $\sim$ \[ht\]. We use $h^{j}$ to denote the derived long-term representation after visiting the locations in first $j$ subtrajectories. We present an illustrative example for the combination of short-term and long-term contexts in Fig. \[fig:seq\].
![An illustrative figure for modelling both short-term and long-term sequential contexts. The locations in a rounded rectangular indicates a subtrajectory. The locations in red and blue rectangular are used for long-term and short-term sequential contexts respectively. “?” is the next location for prediction.[]{data-label="fig:seq"}](seq.pdf){width="0.8\columnwidth"}
So far, given a target location $l_i^{(v,j)}$, we have obtained four representations corresponding to the four factors: networking representation ([*i.e.,*]{}$F_{v}$), visit interest representation ([*i.e.,*]{}$P_{v}$), short-term context representation $S_{i-1}^j$, and long-term context representation $h^{j-1}$. We concatenate them into a single context representation $R_v^{(i,j)}=[F_{v};P_{v};S_{i-1}^j;h^{j-1}]\in \mathbb{R}^{4d}$ and use it for next-location generation. Given the context representation $R_v^{(i,j)}$, we define the probability of $l_i^{(v,j)}$ as
$$\begin{aligned}
&&\Pr[l_i^{(v,j)}|l_1^{(v,j)}:l_{i-1}^{(v,j)}, T_v^1 : T_v^{j-1} , v, \Phi] \nonumber\\
&=&\Pr[l_i^{(v,j)}|R_v^{(i,j)}] \nonumber\\
&=& \frac{\exp(R_v^{(i,j)} \cdot U'_{l_i^{(v,j)}})}{\sum_{l\in L}\exp(R_v^{(i,j)} \cdot U'_{l})}
\label{condp}\end{aligned}$$
where parameter $U'_{l}\in \mathbb{R}^{4d}$ is location representation of location $l\in L$ used for prediction. Note that this location representation $U'_{l}$ is totally different with the location representation $U_l\in \mathbb{R}^{d}$ used in short-term and long-term context modelling. The overall log likelihood of trajectory generation can be computed by adding up all the locations.
The Joint Model
---------------
Our general model is a linear combination between the objective functions for the two parts. Given the friendship network of $G=(V,E)$ and user trajectory $T$, we have the following log likelihood function $$\begin{aligned}
\begin{aligned}
\mathcal{L}(G,T)&= \mathcal{L}_{\text{network}}(G) + \mathcal{L}_{\text{trajectory}}(T)\\
&=\mathcal{L}(G)+\sum_{v\in V}\mathcal{L}(T_v).
\end{aligned}
\label{loglikelihood}\end{aligned}$$
where $\mathcal{L}_{\text{network}}(G)$ is defined in Eq. \[network\] and $\mathcal{L}_{\text{trajectory}}(T)=\sum_{v\in V}\mathcal{L}(T_v)$ is defined in Eq. \[chain2\] respectively. We name our model as *Joint Network and Trajectory Model (**JNTM**)*.
We present an illustrative architecture of the proposed model JNTM in Fig \[fig:general\]. Our model is a three-layer neural network for generating both social network and user trajectory. In training, we require that both the social network and user trajectory should be provided as the objective output to the train the model. Based on such data signals, our model naturally consists of two objective functions. For generating the social network, a network-based user representation was incorporated; for generating the user trajectory, four factors were considered: network-based representation, general visiting preference, short-term and long-term sequential contexts. These two parts were tied by sharing the network-based user representation.
![An illustrative architecture of the proposed model JNTM.[]{data-label="fig:general"}](general.pdf){width="0.8\columnwidth"}
Parameter Learning
------------------
Now we will show how to train our model and learn the parameters, [*i.e.,*]{}user interest representation $P\in \mathbb{R}^{|V|\times d}$, user friendship representation $F,F'\in \mathbb{R}^{|V|\times d}$, location representations $U\in \mathbb{R}^{|L|\times d},U'\in \mathbb{R}^{|L|\times 4d}$, initial short-term representation $S_{0}\in \mathbb{R}^{d}$, transition matrix $W\in \mathbb{R}^{d\times d}$, initial GRU state $C_0\in \mathbb{R}^{d}$ and GRU parameters $W_{i_1},W_{i_2},W_{f_1},W_{f_2},W_{c_1},W_{c_2}\in \mathbb{R}^{d\times d},b_i,b_f,b_c\in \mathbb{R}^{d}$ .
[**Negative Sampling.**]{} Recall that the log likelihood of network generation equation \[network\] includes $|V|\times |V|$ terms. Thus it takes at least $O(|V|^2)$ time to compute, which is time-consuming. Therefore we employ negative sampling technique which is commonly used in NLP area [@mikolov2013distributed] to accelerate our training process.
Note that real-world networks are usually sparse, [*i.e.,*]{}$O(E)=O(V)$. The number of connected vertex pairs (positive examples) are much less than the number of unconnected vertex pairs (negative examples). The core idea of negative sampling is that most vertex pairs serve as negative examples and thus we don’t need to compute all of them. Instead we compute all connected vertex pairs and $n_1$ random unconnected vertex pairs as an approximation where $n_1\ll|V|^2$ is the number of negative samples. In our settings, we set $n_1=100|V|$. The log likelihood can be rewritten as $$\mathcal{L}(G|F,F')=\sum_{(v_i,v_j)\in E}\log\Pr[(v_i,v_j)\in E]+\sum_{k=1,(v_{ik},v_{jk})\not\in E}^{n_1} \log\Pr[(v_{ik},v_{jk})\not\in E].
\label{network2}$$ Then the computation of likelihood of network generation only includes $O(E+n_1)=O(V)$ terms.
On the other hand, the computation of equation \[condp\] takes at least $O(|L|)$ time because the denominator contains $|L|$ terms. Note that the computation of this conditional probability need to be done for every location. Therefore the computation of trajectory generation needs at least $O(|L|^2)$ which is not efficient. Similarly, we don’t compute every term in the denominator. Instead we only compute location $l_i^{(v,j)}$ and other $n_2$ random locations. In this paper we use $n_2=100$. Then we reformulate equation \[condp\] as $$\Pr[l_i^{(v,j)}|R_v^{(i,j)}] = \frac{\exp(R_v^{(i,j)} \cdot U'_{l_i^{(v,j)}})}{\exp(R_v^{(i,j)} \cdot U'_{l_i^{(v,j)}})+\sum_{k=1,l_k\neq l_i^{(v,j)}}^{n_2}\exp(R_v^{(i,j)} \cdot U'_{l_k})}.
\label{condp2}$$ Then the computation of the denominator only includes $O(n_2+1)=O(1)$ terms.
We compute the gradients of the parameters by back propagation through time (BPTT) [@werbos1990backpropagation]. Then the parameters are updated with AdaGrad [@duchi2011adaptive], a variant of stochastic gradient descent (SGD), in mini-batches.
In more detail, we use pseudo codes in algorithm \[alg:net\] and \[alg:tra\] to illustrate training process of our model. The network iteration and trajectory iteration are executed iteratively until the performance on validation set becomes stable.
Update $F$ and $F'$ according to the gradients
Update all parameters according to their gradients
[**Complexity Analysis.**]{} We first given the complexity analysis on time cost. The network generation of user $v$ takes $O(d)$ time to compute log likelihood and gradients of $F_v$ and corresponding rows of $F'$. Thus the complexity of network generation is $O(d|V|)$. In trajectory generation, we denote the total number of check-in data as $|D|$. Then the forward and backward propagation of GRU take $O(d^2|D|)$ time to compute since the complexity of a single check-in is $O(d^2)$. Each step of RNN takes $O(d^2)$ time to update local dependency representation and compute the gradients of $S_0,U,W$. The computation of log likelihood and gradients of $U',F_v,P_v,S_{i-1}^j$ and $h^{j-1}$ takes $O(d^2)$ times. Hence the overall complexity of our model is $O(d^2|D|+d|V|)$. Note that the representation dimension $d$ and number of negative samples per user/location are much less than the data size $|V|$ and $|D|$. Hence the time complexity of our algorithm JNTM is linear to the data size and scalable for large datasets. Although the training time complexity of our model is relatively high, the test time complexity is small. When making location recommendations to a user in the test stage, it takes $O(d)$ time to update the hidden states of RNN/LSTM, and $O(d)$ time to evaluate a score for a single location. Usually, the hidden dimensionality $d$ is a small number, which indicates that our algorithm is efficient to make online recommendations.
In terms of space complexity, the network representations $F$ and location representations $U$ take $O((|V|+|L|)d)$ space cost in total. The space cost of other parameters is at most $O(d^2)$, which can be neglected since $d$ is much less than $|V|$ and $|L|$. Thus the space complexity of our model is similar to that of previous models such as FPMC [@rendle2010factorizing], PRME [@feng2015personalized] and HRM [@wang2015learning].
Experimental Evaluation
=======================
In this section, we evaluate the performance of our proposed model JNTM. We consider two application tasks, namely next-location recommendation and friend recommendation. In what follows, we will discuss the data collection, baselines, parameter setting and evaluation metrics. Then we will present the experimental results together with the related analysis.
Data Collection
---------------
We consider using two publicly available LBSN datasets[^4] [@cho2011friendship], [*i.e.,*]{}Gowalla and Brightkite, for our evaluation. Gowalla and Brightkite have released the mobile apps for users. For example, with Brightkite you can track on your friends or any other BrightKite users nearby using a phone’s built in GPS; Gowalla has a similar function: use GPS data to show where you are, and what’s near you.
These two datasets provide both connection links and users’ check-in information. A connection link indicates reciprocal friendship and a check-in record contains the location ID and the corresponding check-in timestamp. We organized the check-in information as trajectory sequences. Following [@cheng2013you], we split a trajectory wherever the interval between two successive check-ins is larger than six hours. We preformed some preprocessing steps on both datasets. For *Gowalla*, we removed all users who have less than $10$ check-ins and locations which have fewer than $15$ check-ins, and finally obtained $837,352$ subtrajectories. For *Brightkite*, since this dataset is smaller, we only remove users who have fewer than $10$ check-ins and locations which have fewer than $5$ check-ins, and finally obtain $503,037$ subtrajectories after preprocessing. Table \[tab:statistics\] presents the statistics of the preprocessed datasets. Note that our datasets are larger than those in previous works [@cheng2013you; @feng2015personalized].
\[tab:statistics\]
A major assumption we have made is that there exists close association between social links and mobile trajectory behaviors. To verify this assumption, we construct an experiment to reveal basic correlation patterns between these two factors. For each user, we first generate a location set consisting of the locations that have been visited by the user. Then we can measure the similarity degree between the location sets from two users using the overlap coefficient[^5]. The average overlap coefficients are $11.1\%$ and $15.7\%$ for a random friend pair ([*i.e.,*]{}two users are social friends) on Brightkite and Gowalla dataset, respectively. As a comparison, the overlap coefficient falls to $0.5\%$ and $0.5\%$ for a random non-friend pair ([*i.e.,*]{}two users are not social friends) on Brightkite and Gowalla dataset, respectively. This finding indicates that users that are socially connected indeed have more similar visit characteristics. We next examine whether two users with similar trajectory behaviors are more likely to be socially connected. We have found that the probabilities that two random users are social friends are $0.1\%$ and $0.03\%$ on Brightkite and Gowalla dataset, respectively. However, if we select two users with more than $3$ common locations in their location set, the probabilities that they are social friends increase to $9\%$ and $2\%$, respectively. The above two findings show social connections are closely correlated with mobile trajectory behaviors in LBSNs.
Evaluation Tasks and Baselines
------------------------------
### Next-Location Recommendation {#next-location-recommendation .unnumbered}
For the task of next-location recommendation, we consider the following baselines:
- **Paragraph Vector (PV)** [@Le2014Distributed] is a representation learning model for both sentence and documents using simple neural network architecture. To model trajectory data, we treat each location as a word and each user as a paragraph of location words.
- **Feature-Based Classification (FBC)** solves the next-location recommendation task by casting it as a multi-class classification problem. The user features are learned using DeepWalk algorithm [@Perozzi:2014:DOL:2623330.2623732], and the location features are learned using word2vec [@mikolov2013distributed] algorithm (similar to the training method of PV above). These features are subsequently incorporated into a a softmax classifier, [*i.e.,*]{}a multi-class generalization of logistic regression.
- **FPMC** [@rendle2010factorizing], which is a state-of-the-art recommendation algorithm, factorizes tensor of transition matrices of all users and predicts next location by computing the transition probability based on Markov chain assumption. It was originally proposed for product recommendation, however, it is easy adapt FPMC to deal with next-location recommendation.
- **PRME** [@feng2015personalized] extends FPMC by modeling user-location and location-location pairs in different vector spaces. PRME achieves state-of-the-art performance on next-location recommendation task.
- **HRM** [@wang2015learning] is a latest algorithm for next-basket recommendation. By taking each subtrajectory as a transaction basket, we can easily adapt HRM for next-location recommendation. It is the first study that distributed representation learning has been applied to the recommendation problem.
We select these five baselines, because they represent different recommendation algorithms. PV is based on simple neural networks, FBC is a traditional classification model using embedding features, FPMC is mainly developed in the matrix factorization framework, PRME makes specific extensions based on FPMC to adapt to the task of next-location recommendation, and HRM adopts the distributed representation learning method for next-basket modelling.
Next, we split the data collection into the training set and test set. The first $90\%$ of check-in subtrajectories of each user are used as the training data and the remaining $10\%$ as test data. To tune the parameters, we use the last $10\%$ of check-ins of training data as the validation set. Given a user, we predict the locations in the test set in a sequential way: for each location slot, we recommend five or ten locations to the user. For JNTM, we naturally rank the locations by the log likelihood as shown in equation \[condp\]. Note that negative sampling is not used in evaluation. For the baselines, we rank the locations by the transition probability for FPMC and HRM and transition distance for PRME. The predictions of PV and FBC can be obtained from the output of softmax layer of their algorithms. Then we report Recall@5 and Recall@10 as the evaluation metrics where Recall@K is defined as
$$\begin{aligned}
Recall@K=\frac{\text{\# ground truth locations in the $K$ recommended locations}}{\text{\# ground truth locations in test data}}. \nonumber\end{aligned}$$
Note that another common metric Precision@$K$ can be used here, too. In our experiments, we have found it is positively correlated with Recall@$K$, [*i.e.,*]{}if method $A$ has a higher Recall@$K$ score than method $B$, then method $A$ also has a higher Precision@$K$ score then method $B$. We omit the results of Precision@$K$ for ease of presentation.
### Friend Recommendation {#friend-recommendation .unnumbered}
For the task of friend recommendation, we consider three kinds of baselines based on the used data resources, including the method with only the networking data ([*i.e.,*]{}DeepWalk), the method with only the trajectory data ([*i.e.,*]{}PMF), and the methods with both networking and trajectory data ([*i.e.,*]{}PTE and TADW).
- **DeepWalk** [@Perozzi:2014:DOL:2623330.2623732] is a state-of-the-art NRL method which learns vertex embeddings from random walk sequences. It first employs the random walk algorithm to generate length-truncated random paths, and apply the word embedding technique to learn the representations for network vertices.
- **PMF** [@mnih2007probabilistic] is a general collaborative filtering method based on user-item matrix factorization. In our experiments, we build the user-location matrix using the trajectory data, and then we utilize the user latent representations for friend recommendation.
- **PTE** [@tang2015pte] develops a semi-supervised text embedding algorithm for unsupervised embedding learning by removing the supervised part and optimizing over adjacency matrix and user-location co-occurrence matrix. PTE models a conditional probability $p(v_j|v_i)$ which indicates the probability that a given neighbor of $v_i$ is $v_j$. We compute the conditional probabilities for friend recommendation.
- **TADW** [@yang2015network] further extends DeepWalk to take advantage of text information of a network. We can replace text feature matrix in TADW with user-location co-occurrence matrix by disregarding the sequential information of locations. TADW defines an affinity matrix where each entry of the matrix characterizes the strength of the relationship between corresponding users. We use the corresponding entries of affinity matrix to rank candidate users for recommendation.
To construct the evaluation collection, we randomly select $20\sim 50$ of the existing connection links as training set and leave the rest for test. We recommend $5$ or $10$ friends for each user and report Recall@5 and Recall@10. The final results are compared by varying the training ratio from $20$ to $50$ percent. Specifically, for each user $v$, we take all the other users who are not her friends in the training set as the candidate users. Then, we rank the candidate users, and recommend top $5$ or $10$ users with highest ranking scores. To obtain the ranking score of user $v_j$ when we recommend friends for user $v_i$, DeepWalk and PMF adopt the cosine similarity between their user representations. For PTE, we use the conditional probability $p(v_j|v_i)$ which indicates the probability that a given neighbor of $v_i$ is $v_j$ as ranking scores. For TADW, we compute the affinity matrix $A$ and use the corresponding entry $A_{ij}$ as ranking scores. For our model, we rank users with highest log likelihood according to Equation \[edge1\].
The baselines methods and our model involves an important parameter, [*i.e.,*]{}the number of latent (or embedding) dimensions. We use a grid search from $25$ to $100$ and set the optimal value using the validation set. Other parameters in baselines or our model can be tuned in a similar way. For our model, the learning rate and number of negative samples are empirically set to $0.1$ and $100$, respectively. We randomly initialize parameters according to uniform distribution $U(-0.02,0.02)$.
All the experiments are executed on a $12$-core CPU server and the CPU type is Intel Xeon E5-2620 @ 2.0GHz.
Experimental Results on Next-location Recommendation.
-----------------------------------------------------
Table \[tab:nlr\] shows the results of different methods on next-location recommendation. Compared with FPMC and PRME, HRM models the sequential relatedness between consecutive subtrajectories while the sequential relatedness in a subtrajectory is ignored. In the Brightkite dataset, the average number of locations in a subtrajectory is much less than that in the Gowalla dataset. Therefore short-term sequential contexts are more important in the Gowalla dataset and less useful in the Brightkite dataset. Experimental results in Table \[tab:nlr\] demonstrate this intuition: HRM outperforms FPMC and PRME on Brightkite while PRME works best on Gowalla.
As shown in Table \[tab:nlr\], our model JNTM consistently outperforms the other baseline methods. JNTM yields $4.9\%$ and $4.4\%$ improvement on Recall@5 as compared to the best baseline HRM on the Brightkite dataset and FBC on the Gowalla dataset. Recall that our model JNTM has considered four factors, including user preference, influence of friends, short-term and long-term sequential contexts. All the baseline methods only characterize user preference (or friend influence for FBC) and a single kind of sequential contexts. Thus, JNTM achieves the best performance on both datasets.
\[tab:nlr\]
The above results are reported by averaging over all the users. In recommender systems, an important issue is how a method performs in the cold-start setting, [*i.e.,*]{}new users or new items. To examine the effectiveness on new users generating very few check-ins, we present the results of Recall@5 for users with no more five subtrajectories in Table \[tab:nllow\]. In a cold-start scenario, a commonly used way to leverage the side information ([*e.g.,*]{}user links [@cheng2012fused] and text information [@gao2015content; @li2010contextual; @zhao2015sar]) to alleviate the data sparsity. For our model, we characterize two kinds of user representations, either using network data or trajectory data. The user representations learned using network data can be exploited to improve the recommendation performance for new users to some extent. Indeed, networking representations have been applied to multiple network-independent tasks, including profession prediction [@tu2015prism] or text classification [@yang2015network]. By utilizing the networking representations, the results indicate that our model JNTM is very promising to deal with next-location recommendation in a cold-start setting.
\[tab:nllow\]
Note that the above experiments are based on general next-location recommendation, where we do not examine whether a recommended location has been previously visited or not by a user. To further test the effectiveness of our algorithm, we conduct experiments on next new location recommendation task proposed by previous studies [@feng2015personalized]. In this setting, we only recommend new locations when the user decide to visit a place. Specifically, we rank all the locations that a user has never visited before for recommendation [@feng2015personalized]. We present the experimental results in Table \[tab:nlrnew\]. Our method consistently outperforms all the baselines on next new location recommendation in both datasets. By combining results in Table \[tab:nlr\] and \[tab:nllow\], we can see that our model JNTM is more effective in next-location recommendation task compared to these baselines.
\[tab:nlrnew\]
In the above, we have shown the effectiveness of the proposed model JNTM on the task of next-location recommendation. Since trajectory data itself is sequential data, our model has leveraged the flexibility of recurrent neural networks for modelling sequential data, including both short-term and long-term sequential contexts. Now we study the effect of sequential modelling on the current task.
We prepare three variants for our model JNTM
- JNTM$_{base}$: it removes both short-term and long-term contexts. It only employs the user interest representation and network representation to generate the trajectory data.
- JNTM$_{base+long}$: it incorporates the modelling for long-term contexts to JNTM$_{base}$.
- JNTM$_{base+long+short}$: it incorporates the modelling for both short-term and long-term contexts to JNTM$_{base}$.
\[tab:jntm\]
\[tab:jntmnew\]
Table \[tab:jntm\] and \[tab:jntmnew\] show the experimental results of three JNTM variants on the Brightkite and Gowalla dataset. The numbers in the brackets indicate the relative improvement against JNTM$_{base}$. We can observe a performance ranking: JNTM$_{base} $ $<$ JNTM$_{base+long}$ $<$ JNTM$_{base+long+short}$. The observations indicate that both kinds of sequential contexts are useful to improve the performance for next-location recommendation. In general next location recommendation ([*i.e.,*]{}both old and new locations are considered for recommendation), we can see that the improvement from short and long term context is not significant. The explanation is that a user is likely to show repeated visit behaviors ([*e.g.,*]{}visiting the locations that have been visited before), and thus user preference is more important than sequential context to improve the recommendation performance. While for next new location recommendation, the sequential context especially short-term context yields a large improvement margin over the baseline. These results indicate that the sequential influence is more important than user preference for new location recommendation. Our finding is also consistent with previous work [@feng2015personalized], [*i.e.,*]{}sequential context is important to consider for next new location recommendation.
Experimental Results on Friend Recommendation
---------------------------------------------
\[tab:frb\]
\[tab:frg\]
We continue to present and analyze the experimental results on the task of friend recommendation. Table \[tab:frg\] and \[tab:frb\] show the results when the training ratio varies from $20\%$ to $50\%$.
Among the baselines, DeepWalk performs best and even better than the baselines using both networking data and trajectory data ([*i.e.,*]{}PTE and TADW). A major reason is that DeepWalk is tailored to the reconstruction of network connections and adopts a distributed representation method to capture the topology structure. As indicated in other following studies [@Perozzi:2014:DOL:2623330.2623732; @tang2015line], distributed representation learning is particularly effective to network embedding. Although PTE and TADW utilize both network and trajectory data, their performance is still low. These two methods cannot capture the sequential relatedness in trajectory sequences. Another observation is that PMF ([*i.e.,*]{}factorizing the user-location matrix) is better than PTE at the ratio of $20\%$ but becomes the worst baseline. It is because that PMF learns user representations using the trajectory data, and the labeled data ([*i.e.,*]{}links) is mainly used for training a classifier. Our algorithm is competitive with state-of-the-art network embedding method DeepWalk and outperforms DeepWalk when network structure is sparse. The explanation is that trajectory information is more useful when network information is insufficient. As network becomes dense, the trajectory information is not as useful as the connection links. To demonstrate this explanation, we further report the results for users with fewer than five friends when the training ratio of $50\%$. As shown in Table \[tab:frlow\], our methods have yielded $2.1\%$ and $1.5\%$ improvement than DeepWalk for these inactive users on the Brightkite and Gowalla datasets, respectively. The results indicate that trajectory information is useful to improve the performance of friend recommendation for users with very few friends.
\[tab:frlow\]
In summary, our methods significantly outperforms existing state-of-the-art methods on both next-location prediction and friend recommendation. Experimental results on both tasks demonstrate the effectiveness of our proposed model.
Parameter Tuning
----------------
In this section, we study on how different parameters affect the performance of our model. We mainly select two important parameters, [*i.e.,*]{}the number of iterations and and the number of embedding dimensions.
We conduct the tuning experiments on the training sets by varying the number of iterations from $5$ to $50$. We report the log likelihood for the network and trajectory data on the training sets and Recall@5 and Recall@10 of next location recommendation on validation sets.
Fig. \[fig:pti1\] and \[fig:pti2\] show the tuning results of the iteration number on both datasets. From the results we can see that our algorithm can converge within $50$ iterations on both datasets, and the growth of log likelihood slows down after 30 iterations. On the other hand, the performance of next location recommendation on validation sets is relatively stable: JNTM can yield a relatively good performance after $5$ iterations. The recall values increase slowly and reach the highest score at $45$ iteration on Brightkite and $15$ iteration on Gowalla dataset. Here Gowalla dataset converges more quickly and smoothly than Brightkite. It is mainly because Gowalla dataset contains $3$ times more check-in data than that of Brightkite and has more enough training data. However the model may overfit before it gets the highest recall for next-location recommendation because the recall scores are not always monotonically increasing. As another evidence, the performance on new location prediction begins to drop after about $10$ iterations. To avoid the overfitting problem, we reach a compromise and find that an iteration number of $15$ and $10$ is a reasonably good choice to give good performance on Brightkite and Gowalla, respectively. The number of embedding dimensions is also vital to the performance of our model. A large dimension number will have a strong expressive ability but will also probably lead to overfitting. We conduct experiments with different embedding dimension numbers on next location recommendation and measure their performance on validation sets. In Fig. \[dimension\], we can see that the performance of our algorithm is relatively stable when we vary the dimension number from $25$ to $100$. The recall values start to decrease when the dimension number exceeds $50$. We finally set the dimension number to $50$.
Scalability
-----------
In this part, we conduct experiments on scalability and examine the time and space costs of our model. We perform the experiments on Gowalla dataset, and select the baseline method PRME [@feng2015personalized] for comparison. We report the memory usage of both methods for space complexity analysis and running time on a single CPU for time complexity analysis. Since both methods have the same iteration number for convergence, we report the average running time per iteration for each algorithm. Running time for training and testing is presented separately. A major merit of neural network models is that they can be largely accelerated by supported hardware ([*e.g.,*]{}GPU). Therefore we also report the running time of a variation of our model using the GPU acceleration. Specifically, we use a single Tesla K40 GPU for training our model in the <span style="font-variant:small-caps;">TensorFlow</span>[^6] software library. The experimental results are shown in Table \[tab:scalability\].
\[tab:scalability\]
From Table \[tab:scalability\], we can see that our memory usage is almost twice as much as PRME. This is mainly because we set two representations for each user ([*i.e.,*]{}friendship and preference representations), while PRME only has a preference representation. The time complexity of JNTM is $O(d^2|D|+d|V|)$, while the time complexity of PRME is $O(d|D|)$, where $d$ is the embedding dimensionality ($d=50$ in our experiments). Hence the running time of JNTM is about $d$ times as much as that of PRME. Although our model has a longer training time than PRME, the time cost of JNTM for testing is almost equivalent to that of PRME. On average, JNTM takes less than $30ms$ for a single location prediction, which is efficient to provide online recommendation service after the training stage. Moreover, the GPU acceleration offers $12$x speedup for the training process, which demonstrates that our model JNTM can be efficiently learned with supported hardware.
Conclusion and Future Work
==========================
In this paper, we presented a novel neural network model by jointly model both social networks and mobile trajectories. In specific, our model consisted of two components: the construction of social networks and the generation of mobile trajectories. We first adopted a network embedding method for the construction of social networks. We considered four factors that influence the generation process of mobile trajectories, namely user visit preference, influence of friends, short-term sequential contexts and long-term sequential contexts. To characterize the last two contexts, we employed the RNN and GRU models to capture the sequential relatedness in mobile trajectories at different levels, i.e., short term or long term. Finally, the two components were tied by sharing the user network representations. On two important application tasks, our model was consistently better than several competitive baseline methods. In our approach, network structure and trajectory information complemented each other. Hence, the improvement over baselines was more significant when either network structure or trajectory data is sparse.
Currently, our model does not consider the GPS information, [*i.e.,*]{}a check-in record is usually attached with a pair of longitude and latitude values. Our current focus mainly lies in how to jointly model social networks and mobile trajectories. As the future work, we will study how to incorporate the GPS information into the neural network models. In addition, the check-in location can be also attached with categorical labels. We will also investigate how to leverage these semantic information to improve the performance. Such semantic information can be utilized for the explanation of the generated recommendation results. Our current model has provides a flexible neural network framework to characterize LBSN data. We believe it will inspire more follow-up studies along this direction.
The authors thank the anonymous reviewers for their valuable and constructive comments.
[^1]: https://en.wikipedia.org/wiki/Gowalla
[^2]: https://en.wikipedia.org/wiki/Brightkite
[^3]: Note that online social relationship does not necessarily indicate offline friendship in real life.
[^4]: http://snap.stanford.edu/data/
[^5]: https://en.wikipedia.org/wiki/Overlap\_coefficient
[^6]: https://www.tensorflow.org
|
---
author:
- |
\
Department of Physics, Columbia University, New York, NY 10027, USA\
E-mail:
- |
Elaine Goode\
University of Southampton, School of Physics and Astronomy, Highfield, Southampton, SO17 1BJ, United Kingdom\
E-mail:
- RBC and UKQCD collaborations
title: '$\Delta I=3/2$, $K\to\pi\pi$ Decays with Light, Non-Zero Momentum Pions'
---
Introduction
============
Interest in precise lattice calculations of $K\to\pi\pi$ decays stems from the possibility of gaining insight into the origin of the $\Delta I=1/2$ rule, and of putting constraints on CKM matrix elements. In particular, $\epsilon'/\epsilon$ and the CP violating phase can be computed when lattice calculations are compared with experimental results [@CPPACS; @RBC].
Several quenched $K\to\pi\pi$ calculations have been performed in the past [@RBC; @Changhoan_thesis; @Changhoan1; @Changhoan2; @Takeshi]. The calculation presented here is a quenched calculation using domain wall fermions (DWF) on a $24^3\times 64$, $L_s=16$ lattice with strong coupling ($a^{-1}=1.3\text{ GeV}$). This represents a larger spatial volume, $(3.6\text{ fm})^3$, than in previous works, allowing for a lighter, near physical pion mass of $m_\pi=227.6(6)\text{ MeV}$. As in Refs. [@Changhoan_thesis; @Changhoan1; @Changhoan2; @Takeshi] we are directly calculating the two pion decay amplitudes and not relying on chiral perturbation theory to determine these amplitudes from the simpler $K\to\pi$ and $K\to |0\rangle$ amplitudes. This use of chiral perturbation theory, upon which Refs. [@CPPACS; @RBC] were based, now appears to introduce large, uncontrolled systematic errors [@Christ_Lat_08]. This quenched study is a pilot project that will be followed by a full QCD calculation using the RBC/UKQCD $32^3\times 64$, $L_s=32$ lattices with 2+1 flavors of DWF and strong coupling ($a^{-1}=1.4$ GeV) that are currently being generated, with light pion masses ($m_\pi \le 250$ MeV).
Four Quark Operators and the Effective Hamiltonian
==================================================
The weak interactions and the effects of the heavier quarks can be included in the lattice QCD simulation by evaluating matrix elements of an effective Hamiltonian [@Ciuchini; @Buchalla]. In particular we use the conventions of equation 3 in [@RBC]. In this formalism we are interested in calculating matrix elements of four quark operators between $K$ and $\pi\pi$ states. Here we calculate matrix elements with $\Delta I=3/2$ operators, where $\Delta I$ is the change in isospin. The $\Delta I=3/2$ operators are further classified by how they transform under $SU(3)_L\times SU(3)_R$ into “(27,1)”, “(8,8)” and “(8,8) mixed” parts. [@CPPACS; @RBC].
Twisted Boundary Conditions
===========================
The kinematics of the physical $K\to\pi\pi$ decay are such that the pions have significant momentum, $\sim 200$ MeV each, so that it is required to simulate a two pion excited state on the lattice. Extracting an excited state requires multi-exponential fitting making it difficult to obtain a signal. One remedy is to use *twisted boundary conditions* [@Changhoan_thesis; @Changhoan1; @Changhoan2; @Sachrajda_Villadoro], in which the boundary conditions on a quark field, instead of simply being periodic, cause it to change by a phase $e^{i\phi}$ when going through the boundary. We say that this spatial direction is “twisted” by an amount $\phi$.
Twisting a quark by $\phi$ will cause it to have momentum $$p_n=\frac{\phi+2\pi n}{L}$$ where $L$ is the spatial extent of the lattice. In particular, we use twists of amount $\pi$ (corresponding to *antiperiodic* boundary conditions in the twisted direction) to obtain a two pion system with 0 total momentum. For example, if only the $x$ direction is twisted then we can have a two pion ground state in which one pion has momentum $p_x=\frac{\pi}{L}$ and the other pion has momentum $p_x=-\frac{\pi}{L}$. We can obtain two pion ground states with pion 3-momentum of magnitude $$p_\pi=\frac{\pi}{L},\frac{\sqrt{2}\pi}{L},\frac{\sqrt{3}\pi}{L}$$ by twisting one, two, and three spatial directions respectively. Since we actually perform the calculation for the decay $K^+\to\pi^+\pi^+$, and relate it to the physical decay via the Wigner-Eckhart theorem [@Changhoan_thesis], we can twist the down quark only, thus giving the two pions momentum and giving no momentum to the kaon. By rotational symmetry we can average over results for pion momenta of the same magnitude but different directions, i.e. we can average over results with the same number of twists.
Details of the Calculation
==========================
The calculation was carried out on 68 configurations of quenched $24^3\times 64$ lattices using the DBW2 action, and domain wall fermions with $L_s=16$. We used the QCDOC computers at Brookhaven and Columbia University for both configuration generation and propagator inversion. The inverse lattice spacing is $a^{-1}=1.3\text{ GeV}$, the physical volume is $(3.6\text{ fm})^3$, and we set the light and strange quark masses to $m_l=0.0055$ (chosen so that $m_\pi L\approx 4$) and $m_s=0.08$ in lattice units respectively. This yielded a pion mass of $m_\pi=227.6(6)\text{ MeV}$ and a kaon mass of $m_K=564(2)\text{ MeV}$ in physical units. The pion momentum corresponding to one twist for these lattices is $p_\pi=\pi/L=170\text{ MeV}$ so that the decay is expected to be nearly energy conserving ($E_{\pi\pi}-m_K\approx 2\sqrt{m_\pi^2+p_\pi^2}-m_K=4\text{ MeV}$).
We added and subtracted quark propagators with periodic and antiperiodic boundary conditions in the *time* direction from each other in order to double the effective time length. Periodic plus antiperiodic (P+A) propagators thus provide a source at $t=0$ and periodic minus antiperiodic (P-A) propagators provide an effective source at $t=64$. These provide the left and right walls for the kaon and two pions and the time $t$ of the four quark operator is varied. The kaon mass, two pion energy, and $K\to\pi\pi$ matrix element were extracted from the asymptotic behavior of the corresponding correlation functions as in [@My_Lat_08], the only difference being that the $K\to\pi\pi$ correlator was fit directly rather than fitting the quotient of correlators defined in equation 3.5 of [@My_Lat_08].
For the kaon and the two pions with zero momentum we used propagators with Coulomb gauge fixed wall sources. For the two pions with non-zero momentum, we used the same type of propagators for the $u$ quark, but used propagators with twisted (antiperiodic spatial) boundary conditions for the $d$ quark with Coulomb gauge fixed momentum wall sources of the “cosine” type $$\label{cosine_src}
s_{{\bf p},\text{cosine}}({\bf x})=\cos(p_x x)\cos(p_y y)\cos(p_z z)$$
Had we used sources of the “pure momentum” type $$s_{\bf p}({\bf x})=e^{i{\bf p}\cdot {\bf x}} .$$ it would have been necessary to do an inversion with source momentum $+{\bf p}$ for one $d$ quark and an inversion with source momentum $-{\bf p}$ for the other $d$ quark in order to couple to a two pion state with zero total momentum. Using the same cosine source propagator for each $d$ quark eliminates the need for this extra inversion, but this leads to cross terms that couple to two pion states with non-zero total momentum. For example, with a single twist in the $x$ direction, we hope to couple to a two pion state with individual pion momenta ${\bf p}_1=-\frac{\pi}{L}{\bf \hat{x}}$ and ${\bf p}_2=\frac{\pi}{L}{\bf \hat{x}}$, and thus the product of the sources of the two $d$ quarks according to Equation \[cosine\_src\] should be $$\begin{aligned}
s_{{\bf p}_1,\text{cosine}}({\bf x}_1)s_{{\bf p}_2,\text{cosine}}({\bf x}_2)&=\cos\left(\frac{\pi}{L}x_1\right)\cos\left(\frac{\pi}{L}x_2\right) \nonumber \\
&=\frac{1}{4}\left(e^{i\frac{\pi}{L}x_1}e^{i\frac{\pi}{L}x_2}+e^{i\frac{\pi}{L}x_1}e^{-i\frac{\pi}{L}x_2}+e^{-i\frac{\pi}{L}x_1}e^{i\frac{\pi}{L}x_2}+e^{-i\frac{\pi}{L}x_1}e^{-i\frac{\pi}{L}x_2}\right) \label{src_prod}\end{aligned}$$ But in Equation \[src\_prod\] the first and last term couple to two pion states with total momentum $2\frac{\pi}{L}$ and $-2\frac{\pi}{L}$ in the $x$ direction respectively. To circumvent this difficulty, we calculate two pion correlators with cosine sources but pure momentum sinks. The unwanted terms from the source vanish in the limit of an infinite number of configurations since the two pion final state is constrained to have zero total momentum by the pure momentum sink, but since these are sinks they require no extra inversions. In the $K\to\pi\pi$ correlator, the 0 momentum kaon in the initial state has a similar effect on the cosine sources of the two pions in the final state.
Results of the First Calculation
================================
The calculation was carried out as described, and the results for the matrix elements are shown in Table \[tb:bad\_data\]. The errors on the (8,8) and (8,8) *mixed* matrix elements with momentum $\pi/L$ (where $L=24$) are of order 50%, and the data was too noisy to extract a signal for the (27,1) matrix element with momentum $\pi/L$ and for all the matrix elements with momentum $\sqrt{2}\pi/L$ and $\sqrt{3}\pi/L$. This is due to the fact that a time separation of 64 is too large, especially for the higher momenta where the higher energy of the two pion state causes its signal to decay into noise well before it can reach the four quark operator.
$p$ $\mathcal{M}_{(27,1)}$ $\mathcal{M}_{(8,8)}$ $\mathcal{M}_{(8,8)\text{ mixed}}$
--------- ------------------------ ----------------------- ------------------------------------ -- -- --
0 0.00408(40) 0.0950(77) 0.294(34)
$\pi/L$ - 0.161(85) 0.55(30)
: Lattice matrix elements for a time separation of 64 between the kaons and the two pions (first calculation). Note that these values differ from those presented in the talk where a normalization factor was omitted.[]{data-label="tb:bad_data"}
Results of the Second Calculation
=================================
The calculation was redone with smaller time separations between the kaon and the two pions.[^1] Specifically, propagators with sources at $t_K=$20, 24, 28, 32, 36, 40, and 44 were generated in order to calculate $K\to\pi\pi$ correlators with kaon sources at these times, while the two pion sources remained at either the left or the right walls. Thus we could achieve time separations between the kaon and the two pions of 20, 24, 28, and 32 in two different ways in order to double the statistics (for example a time separation of 20 results when the two pions are at the left wall, i.e. $t_\pi=0$, and when $t_K=20$, and also results when the two pions are at the right wall, i.e. $t_\pi=64$, and when $t_K=44$). These separations were chosen so that the kaon and two pion sources were close enough to the four quark operator (with the operator between the sources) so that the signals from the sources do not decay into noise before reaching the operator, but were sufficiently distant from each other that a plateau region can be found where the effects of possible excited states have died away.
Effective mass plots for the kaon correlator, the two pion correlator with momentum $\pi/L$, and the $K\to\pi\pi$ correlator with the (27,1) operator, momentum $\pi/L$ and time separation 24 between the two sources, are shown in Figures \[fig:kaon\_pipi\_meff\] and \[fig:kpipi\_meff\]. The two pion energies that were extracted from these effective mass plots are given in Table \[tb:EPiPi\]. Table \[tb:Mvals\] shows the lattice matrix elements obtained for different time separations of the two sources, as well as an error weighted average of the results from the four different time separations. The error of the error weighted average is smaller than the error of the matrix element for any single time separation in all cases. For momentum 0 and $\pi/L$ the error of the error weighted average is of order 3-4% (for all of the operators), for momentum $\sqrt{2}\pi/L$ it is of order 7%, and for momentum $\sqrt{3}\pi/L$ it is of order 15%.
![Effective mass plots for the kaon correlator (left) and the two pion correlator for pions with momentum $\pi/L$ (right). The effective mass of a correlator C(t) is defined as $m_\mathrm{eff}(t)=-\ln\left(C(t)/C(t-1)\right)$.[]{data-label="fig:kaon_pipi_meff"}](kaon_meff_comb "fig:") ![Effective mass plots for the kaon correlator (left) and the two pion correlator for pions with momentum $\pi/L$ (right). The effective mass of a correlator C(t) is defined as $m_\mathrm{eff}(t)=-\ln\left(C(t)/C(t-1)\right)$.[]{data-label="fig:kaon_pipi_meff"}](pipi_meff_comb_momnum1 "fig:")
![Effective mass plot for the $K\to\pi\pi$ correlator with the (27,1) four quark operator, pions with momentum $\pi/L$, the two pion source located at $t=0$ and the kaon source located at $t=24$.[]{data-label="fig:kpipi_meff"}](O27_unmixed_meff_comb_tK24_momnum1)
$2\sqrt{m_\pi^2+p^2}$ (lattice units) $E_{\pi\pi}$ (lattice units) $2\sqrt{m_\pi^2+p^2}$ (MeV) $E_{\pi\pi}$ (MeV)
------------------- --------------------------------------- ------------------------------ ----------------------------- --------------------
$p=0$ 0.3501 0.3509(13) 455 456(2)
$p=\pi/L$ 0.4372 0.4447(19) 568 578(2)
$p=\sqrt{2}\pi/L$ 0.5096 0.5316(50) 662 691(6)
$p=\sqrt{3}\pi/L$ 0.5729 0.577(20) 745 750(27)
: Values of the two pion energies ($E_{\pi\pi}$) extracted from the effective mass plots. These are compared to the value one would expect in the absence of interactions between the two pions, $2\sqrt{m_\pi^2+p^2}$, as a sanity check.[]{data-label="tb:EPiPi"}
[|c|c|c|c|c|c|]{}\
&$\Delta t=20$&$\Delta t=24$&$\Delta t=28$&$\Delta t=32$&Error Weighted Ave.\
$p=0$&0.00529(29)&0.00540(28)&0.00490(26)&0.00550(34)&0.00526(17)\
$p=\pi/L$&0.00882(55)&0.00923(59)&0.00813(57)&0.00924(68)&0.00884(37)\
$p=\sqrt{2}\pi/L$&0.0140(11)&0.0150(16)&0.0137(17)&0.0157(19)&0.01449(99)\
$p=\sqrt{3}\pi/L$&0.0249(47)&0.0278(62)&0.0191(49)&0.0262(91)&0.0240(38)\
\
&$\Delta t=20$&$\Delta t=24$&$\Delta t=28$&$\Delta t=32$&Error Weighted Ave.\
$p=0$&0.0864(34)&0.0917(37)&0.0863(40)&0.0947(45)&0.0895(28)\
$p=\pi/L$&0.0898(40)&0.0918(41)&0.0861(51)&0.0951(57)&0.0906(31)\
$p=\sqrt{2}\pi/L$&0.0895(69)&0.0912(76)&0.087(11)&0.113(15)&0.0931(61)\
$p=\sqrt{3}\pi/L$&0.092(19)&0.098(19)&0.120(32)&0.171(43)&0.111(14)\
\
&$\Delta t=20$&$\Delta t=24$&$\Delta t=28$&$\Delta t=32$&Error Weighted Ave.\
$p=0$&0.297(11)&0.315(12)&0.295(13)&0.324(15)&0.3068(91)\
$p=\pi/L$&0.319(14)&0.327(14)&0.302(17)&0.333(19)&0.320(11)\
$p=\sqrt{2}\pi/L$&0.336(25)&0.342(27)&0.319(37)&0.416(51)&0.347(21)\
$p=\sqrt{3}\pi/L$&0.377(68)&0.385(74)&0.45(14)&0.66(15)&0.437(51)\
Conclusion
==========
Lattice matrix elements were computed for $\Delta I=3/2$, $K\to\pi\pi$ decays on quenched $24^3\times 64$ lattices, with a pion mass of $m_\pi=227.6(6)\text{ MeV}$, a kaon mass of $m_K=564(2)\text{ MeV}$, and pion momentum $p_\pi=\sqrt{n}\pi/L=170\sqrt{n}\text{ MeV}$ where $n$ is the number of twists (0, 1, 2, or 3). It was necessary to reduce to the time separation between the kaon and the two pion sources to 20, 24, 28, and 32 units. An error weighted average of the results from all four separations produced an overall result with a smaller error. This error was of order 3-4% for pion momentum 0 and $\pi/L$, 7% for pion momentum $\sqrt{2}\pi/L$, and 15% for pion momentum $\sqrt{3}\pi/L$.
Future plans include an expansion of this study (on the same lattices) to include a larger range of pion and kaon masses in order to perform an extrapolation to the physical pion and kaon mass, and to the physical (on-shell) pion momentum using the range of momenta obtained from the different twists. The Lellouch-Luscher factor [@Lellouch_Luscher] will be used to convert the lattice matrix elements to physical decay amplitudes. The calculation will also be performed on the RBC/UKQCD $32^3\times 64$, $L_s=32$ lattices with 2+1 flavors of DWF that are currently being generated, with a pion mass of $m_\pi\le 250\text{ MeV}$, in order to obtain results in the full dynamical theory which can be compared with the results of the quenched approximation.
Acknowledgements
================
Thanks to Tom Blum, Norman Christ, Chris Dawson, Chulwoo Jung, Changhoan Kim, Qi Liu, Robert Mawhinney, Chris Sachrajda, and all of our colleagues in the RBC and UKQCD collaborations for helpful discussions and the development and support of the QCDOC hardware and software infrastructure which was essential to this work. In addition we acknowledge Columbia University, RIKEN, BNL and the U.S. DOE for providing the facilities on which this work was performed. This work was supported in part by U.S. DOE grant number DE-FG02-92ER40699. EG is supported by an STFC studentship and grant ST/G000557/1 and by EU contract MRTN-CT-2006-035482 (Flavianet).
[99]{}
CP-PACS Collaboration, J.I. Noaki et al., *Phys. Rev. D* [**68**]{} (2003) 014501 \[[hep-lat/0108013]{}\]. RBC Collaboration, T. Blum et al., *Phys. Rev. D* [**68**]{} (2003) 114506 \[[hep-lat/0110075]{}\]. Changhoan Kim, Ph.D. Thesis, Columbia University, 2004. Changhoan Kim and Christ, Norman H., *Nucl. Phys. Proc. Suppl.* [**119**]{} (2003) 365 \[[hep-lat/0210003]{}\]. C.H. Kim, *Nucl. Phys. Proc. Suppl.* [**140**]{} (2005) 381. T. Yamazaki, the RBC and UKQCD Collaborations, *Phys. Rev. D* [**79**]{} (2009) 094506 \[[hep-lat/0807.3130]{}\]. N.H. Christ and Li, S., *Proceedings of The XXVI International Symposium on Lattice Field Theory*, PoS(LATTICE 2008)272 \[[hep-lat/0812.1368]{}\]. M. Ciuchini et al., *Z. Phys. C* [**68**]{} (1995) 239 \[[hep-ph/9501265]{}\]. G. Buchalla et al., *Rev. Mod. Phys.* [**68**]{} (1996) 1125 \[[hep-ph/9512380]{}\]. C.T. Sachrajda and Villadoro, G., *Phys. Lett. B* [**609**]{} (2005) 73 \[[hep-lat/0411033]{}\] M. Lightman, *Proceedings of The XXVI International Symposium on Lattice Field Theory*, PoS(LATTICE 2008)273 \[[hep-lat/0906.1847]{}\]. L. Lellouch and Luscher, M., *Commun. Math. Phys.* [**219**]{} (2001) 31 \[[hep-lat/0003023]{}\].
[^1]: These results were not presented in the talk.
|
---
abstract: 'The generation of entangled photon pairs by parametric down–conversion from solid state CW lasers with small coherence time is theoretically and experimentally analyzed. We consider a compact and low-cost setup based on a two-crystal scheme with Type-I phase matching. We study the effect of the pump coherence time over the entangled state visibility and over the violation of Bell’s inequality, as a function of the crystals length. The full density matrix is reconstructed by quantum tomography. The proposed theoretical model is verified using a purification protocol based on a compensation crystal.'
author:
- Simone Cialdi
- Fabrizio Castelli
- 'Matteo G. A. Paris'
title: 'Properties of entangled photon pairs generated by a CW laser with small coherence time: theory and experiment'
---
Introduction
============
Generation of entanglement is the key ingredient of quantum information processing. In optical implementation with discrete variables the standard source of entangled photon pairs is parametric down–conversion in nonlinear crystals pumped by single-mode laser [@kwi99]. Recent advances in laser diodes technology allow the realization of simpler and cheaper apparatuses for the entanglement generation [@deh02; @cia], though the quality of the resulting photon pairs is degraded by the small coherence time of the pump laser.
In this paper we address theoretically and experimentally the generation of entanglement using laser diode pump as well as its application to visibility and nonlocality tests. We focus on the effects of the small coherence time and implement a purification protocol based on a compensation crystal [@nambu02] to improve entanglement generation. We reconstruct the full density matrix by quantum tomography and analyze in details the properties of the generated state, including purity and visibility, as a function of the crystals length and the coherence time of the pump. The topics is relevant for applications for at least two reasons. On one hand quantifying the degree of entanglement is of interest in view of large scale application. On the other hand a detailed characterization of the generated state allows one to suitably tailor entanglement distillation protocols.
The paper is structured as follows: In Section \[s:exp\] we describe the experimental apparatus used to generate entanglement, whereas Section \[s:wav\] is devoted to illustrate in details the quantum state of the resulting photon pairs in the ideal case. The effects of small coherence time are analyzed in Section \[s:pol\] and the experimental characterization of the generated states is reported in Section \[s:tom\]. Section \[s:bel\] is devoted to nonlocality test whereas Section \[s:out\] closes the paper with some concluding remarks.
The experimental apparatus {#s:exp}
==========================
A scheme of the experimental apparatus is shown in Fig. \[f:exp\]. The “state generator” consists of two identical BBO crystals, each cut for Type-I down-conversion, one half-wave plate (HWP) and one quarter-wave plate (QWP) as implemented in [@gog05]. The crystals are stacked back-to-back, with their axes oriented at $90^o$ with respect to each other [@kwi99; @deh02]. The balancing and the phase of the entangled states are selected by changing the HWP and QWP orientation.
The crystals are pumped using a 40mW, 405-nm laser diode (Newport LQC405-40P), with a spectral line that is typically broadened by phonon collisions. The coherence time of the pump light $\tau_c$, which is a fundamental parameter for our experiment, results 544 fs and correspond to a spectral width around 0.3 nm. We obtained this important information with a standard measurement of the first order correlation function.
![Experimental apparatus for generating and analyzing entangled states.[]{data-label="f:exp"}](schema2.eps){width="44.00000%"}
The generated photons are analyzed using adjustable QWP, HWP and a polarizer [@kwi01]. Finally light signals are focused into multimode fibers which are used to direct the photons to the detectors. The detectors are home-made single photon counting modules (SPCM), based on an avalanche photodiode operated in Geiger mode with active quenching. For the coincidence counting we use a TAC/SCA.
The nonlinear crystals are properly cut to generate photons into a cone of half-opening angle $3.0^o$ with respect to pump. The first crystal converts horizontally polarized pump photons into vertically polarized ($V$) signal and idler photons, while the second crystal converts vertically polarized pump photons into horizontally polarized ($H$) signal and idler photons. This configuration introduces a delay time $\Delta\tau$, depending on the crystal length, between the $V$ and the $H$ part of the entangled state, as discussed in the following Sections.
The state vector of the generated entangled photons {#s:wav}
===================================================
The pair of photons generated by SPDC of Type I from a single nonlinear crystal, having wave vectors $\vec{k}_{s}$ and $\vec{k}_{i}$, are represented by state vectors $|\vec{k}_{s}\rangle_s$ and $|\vec{k}_{s}\rangle_i$ for the signal and idler, respectively. The wavefunction appropriate to the system can be written as a superposition of these state vectors [@hong; @joo94; @joo96]: $$|\Psi\rangle = \int d^3\vec{k}_{s} \, d^3\vec{k}_{i} \,\,
A(\omega_p - \Omega_p^0) \,
F(\Delta k_{\perp}) \, f(\Delta k_{\parallel}) \,
|\vec{k}_{s}\rangle_s \, |\vec{k}_{i}\rangle_i$$ where $A(\omega_p - \Omega_p^0)$ is the spectral complex amplitude of the pump laser, which is a function of the pump frequency $\omega_p(k_p) = \omega_s(k_s) + \omega_i(k_i)$, assuming as usual the validity of the energy conservation in the generation process, and it is centered around the reference frequency $\Omega_p^0$. The factors $F$ and $f$ are mismatch functions depending on the variation of the transverse and longitudinal part of the pump wave vector with respect to the reference of momentum conservation, and are described in detail in the following.
![Geometry for the generation of photon pairs.[]{data-label="angoli"}](angoli2.eps){width="44.00000%"}
The function $F(\Delta k_{\perp})$ comes from a spatial integration over all the possible processes of photon generation within the pump transverse profile in the crystal, taking the first order approximation of the nonlinear interaction. For a Gaussian pump profile we obtain again a Gaussian function, with a width varying as the inverse of the beam waist $w$: $$\label{ftrasv}
F(\Delta k_{\perp}) = e^{-w^2 \, \Delta k_{\perp}^2 / 4}$$ where, referring to Fig. \[angoli\], one has $$\Delta k_{\perp} = k_s(\omega_s) \sin(\theta_s)
- k_i(\omega_i) \sin(\theta_i)$$ with the internal generation angles $\theta_s$ and $\theta_i$ for signal and idler, respectively. In our case the pump beam waist is near 2 mm, therefore we can consider exact transverse momentum conservation to a good approximation. In fact it is easy to verify that with this beam waist we have an angular gaussian width of $0.006^o$ around the reference internal angles $\Theta_s = \Theta_i = 1.8^o$ (derived from external angles $\Theta^{ext}_s = \Theta^{ext}_i = 3.0^o$ using Snell’s law), very small with respect to the acceptance angle of $0.074^o$ FWHM of the optical coupling devices. The conservation of the transverse wave vector permits to simplify the geometry of the system, by considering in the following a generation angle, say $\theta_i$, as a function of the other quantities $\omega_s, \omega_i, \theta_s$.
The mismatch function $f(\Delta k_{\parallel})$ has the same meaning of $F$, but derives from an integration along the crystal length $L_C$, and reads: $$\label{ff}
f(\Delta k_{\parallel}) =
\frac{\sin(\Delta k_{\parallel} L_C /2)}{\Delta k_{\parallel} L_C /2}$$ where $$\Delta k_{\parallel} = k_{p}(\omega_p)
-k_{s}(\omega_s) \cos(\theta_s)
- k_{i}(\omega_i) \cos\left[\theta_i(\omega_s, \omega_i, \theta_s)\right] \, .$$ As a matter of fact the pump spectrum width, yet determining the visibility effects, is very small with respect to the spectral width of the down–conversion; this means that $f$ is slightly dependent on $\omega_p = \omega_s + \omega_i$, as can be verified numerically. We will not consider such a dependence by substituting $\omega_p$ with the reference pump frequency $\Omega^0_p$ as the argument of $f$. This approximation turns out to be very good for crystal lengths below a few mm, but around 3 mm (our maximum crystal length) the conservation of the longitudinal wave vector starts to shrink the down–conversion spectrum. A similar consideration can be done over the dependence of $f$ over the internal angle $\theta_s$; being the experimental configuration highly collinear, the optical couplers are practically insensible to its variation (within the acceptance cone). Therefore we can substitute $\theta_s$ with the fixed reference angle $\Theta_s$, and the mismatch function becomes: $$\label{fo}
f (\omega_p, \omega_s, \theta_s) \approx f (\Omega^0_p, \omega_s, \Theta_s)
\equiv f( \omega_s )$$ The wavefunction of the photon pair can now be written in the simpler form: $$\begin{aligned}
|\Psi\rangle = & \int d\omega_p \, d\omega_s \, d\theta_s \,
A(\omega_p - \Omega^0_p) \,f(\omega_s) \nonumber \\
& \times |\omega_s, \theta_s \rangle_s \,\,
|\omega_p - \omega_s, \theta_i (\omega_p, \omega_s, \theta_s) \rangle_i\end{aligned}$$
In the first approximation we can solve for the integral over the internal generation angle $\theta_s$ because neither $A$ nor $f$ depend on it, but a more refined reasoning put forward the fact that the conservation of the transverse wave vector introduces a limitation in the effective spectral width of the mismatch function, hence affecting the integration over $\omega_s$. This happens because by varying $\omega_s$ around the down–converted reference $\Omega^0_p/2$, the idler angle $\theta_i$ may go outside from the optical coupler acceptance limit, as verified by means of the experimental data discussed in Appendix A. This problem does not affect the integral over $\omega_p$ for the smallness of the pump spectral width. To take care of this spectral limitation we introduce a correction factor $R(\Omega^0_p/2, \Delta\omega_s)$ centered around the reference $\Omega^0_p/2$ and having the limited spectral width $\Delta\omega_s$ (see Appendix A). Defining $\tilde{f}(\omega_s) = f(\omega_s) \cdot R$, we arrive at this wavefunction for the photon pairs: $$\label{wf1}
|\Psi\rangle = \int d\omega_p \, d\omega_s \, A(\omega_p - \Omega^0_p) \,
\tilde{f}(\omega_s) \,\, |\omega_s \rangle_s \,\, |\omega_p - \omega_s \rangle_i$$
This expression is used to construct the proper wavefunction (or the proper state vector) for the entangled state generated in our experiment using the pair of oriented crystals [@kwi99; @deh02], as described in the previous Section. In particular we consider a suitable superposition of the single crystal wavefunctions of eq. (\[wf1\]), introducing the degree of freedom of polarization on state vectors, because the first crystal generates a vertical polarized $(VV)$ and the second crystal generates a horizontal polarized $(HH)$ photon pairs, respectively. Moreover we have a delay time between these pairs, due to the different optical length of the photon trajectories in the inner of crystals. This can be represented in the model by assuming photon generation in the crystals middle [@brida1; @brida2] (in Appendix B we show that this is a very good approximation) and introducing propagation factors for the internal state transport.
![Entangled photon generation and propagation inside the crystals. For clarity, only signal photon trajectories (red lines) are drawn (idler ones are symmetrically upset). The horizontal blue line is the pump ray. $(o)$ and $(e)$ indicates ordinary and extraordinary rays, respectively. $L_c$ is the length of both crystals.[]{data-label="f:delay"}](delay2.eps){width="44.00000%"}
In the Fig. \[f:delay\] a sketch of the geometry for entangled photon generation is shown, limited for clarity to the signal trajectories. In the first crystal $V$ photons are generated, the state is $|V, \omega_s \rangle_s \,\, |V, \omega_p - \omega_s \rangle_i $, and the complex exponential for the product of the signal and idler propagation factor (as required by the form of eq. (\[wf1\])) till to exiting the crystal is given by:
$$P(V) = \exp \left\{ i L_C \left[ k^o_s(\omega_s)\frac{1}{2 \cos(\phi_1)} +
k^o_i(\omega_p - \omega_s) \frac{1}{2 \cos(\phi_1)} +
k^e_s(\omega_s) \frac{1}{\cos(\phi_2)} + k^e_i(\omega_p - \omega_s)
\frac{1}{\cos(\phi_2)} \right] \right\}$$
where the superscripts $(o)$ and $(e)$ on wave vectors indicates ordinary and extraordinary propagation, and the angles $\phi_1 = 1.807, \, \phi_2 = 1.84$ can be found using the laws of wave rays in birefringent crystals [@yar] and Snell’s law, under the request of an exit angle of $3^o$. For the second crystal, in which $H$ photons are generated and the state is $|H, \omega_s \rangle_s \,\, |H, \omega_p - \omega_s \rangle_i $, the respective propagation factor is: $$P(H) = \exp \left\{ i L_C \left[\frac{k^o_p(\omega_p)}{2} +
\frac{k^e_p(\omega_p)}{2} + k^o_s(\omega_s) \frac{1}{2\cos(\phi_3)} +
k^o_i(\omega_p - \omega_s) \frac{1}{2\cos(\phi_3)} \right] \right\}$$ where $\phi_3 = 1.806$, and it has been included the propagation of the pump ray from the generation point of the $(VV)$ pair (note that $\phi_3$ is slightly different from $\phi_2$ due the different refraction index for $o$ and $e$ propagation). The entangled state wavefunction is therefore: $$| \Psi \rangle = \, \int \, d\omega_p \, d\omega_s \,
A(\omega_p - \Omega^0_p) \, \tilde{f}(\omega_s)
\frac{1}{\sqrt{2}} \left\{ P(H) \, |H, \omega_s \rangle_s |H, \omega_p-\omega_s \rangle_i
+ P(V) \,
|V, \omega_s \rangle_s |V, \omega_p-\omega_s \rangle_i \right\}
\label{propaga}$$
This expression can be recast in a more useful form in the following way. Let’s write the frequencies as $\omega_p = \Omega_p^0 + \Omega_p$, $\omega_s =
\Omega^0 + \Omega$ (with of course $\Omega_p^0 = 2 \Omega^0$), where $\Omega_p$ and $\Omega$ represent the frequency shift with respect to reference for the pump and for the down conversion, respectively. Now, in the propagation factors we introduce a first order approximation for the wave vectors putting: $$k_p(\omega_p) \approx k(\Omega_p^0) + \Omega_p / V_p \,\,, \hspace{1.cm}
k_s(\omega_s) \approx k(\Omega^0) + \Omega / V \,, \hspace{1.cm}
k_i(\omega_p - \omega_s) \approx k(\Omega^0)
+ \Omega_p / V - \Omega / V$$ where $V_p$ and $V$ are the proper group velocities of the pump and of the down converted signal, and these relations must be considered both for the ordinary wave and for the extraordinary wave. With these substitutions, and rewriting for future convenience the quantum states by factorizing the polarization part from the frequency one, the final form of the wavefunction eq. (\[propaga\]) read: $$\begin{aligned}
| \Psi \rangle &=& \, \int \, d\Omega_p \, d\Omega \, A(\Omega_p) \,
\tilde{f}(\Omega^0 + \Omega) \, \frac{1}{\sqrt{2}} \cdot \nonumber \\
& & \hspace{1.cm} \cdot \left\{ e^{i(\phi_H + \tau_H \Omega_p)} \,
|H \rangle_s |H \rangle_i |\Omega \rangle_s |\Omega_p-\Omega \rangle_i
+ e^{i(\phi_V + \tau_V \Omega_p)} \, |V \rangle_s |V \rangle_i
|\Omega \rangle_s |\Omega_p-\Omega \rangle_i \right\}
\label{propaga2}\end{aligned}$$ where the phase terms coming from propagation factors are the sum of a constant phase:
$$\begin{aligned}
\phi_H &=& \left\{ k^o(\Omega_p^0) + k^e(\Omega_p^0) +
\frac{2 \, k^o(\Omega^0)}{\cos(\phi_3)} \right\} \frac{L_C}{2} \,\, , \\
\phi_V &=& \left\{ \frac{2 \, k^o(\Omega^0)}{\cos(\phi_1)} +
\frac{4 \, k^e(\Omega^0)}{\cos(\phi_2)} \right\} \frac{L_C}{2}\end{aligned}$$
and frequency dependent terms $\tau_H \Omega_p$, $\tau_V \Omega_p$ containing the total propagation time inside the crystals: $$\begin{aligned}
\tau_H &=& \, \left\{ \frac{1}{V_p^o} + \frac{1}{V_p^e} + \
\frac{1}{V^o \cos(\phi_3)} \right\} \frac{L_C}{2} \,, \\
\tau_V &=& \left\{ \frac{1}{V^o \cos(\phi_1)} + \frac{2}{V^e \cos(\phi_2)}
\right\} \frac{L_C}{2}\end{aligned}$$
It is important to note that these delay factors depend on pump frequency (not on the down converted frequency); this fact can be interpreted saying that the states $(HH)$ and $(VV)$ exiting the crystals are generated from the pump in two different temporal events in the past, depending on the different trajectories across the crystals. For all these four phase factors, their numerical value are determined from the data on refraction indexes and group velocities taken from ref. [@snlo], and listed in the following Table:
----- ----------- ----------- ----------- -----------
(o) (e) (o) (e)
$n$ 1.691719 1.659273 1.659984 1.632171
$V$ c/1.77878 c/1.73901 c/1.68376 c/1.65483
----- ----------- ----------- ----------- -----------
As a final observation, we note that in writing the final expression for the wavefunction eq. (\[propaga2\]), it has been discarded the variation of the propagation factors with respect to the propagation angles. Due to small angular acceptance of the detectors, it is possible to verify that, with excellent approximation, this dependence does not introduce any relevant effect.
The polarization density matrix {#s:pol}
===============================
For the calculation of the density matrix and the complete characterization of the wavefunction it is important to define at best the statistical properties of the CW pump radiation, because our experimental data depend strongly on its coherence length. In the temporal domain, this light is characterized by a (real) constant amplitude $A_0$ and a rapidly varying phase with a characteristic time equal to the coherence time of the pump $\tau_c$. Therefore we can write: $$\int \, d\omega \, A(\omega) \, e^{i \omega t} \, = \,
A_0 \, e^{i\delta(t)}
\label{laser}$$ where $\delta(t)$ is a proper fluctuating phase. The pump amplitude in the temporal domain can be considered as the Fourier transform of the complex spectral amplitude over a large time interval $\Delta T$: $$A(\omega) \, = \, \frac{1}{2 \pi} \, \int_{\Delta T} \, d t \,
A_0 \, e^{i\delta(t)} \, e^{- i \omega t}
\label{laser2}$$
Our experiment mainly concerns the reconstruction of the density matrix of the entangled system on the basis composed by the four signal and idler polarization combinations $HH, HV, VH, VV$. The relative density operator $\rho$, from which we derive the reduced density matrix on this polarization basis, is obtained from the full density operator $\rho_{\mbox{tot}} = | \Psi \rangle \langle \Psi|$ by tracing over frequencies, *e.g.* by integrating over the frequency state matrix elements:
$$\rho = \int d\omega'_p \, d\omega' \, _i\langle \omega'_p - \omega'
| \, _s\langle \omega' | \Psi \rangle \langle \Psi | \omega'
\rangle_s | \omega'_p - \omega' \rangle_i \label{densita1}$$
corresponding to the fact that we do not perform frequency measurements.
The form of the wavefunction in eq. (\[propaga2\]) implies that only four elements of the $4 \times 4$ reduced density matrix are different from zero. Using the general relation $\langle \omega | \omega^{'} \rangle = \delta (\omega - \omega^{'})$, we straightforwardly obtain for the first diagonal element: $$\rho_{HH,HH} = \frac{1}{2} \int d\omega \, | f(\omega) |^2 \, \int
d\omega_p \, | A(\omega_p) |^2 \, = \, \frac{1}{2} \, \epsilon \,
A_0^2 \, \frac{\Delta T}{2\pi}
\label{rhoHHHH}$$ where we put $\epsilon = \int |f(\omega)|^2$ and $\int d\omega_p \,
|A(\omega_p)|^2 = A_0^2 \, \Delta T / 2\pi$ from eq. (\[laser2\]). With similar calculation the other nonzero diagonal element results $\rho_{VV,VV} = \rho_{HH,HH}$, as expected by symmetry arguments.
For the two off diagonal elements one has $\rho_{HH,VV} = \rho^*_{VV,HH}$, and in particular:
$$\begin{aligned}
\rho_{HH,VV} &=& \frac{1}{2} \, \int d\omega | f(\omega) |^2 \, \int d\omega_p
\, | A(\omega_p) |^2 \, e^{-i(\phi_H - \phi_V)} \, e^{-i \omega_p(\tau_H - \tau_V)}
\nonumber \\
&=& \frac{1}{2} \, \epsilon \, e^{-i\phi} \int d\omega_p \, |A(\omega_p)|^2
\, e^{-i\omega_p (\tau_H - \tau_V)}\end{aligned}$$
where we put $\phi = \phi_H - \phi_V$. With the Wiener-Khinchine theorem this frequency integral can be recast as a two time correlation function over the interval $\Delta
T$, which can be taken very large with respect to the coherence time of the pump, and smaller than the detector response time:
$$\int d\omega_p \, | A(\omega_p) |^2 \, e^{-i \omega_p (\tau_H - \tau_V)}
\, = \, A_0^2 \frac{\Delta T}{2\pi} \left( \frac{1}{\Delta T} \int
_{\Delta T} dt \, e^{-i \delta(t) + i \delta \left( t -(\tau_H - \tau_V)\right)}
\right) = A_0^2 \frac{\Delta T}{2\pi} \, e^{- \Delta\tau / \tau_c}$$
where $\Delta \tau = |\tau_H - \tau_V|$, and the result is taken from Ref. [@blu04]. If $\Delta \tau \gg \tau_c$ we have an incoherent superposition of random phases and the average of the complex exponentials tends to zero, otherwise we have a coherent sum, and the integral tends to one.
Finally, setting the state generator QWP in order to have $\phi = 0$ (see Ref. [@kwi99]) and putting for simplicity $p = e^{-\Delta
\tau /\tau_c}$, the reduced density matrix is:
$$\label{matrix2}
\begin{array}{cc}
& HH \hspace{0.7cm} HV \hspace{0.7cm} VH \hspace{0.7cm} VV \\
\begin{array}{c}
HH \\
HV \\
VH \\
VV \\
\end{array} &
\left(%
\begin{array}{cccc}
\frac{1}{2} & \hspace{0.9cm} 0 \hspace{0.7cm} &
\hspace{0.7cm} 0 \hspace{0.9cm} & \frac{1}{2} \, p \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
\frac{1}{2} \, p & 0 & 0 & \frac{1}{2}\\
\end{array}%
\right)
\end{array}$$
which can also be conveniently derived from a sum of two distinct density matrix, one of a pure entangled state and the other of a statistical mixture: $\rho = p \: \rho_e + (1-p) \rho_m$ where $\rho_e =
| \Psi_e \rangle \langle \Psi_e |$ with $| \Psi_e \rangle =
\frac{1}{\sqrt{2}}(|HH\rangle + |VV\rangle)$ and $\rho_m = \frac{1}{2} |
HH \rangle \langle HH | + \frac{1}{2} | VV \rangle \langle VV |$. This is the more suitable form used for a comparison between theory and experimental data.
Experimental tomographic reconstruction of the density matrix and correlation visibility {#s:tom}
========================================================================================
In order to fully characterize the generated states at the quantum level we employ quantum tomography of their density matrices [@par04]. The experimental procedure goes as follows: upon measuring a set of independent two-qubit projectors $P_\mu= |\psi_\mu\rangle
\langle\psi_\mu |$ $(\mu=1,...,16)$ corresponding to different combinations of polarizers and phase-shifters, the density matrix may be reconstructed as $\varrho=\sum_\mu p_\mu\,
\Gamma_\mu$ where $p_\mu = \hbox{Tr}[\varrho \, P_\mu]$ are the probabilities of getting a count when measuring $P_\mu$ and $\Gamma_\mu$ the corresponding dual basis, [*i.e.*]{} the set of operators satisfying $\hbox{Tr}[P_\mu\,\Gamma_\nu] =
\delta_{\mu\nu}$ [@dar01]. Of course in the experimental reconstruction the probabilities $p_\mu$ are substituted by their experimental samples [*i.e.*]{} the frequencies of counts obtained when measuring $P_\mu$. In order to minimize the effects of fluctuations and avoid non physical results we use maximum-likelihood reconstruction of two-qubit states [@kwi01; @ban00]. At first we write the density matrix in the form $$\label{Eq:rhoTT}
\hat{\varrho} \, = \, \hat{T}^{\dagger} \, \hat{T}\;,$$ which automatically guarantees that $\hat{\varrho}$ is positive and Hermitian. The remaining condition of unit trace $\hbox{Tr}\hat{\varrho} = 1$ will be taken into account using the method of Lagrange multipliers. In order to achieve the minimal parametrization, we assume that $\hat{T}$ is a complex lower triangular matrix, with real elements on the diagonal. This form of $\hat{T}$ is motivated by the Cholesky decomposition known in numerical analysis [@Cholesky] for arbitrary non negative Hermitian matrix. For an $M$-dimensional Hilbert space, the number of real parameters in the matrix $\hat{T}$ is $M+2M(M-1)/2=M^2$, which equals the number of independent real parameters for a Hermitian matrix. This confirms that our parametrization is minimal, up to the unit trace condition.
In numerical calculations, it is convenient to replace the likelihood functional by its natural logarithm, which of course does not change the location of the maximum. Thus the function subjected to numerical maximization is given by $$%\label{eq:lt}
L(\hat{T}) = \sum_{k=1}^N \ln \hbox{Tr}(\hat{T}^\dagger
\hat{T} P_{\mu_k}) - \lambda \hbox{Tr}(\hat{T}^\dagger
\hat{T})\;,
\label{loglik}$$ where $\lambda$ is a Lagrange multiplier accounting for normalization of $\hat \varrho$ that equals the total number of measurements $N$. This may be easily proved upon writing $\hat{\varrho}$ in terms of its eigenvectors $| \phi_\mu
\rangle $ as $
\hat{\varrho} = \sum_{\mu } y_{\mu}^{2} | \phi_\mu
\rangle \langle \phi_\mu |
$, with real $y_{\mu}$, the maximum likelihood condition $\partial L/\partial y_{\nu} = 0$ reads $$\lambda y_{\nu} = \sum_{k=1}^{N} \frac{y_\nu \langle \phi_\nu
| P_{\mu_k} | \phi_\nu \rangle}
{\hbox{Tr}(\hat\varrho P_{\mu_k}}\;,$$ which, after multiplication by $y_{\nu}$ and summation over $\nu $, yields $\lambda = N$.
The above formulation of the maximization problem allows one to apply standard numerical procedures for searching the maximum over the $M^2$ real parameters of the matrix $\hat{T}$. The examples presented below use the downhill simplex or the simulated annealing methods [@Ameba]. Results of the reconstruction are reported for crystals with three different thicknesses, precisely $0.5$, $1$ e $3$ mm, and in the case of compensation of the delay time between generated photons, as discussed later. Moreover we present an analysis on the direct measurement of the visibility of the entangled state.
![Entangled state visibility as a function of the polarizer angle, for generating crystals of $0.5$, $1$ e $3$ mm thickness.[]{data-label="f:corr"}](correlazioni.eps){width="44.00000%"}
Data on correlation visibility are simply obtained by removing the HWP and QWP plates of the tomographic analyzer (see Fig. \[f:exp\]) and detecting the signal and idler coincidence counts in a time interval, as a function of the signal polarizer angle, and having fixed the idler polarization angle at $45^o$. The theoretical prediction is: $$\begin{aligned}
P(\xi_s, 45^o_i) & = \:
_i\langle 45^o | \:\, _s\langle \xi_s | \: \rho \:
| \xi_s \rangle_s \, | 45^o \rangle_i \, \nonumber \\ & = \,
\frac{1}{2} p \: (\cos(\xi_s - 45^o))^2 + \frac{1}{4} (1-p)
\label{correlazioni}\end{aligned}$$ where $\xi_s$ is the angle of the signal polarizer in the counter–clockwise direction, with the horizontal axis as the $0^o$ reference. As it is apparent from this formula, when $p$ is near the unity (delay time smaller with respect the coherence time of the pump) the oscillating contribute due to the non–local correlations is dominant. On the contrary, with greater delay time (and small $p$) the correlations are washed out and the result is that of a statistical mixture which does not depend on the angle. In particular, the maximum of $P(\xi_s, 45^o_i)$ is at $45^o$, while the minimum is at $135^o$, hence we can write explicitly the visibility $\mathcal{V}$ of the oscillation as: $$\mathcal{V }= \frac{P(45^o_s, 45^o_i) - P(135^o_s, 45^o_i)}
{P(45^o_s, 45^o_i) + P(135^o_s, 45^o_i)} = \, p
\label{visibilita}$$
![Tomographic reconstruction of the generated state for three different crystals. The measured and calculated visibility are shown in table.[]{data-label="f:rec"}](tomo2.eps){width="44.00000%"}
![Tomographic reconstruction with a delay compensation crystal (see text). (a) Crystal angle set for maximum compensation, visibility 0.66. (b) Crystal angle at $90^o$ with respect to (a), visibility 0.17.[]{data-label="f:comp"}](comp2.eps){width="44.00000%"}
In Fig. \[f:corr\] we show the visibility measurements as a function of the signal polarizer angle for the three different crystal pairs, with the theoretical prediction of eq. (\[visibilita\]) indicated by a full line. The comparison between the theoretical density matrix elements of eq. (\[matrix2\]) and their tomographic reconstruction from experimental data is shown in Fig. \[f:rec\]. It is confirmed that the off diagonal elements tend to reduce in magnitude for larger crystal thickness; in particular for $3$ mm crystals we obtain the density matrix of a statistical mixture.
In our model the lack of visibility of the entangled state is fully ascribed to the decoherence effect due to the fluctuating phase difference between $H$ and $V$ parts of the SPDC, depending on the delay $\Delta\tau$. Having a very small area of the fiber collimator, we have neglected any decoherence of spatial origin, which introduces a phase variation depending on the detector viewing angle. In order to verify this statement, we have performed a series of measurements with the 3 mm crystal, putting windows of 0.5 mm linear aperture in front of the collimators: if the decoherence had a spatial contribution, we would have expected an increasing in the state purity. In fact, the results of the state reconstruction were the same as the original configuration, thus supporting our hypothesis.
This fact also suggests how to improve the purity of the entangled state by a phase retardation on the $H$ polarized part of the pump with respect to the $V$ polarized part, to get $\delta_H(t +
\Delta\tau) = \delta_V (t)$. This can be approximatively accomplished by inserting, along the pump ray and before the state generator, a properly oriented BBO crystal with a suitable length. We performed a series of measurements using the 1 mm double crystal as state generator, and a 3 mm single crystal as pump phase retarder. By varying the orientation of the axis of this crystal, we have compensation or enhancement of the effect of the time delay between the parts of the generated entangled state. In particular, the visibility is expected to vary from a maximum to a minimum for a $90^o$ change in orientation, as confirmed by the tomographic reconstruction shown in Fig. \[f:comp\]. Notice that the maximum visibility of 0.66 is larger than the corresponding visibility without the auxiliary crystal (see Fig. \[f:rec\]), thus demonstrating a partial time delay compensation.
Measurements on the violation of Bell’s inequality {#s:bel}
==================================================
We have also performed a series of measurements of the S parameter, characterizing the Bell’s inequality in the CHSH version [@chsh], for a comparison with the prediction of our theoretical model. To obtain reliable data on the S parameter we used the same experimental apparatus previously employed for correlation measurements. We considered as usual the 16 different configuration of the polarization angles on the signal an and on the idler [@deh02].
The Bell S parameter is theoretically defined as: $$S = E(a,b) - E(a,b') + E(a',b) + E(a',b')
\label{parS}$$ where the arguments $a, a'$ and $b, b'$ are the selected angles for signal polarizer and idler polarizer, respectively. The function $E$ is defined as $E(\alpha,\beta) = P(\alpha,\beta) + P(\alpha^{\perp},\beta^{\perp}) - P(\alpha,\beta^{\perp}) -
P(\alpha^{\perp},\beta)$, where $\alpha^{\perp} = \alpha + 90^o$ e $\beta^{\perp} = \beta + 90^o$. The function $P$ is exactly that described in eq. \[correlazioni\], but with the idler angle specified by the argument. For any realistic local theory one has $|S| \leq 2$, while for quantum mechanics $|S|$ can be greater than $2$, reaching a maximum value of $2 \sqrt{2}$. The following choice for the angles is used: $a = 0^o$, $b = \theta$, $a' = b + \theta$ and $b' = a' + \theta$. In this way $S$ is a function of the angle $\theta$ alone. In Fig. \[f:S-vs-theta\] we show the calculated $S(\theta)$ for three states with different visibility $\mathcal{V} = p = (1, 0.7, 0.5)$; by decreasing $p$, the values of $S$ tend to return in the limit of a local theory.
![S parameter as a function of $\theta$ for three different visibility values. Full line: $\mathcal{V} = 1$; dashed line; $\mathcal{V} = 0.7$; dotted line $\mathcal{V} = 0.5$.[]{data-label="f:S-vs-theta"}](S-vs-theta.eps){width="44.00000%"}
![Experimental results for S parameter for three values of $\theta$, compared with the theoretical S for a visibility of 0.77.[]{data-label="f:Smisurato"}](Smisurato.eps){width="44.00000%"}
For a comparison with these results of our model, we have measured the $S$ parameter for three different angles using as state generator the pair of $0.5$ mm crystals, that is the case with higher visibility. In Fig. \[f:Smisurato\] we show the theoretical curve of the $S$ parameter for a visibility equal to $0.77$ (full line), together with two other curves (dashed lines) indicating the extremal of the experimental errors, relative to the limited number of count during data acquisition. The three measurements of $S$ for the angles of $16^o, 24^o, 40^o$ are indicated with error bars. In particular we get $S(16^o) = 2.38 \pm 0.03$, $S(24^o) = 2.417 \pm 0.025$ and $S(40^o) = 0.80 \pm 0.05$. From these data we can conclude that in the case of $24^o$ the Bell’s inequality is violated for more than $17$ standard deviations.
Conclusions {#s:out}
===========
We have analyzed, both theoretically and experimentally, the generation of polarization-entangled photon pairs by parametric down–conversion from solid state CW lasers with small coherence time. In particular, we have analyzed in some details a compact and low-cost setup based on a two-crystal scheme with Type-I phase matching. The effect of pump coherence time on the entanglement and the nonlocality has been studied as a function of the crystals length. The full density matrix has been reconstructed by quantum tomography and the proposed theoretical model is verified using a purification protocol based on a compensation crystal. We conclude that laser diodes technology is of interest in view of large scale application and that its that the characterization of the generated state allows one to suitably tailor entanglement distillation protocols.
Acknowledgements {#acknowledgements .unnumbered}
================
MGAP thanks Maria Bondani e Marco Genovese for useful discussions. We are also indebted with Stefano La Torre for his help in detector realization.
Measurement of the coherence length
===================================
In this Appendix we experimentally verify that the spectrum of the down–converted signal is reduced when coincidence photon counts are performed, as a consequence of the trasverse momentum conservation.
If we observe only a single photon of the generated pair, the part of the spectrum incident on the coupling device is described in practice by the mismatch function $f(\omega_s)$ defined in Eqs. (\[ff\]) and (\[fo\]). But if we observe both photons and measure the simultaneous counts between signal and idler, we will detect a spectrum with a smaller width, and therefore we have a greater coherence length of the radiation. This because if we have a very wide spectrum for the signal at a fixed angle of observation, the idler photons, correlated with the signal photons also by transverse momentum conservation, will be dispersed over an angle that can be wider with respect to the acceptance angle of the coupling device. Hence the pair of coupling devices work as a filters limiting the spectral window for observation. To the purpose of a determination of this effective spectral width, we present here some measurements using interference methods. In particular we performed two series of measurements, the first relative to the direct counts on a single detector to find the width associated with $f(\omega_s)$, the second relative to the coincidence counts on the two detectors to determine the width of the corrected mismatch function $\tilde{f}(\omega_s)$ used in eq. (\[wf1\]).
![Sketch of the experimental apparatus for the measure of the coherence length.[]{data-label="f:schema-coerenza"}](schema-coerenza.eps){width="44.00000%"}
In Fig. \[f:schema-coerenza\] we show the experimental scheme (based on a single BBO crystal) employed for these types of measurements. An interferometer equal to that described in [@gog05] is placed among the signal ray. This interferometer is easy to align and is a very stable device. In both types of counting measurements we expect to see interference fringes as a function of the delay time introduced by the interferometer between two optical paths, and within the radiation coherence time. In particular we would determine a greater coherence length in the case of coincidence counts with respect to the case of signal single counts.
The theoretical description of the interferometric experiment is as follows. In the case of a single photon observation, and with a crystal generating horizontal photons, the density matrix for the signal before the interferometer can be built with the wavefunction of eq. (\[propaga2\]) of Section \[s:wav\] by neglecting vertical polarization states, and tracing over the idler frequency:
$$\rho_1 \, = \, \int \, d\Omega \,
|f(\Omega^0 + \Omega)|^2 \,\, |H, \Omega \rangle_s \,\, _s \langle
H, \Omega |$$
where we do not consider the immaterial propagation factor and use the original mismatch function of eq. (\[fo\]).
The density matrix after the interferometer follows by considering: (a) a polarization rotation of $45^o$ due to the HWP plate placed before the first calcite crystal; (b) the delay time $\tau$ introduced by the interferometer between the $H$ and $V$ parts; (c) the projection of these states over the axis of the final polarizer oriented at $45^o$, placed before the coupling device. The final density matrix is easy obtained as:
$$\rho_1 \, = \, \int \, d\Omega \,
|f(\Omega^0 + \Omega)|^2 \, \frac{1}{4} \,
\left| 1 + e^{i \Omega \tau} \right|^2 \,
|45^o, \Omega \rangle_s \,\, _s \langle
45^o , \Omega |$$
The probability to observe a count on the detector is then proportional to:
$$\begin{aligned}
P_1(\tau) \, i& = \, \int \, d\Omega^{'} \, _s\langle 45^o,
\Omega^{'}| \rho_1 | 45^o, \Omega^{'} \rangle_s
\nonumber \\ & = \,
\int \, d\Omega \,
|f(\Omega^0 + \Omega)|^2 \, \frac{1}{4} \, \left| 1 + e^{i \Omega \tau} \right|^2
\label{prob1}\end{aligned}$$
The width of the interference pattern representing count numbers as a function of the delay $\tau$, is given by a factor similar to a Fourier transform of the down-converted power spectrum; hence this width scales as the inverse of the spectral power width of the function $f$.
In the case of signal and idler coincidence counting, the state vector is again derived from eq. (\[propaga2\]), by taking only the $H$ part and discarding the propagation factor. After the passage in the interferometer, it is straightforward to see that the coincidence probability is the same as for the single count probability by replacing $f(\Omega^0 + \Omega)$ with the modified mismatch function $\tilde{f}(\Omega^0 + \Omega)$. Hence in this case the width of the interference curve is governed by the modified spectral power width $\Delta \omega_s$ of $\tilde{f}$.
![Interference patterns: (left) single signal counts; (right) signal and idler coincidence counts.[]{data-label="f:confronto-corr"}](confronto-corr.eps){width="44.00000%"}
In Fig. \[f:confronto-corr\] we show on the left the interference pattern obtained with signal single counts, using the BBO crystal of 3 mm length. The width of the curve is near $30$ fs, corresponding to a down converted spectrum of about $64$ nm. On the right we show the pattern in the case of signal and idler coincidence counts: the coherence time is enlarged to $70$ fs, corresponding to a spectral width of $27$ nm. In both cases the coherence length is well below that of the pump light. These data are used to determine the appropriate correction factor $R(\Delta \omega_s) = \tilde{f}(\omega_s) / f(\omega_s)$ in the definition of the wavefunction eq. (\[wf1\]).
Complete calculation of the delay times in state generation
============================================================
In deriving the delay time between $(HH)$ and $(VV)$ photons, we assumed state generation in the crystals middle. But in fact these states can be generated in any point in the inner of the crystals, therefore the propagation factors $P(H)$ and $P(V)$ must be position dependent. Let’s indicate with $z_1$ and $z_2$ the longitudinal coordinates of the internal generation points for the first crystal and for the second crystal, respectively. Referring to the Fig. \[f:delay\], we now have the following two equations for the propagation times $\tau_H$ and $\tau_V$: $$\tau_H(z_1,z_2) = \frac{L_C - z_1}{V_p^o} + \frac{z_2}{V_p^e} +
\frac{L_C - z_2}{V^o \cos(\phi_3)}$$
$$\tau_V(z_1,z_2) = \frac{L_C - z_1}{V^o \cos(\phi_1)} + \frac{L_C}{V^e \cos(\phi_2)}$$
Generally speaking, the state visibility $p$ would depend on the delay time $\Delta \tau(z_1,z_2) = \tau_H(z_1,z_2) - \tau_V(z_1,z_2)$. Because we do not have any information about the effective position in which a particular photon pair is generated, we consider an average over the possible positions, by integrating with a flat distribution probability: $$p_z = \frac{1}{{L_C}^2} \, \int dz_1 \, dz_2 \,\,
e^{- \Delta \tau(z_1,z_2) / \tau_c}$$
In Fig. \[f:delay-z\] we show a comparison between the visibility for state generation in the crystals middle, and that obtained from the above formula. It is clear that there is some difference only for very small visibilities obtained with very long crystals.
![Entangled state visibility as a function of the crystal length for the different assumptions on the state generation position. Full line: in the crystals middle; Dotted line: in the whole crystal length.[]{data-label="f:delay-z"}](delay-z.eps){width="44.00000%"}
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|
---
author:
- |
M. Auger, A. Ereditato, D. Goeldi , I. Kreslo[^1], D. Lorca, M. Luethi, C. Rudolf von Rohr, J. Sinclair, and M. S. Weber\
Laboratory for High Energy Physics\
Albert Einstein Center for Fundamental Physics\
Universität Bern, Switzerland\
E-mail:
title: 'Multi-channel front-end board for SiPM readout'
---
=1
Introduction
============
Geiger-mode multi-pixel avalanche photo-diodes, also known as Silicon Photo-multipliers (SiPMs) are gradually taking over the market of photon-counting sensors from the vacuum photo-multiplier tubes (PMT). Compared to PMTs, SiPMs are robust, insensitive to vibrations and magnetic field, faster, require much less electric power and provide excellent single-photon resolution. Several examples of charged particle detectors utilizing SiPMs to detect scintillating light can be seen here [@T2K; @SBND; @DUNE].
The front-end electronic board (FEB) design is driven by requirements of the Cosmic-Ray Tracking subsystem of the Short Baseline Near Detector (SBND) Liquid Argon TPC [@SBND].
The subsystem is constructed of 142 panels with strips of scintillator. The panel contains 16 strips, each equipped with 2 wavelength-shifting fibers. The fibers guide converted scintillation light to the end of the strip, where it is read by a pair of SiPMs. The module contains 32 SiPMs to read out. The FEB is designed to serve one such module. The coincidence between signals from two SiPMs from the same strip allows significant reduction of the dark pulse rate.
The summary of the functionality implemented in the FEB is listed below:
1. Provides bias voltage in the range of 62-82 V (extendable to 20-90 V) individually adjustable for each of 32 MPPCs;
2. Amplifies and perform shaping of the MPPC output pulse on each of 32 channels;
3. Performs discrimination of shaped signal at configurable level from 0 to 50 SiPM photo-electrons;
4. Provides basic coincidence of signals from each pair of adjacent channels (optional);
5. Allows to trigger only on events that are validated by external signals, such as, by an event in a group of other FEBs;
6. Generates trigger for digitization of the signal amplitude;
7. Generates time stamp w.r.t. the input reference pulse with an accuracy of 1.3 ns;
8. Performs digitization of signal amplitude of each of the 32 channels;
9. Provides on-board data buffering;
10. Provides efficient back-end communication based on Ethernet standard;
11. Allows firmware upgrade via back end Ethernet link.
The power requirements are: +5V with the consumption ranging from 450 mA to 550 mA depending on channel configuration.
General view and block-scheme
=============================
The board is realized in a custom compact form-factor suitable for mounting at the edge of a scintillating detector as shown in Figure \[fig-view\]. At one side a 72-pin SiPM connector is located, the other side hosts a power connector, two RJ45 network jacks and four LEMO connectors for reference and control signals.
The block-scheme of the board is shown in Figure \[fig-block\]. The analog input signal is processed by a CITIROC 32-channel ASIC from Omega [^2][@citiroc]. For each channel the chip provides charge amplifier with configurable gain, fast shaping with a peaking time of 15 ns and slow shaping with configurable peaking time from 12.5 ns to 87.5 ns. Signals from the fast shaper are discriminated at configurable level and produce digital signals (T0-T31) for triggering an event. These 32 signals are routed to the XILINX Spartan-6 FPGA chip, where the basic input coincidence and event triggering logic is realized. The analog signals for all channels can be stored in the ASIC Sample-and-Hold (S/H) circuit and multiplexed to a single analog output. This output is then routed to the ADC (part of NXP LPC4370 ARM micro-controller chip).
![General view of the Front-End Board.[]{data-label="fig-view"}](GenView.png){width="\textwidth"}
![Block-scheme of the Front-End Board.[]{data-label="fig-block"}](block-scheme.png){width="\textwidth"}
Triggering logic
================
In the Figure \[fig-trig\] a block-scheme of the trigger formation circuit is shown. Within the CITIROC for each of the 32 channels a charge amplifier with configurable gain and dynamic range from 1 photo-electron (p.e.) to 2000 p.e. (at MPPC gain of 10$^6$) is followed by a fast RC-CR shaper with peaking time of 15 ns. The shaped signal is binarized by a discriminator. The discriminator threshold is supplied by a common 10-bit DAC plus a 4-bits DAC for the fine adjustment of individual channels. Each of these 32 digital trigger signals (C0 to C31) are routed to the FPGA where they are paired with an AND logic to form coincidence signals for each SiPM pair (C0&C1, C2&C3 etc.). The OR of the resulting 16 signals, together with each individual channel trigger signal C0-C31, is fed to the trigger selection logic, which is configurable via back-end interface. Any combination of these signals can be selected to produce a primary event trigger for the CPU interrupt input. Within the FPGA this signal also triggers the generation of the event time stamp. Each of 32 channels can be individually enabled or disabled by the CITIROC configuration bit stream. This bit stream is produced by the on-board CPU on the basis of a configuration command received via Ethernet link from the host computer.
![Block-scheme of Front-End Board triggering circuit.[]{data-label="fig-trig"}](Trigger.png){width="\textwidth"}
The timing diagram of the circuit is shown in Figure \[fig-trig1\]. The time window of generated odd-even channel coincidence varies from 0 to 30 ns depending on the amplitude of the input pulse w.r.t the discriminator threshold. After a delay of 50 ns the primary event trigger sets the HOLD signal to the Sample-and-Hold circuit to memorize instantaneous signal levels at all 32 channels at the moment of the top of the peak (see next Section). The same signal is output at the “TOUT” LEMO connector (Figure \[fig-view\]). The HOLD signal is kept for at least 150 ns to define the coincidence window. If during this period the circuit detects a high level at the input “TIN” LEMO connector, the event is considered valid and the HOLD signal is kept high until the CPU finishes the digitization cycle and resets it to initial state. If no signal is received in “TIN” during 150 ns the HOLD signal is reset by the FPGA and the event is discarded. The “TOUT” signal is reset to zero in any case after 150 ns. This input has an optional on-board weak pull-up resistor, allowing operation without the external signal. Such functionality allows for a hardware coincidence between the event generated by the FEB and other external events.
![Timing diagram of the triggering circuit. The first event on ch0 (red) has no coincidence with ch1 (blue), so does not produce a trigger. The Second event on ch0 in coincidence with ch1 triggers a readout cycle.[]{data-label="fig-trig1"}](Trigger1.jpeg){width="80.00000%"}
Bias generator and analog signal readout
========================================
The block-scheme of the analog signal processing circuit is shown in Figure \[fig-analog\]. The bias voltage is generated by switching a stabilized power supply circuit operating at 100 MHz frequency. The voltage can be adjusted from 62V to 82V by the trimmer resistor. The range can be extended to 20V-90V by replacing divider resistors in the circuit. This voltage is common for all 32 SiPMs connected to the FEB inputs. The output V-A characteristics of the bias power supply are shown in Figure \[fig-outva\]. The power supply can be enabled or disabled by a dedicated signal generated by the on-board CPU. The individual bias voltage adjustment is performed by an 8-bits DACs within the CITIROC. The DAC’s positive output levels are supplied to the signal lines as a DC-offset; therefore, increasing this voltage reduces the effective bias for individual SiPMs. The full DAC range is +0.5V to +4.5 V.
Amplified charge pulses from SiPMs are shaped by a slow RC-CR shaper with configurable peaking time (12.5 ns to 87.5 ns). This time is adjusted in such a way that the HOLD signal, delayed by 50 ns w.r.t. event, has its rise flank at the flat top of the peak of the shaper output pulse, minimizing the noise due to time jitter. The amplitudes at the 32 shaper outputs are latched at S/H circuit and routed to an analog multiplexer. When the CPU receives the trigger interrupt it initiates the readout cycle. The timing diagram is shown in Figure \[fig-analog1\]. The solid colored lines represent the shaper outputs, the dotted lines the outputs of the S/H circuits. The event illustrated in Figure \[fig-analog1\] is triggered by the coincidence between two neighboring channels (x and x+1). The CPU controls multiplexing of all 32 outputs via a single line that is routed to the 12-bits ADC input. In Figure \[fig-analog1\] only 8 channels multiplexing is shown for simplicity. Once all 32 channels are digitized and stored in the event buffer, the CPU sends the reset signal to the FPGA and completes the readout. Each of the 32 charge ampthe lifiers can be individually enabled or disabled by the CITIROC configuration bit stream.
The performance of the analog readout circuit is illustrated in Figure \[fig-analog2\], where the amplitude spectrum for dark counts with a discrimination threshold at 0.5 p.e. is shown in blue. The red spectrum shows the location of the pedestal when triggered by some other channel.
![Block-scheme of analog signal processing circuit.[]{data-label="fig-analog"}](Analog.png){width="\textwidth"}
![Output Volt-Ampere characteristics of the bias power supply for 62V (yellow), 72V (red) and 82V (blue) bias settings.[]{data-label="fig-outva"}](Output_VA.png){width="70.00000%"}
![Timing diagram of the analog signal processing circuit. The first event on ch0 (red) has no coincidence with ch1 (blue) and does not produce a trigger. The second event on ch0 in coincidence with ch1 triggers readout cycle. Peak values on all channels are stored by a track-and-hold circuit for a period needed to multiplex them to a common analog output and digitization (R/O window).[]{data-label="fig-analog1"}](Analog1.jpeg){width="80.00000%"}
![Typical performance of the analog signal processing circuit (with a Hamamatsu S12825-050P MPPC). The amplitude spectrum for dark counts with the discrimination threshold at 0.5 p.e. is shown in blue. The red spectrum shows the location of the pedestal (it comes from the non-triggering channel).[]{data-label="fig-analog2"}](Analog2.jpeg){width="60.00000%"}
Time stamp generator
====================
The Time-to-Digit Converter (TDC) of the time stamp generator is composed of the coarse counter, working at the clock frequency of 250 MHz, and the delay-chain interpolator improving accuracy down to 1 ns. The solution is based on the approach published in [@tdc1].
For each event the FEB is capable of recording two independent time stamps w.r.t the positive flank on “T0” and “T1” LEMO inputs (Figure \[fig-tdc\]). Each time stamp is a 32-bits word, having time information in 30 Least Significant Bits (LSB) represented in Gray code. The resolution of the circuit is illustrated in Figure \[fig-tdc2\], where the events arrive 100 ns after the T0 reference signal.
Two Most Significant Bits (MSB) are used for flagging special events. Two special events are foreseen. The first is the arrival of the reference signal at either “T0” or “T1” inputs (time reference event). For such an event the time passed since the previous time reference event is recorded, and the timing circuit is reset to zero. This allows to measure the period between reference signals and, in case the real period is highly stable and accurate, the measured period allows to derive a deviation of the internal on-board oscillator frequency from its nominal value. If this deviation is known it can be applied offline to scale all time stamps between two reference pulses to recover accuracy.
The second special event happens in the absence of the reference pulse for more than 1074 ms, which leads to overflow of the coarse counter. This situation is flagged and can be used by offline software to invalidate the time stamp until the presence of the reference pulse is restored.
The 20 MHz temperature-compensated voltage-controlled crystal oscillator (VCXO) is used as a source of the reference clock for the FPGA and timing circuit. The feedback voltage for the VCXO is generated by a 10-bits DAC under control of the on-board CPU (Figure \[fig-tdcloop\]). The frequency correction signal is derived on-board, based of measured periods between reference pulses supplied to the “T0” input from the high-stability GPS-disciplined pulse-per-second (PPS) generator. The performance of the timer and oscillator control loop can be seen at Figure \[fig-tdc1\] where the measured period between PPS reference pulses is shown. The PPS pulse for this measurement is supplied by an external high-stability oscillator disciplined with the PPS from U-Blox M8T[^3] high-accuracy GPS receiver for timing applications. The correction signal is derived after each 20 successive measurements of the PPS period. After about a minute the control loop is stabilized and maintains stability of the on-board oscillator within $1.9\times10^{-9}$ 1-s Alan deviation.
Together with two 32-bits words from the two time stamp generators, an additional word is present in time stamp data. This word contains the number of extra primary event triggers that occurred during the readout cycle of one event. These events will be missed by the FEB and knowledge of their number allows to measure the current acquisition inefficiency.
Once the time stamp for the event is latched in the internal FPGA register the CPU initiates the transmission of this data via a Serial Peripheral Interface (SPI).
![Block-scheme of time stamp generation circuit.[]{data-label="fig-tdc"}](Timer.png){width="\textwidth"}
![Accuracy of the time-stamp generator. Events are delayed by 100 ns w.r.t. the reference signal.[]{data-label="fig-tdc2"}](TimeResolution.png){width="60.00000%"}
![Block-scheme of the oscillator control loop of the time stamp generation circuit.[]{data-label="fig-tdcloop"}](VCXO.png){width="\textwidth"}
![Stability of the time-stamp generator disciplined oscillator.[]{data-label="fig-tdc1"}](TimerStability1.png){width="95.00000%"}
Event buffer and back-end Ethernet interface
============================================
Once the CPU completes digitization of all 32 analog channels and transmission of the time stamp from the FPGA it combines this data into an event and stores it in the internal ring buffer, which has a capacity of 1024 events. The on-board micro-controller contains three CPU cores, each working at 160 MHz clock frequency. One core is taking care of filling the ring event buffer, while the second one is emptying it sending data out via an on-board Ethernet switch when it is requested to do so by the host PC. If the incoming data rate exceeds the capacity of the back-end interface, the events are overwritten in the ring buffer. This situation is detected by the CPU and the number of overwritten (and therefore lost) events is stored, to be transmitted to the host PC. This number together with the “Missed event” counter in the FPGA contributes to the measured acquisition inefficiency of the FEB.
The second CPU core communicates with the host computer via three-port Ethernet switch IC. The switch to the CPU interface (port 3) is a 25 MHz 4-bits MII bus with throughput of 100 Mbps. The other two ports (1 and 2) of the switch have Physical Layer (PHY) transmitters and are connected to two RJ45 Ethernet jacks. The switch is configured to forward ports 1 and 2 in both directions, to accept control commands on these ports and to forward these commands via the port 3 to the on-board CPU (Figure \[fig-backend\]. The event data from the CPU is forwarded only to that port from which the FEB received a data transmission request.
Both PHY ports of the switch have MDI/MDIX Auto Cross functionality. Therefore, the user does not need to care about the structure of the connecting Ethernet cables (straight or cross-wired). The switch detects the type of the cable on connection and sets its configuration accordingly.
![Block-scheme of the back-end data transmission and control interface.[]{data-label="fig-backend"}](Backend.png){width="80.00000%"}
A test was conducted with several FEBs daisy-chained to one host PC running development data acquisition code. Events were simulated by sending reference pulses to the “T1” trigger input at a variable rate. As the event rate increased, the time to transfer data from all FEBs (polling period) also increased. The maximum number of events accumulated in each FEB internal buffer increases accordingly. This number is limited by the buffer capacity (1024 events). Variations of these two parameters as a function of the number of FEBs in a chain at 3.5 kHz event rate is shown on the left of Figure \[fig-flux\]. In the right plot of Figure \[fig-flux\] the dependence on the event rate with a fixed number of FEBs (18) in the chain is shown.
![Measured polling period, in ms, and buffer use as a function of the number of FEBs in the chain for a fixed event rate of 3.5 kHz (left) and as a function of the event rate for 18 FEBs in a chain (right).[]{data-label="fig-flux"}](Throughput-NFEBs.png "fig:"){width="0.50\linewidth"} ![Measured polling period, in ms, and buffer use as a function of the number of FEBs in the chain for a fixed event rate of 3.5 kHz (left) and as a function of the event rate for 18 FEBs in a chain (right).[]{data-label="fig-flux"}](Throughput-Freq18FEBs.png "fig:"){width="0.48\linewidth"}
The MAC address of each board is set by firmware to 00:60:37:12:34:XX where last XX byte can be set from the hardware 8-bits switch array. In Figure \[fig-chain\] an example communication scheme between several FEBs and a host PC is shown. FEBs are daisy-chained and connected to a single Ethernet port of the host PC.
![Scheme of daisy-chain communication between several FEBs and host PC.[]{data-label="fig-chain"}](Chain.png){width="60.00000%"}
Conclusions
===========
A novel high-speed front-end electronic board (FEB) for interfacing an array of 32 Silicon Photo-multipliers (SiPM) to a computer has been developed. The FEB incorporates analog and digital processing circuits, provides bias supply, signal processing and digitization, buffering and communication with the host computer via a 100 Mbps Ethernet link. The FEB provides event time stamping with 1.3 ns accuracy, w.r.t. an external reference signal, such as a GPS Pulse-Per-Second. The design provides capability of daisy-chaining of up to 256 boards into one network interface cable. The FEB design enables low-power, compact and efficient readout schemes for multi-channel instrumentation with SiPMs as photo-sensors. The FEB is commercialized and available for purchasing from CAEN[^4] (A1702[^5]).
Acknowledgments
===============
We acknowledge financial support of the Swiss National Science Foundation.
[10]{}
url\#1[`#1`]{}urlprefix S.Aoki et al. (T2K collaboration), *The T2K Side Muon Range Detector (SMRD)*, Nucl.Instrum.Meth. A698 (2013) 135-146.
M. Auger et al., *Scintillator-based Cosmic Ray Tagger system for Short Baseline Neutrino detectors*, to be published in JINST, 2016.
R. Acciarri et al. (DUNE collaboration), *Long-Baseline Neutrino Facility (LBNF) and Deep Underground Neutrino Experiment (DUNE) Conceptual Design Report, Volume 4: The DUNE Detectors at LBNF*, arxiv:1601.02984.
J. Fleury et al., *Petiroc and Citiroc: front-end ASICs for SiPM read-out and ToF applications*, JINST 9 (2014) C01049.
A. Aloisio et al., *FPGA Implementation of a High-Resolution Time-to-Digital Converter*, 2007 IEEE Nuclear Science Symposium Conference Record, N15-137.
[^1]: Corresponding author.
[^2]: `http://omega.in2p3.fr/`
[^3]: `http://www.u-blox.com`
[^4]: CAEN - Costruzioni Apparecchiature Elettroniche Nucleari S.p.A
[^5]: http://www.caen.it/servlet/checkCaenDocumentFile?Id=11427
|
---
abstract: 'In many contexts the modal properties of a structure change, either due to the impact of a changing environment, fatigue, or due to the presence of structural damage. For example during flight, an aircraft’s modal properties are known to change with both altitude and velocity. It is thus important to quantify these changes given only a truncated set of modal data, which is usually the case experimentally. This procedure is formally known as the generalised inverse eigenvalue problem. In this paper we experimentally show that first-order gradient-based methods that optimise objective functions defined over a modal are prohibitive due to the required small step sizes. This in turn leads to the justification of using a non-gradient, black box optimiser in the form of particle swarm optimisation. We further show how it is possible to solve such inverse eigenvalue problems in a lower dimensional space by the use of random projections, which in many cases reduces the total dimensionality of the optimisation problem by 80% to 99%. Two example problems are explored involving a ten-dimensional mass-stiffness toy problem, and a one-dimensional finite element mass-stiffness approximation for a Boeing 737-300 aircraft.'
address:
- '$^1$ School of AMME, The University of Sydney, NSW 2006, Australia'
- '$^2$ School of Civil and Environmental Engineering, University of New South Wales, Sydney, NSW 2052, Australia'
author:
- 'Prasad Cheema$^1$, Mehrisadat M. Alamdari$^2$, Gareth A. Vio$^1$'
bibliography:
- '4241\_che.bib'
title: New Approaches to Inverse Structural Modification Theory using Random Projections
---
Inverse Eigenvalue Problems ,Modal Analysis ,Random Projections ,Particle Swarm Optimisation ,Finite Element Analysis
Introduction {#S:1}
============
Eigenvalue problems are common in the engineering context [@bathe1973solution; @elishakoff1991some]. As such, they have been used in a plethora of applications such as in analysing the state matrix of an electronic power system [@lima2000assessing], in studying the aeroelastic instability for wind turbines [@hansen2007aeroelastic], for determining the spectral radius of Jacobian matrices [@day1984run], and in the operational modal analysis of a structures [@sun2017automated]. The most common eigenvalue problem, known as the the *direct* or *forward* problem involves determining the impact of a known set of modifications to a group of matrices, either by computing the eigenvalues, eigenvectors, singular values, or singular vectors of the group of matrices. The direct problem is well studied, and is the subject of many elementary courses in linear algebra, but the *inverse* problem is much more complex.
The inverse problem tries to find or infer a particular type of modification which was applied to a set of matrices, from a larger set of possible modifications, using mainly spectral information [@chu2005inverse]. It is clear that this problem would be trivial if all the spectral information of the system before and after any modifications were known (that is, we are not dealing with a truncated modal system), or if the desired modifications were completely unstructured (that is, they are allowed to be any value). Thus in order to strive for more physical, and mathematical solutions we often try to restrict the group of possible matrices for the inverse eigenvalue problem. In a recent review article, Chu [@chu1998inverse] devised a collection of thirty-nine possible inverse eigenvalue problems. These problems were roughly categorized based on their: paramterisation, underlying structure, and the partiality of the system description (that is, whether or not we have complete modal information). A summary of the most common kinds of inverse eigenvalue problems are given in Figure \[fig:inverse\_eig\_value\_probs\], where the following terminology is used:
- MVIEP: Multivariate inverse eigenvalue problem
- LSIEP: Least square inverse eigenvalue problem
- PIEP: Parameterised inverse eigenvalue problem
- SIEP: Structured inverse eigenvalue problem
- PDIEP: Partially described inverse eigenvalue problem
- AIEP: Additive inverse eigenvalue problem
- MIEP: Multiplicative inverse eigenvalue problem
In this paper we aim to explore AIEPs which have a highly general parameterisation, in order to demonstrate the potential capabilities of random projections for the inverse eigenvalue problem.
![An overview of some of the general classes of inerse eigenvalue problems as defined by Chu [@chu1998inverse].[]{data-label="fig:inverse_eig_value_probs"}](4241_che_Fig1.PNG){width="0.45\linewidth"}
In order to solve for the AIEP we shall define an optimisation problem. Although there are many optimisation procedures available for solving such problems, a particle swarm optimiser (PSO) is used in this paper. PSO is an optimisation procedure first introduced by Eberhart and Kennedy in 1995 [@eberhart1995new]. It is a stochastic, population-based optimisation procedure modeled on the observed behaviour of animals which exhibit swarm-like tendencies, such in the social behaviour of birds or insects. Because of this PSO, tries to mimic swarm-like behaviour with each *particle* having access to both: a personal best solution, *and* access to the global optimum, thereby enabling the sharing of information across the swarm. This introduces the idea of the classic *exploration-exploitation* trade-off since the particles are either allowed to *exploit* the current global optimum, or *explore* further if their local optimum is far away from the current global [@eberhart1995new; @van2010convergence].
PSO is used in this paper to perform the optimisation for two main reasons. Firstly, to the knowledge of the authors, it has not previously been used in the context of this problem (AIEP), hence there is novelty in doing so. Secondly, it is a known black-box, gradient-free optimiser which makes it simple to work with since there is no requirement to compute the Jacobian, or calculate analytical gradients. [@van2010convergence] Moreover as we shall demonstrate experimentally, gradient step sizes are required to be very small in the inverse eigenvalue problem, with the issue exacerbating in higher dimensions. PSO as an algorithm has been successfully used in many different areas, including but not limited to reactive power and voltage control problems [@yoshida2000particle], in the study of material degradation for aeroelastic composites [@vishwanathan2017multi], composites structures with robustness [@vishwanathan2017robust], and in the optimum design of Proportional-Integral-Derivative (PID) control [@gaing2004particle] which additionally helps in justifying its potential to work well in this context.
Regardless, of the choice of optimiser, all optimisation procedures are known to suffer from the *curse of dimensionality*, and since structural problems are generally able to grow without bound in terms of degrees of freedom (for example, a finite element model can keep growing in the number of elements), it is important to devise methods which can help reduce, or at least limit the rate of growth of these dimensions. Random projections have recently emerged as a powerful method to address the problem of dimensionality reduction [@bingham2001random]. This is because theory (in particular the Johnson–Lindenstrauss lemma) suggests that certain classes of random matrices are able to preserve Euclidean distances to within a tolerable error bound in the lower dimensional space [@johnson1984extensions]. As a result, random projections have been used successfully to reduce the dimensionality of the underlying optimisation problem, consequently allowing for optimisation to be performed in this lower dimensional space [@wang2016bayesian; @gardner2014bayesian; @krummenacher2014radagrad].
Thus, it is ultimately the aim of this paper to explore the impact of random projections and how they may be used in connection with the PSO algorithm for the generalised inverse eigenvalue problem of an additive nature. In the following sections we demonstrate experimentally that gradient-based approaches lose accuracy in even moderate step sizes, and clarify the theory that we shall use from random projections to help in lowering the dimensionality of the underlying optimisation problem. Lastly we showcase a gamut of positive results on a 10 dimensional (meaning a matrix of size 10) toy problem, and 1 dimensional finite element model based on aircraft data for the Boeing 737-300 aircraft.
Background and Methodology
==========================
The Generalised Eigenvalue Problem
----------------------------------
Ultimately it is the aim of this paper to use particle swarm optimisation (PSO) in order to try and solve the generalised *inverse* eigenvalue problem, with the use of random projections. We thus commence by formalising the notion of the generalised eigenvalue problem here.
Suppose we have the generalised eigenvalue problem as shown in Equation \[eqn:gen\_eig\_prob\], which represents an undamped mechanical vibration system.
$$\label{eqn:gen_eig_prob}
\mathbf{M}\Ddot{\mathbf{x}} + \mathbf{Kx} = 0$$
where $\mathbf{M},\mathbf{K} \in \mathbb{R}^{N \times N}$, and $\mathbf{x}\in\mathbb{R}^N$. The eigensystem for Equation \[eqn:gen\_eig\_prob\] defines the following set of eigenvalue, eigenvector pairs: $\mathbb{A} = \{(\lambda_i,\mathbf{v}_i)|i = 1,..,N; \mathbf{v}_i \in \mathbb{R}^N, \lambda \in \mathbb{R}\}$. Furthermore we assume that the eigenvalue problem is perturbed via the addition of some arbitrary matrices we denote as $\mathbf{\Delta}\in (\mathbb{R}^{N\times N}, \mathbb{R}^{N\times N}$). That is, from here onwards whenever the $\mathbf{\Delta}$ symbol is written in isolation, in a bold-type font it denotes a 2-tuple of perturbation matrices of the system, that is, $\mathbf{\Delta} \coloneqq (\Delta \mathbf{M}, \Delta \mathbf{K})$, where $\Delta \mathbf{M} \in \mathbb{R}^{N\times N}$ and $\Delta \mathbf{K} \in \mathbb{R}^{N\times N}$.
$$\label{eqn:mod_eig_prob}
( \mathbf{M} + \Delta \mathbf{M})\Ddot{\mathbf{x}} + (\mathbf{K}+\Delta \mathbf{K})\mathbf{x} = 0$$
The eigenpairs for the modified system shown in Equation \[eqn:mod\_eig\_prob\] may be represented via the following set of eigenpairs: $\mathbb{B} = \{(\sigma_i,\mathbf{w}_i)|i = 1,..,N; \mathbf{w}_i \in \mathbb{R^N}, \sigma \in \mathbb{R}\}$. However, if both the initial and modified systems are full rank systems, then it would be trivial to obtain the $\Delta \mathbf{M}$ and $\Delta \mathbf{K}$ matrices. Thus for this paper we assume that we only have access to a truncated eigensystem for the modified system. That is, we only have access to some subset of the pairs: $\mathbb{C} \subset \mathbb{B}$, where $|\mathbb{C}| = n < N$. Thus our objective function in the search for the optimal $\mathbf{\Delta}$ matrices reflects this, in Equation \[eqn:obj\_fun\_init\].
$$\label{eqn:obj_fun_init}
\Delta \mathbf{M}^{\star}, \Delta \mathbf{K}^{\star} = \operatorname*{argmin}(||\bm{\sigma}^{\dagger}_{1:n} - \bm{\sigma}_{1:n}(\bm{\Delta})||_2^2),$$
where $||\cdot ||_2^2 $ denotes the square of the standard 2-norm, $\bm{\sigma}^{\dagger}$ denotes the desired eigenvalues, and $\bm{\sigma}$ refers to the eigenvalues as calculated from applying the $\bm{\Delta}$ matrices (clarified in the prior paragraph in reference to set $\mathbb{B}$). As is made clear in Equation \[eqn:obj\_fun\_init\], we only consider the first $n < N$ dimensions, since we are dealing with a reduced set of eigenvalues.
In this paper we propose investigating the solutions for the objective function shown in Equation \[eqn:obj\_fun\_init\] via PSO. We aim to use a non-gradient based, black-box optimisation since a first order perturbation analysis of the modified eigenvalues seem to suggest that for higher dimensional problems the step-sized used by gradient-based approaches may become prohibitively small. We establish this idea by first developing Lemma \[lem:gen\_perturb\] as follows.
\[lem:gen\_perturb\] Suppose we have the two following generalised eigenvalue problems, $$\begin{aligned}
\lambda \mathbf{M} \mathbf{v} &= \mathbf{K} \mathbf{v} \label{eqn:not_pertubed}\\
(\lambda+\delta \lambda) (\mathbf{M} +\Delta \mathbf{M}) (\mathbf{v}+\delta \mathbf{v}) &= (\mathbf{K} + \Delta \mathbf{K}) (\mathbf{v} + \delta \mathbf{v}) \label{eqn:pertubed},\end{aligned}$$ where $\bm{\Delta}$ perturbations are controlled system inputs, and the $\delta$ perturbations are a consequence of applying $\bm{\Delta}$. Then, $$\begin{aligned}
\label{eqn:pert_gen}
\delta \lambda_i = \frac{\mathbf{v}_i^{\intercal}\left(\Delta \mathbf{K} - \lambda_i \Delta \mathbf{M}\right)\mathbf{v}_i}{\mathbf{v}_i^{\intercal}\Delta \mathbf{M} \mathbf{v}_i},\end{aligned}$$ if $\mathbf{M}$ and $\mathbf{K}$ are Hermitian.
By expanding Equation \[eqn:pertubed\], removing higher order terms (that is, keeping only linear terms), and considering the $i^{\text{th}}$ eigenvalue-eigenvector pairs we arrive at, $$\begin{aligned}
\label{eqn:proof_one}
\mathbf{K}\delta \mathbf{v}_i + \Delta \mathbf{K} \mathbf{v}_i = \lambda_i \mathbf{M} \delta \mathbf{v}_i + \delta \lambda_i \mathbf{M} \mathbf{v}_i + \lambda_i \Delta \mathbf{M} \mathbf{v}_i.\end{aligned}$$
Since $\mathbf{M}$ and $\mathbf{K}$ are Hermitian it implies that that the eigenvectors of Equation \[eqn:not\_pertubed\] are mutually $\mathbf{M}$-orthogonal. Moreover since they are assumed diagonsliable, these eigenvectors form a complete basis. Hence we can express each perturbation vector, $\delta \mathbf{v}_i$ as a sum of the eigenvectors of Equation \[eqn:not\_pertubed\]. As an equation this is,
$$\begin{aligned}
\label{eqn:eig_basis}
\delta \mathbf{v}_i = \sum_{k=1}^N c_k \mathbf{v}_k,\end{aligned}$$
for some arbitrary constants $c_k \in \mathbb{R}$. Thus, substituting Equation \[eqn:eig\_basis\] into Equation \[eqn:proof\_one\], and using Equation \[eqn:not\_pertubed\]:
$$\begin{aligned}
\label{eqn:proof_two}
\sum_{k=1}^N c_k \lambda_k \mathbf{M} \mathbf{v}_k + \Delta \mathbf{K} \mathbf{v}_i = \lambda_i \mathbf{M} \sum_{k=1}^N c_k \mathbf{v}_k + \delta \lambda_i \mathbf{M} \mathbf{v}_i + \lambda_i \Delta \mathbf{M} \mathbf{v}_i. \end{aligned}$$
Finally, left multiplying Equation \[eqn:proof\_two\] by $\mathbf{v}_i^{\intercal}$, and re-arranging results in,
$$\begin{aligned}
\delta \lambda_i = \frac{\mathbf{v}_i^{\intercal}\left(\Delta \mathbf{K} - \lambda_i \Delta \mathbf{M}\right)\mathbf{v}_i}{\mathbf{v}_i^{\intercal}\Delta \mathbf{M} \mathbf{v}_i},\end{aligned}$$
since the eigenvectors $\mathbf{v}_i$ are $\mathbf{M}$-orthogonal.
\[cor:cor\_perturb\] Suppose we have the two following standard eigenvalue problems, $$\begin{aligned}
\lambda \mathbf{v} &= \mathbf{K} \mathbf{v} \label{eqn:not_pertubed_2}\\
(\lambda+\delta \lambda) (v+\delta v) &= (\mathbf{K} + \Delta \mathbf{K}) (\mathbf{v} + \delta \mathbf{v}) \label{eqn:pertubed_2}.\end{aligned}$$ Then, $$\begin{aligned}
\label{eqn:pert_K}
\delta \lambda_i &= \frac{\mathbf{v}_i^{\intercal}\Delta \mathbf{K} \mathbf{v}_i}{\mathbf{v}_i^{\intercal} \mathbf{v}_i},\end{aligned}$$ if $\mathbf{K}$ is Hermitian.
The proof follows similarly from repeating the steps shown in Lemma \[lem:gen\_perturb\], in the absences of the $\mathbf{M}$ and $\Delta \mathbf{M}$ matrices.
Note that in Equations \[eqn:pert\_gen\] and \[eqn:pert\_K\], it is not necessarily true that $\mathbf{v}_i^{\intercal}\Delta \mathbf{M} \mathbf{v}_i = 1$ since each eigenvector $\mathbf{v}_i$ is only orthogonal with respect to the $\bm{M}$ matrix, and not $\Delta \bm{M}$. A similar argument may be made with the $\bm{K}$ and $\Delta \mathbf{K}$ matrices. In addition, Equations \[eqn:pert\_gen\] and \[eqn:pert\_K\] make clear how the perturbation matrices, $\bm{\Delta}$, impact the changes in the eigenvalues, $\delta \lambda$, up to a linear approximation (since in the derivation the higher order effects were ignored). As a result this relationship may be used in better understanding gradient-based relationships for eigenvalue problems. That is, we may use these to analyse the potential accuracy and or quality of gradient-based approaches for such problems. An investigation of these ideas is made clear in Figure \[fig:step\_sizes\].
Equation \[eqn:perc\_err\] is used order to calculate the percentage errors for the $|\Delta \delta \lambda_1|$ values in Figure \[fig:step\_sizes\], where $\delta \lambda_1^{\dagger}$ refers to the change in first eigenvalue as obtained from Equation \[eqn:pert\_gen\], and $\delta \lambda_1$ is obtained using Equations \[eqn:gen\_eig\_prob\] and \[eqn:mod\_eig\_prob\], which refers to the benchmark correct value.
$$\begin{aligned}
\label{eqn:perc_err}
|\Delta \delta \lambda_1|_{\mu} &\coloneqq \mathbb{E}(|\Delta \delta \lambda_1|) \nonumber\\ &\hspace{1mm}= \mathbb{E}\left(\frac{|\delta \lambda_1^{\dagger}-\delta \lambda_1|}{\delta \lambda_1}\right)\end{aligned}$$
In order to calculate the size of terms inside the $\bm{\Delta}$ matrices, Equation \[eqn:p\_dist\] is used, where $\mathcal{U}$ refers to the uniform distribution, the $i,j = 1,...,d$ indices refer to each term in the matrix, and $d = 1,...,20$ defines the dimensionality (size) of the matrices. Through this definition there are $d^2$ degrees of freedom inside the matrix at any time.
$$\label{eqn:p_dist}
\bm{\Delta}_{i,j} \sim p\cdot\mathcal{U}[0,1],$$
The value of $d$ changes because we are considering the effect of dimensionality on the quality of the linear step size, $\delta \lambda$. Moreover the scalar $p \in \{1/100,1/10,1,10\}$ defines the magnitude of the terms in the $\bm{\Delta}$ matrices. Thus $p$ in a practical sense (that is, in reference to a gradient-based optimisation algorithm) can be interpreted as the *step size* of the algorithm.
From Figure \[fig:step\_sizes\] it can be seen that as the average magnitude of the elements inside the $\bm{\Delta}$ matrices increase, the absolute difference between the theoretical $\delta \lambda_1$ value, and those calculated via Equation \[eqn:pert\_gen\], that is, $\delta \lambda^{\dagger}_1$, becomes larger. Even when the average step size takes value p = 1/100, there appears to be a bias in the magnitude of the error, which is made clear in the *zoomed in* subplot of Figure \[fig:step\_sizes\_zoomed\]. Moreover as the dimensionality of the $\bm{\Delta}$ matrices increase, the $|\Delta \delta \lambda_1|$ errors appear to increase slightly. Hence in summary as this experimental analysis of Equation \[eqn:pert\_gen\] seems to suggest, gradient based methods are potentially difficult to implement. In particular, it would appear that we would require $p< 1/100$ at a minimum, and that this value would need to continually decrease as dimensionality increases. For this reason, we opt to explore the viability of particle swarm optimsation as a means for optimisation since it is a non-gradient based approach, and gradient-based approaches seem to require very small step sizes for accurate gradients.
Particle Swarm Optimisation
---------------------------
Particle Swarm Optimisation (PSO) is a stochastic, evolutionary optimisation first proposed by Kenedy and Eberhart[@eberhart1995new]. The algorithm works by generating an array of candidate particles (the swarm) across an objective space. Within this space each particle searches for the global optimum through the sharing of information within the swarm in a classic exploration-exploitation trade-off. This is clarified in Equations \[eqn:PSO\] and \[eqn:PSO2\].
$$\begin{aligned}
v_{i}^{k+1} &= \omega v_{i}^{k} + c_1r_1(p_i - x_i^k) + c_2r_2(p_G - x_i^k) \label{eqn:PSO} \end{aligned}$$
$$\begin{aligned}
x_{i}^{k+1} &= x_{i}^{k} + \alpha v_{i}^{k+1} \label{eqn:PSO2}\end{aligned}$$
where $\alpha$ denotes step size, $\omega$ controls the particle’s inertia, $c_1$ and $c_2$ (known as the acceleration coefficients) are constants which control the degree of the *exploration*-*exploitation* trade-off, $p_i$ and $p_G$ are the local optima, and global optimum, for each, and across all particles respectively (that is each particle stores their own local optimum, but shares knowledge of the current global optimum), and $r_1, r_2 \sim \mathcal{U}(0,1)$. In this way, every particle is made aware of the current global optimum, and explores the objective space accordingly (as specified through the $c_1$ and $c_2$ constants).
Empirical studies in PSO theory have shown that the correct choice of inertia weight is critical in ensuring convergent behaviour of the algorithm [@van2010convergence]. Prior investigations have suggested that the choice of inertia is driven by the acceleration coefficients through: $2\omega > (c_1 + c_2) - 2$. This region describes the set of all $c_1$, and $c_2$ values which guarantee convergent behaviour based on the spectral analysis of the matrix describing the PSO dynamics [@van2006study]. However this inequality should be only be taken as a rough guide since it was derived assuming the PSO system has one particle, and one dimension. Empirically however, Eberhart and Shi suggest using values of $\omega=0.7298$ and $c_1 = c_2 = 1.49618$ for *good* convergent behaviour in general [@eberhart2000comparing].
Random Embedding
----------------
An aim of this paper is to explore the capability of random projections to reduce the underlying dimensionality of the generalised eigenvalue problem. In particular, we propose an extremely general parameterisation of the $\Delta \mathbf{M}$, and $\Delta \mathbf{K}$ matrices, and explore whether or not it is possible to solve this problem in a lower dimensional space. The lowest dimensional space in which the problem may be solved completely is known as the *effective dimension*, and is denoted by $d_e$. In particular the following definition is used to strictly define the notion of $d_e$, where Definition 4.1 is based on Definition 1 of Wang et al. [@wang2016bayesian].
Suppose there exists a linear subspace $\mathcal{T} \subset \mathbb{R}^D$, where $\text{dim}(\mathcal{T}) = d_e < D$. A function $f : \mathbb{R}^D \to \mathbb{R}$ is said to have **effective dimensionality** $d_e$, if $d_e$ is the smallest integer such that $\forall x \in \mathcal{T}$ and $x^{\bot} \in \mathcal{T}^{\bot} \subset \mathbb{R}^D$, where $\mathcal{T} \small{\oplus} \mathcal{T}^{\bot} = \mathbb{R}^D$, we have $f(x + x^{\bot}) = f(x)$.
A simple example to clarify this defintion for the reader may be seen if we define the following function: $f(x_1,x_2) = x_1^2 + x_2^2, $ $\text{where, } f : \mathbb{R}^{10} \to \mathbb{R}, \text{ and } x_1, x_2 \in \mathbb{R} $. In this example, although the original space of $f$ is assumed to be 10-dimensional, one may easily observe that it has an *effective dimension* of 2 (that is, $d_e = 2$), since it clearly only makes use of 2 dimensions, of the 10 possible dimesions it has access to. The remaining 8 dimensions are *ineffective dimensions*. Unfortunately, in practice we never really know the actual value of $d_e$, but we either know or assume from prior knowledge that our problem may have a lower dimensional representation. In other words, in practice we only ever know or use $d\in \mathbb{Z}$ dimensions in total, where $D \geq d \geq d_e$, and thus our random embedding generally occurs via random matrices with dimensionality $D\times d$.
Although the notion of effective dimensionality is developed, it does not explain how such random projections to lower dimensional spaces should occur. Ideally when projecting to a lower dimensional space we desire $||T(x_i - x_j) || \approx ||(x_i - x_j) ||$, where $T: \mathbb{R}^n \to \mathbb{R}^m$ is some linear operator. That is, we aim to reduce the dimensionality of a set of points in some Euclidean space, which approximately retains these pair-wise distances measures in this new, lower-dimensional subspace. A bound on the degree of *distortion* that occurs to the original space when we project to a lower dimensional space is famously shown through the Johnson-Lindenstrauss (JL) Lemma [@johnson1984extensions], stated in Lemma \[lem:rand\_emb\].
In effect the JL Lemma tells us that the quality of the projection down to some dimension $k$, is a function of some allowable error tolerance, ${{\varepsilon}}$, and the amount of points invovled in the projection $n$. In particular the relative Euclidean distances will be distorted by a factor of no more than $(1\pm {{\varepsilon}})$, where ${{\varepsilon}}\in (0,1)$. Note that this makes no reference to the initial dimension of the points existed in before the projection occured.
Although the JL Lemma is used commonly with large datasets, we are only working with indiviudal, possibly high-dimensional points, which are used as inputs into functions used in an objective function. Thus in the case of random projections of data points into functions we also must consider the effective dimension $d_e$. Theorem 2 of Wang et al.[@wang2016bayesian] implies that no matter the degree of distortion which may occur, there shall always exist a solution in this lower dimensional space. This theorem is restated here in order to self-contain the paper.
Assume we are given a function $f : \mathbb{R}^D \to \mathbb{R}$ with effective dimensionality $d_e$ and a random matrix $A\in\mathbb{R}^{D\times d}$ with independent entries sampled according to $\mathcal{N}(0,1)$, where $d\geq d_e$. Then with probability 1, for any $x \in \mathbb{R}^D$, there exists a $y\in \mathbb{R}^d$ such that $f(x) = f(Ay)$
That is, we should always be able to find some $y$ such that $f(x) = f(Ay)$, where $A \in \mathbb{R}^{D \times d}$ is some random matrix. And thus the distortion of the projection as predicted by the JL Lemma is not as important of a factor if we can ensure $d \geq d_e$, since with a good enough optimisation algorithm, if there exists a $x^{\star} \in \mathbb{R}^D$ which is optimal, then there exists a $y^{\star} \in \mathbb{R}^d$ such that $f(x^{\star}) = f(Ay^{\star})$. In practice however we may by chance select some $d < d_e$, and so in these cases it may become informative to use JL Lemma as a guide to assist in undesrtanding the degree of distortion which did indeed occur by projecting into this new subspace.
In order to ensure that the point-wise distances in this new subspace abides by the JL Lemma we consider random matrices of the form: $A_{i,j} \sim \mathcal{N}(0,1/\sqrt{d})$. This is simply a scaled version of the random Gaussian matrices proposed by Wang et al. in Threom 2, but a Gaussian matrix of this form is known to better preserves distance properties in this new subspace [@johnson1984extensions].
An issue which may occur when trying to use random embeddings for the purpose of optimisation is that the optimisation bounds which are defined in the larger $D$-dimensional space may be violated in the lower $d$-dimensional space. Thus, it is suggested to use a convex projection method to ensure that any variables $y \in \mathbb{R}^d$ fall the into bounding constraints defined by variables $x \in \mathbb{R}^D$. This idea is shown in Figure \[fig:embed\_prop\].
![Need to perform a convex projection [@wang2016bayesian].[]{data-label="fig:embed_prop"}](4241_che_Fig4.PNG){width="0.45\linewidth"}
Assuming that the feasible set for $\mathbf{x}$ is defined with box constraints, denoted by $\mathcal{X} \coloneqq [-c,c]^D$ where $c\in\mathbb{R}^{+}$, a simple way to ensure that $\mathbf{Ay}$ maps to the range defined by $\mathcal{X}$ may be achieved through a least squares method [@wang2016bayesian]. Mathematically, we define: $p_{\mathcal{X}}(\mathbf{Ay})= \mathrm{argmin}_{\mathbf{z}}||\mathbf{z-Ay}||_2^2$, where $\mathbf{Ay} \in \mathbb{R}^D$, $\mathbf{z} \in \mathcal{X}$, and $p_{\mathcal{X}} : \mathbb{R}^D \to \mathcal{X}$ denotes *projection*. This least squares method effectively drops the perpendicular from points outside the bounding box, $\mathcal{X}$, which can be located aribtrarily in $\mathbb{R}^D$, towards the nearest point on the boundary of $\mathcal{X}$. This is made clear in Figure \[fig:embed\_prop\].
Results
=======
Ten-Dimensional Toy Problem
---------------------------
In this section the results of applying the combination of PSO and random embedding for inverse eigenvalue problems in structural engineering are explored. We begin by considering the 10 degree of freedom (DoF) system defined in Equations \[eqn:Mass\_Sivan\] and \[eqn:Stiff\_Sivan\]. The stiffness matrix is based on that of Sivan & Ram [@sivan1996mass], with the mass matrix being modified from a diagonal of ones to allow for a more complex scenario.
$$\begin{aligned}
\label{eqn:Mass_Sivan}
\mathbf{K} =
\begin{pmatrix}
200 & -10 & -20 & -5 & -5 & -10 & 0 & 0 & -50 & -50\\ -10 & 100 & 0 & 0 & 0 & 0 & -20 & -10 & -20 & -10\\ -20 & 0 & 300 & -40 & -30 & -60 & -10 & 0 & -20 & -10\\ -5 & 0 & -40 & 400 & -30 & -40 & -50 & -20 & -10 & -70\\ -5 & 0 & -30 & -30 & 150 & -10 & -5 & -5 & -20 & 0\\ -10 & 0 & -60 & -40 & -10 & 250 & 0 & 0 & 0 & -80\\ 0 & -20 & -10 & -50 & -5 & 0 & 120 & -5 & 0 & -10\\ 0 & -10 & 0 & -20 & -5 & 0 & -5 & 250 & 0 & -100\\ -50 & -20 & -20 & -10 & -20 & 0 & 0 & 0 & 350 & -40\\ -50 & -10 & -10 & -70 & 0 & -80 & -10 & -100 & -40 & 400
\end{pmatrix}\end{aligned}$$
$$\begin{aligned}
\label{eqn:Stiff_Sivan}
\mathbf{M} = \text{diag}(1,...,10)\end{aligned}$$
The first two eigenvalues of the generalised eigenvalue problem defined through these particular mass and stiffness matrices are given in Equation \[eqn:eig\_Sivan\].
$$\begin{aligned}
\label{eqn:eig_Sivan}
\bm{\Lambda} = \begin{pmatrix} 10.99, 19.12 \end{pmatrix}\end{aligned}$$
Our aim for this problem will be to find some $\Delta \bm{M}$ and $\Delta \bm{K}$ matrices which will transform the system eigenvalues into those specified by Equation \[eqn:eig\_Sivan\_mod\].
$$\begin{aligned}
\label{eqn:eig_Sivan_mod}
\bm{\Lambda}^{\star} = \begin{pmatrix} 2.00, 5.00 \end{pmatrix}\end{aligned}$$
In particular we shall work to minimise the objective function defined previously in Equation \[eqn:obj\_fun\_init\] in order to find the associated the $\bm{\Delta}^{\star}$ matrices. We assume without loss of generality that the $\bm{\Delta}$ matrices are upper triangular, and real valued since the underlying $\bm{M}$ and $\bm{K}$ matrices are Hermitian and so are by definition symmetric. That is, it suffices to perturb only the upper (or lower) triangular portion of the $\bm{M}$ and $\bm{K}$ matrices. This means that for each of $\Delta \bm{K} \in \mathbb{R}^D$and $ \Delta \bm{M}\in \mathbb{R}^D $, there are $D(D+1)/2$ free parameters. In this way the amount of free parameters grows on the order of $\mathcal{O}(D^2)$. We thus use random projections to reduce the effect of this quadratic complexity. One convergence was run for a random projection which reduced the overall dimensional size by an order of magnitude, and another was reducing it by a factor of a half. Convergence results are shown in Figure \[fig:toy\_data\_conv\].
From Figure \[fig:toy\_data\_conv\_a\], it would appear that the initial rate of decrease of the random embedding space is slightly faster than the full dimensional space. However, as Figure \[fig:toy\_data\_conv\_a\] then further suggests, although a low dimensional space may give a faster initial convergence rate, if the dimensionality reduction is too great, the optimisation routine can become plateau after a certain amount of iterations. This is seen in Figure \[fig:toy\_data\_conv\_a\] as the random embedding space remains at approximately $10^0$ after iteration 9. Thus it is clear that with this level of dimensionality reduction (for this particular problem), decreasing the dimensions by an order of magnitude seems to introduce a bias into the optimistion problem. It is conjectured that this is because the value of ‘10’ lies well below the *effective dimension* (the $d_e$ value) for this problem. This hypothesis is supported if we examine the same optimisation problem but instead reduce the exploration space to 50 dimensions as in the case of Figure \[fig:toy\_data\_conv\_b\]. We notice that not only do we have the faster initial decrease in optimisation rate, but also we converge to better values, faster.
The reader may additionaly notice that in Figure \[fig:toy\_data\_conv\_a\] the 110 dimensional space reaches values as low as $10^{-6}$, but in Figure \[fig:toy\_data\_conv\_b\] it apparently stalls at $10^0$. However this is only due to the truncation of the plotting, since at has been hinted at approximately iteration 98 the full dimensional space starts to decrease its magnitude value again. But this only serves to demonstrate the notion within a 100 iteration limit the random projection has allowed to the optimisation to converge better values, significantly faster on average. Indeed, the full dimensional space will *eventually* reach values as low as $10^{-6}$, however this toy problem suggests that on average it will not be as fast. Also note that since Figure \[fig:toy\_data\_conv\_b\] does not exhibit the same bias problems as in Figure \[fig:toy\_data\_conv\_a\], it would appear that $10 \leq d_e \leq 50$. Hence although we have not been able to determine the *effective dimension*, we can infer a range of existance for it.
Also important to take note off is that for this problem the JL Lemma is not readily applicable since this 110 dimensional space is too low for the Lemma to take practical significance. In particular, if we set $n=110$ and assume error values of ${{\varepsilon}}\in \{0.1,0.3,0.7,1.0\}$ we arrive at Table \[tab:JL\_toy\].
Distortion Error (%) **10** **30** **70** **100**
---------------------- -------- -------- -------- ---------
Dimension 4029 523 143 112
: How the distortion error effects the corresponding dimension of the mapped subspace, for $n=110$ in accordance with the JL Lemma.[]{data-label="tab:JL_toy"}
From Table \[tab:JL\_toy\] we see that for our toy problem, for an initial dimension of 110, the *sufficient* dimension to guarantee no more than a 10% error in the Euclidean distances between points in the new space is $4029 >> 110 > 50$. This number seems unreasonable because in lower dimensions the bounds predicted by the JL Lemma are not tight. That is these bounds serve give an idea of *sufficiency*. Consequently by inspecting the mathematical equation of this bound, if the initial number of points is relatively low (as is the case for this toy example) we will not achieve a practically meaningful answer. However an important takeaway from the JL Lemma is that its formulation makes no reference to the initial dimension of projection, only the number of points considered, and so whether or not we began with $110$ points, or $110^{110}$ points, the *sufficient* dimension to guarantee no more than a 10% error between points after a random embedding is $110 < 4029 << 110^{110}$. Thus in this case it can be concluded that the initial $n=110$ value is too small for the JL Lemma to be directly useful.
One-Dimensional Boeing-737 Finite Element Problem
-------------------------------------------------
Here we shall explore how the PSO algorithm coupled with random embeddings may be exploited to assist in solving a truncated version of the generalised inverse eigenvalue problem for a 1D Boeing 737-300 (B737) Finite Element (FE) problem. In particular the model used to analyse the B737 plane is outlined in Figure \[fig:B737\]. This model is based upon one found in Theory of Matrix Structural Analysis [@przemieniecki1985theory].
\
In Figure \[fig:B737\] it is assumed that the total wing mass is uniformly distributed over the wing span of length $2L$, and its mass is $2M_w$. Moreover the total mass of the fuselage is $2M_F$. The wing elements are approximated as being finite element beam structures and have flexural stiffness given by $EI$, with the effects of shear deformations and rotary inertia neglected. Assuming only two FE nodes were used in this model, then the one element FE beam matrix for this problem is outlined in Equation \[eqn:FE\_element\], where $R$ denotes the mass ratio between the fuselage and the wing, that is, $R = M_F/M_w$, and $\mathbf{q} = \begin{bmatrix} w_1, \frac{\partial w_1}{\partial x},w_2, \frac{\partial w_2}{\partial x} \end{bmatrix}^{\intercal}$. In order to estimate a value for $R$ the parameters for a B737 where obtained from literature, and summarised in \[tab:B737\_general,tab:B737\_fuselage,tab:B737\_wing\]. In order for the FE model to increase its accuracy more nodes must used, which involves constructing a large block matrix using the elements defined in Equation \[eqn:FE\_element\].
$$\begin{aligned}
\label{eqn:FE_element}
\left( \frac{EI}{L^3}
\begin{bmatrix}
12 & \text{symmetric} & & \\
6L & 4L^2 & & \\
-12 & -6L & 12 &\\
6 & 2L^2 & -6L & 4L^2
\end{bmatrix} - \lambda M_w
\begin{bmatrix}
\frac{13}{35}+R & \text{symmetric} & \\
\frac{11}{210}L & \frac{L^2}{105} & & \\
\frac{9}{70} & \frac{13}{240}L & \frac{13}{35} & \\
-\frac{13L}{420} & \frac{-L^2}{140} &-\frac{11L}{210} & \frac{L^2}{105}
\end{bmatrix} \right) \mathbf{q} = \mathbf{0}\end{aligned}$$
As per the recommendation of Przemieniecki[@przemieniecki1985theory] the modal analysis of this structure may be separated into its symmetric and asymmetric counterparts, since this wing is symmetric around its fuselage centre. In order to enforece a symmetric condition, the second row and column of Equation \[eqn:FE\_element\] needs to be removed since it represents a degree of rotational freedom of the fuselage mass. For symmetry the fuselage mass is only allowed the move in a translational sense (up an down), and as such should not have any gradient (that is, not be able to rotate about some axis). The opposite is true in the case of anti-symmetry where instead the first row and column of the mass and stiffness matrices were removed, since in the case of asymmetry, the fuselage is allowed to rotate about an axis and thus have a defined gradient. Note that although doing this will not dramatically change the eigenvector response of the full system (if they are scaled properly), it can shift the eigenvalues appreciably.
In order to assess the validity of this symmetric - asymmetric separation, Figures \[fig:sym\_bending\] and \[fig:asym\_bending\] representing the modal responses were constructed. Firstly, the mode shapes are consistent with those formulated by Przemieniecki [@przemieniecki1985theory], and clearly there exists symmetry and asymmetry for the two shapes. Moreover we notice that the effect of the fuselage mass does have an appreciable albeit small effect on the symmetric modes, and no visible effect on the asymmetric modes, which agrees with Przemieniecki’s analysis, and general intuition. This is because the removal of the first row and columns of the elemental beam matrix results also removes the $R$ variable. Notice also that in both cases (symmetric and asymmetric) there is an unconstrained mode, which mathematically exists due to the FE model having no fixed boundary conditions.
In order to simplify analysis only the symmetric bending mode cases are considered, since as Figure \[fig:sym\_bending\_a\] demonstrates, it takes into account the effect of the B737 model through the $R$ variable. As before with the toy example, without loss of generality we aim to formulate the $\bm{\Delta}$ matrices as upper triangular matrices, and perform the random embedding on these upper triangular matrices. However, different to the former case is that due to the reformulation of the problem as an FE model, it is now possible to arbitrarily grow the dimensionality of the problem by increasing the amount of elements of the FE model so that the optimisation problem can grow arbitrarily large, allowing for a more rigorous analysis of the potential usefulness and effects of random projections.
The first three non-dimensional symmetric-mode frequency values of the system under investigation are given in Table \[tab:nondim\_freq\]. The frequency values given are non-dimensionalised according to $\lambda^2\sqrt{M_w L/(EI)}$, and $R=0$ refers to the base-line reference (if the aircraft purely consisted of beam elements and no fuselage mass), whereas $R=1.35$ refers to the B737 aircraft parameters, which are the values we will use in the optimisation procedure. As was mentioned earlier, the existence of the fuselage mass does indeed alter the eigenvalues in an appreciable manner. For the optimisation problem, we aim to alter the first three non-dimensional frequencies of the symmetric bending mode to become: $\omega = \begin{bmatrix} 2, 7, 22 \end{bmatrix}$. That is we would like the following mapping to occur between the eigenvalues, $\begin{bmatrix} 0,4.09,23.36 \end{bmatrix} \xmapsto{\bm{\Delta}} \begin{bmatrix} 2, 7, 22 \end{bmatrix}$. This is why only the first three eigenvalues are shown in Table \[tab:nondim\_freq\]. Note however any number of eigenvalues may be used, and that from a physical point of view, it does not necessarily make sense to be transforming the first eigenvalue from ‘0’ to ‘2’ since this changes the constraints of the system (as ‘0’ represents rigid body motion). However, the emphasis of this paper is to explore PSO as applied to inverse structural eigenvalue problems in parallel with random projections, and so the objective functions were chosen arbitrarily.
**Frequency Number** **R = 0** **R = 1.35**
---------------------- ----------- --------------
1 0 0
2 5.59 4.09
3 30.23 23.36
: First three non-dimensional symmetric-mode frequency values, of the FE model aircraft, non-dimensionalised by $\lambda^2\sqrt{M_w L/(EI)}$, where $R=M_F/M_w$ is the fuselage-to-wing mass ratio.[]{data-label="tab:nondim_freq"}
The convergence behaviour for this optimisation problem is shown in Figure \[fig:conv\_FE\_Model\]. In all three cases we note extremely similar behaviour as compared to the toy example. That is, convergent behaviour in the lower dimensional space is initially and consisitently much faster in the sense that (faster in the sense that with less iterations, the random embedding method tends to have a much lower objective function magnitude). Eventually however, the full dimensional space does *tend* to approach similar values to the random embedding but this is to be expected, since the full system always perfectly describes the problem, and the problem at hand also does seem to possess a very low effective dimensionality, implying that the optimisation procedure may not need to actively explore all possible dimensions. That is, although the full dimensional space seems high, the particle swarm doesn’t need to explore it fully to obtain a good solution. Regardless of the conjectured advantages of this particular problem, the random projection assists in the notion of *faster* convergence behaviuor across all areas.
\
In order to explore the capabilities of random embedding even further, it was applied to a case of a 100 element model for the aircraft. This resulted in a large search space of 40602 free parameters to explore for optimisation. It was proposed to reduce the dimensionality of the problem by 99.3% resulting in a random embedding space of only 300 free parameters. In addition to this, the total amount of particles used in the swarm was reduced by a factor 2 (from 500 to 250). The result of this is shown in Figure \[fig:conv\_FE\_massive\].
![An extreme example showcasing the convergence of a 100 element discretisation resulting in 40602 free paramters. A dimension reduction of 99.3% was used.[]{data-label="fig:conv_FE_massive"}](4241_che_Fig16){width="0.70\linewidth"}
From Figure \[fig:conv\_FE\_massive\] once again extremely similar behaviour can be observed as in the previous problems. That is, the lower dimensional space is able to achieve much lower objective function magnitudes, a lot more rapidly. Moreover in this example, it was shown to be able to do this not only in less iterations, but also with less overall particles. In addition, the overall converged solution of this lower dimensional space is much better than the full dimensional solution which simply found it very difficult on average to converge to good values due to the enormous search space. The full dimensional solution could only converge on the order of $10^0$ on average, whereas the reduced dimension solution is able to converge to a value on the order of $10^{-2}$ on average.
Thus as this paper has consistently demonstrated, through the use of random embedding we are able to significantly increase the speed and quality of convergence of a PSO optimiser, in terms of using less overall particles, coupled with less total iterations, ultimately leading to greater overall computational efficiency, in less total time. Note in this case we say ‘a’ solution since the inverse eigenvalue problem with incomplete modal information is in a well known ill-conditioned problem and there does exist many locally optimal solutions. However the main purpose of this paper was not to explore the ability to achieve the global optima (which for the inverse eigenvalue problem, may not necessarily even be the best solution depending on context), but to analyse the applicability of using random projections alongside PSO in order to study the efficiency of an underlying optimisation procedure in the field of structural engineering.
A further point of discussion for the extremely high dimensional problem explored in Figure \[fig:conv\_FE\_massive\] is to consider the level of distortion that has occured to the original surface when projecting down to a surface which has 99.3% less overall dimensions. Table \[tab:JL\_FE\_massive\] summarises the relationship for $n=40602$ for the JL Lemma. Here we note that for a 70% average discrepancy between the pairwise Euclidean distancs of the points in the new lower dimensional space, 325 dimensions are *sufficient*, which is comparable to what was used in Figure \[fig:conv\_FE\_massive\]. Thus the geometry between adjacent points in this new subspace are most likely significantly different to what was in the original high dimensional space. Regardless however, it was nevertheless possible to converge to extremely good values suggesting that even though there may be a large geometric distortion, $d_e \leq 300$. And thus by Theorem 4.1 there exists a solution (or several), which we are able to find due to the strength of the black-box PSO algorithm. Note however that if we did not opt to reduce the dimensions by 99.3%, but by a factor of $\approx 80\%$ we could still optimise with 9096 dimensions and achieve no more than 10% error in the pairwise Euclidean distances between points. However the surface distortion issue does not appear to be a huge problem given that the problem has a low underlying effective dimension, and a good optimiser is used (as is the case of PSO).
Distortion Error (%) **10** **30** **50** **70** **100**
---------------------- -------- -------- -------- -------- ---------
Dimension 9096 1180 510 325 255
: How the distortion error effects the corresponding dimension of the mapped subspace, for $n=40602$ in accordance with the JL Lemma.[]{data-label="tab:JL_FE_massive"}
Lastly it is important for the reader to note that the solutions obtained in this paper will be nonphysical. This is because the $\bm{\Delta}$ matrices are assumed to be full rank, upper-triangular matrices, without physical constraints applied to them (apart from symmetry being enforced via the upper-triangular nature of $\bm{\Delta}$). In order to enforce complete physicality of the solution it would be necessary to place constraints in the search space (either through equality and or inequality constraints). This idea has been explored partly by previous authors [@sivan1996mass; @olsson2007inverse], but it remains an open question in the case of truncated modal systems. Nevertheless, although it would be trivial to place constraints on the systems explored in this paper, it reamins that the purpose of this paper is to explore the viability of dimensionality reduction for structural vibration problems, of which the results appear to be extremely promising. The placement of constraints would not allow the justified exploration of spaces as high approximately $40000$ in the case of 1D FE model structures.
Conclusion
==========
Random projection is a popular technique used to reduce the dimensionality of a problem. It has been demonstrated in this paper that by using random projections we were able to successfully perform optimisation in this lower dimensional space which resulted in much faster overall convergence, faster in the sense that on average less iterations were required to achieve much better results. This was demonstrated on an example 10-dimensional toy problem, as well as on a 1-D FE model of a Boeing 737-300 aircraft. Moreover the existence of a moderately small effective dimension was predicted to exist for generalised inverse eigenvalue problems which have Hermitian matrices. Moreover it was demonstrated experimentally that gradient-based approaches for performing optimisation for eigenvalue problems may necessitate prohibitively small step sizes, which tends to suggest that non-gradient, black-box optimisation methods may be preffered for these types of problems.
{#section .unnumbered}
Boeing 737-300 Data and Formulation
===================================
An overview of the data used in modeling the 1D FE model B737 structure is presented in this section, as well as a the technical diagram used to extract some of its lengths.
**Parameters** **Value** **Units**
----------------------- ---------------------- ---------------------
Cruise Velocity 725.43 ft/s
Cruise Altitude 30000 ft
Air Density at Cruise 8.91$\times 10^{-4}$ slugs/$\text{ft}^3$
Dynamic Pressure 234.44 lb/$\text{ft}^2$
Ultimate Load Factor 5.7 -
Design Gross Weight 109269.60 lb
: General Flight Parameters, available from *Jane’s all the World’s Aircraft* [@taylor1976jane].[]{data-label="tab:B737_general"}
**Parameters** **Value** **Units**
------------------------- ----------- ---------------
Length 105.94 ft
Depth 12.33 ft
Wet Area 4104.80 $\text{ft}^2$
Tail Length 15.89 ft
Cabin $\Delta$ Pressure 8.00 Pa
: Fuselage Parameters, available from *Jane’s all the World’s Aircraft* [@taylor1976jane].[]{data-label="tab:B737_fuselage"}
**Parameters** **Value** **Units**
-------------------------- ----------- ---------------
Wet Area 1133.90 $\text{ft}^2$
Weight of Fuel in Wing 35640.00 lb
Aspect Ratio 9.16 -
Wing Sweep at $25\%$ MAC 25.00 degrees
Thickness-to-Chord ratio 8.00 -
: Wing Parameters, available from *Jane’s all the World’s Aircraft* [@taylor1976jane].[]{data-label="tab:B737_wing"}
The equations used for estimating the fuselage and wing masses are available from Roskam [@raymer1999aircraft], in particular the Equations used below refer to Equations (15.46), and (15.49) in Roskam. All terms of the below equatinos are are defined in this reference. $$\begin{aligned}
&W_{\text{wing}} = 0.036S_w^{0.758}W_{fw}^{0.0035}\left(\frac{A}{\cos^2\Lambda}\right)^{0.6}q^{0.006}\lambda^{0.04}\left( \frac{100 t/c}{\cos\Lambda} \right)^{-0.3} (N_z W_{dg})^{0.49} \\
&W_{\text{fuselage}}=0.052S_f^{1.086}(N_z W_{dg})^{0.177} L_t^{-0.051} (L/D)^{-0.072} q^{0.241} + W_{press} \\
&W_{press} = 11.9 + (V_{pr}P_{\delta})\end{aligned}$$
\
|
---
abstract: 'In supervised learning one wishes to identify a pattern present in a joint distribution $P$, of instances, label pairs, by providing a function $f$ from instances to labels that has low risk ${\protect\mathbb{E}}_{P}\ell(y,f(x))$. To do so, the learner is given access to $n$ iid samples drawn from $P$. In many real world problems clean samples are not available. Rather, the learner is given access to samples from a corrupted distribution $\tilde{P}$ from which to learn, while the goal of predicting the clean pattern remains. There are many different types of corruption one can consider, and as of yet there is no general means to compare the relative ease of learning under these different corruption processes. In this paper we develop a general framework for tackling such problems as well as introducing upper and lower bounds on the risk for learning in the presence of corruption. Our ultimate goal is to be able to make informed economic decisions in regards to the acquisition of data sets. For a certain subclass of corruption processes (those that are *reconstructible*) we achieve this goal in a particular sense. Our lower bounds are in terms of the coefficient of ergodicity [@Dobrushin1956], a simple to calculate property of stochastic matrices. Our upper bounds proceed via a generalization of the method of unbiased estimators appearing in [@Natarajan2013] and implicit in the earlier work [@Kearns1998].'
author:
- 'Brendan van Rooyen$^{*, \dagger}$\'
- 'Robert C. Williamson$^{*, \dagger}$\'
- |
\
$^*$The Australian National University $^\dagger$National ICT Australia\
[`{ brendan.vanrooyen, bob.williamson }@nicta.com.au` ]{}\
bibliography:
- 'ref.bib'
title: Learning in the Presence of Corruption
---
Introduction
============
The goal of supervised learning is to find a function in some hypothesis class that predicts a relationship between instances and labels. Such a function should have low average loss according to the true distribution of instances and labels, $P$. The learner is not given direct access to $P$, but rather a training set comprising $n$ iid samples from $P$. There are many algorithms for solving this problem (for example empirical risk minimization) and this problem is well understood.\
\
There are many other *types* of data one could learn from. For example in semi-supervised learning [@Chapelle2010] the learner is given $n$ instance label pairs and $m$ instances devoid of labels. In learning with noisy labels [@Angluin1988; @Kearns1998; @Natarajan2013], the learner observes instance label pairs where the observed labels have been corrupted by some noise process. There are many other variants including, but not limited to, learning with label proportions [@Quadrianto2009], learning with partial labels [@Cour2011], multiple instance learning [@Maron1998] as well as combinations of the above.\
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What is currently lacking is a general theory of learning from corrupted data, as well as means to *compare* the relative usefulness of different data types. Such a theory is required if one wishes to make informed economic decisions on which data sets to acquire. For example, are $n$ clean datum better or worse than $n_1$ noisy labels and $n_2$ partial labels?\
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To answer this question we first place the problem of corrupted learning into the abstract language of statistical decision theory. We then develop general lower and upper bounds on the risk relative to the amount of corruption of the clean data. Finally we show examples of problems that fit into this abstract framework.\
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The main contributions of this paper are:
- Novel, general means to construct methods for learning from corrupted data based on a generalization of the method of unbiased estimators presented in [@Natarajan2013] and implicit in the earlier work [@Kearns1998] (theorems \[MUBE\] and \[Corrupted PAC Bayes\])
- Novel lower bounds on the risk of corrupted learning (theorem \[Relative Lower Bound\]).
- Means to understand *compositions* of corruptions (lemmas \[Alpha Composition\] and \[Loss norm Composition\]).
- Upper and lower bounds on the risk of learning from combinations of corrupted data (theorems \[Collection of Corrupted PAC Bayes\] and \[Collection of Corrupted Lower Bound\]).
- Analyses of the tightness of the above bounds.
In doing so we provide answers to our central question of how to rank different types of corrupted data, through the utilization of our upper or lower bounds. While not the complete story for *all* problems, the contributions outlined above make progress toward the final goal of being able to make informed economic decisions regarding the acquisition of data sets. All proofs omitted in the main text appear in the appendix.
The Decision Theoretic Framework
================================
Decision theory deals with the general problem of decision making under uncertainty. One starts with a set $\Theta$ of possible true hypotheses (only one of which is actually true) as well as set $A$ of actions available to the decision maker. Prior to acting, the decision maker performs an experiment, the outcome of which is assumed to be related to the true hypothesis, and observes $z$ in an observation space ${\protect\mathcal{O}}$. Ultimately the decision maker makes act $a$ and incurs loss $L(\theta,a)$, with $\theta$ the unknown true hypothesis. We model the relationships between unknowns and the results of experiments with *Markov kernels* [@Torgersen1991; @Cam2011; @Morse1966; @Chentsov1982]. The abstract development that follows is necessary in order to place a wide range of corruption processes into a single framework so that they may be compared.
Markov Kernels {#sec:markov-kernels}
--------------
As much of our focus will be on noise on the *labels* and not on the *instances*, henceforth we will assume we are only working with finite sets.\
Denote by ${\protect\mathbb{P}}(X)$ the set of probability distributions on a set $X$. Define a *Markov kernel* from a set $X$ to a set $Y$ (denoted by $X \rightsquigarrow Y$) to be a function $T : X \rightarrow {\protect\mathbb{P}}(Y)$. Denote the set of all Markov kernels from $X$ to $Y$ by $M(X,Y)$. Every function $f : X \rightarrow Y$ defines a Markov kernel $T : X \rightsquigarrow Y$ with $T(X) = \delta_{f(x)}$, a point mass on $f(x)$. Given two Markov kernels $T_1 : X \rightsquigarrow Y $ and $T_2 : Y \rightsquigarrow Z$ we can *compose* them to form $T_2 T_1 : X \rightsquigarrow Z$ by taking $${\protect\mathbb{E}}_{T_2 T_1(x)} f = {\protect\mathbb{E}}_{ y {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}T_1(x)} {\protect\mathbb{E}}_{z {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}T_2(y)} f(z)$$ for all $f: Z \rightarrow {\protect\mathbb{R}}$. One can also combine Markov kernels in parallel. If $P \in {\protect\mathbb{P}}(X)$ and $Q \in {\protect\mathbb{P}}(X)$, denote the product distribution by $P \otimes Q$. If $T_i : X \rightsquigarrow Y$, $ i \in [1;n]$, are Markov kernels then $\otimes_{i=1}^n T_i : X^n \rightsquigarrow Y^n$ with $\otimes_{i=1}^n T_i(x^n) = T_1(x_1) \otimes \dots \otimes T_n(x_n)$. By restricting ourselves to finite sets, distributions can be represented by vectors, Markov kernels by column stochastic matrices (positive matrices with column sum 1) and composition by matrix multiplication. An *experiment* on $\Theta$ is any Markov kernel with domain $\Theta$ and a *learning algorithm* ${\protect\mathcal{A}}$ is any Markov kernel with co-domain $A$. Finally, from any experiment $e : \Theta \rightsquigarrow {\protect\mathcal{O}}$ we define the *replicated experiment* $e_n : \Theta \rightsquigarrow {\protect\mathcal{O}}^n ,\ n\in \{1,2,\dots\}$, with $e_n(\theta) = e(\theta)^n$ the $n$-fold product of $e(\theta)$.
Loss and Risk
-------------
One assesses the *consequence* of actions through a *loss* $L : \Theta \times A \rightarrow {\protect\mathbb{R}}$. It is sometimes useful to work with losses in curried form. From any loss $L$ and action $a \in A$, define $L_a \in {\protect\mathbb{R}}^ \Theta$ with $L_a(\theta) = L(\theta,a)$. We measure the size of a loss function by its supremum norm $\lVert L \rVert_\infty = \sup_{\theta,a} |L(\theta,a)|$. If $P\in{\protect\mathbb{P}}(\Theta)$ and $Q \in {\protect\mathbb{P}}(A)$ we overload our notation with $L(P,Q) = {\protect\mathbb{E}}_{\theta {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}P} {\protect\mathbb{E}}_{a {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}Q} L(\theta,a)$.\
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Normally, we are not interested in the absolute loss of an action, rather its loss relative to the best action, defined formally as the *regret* ${\protect{\Delta L}}(\theta,a) = L(\theta,a) - \inf_{a'} L(\theta,a')$. We measure the performance of an algorithm ${\protect\mathcal{A}}$ by the *risk* $${{\protect{\mathcal{R}}}_{L}}(e, \theta, {\protect\mathcal{A}}) = {\protect\mathbb{E}}_{z {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}e(\theta)} {\protect\mathbb{E}}_{a {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}{\protect\mathcal{A}}(z)} {\protect{\Delta L}}(\theta,a).$$ For the sake of comparison by a single number either the max risk or the average risk with respect to a distribution $P_{\Theta} \in {\protect\mathbb{P}}({\protect\mathcal{O}})$ can be used. We define a *learning problem* to be a pair $(L, e)$ with $L : \Theta \times A \rightarrow {\protect\mathbb{R}}$ a loss and $e : \Theta \rightsquigarrow {\protect\mathcal{O}}$ an experiment. We measure the difficulty of a learning problem by the *minimax risk* $${\protect{\underline{{\protect{\mathcal{R}}}}_{L}}}(e) = \inf_{{\protect\mathcal{A}}} \sup_{\theta} {{\protect{\mathcal{R}}}_{L}}(e, \theta, {\protect\mathcal{A}}).$$ Normally we are not concerned with the quality of a learning algorithm for observation of a single $z\in {\protect\mathcal{O}}$. Rather we wish to know the rate at which the risk decreases as the number of replications of the experiment grows. Hence the prime quantity of interest is ${\protect{\underline{{\protect{\mathcal{R}}}}_{L}}}(e_n)$.
Statistics vs Machine Learning
------------------------------
While the ideas of the previous subsections originated in theoretical statistics [@Torgersen1991; @Cam2011; @Blackwell:1954; @Ferguson1967] they can be readily applied to machine learning problems. The main distinction is that statistics focuses on *parametric families* and loss functions of type $L : \Theta \times \Theta \rightarrow {\protect\mathbb{R}}$. The goal is to accurately *reconstruct parameters*. In machine learning one is interested in *predicting the observations* of the experiment well. There the focus is on problems with $\Theta = {\protect\mathbb{P}}({\protect\mathcal{O}})$ and loss functions of the form $L(\theta, a) = {\protect\mathbb{E}}_{z {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}P_{\theta}} \ell(z,a)$, where $\ell : {\protect\mathcal{O}}\times A \rightarrow {\protect\mathbb{R}}$ measures how well $a$ predicts the observation $z$. Our focus is on problems of the second sort, however abstractly there is no real difference. Both are just different learning problems. When clear we use $\ell(P,a)$ and $L(P,a)$ interchangeably.
### Supervised Learning
In Table 1 we explain the mapping of supervised learning into our abstract language. We focus on the problem of conditional probability estimation of which learning a binary classifier is a special case. Letting $X$ be the instance space and $Y$ the label space we have
Unknowns $\Theta$ Distributions of instance, label pairs, ${\protect\mathbb{P}}(X\times Y)$
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Observation Space ${\protect\mathcal{O}}$ $n$ instance label pairs $(X \times Y)^n$.
Action Space $A$ Function class $\mathcal{F} \subseteq {\protect\mathbb{P}}(Y)^X$
Experiment $e$ Maps each $P \in {\protect\mathbb{P}}(X\times Y)$ to itself
Loss $L$ $L(\theta,f) = {\protect\mathbb{E}}_{(x,y){\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}P_\theta} \ell(y, f(x))$
We have $${{\protect{\mathcal{R}}}_{L}} (e_n, P, {\protect\mathcal{A}}) = {\protect\mathbb{E}}_{S {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}P^n} {\protect\mathbb{E}}_{f{\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}{\protect\mathcal{A}}(S)} {\protect\mathbb{E}}_{(x,y) {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}P} \ell(y,f(x)) - \inf_{f \in \mathcal{F}} {\protect\mathbb{E}}_{(x,y) {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}P} \ell(y,f(x))$$ a standard object of study in learning theory [@Bousquet2004].
Corrupted Learning
------------------
In corrupted learning, rather than observing $z \in {\protect\mathcal{O}}$, one observes a corrupted $\tilde{z}$ in a different observation space $\tilde{{\protect\mathcal{O}}}$. We model the corruption process through a Markov kernel $T : {\protect\mathcal{O}}\rightsquigarrow \tilde{{\protect\mathcal{O}}}$ and define a corrupted learning problem to be the triple $(L,e,T)$. For convenience we define the corrupted experiment $\tilde{e} = T e$. Ideally we wish to compare ${\protect{\underline{{\protect{\mathcal{R}}}}_{L}}}(\tilde{e}_n) $ with ${\protect{\underline{{\protect{\mathcal{R}}}}_{L}}}(e_n)$. By general forms of the information processing theorem [@Reid2009b; @Garca-Garca] ${\protect{\underline{{\protect{\mathcal{R}}}}_{L}}}(\tilde{e}_n) \geq {\protect{\underline{{\protect{\mathcal{R}}}}_{L}}}(e_n)$, however this does not allow one to *rank* the utility of *different* $T$.\
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Even after many years of directed research, in general we can not compute ${\protect{\underline{{\protect{\mathcal{R}}}}_{L}}}(e_n)$ exactly, let alone ${\protect{\underline{{\protect{\mathcal{R}}}}_{L}}}(\tilde{e}_n)$ for general corruptions. Consequently our effort for the remaining turns to upper and lower bounds of ${\protect{\underline{{\protect{\mathcal{R}}}}_{L}}}(\tilde{e}_n)$.
Upper Bounds for Corrupted Learning {#sec:upper-bounds-for-corrupted-learning}
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When convenient we use the shorthand $T(P) = \tilde{P}$. [@Natarajan2013] introduced a method of learning classifiers from data subjected to label noise, termed the *method of unbiased estimators*. Here we show that this method can be generalized to other corruptions. Firstly, ${\protect\mathbb{P}}({\protect\mathcal{O}}) \subseteq ({\protect\mathbb{R}}^{\protect\mathcal{O}})^*$, the dual space of ${\protect\mathbb{R}}^{\protect\mathcal{O}}$. We use the notation $\langle P, f \rangle = {\protect\mathbb{E}}_{z {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}P} f(z)$. From any markov kernel $T : {\protect\mathcal{O}}\rightsquigarrow \tilde{{\protect\mathcal{O}}}$, we obtain a linear map $T : ({\protect\mathbb{R}}^{\protect\mathcal{O}})^* \rightarrow ({\protect\mathbb{R}}^{\tilde{{\protect\mathcal{O}}}} )^*$ with $${\langle T(\alpha), \tilde{f}\rangle} = {\langle \alpha, T^*(\tilde{f})\rangle} ,\ \forall \tilde{f} \in {\protect\mathbb{R}}^{\tilde{{\protect\mathcal{O}}}}$$ where $T^*(\tilde{f})(z) = {\protect\mathbb{E}}_{\tilde{z} {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}T(z)} \tilde{f}(\tilde{z})$ is the *pullback* of $\tilde{f}$ by $T$. In terms of matrices $T^*$ is the transpose or *adjoint* of T.
A Markov kernel $T : {\protect\mathcal{O}}\rightsquigarrow \tilde{{\protect\mathcal{O}}}$ is *reconstructible* if $T$ has a left inverse, there exists a linear map $R : ({\protect\mathbb{R}}^{\tilde{{\protect\mathcal{O}}}})^* \rightarrow ({\protect\mathbb{R}}^{\protect\mathcal{O}})^*$ such that $R T = I$.
Intuitively, $T$ is reconstructible if there is some transformation that “undoes" the effects of $T$. In general $R$ is not a Markov kernel. Many forms of corrupted learning are reconstructible, including semi-supervised learning, learning with label noise and learning with partial labels for all but a few pathological cases. The reader is directed to \[Examples\] for worked examples.\
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We call a left inverse of $T$ a *reconstruction*. For concreteness, one can always take $$R = (T^* T)^{-1} T^*$$ the Moore-Penrose pseudo inverse of $T$. Reconstructible Markov kernels are exactly those where we can *transfer* a loss function from the clean distribution to the corrupted distribution. We have by properties of adjoints $${\langle P, f\rangle} = {\langle R T (P), f\rangle} = {\langle T(P), R^*(f)\rangle}.$$ In words, to take expectations of $f$ with samples from $\tilde{P}$ we use the corruption corrected $\tilde{f} = R^*(f)$.
\[MUBE\] For all reconstructible $T : {\protect\mathcal{O}}\rightsquigarrow \tilde{{\protect\mathcal{O}}}$, loss functions $\ell : {\protect\mathcal{O}}\times A \rightarrow {\protect\mathbb{R}}$ and reconstructions $R$ define the *corruption corrected* loss $\tilde{\ell} : \tilde{{\protect\mathcal{O}}} \times A \rightarrow {\protect\mathbb{R}}$, with $\tilde{\ell}_a = R^*\ell_a$. Then for all distributions $P \in {\protect\mathbb{P}}({\protect\mathcal{O}})$, $\ell(P,a) = \tilde{\ell}(\tilde{P},a)$.
We direct the reader to \[Examples\] for some examples of $\tilde{\ell}$ for different corruptions. Minimizing $\tilde{\ell}$ on a sample $\tilde{S} {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}\tilde{P}$ provides means to learn from corrupted data. Let $\ell(S,a) = \frac{1}{|S|} \sum_{z\in S} \ell(z,a)$, the average loss on the sample. By an application of the PAC Bayes bound ([@McAllester1998; @Zhang2006; @Catoni2007]) one has for all algorithms ${\protect\mathcal{A}}: \tilde{{\protect\mathcal{O}}}^n \rightsquigarrow A$, priors $\pi \in {\protect\mathbb{P}}(A)$ and distributions $P \in {\protect\mathbb{P}}({\protect\mathcal{O}})$ $${\protect\mathbb{E}}_{\tilde{S}{\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}\tilde{P}^n} \tilde{\ell}(\tilde{P},{\protect\mathcal{A}}(\tilde{S})) \leq {\protect\mathbb{E}}_{\tilde{S}{\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}\tilde{P}^n} \tilde{\ell}(\tilde{S},{\protect\mathcal{A}}(\tilde{S})) + \lVert \tilde{\ell}\rVert_\infty \sqrt{\frac{2 {\protect\mathbb{E}}_{\tilde{S}{\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}\tilde{P}^n} D_{KL}({\protect\mathcal{A}}(\tilde{S}),\pi)}{n}}.$$ This bound yields the following theorem.
\[Corrupted PAC Bayes\] For all reconstructible Markov kernels $T : {\protect\mathcal{O}}\rightarrow \tilde{{\protect\mathcal{O}}}$, algorithms ${\protect\mathcal{A}}: \tilde{{\protect\mathcal{O}}}^n \rightsquigarrow A$, priors $\pi \in {\protect\mathbb{P}}(A)$, distributions $P \in {\protect\mathbb{P}}({\protect\mathcal{O}})$ and bounded loss functions $\ell$ $${\protect\mathbb{E}}_{\tilde{S}{\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}\tilde{P}^n} \ell(P,{\protect\mathcal{A}}(\tilde{S})) \leq {\protect\mathbb{E}}_{\tilde{S}{\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}\tilde{P}^n}\tilde{\ell}(\tilde{S},{\protect\mathcal{A}}(\tilde{S})) + \lVert \tilde{\ell}\rVert_\infty \sqrt{\frac{2 {\protect\mathbb{E}}_{\tilde{S}{\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}\tilde{P}^n} D_{KL}({\protect\mathcal{A}}(\tilde{S}),\pi)}{n}}.$$
A similar result also holds with high probability on draws from $\tilde{P}^n$. If ${\protect\mathcal{A}}$ is Empirical Risk Minimization (ERM), $A$ is finite and $\pi$ uniform on $A$ the above analysis yields convergence to the optimum $a\in A$ as $\frac{ \lVert \tilde{\ell}\rVert_\infty}{\sqrt{n}}$ for learning with corrupted data versus $\frac{\lVert \ell\rVert_\infty}{\sqrt{n}}$ for learning with clean data. Therefore, the ratio $\frac{\lVert \tilde{\ell}\rVert_\infty}{\lVert \ell\rVert_\infty}$ measures the relative difficulty of corrupted versus clean learning.
Upper Bounds for Combinations of Corrupted Data {#sec:upper-bounds-for-combinations-of-corrupted-data}
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Recall that our final goal is to be able to make informed economic decisions in regarding the acquisition of data sets. As such, we wish to quantify the utility of a data set comprising different corrupted data. For example in learning with noisy labels out of $n$ datum, there could be $n_1$ clean, $n_2$ slightly noisy and $n_3$ very noisy samples and so on. More generally we assume access to a corrupted sample $\tilde{S}$, made up of $k$ different types of corrupted data, with $\tilde{S}_i {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}\tilde{P}^{n_i}$.
\[Collection of Corrupted PAC Bayes\] Let $T_i : {\protect\mathcal{O}}\rightsquigarrow \tilde{{\protect\mathcal{O}}}_i$ be a collection of $k$ reconstructible Markov kernels. Let $\tilde{Q} = \otimes_{i=i}^k \tilde{P}_i^{n_i}$ and $\tilde{{\protect\mathcal{O}}} = \times_{i=1}^k \tilde{{\protect\mathcal{O}}}_i^{n_i}$, $n = \sum_{i=1}^k n_i$ and $r_i = \frac{n_i}{n}$. Then for all algorithms ${\protect\mathcal{A}}: \tilde{{\protect\mathcal{O}}} \rightsquigarrow A$, priors $\pi \in {\protect\mathbb{P}}(A)$, distributions $P \in {\protect\mathbb{P}}({\protect\mathcal{O}})$ and bounded loss functions $\ell$ $${\protect\mathbb{E}}_{\tilde{S} {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}\tilde{Q}} \ell(P,{\protect\mathcal{A}}(\tilde{S})) \leq {\protect\mathbb{E}}_{\tilde{S}{\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}\tilde{Q}} \sum_{i=1}^k r_i \tilde{\ell}_i(\tilde{S}_i,{\protect\mathcal{A}}(\tilde{S})) + K \sqrt{\frac{2 {\protect\mathbb{E}}_{\tilde{S}{\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}\tilde{Q}} D_{KL}({\protect\mathcal{A}}(\tilde{S}),\pi)}{n}}.$$ where $K = \sqrt{\sum\limits_{i=1}^{k}r_i \lVert \tilde{\ell}_i\rVert_{\infty}^2}$.
A similar result also holds with high probability on draws from $Q$. Theorem \[Collection of Corrupted PAC Bayes\] is a generalization of the final bound appearing in [@Crammer2005] that only pertains to symmetric label noise and binary classification. Theorem \[Collection of Corrupted PAC Bayes\] suggest the following means of choosing data sets. Let $c_i$ be the cost of acquiring data corrupted by $T_i$ and $C$ the maximum total cost. First, choose data from the $T_i$ with lowest $c_i \lVert \tilde{\ell}_i\rVert_{\infty}^2$ until picking more violates the budget constraint. Then choose data from the second lowest and so on.
Lower Bounds for Corrupted Learning
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Thus far we have developed upper bounds for ERM style algorithms. In particular we have found that reconstructible corruption does not effect the *rate* at which learning occurs, it only effects constants in the upper bound. Can we do better? Are these constants *tight*? To answer this question we develop lower bounds for corrupted learning.\
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Here we review Le Cam’s method [@Cam2011] a powerful technique for generating lower bounds for learning problems that very often gives the correct rate and dependence on constants (including being able to reproduce the standard VC dimension lower bounds for classification presented in [@Massart2006]). In recent times it has been used to establish lower bounds for: differentially private learning [@Duchi2013], learning in a distributed set up [@Zhang2013], function evaluations required in convex optimization [@Agarwal2009] as well as generic lower bounds in statistical estimation problems [@Yang1999]. We show how this method can be extended using the strong data processing theorem [@Boyen1998; @Cohen1998] to provide a general tool for lower bounding corrupted learning problems.
Le Cam’s Method and Minimax Lower Bounds
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Le Cam’s method proceeds by reducing a general learning problem to an easier binary classification problem, before relating the best possible performance on this classification problem to the minimax risk. Define the *separation* $\rho : \Theta \times \Theta \rightarrow {\protect\mathbb{R}}$, $\rho(\theta_1,\theta_2) = \inf_a {\protect{\Delta L}}(\theta_1,a) + {\protect{\Delta L}}(\theta_2,a)$. The separation measures how hard it is to act well against both $\theta_1$ and $\theta_2$ simultaneously. We have the following (see section \[sec:le-cam’s-method-and-minimax-lower-bounds PROOF\] for a more detailed treatment).
\[Le Cam Lemma\] For all experiments $e$, loss functions $L$ and $\theta_1, \theta_2 \in \Theta$ $${\protect{\underline{{\protect{\mathcal{R}}}}_{L}}} (e) \geq \rho(\theta_1,\theta_2) \left(\frac{1}{4} - \frac{1}{4} V(e(\theta_1), e(\theta_2)) \right).$$ where $V$ is the variational divergence.
This lower bound is a trade off between distances measured by $\rho$ and statistical distances measured by the variational divergence. A learning problem is easy if proximity in variational divergence of $e(\theta_1)$ and $e(\theta_2)$ (hard to distinguish $\theta_1$ and $\theta_2$ statistically) implies proximity of $\theta_1$ and $\theta_2$ in $\rho$ (hard to distinguish $\theta_1$ and $\theta_2$ with actions).\
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If there exists $\theta_1, \theta_2$ with $e(\theta_1) = e(\theta_2)$ and $\rho(\theta_1,\theta_2) > 0$ we instantly get that the minimax regret must be positive. For corrupted experiments, if $T$ is not reconstructible it may be the case that $T e(\theta_1) = T e(\theta_2)$ for some $\theta_1, \theta_2$. Hence we assume that $T$ is reconstructible.
### Replication and Rates {#Duchi Method}
We wish to lower bound how the risk decreases as $n$ grows. When working with replicated experiments it can be advantageous to work with an $f$-divergence (see section \[sec:a-generic-strong-data-processing-theorem.\]) different to variational divergence and to invoke a generalized Pinkser inequality [@Reid2009b]. Common choices in theoretical statistics are the Hellinger and alpha divergences [@Guntuboyina2011] as well as the KL divergence [@Duchi2013]. Here we use the variational divergence and the following lemma.
\[Variational Divergence for product distribitions\] For all collections of distributions $P_i, Q_i \in {\protect\mathbb{P}}({\protect\mathcal{O}}_i)$, $i \in [1 ; k]$ $$V(\otimes_{i=1}^k P_i, \otimes_{i=1}^k Q_i) \leq \sum_{i=1}^k V(P_i,Q_i)$$
Here we make use of the specific case where $P_i = P$ and $Q_i = Q$ for all $i$.
\[Replicated LeCam\] For all experiments $e$, loss functions $L$, $\theta_1, \theta_2 \in \Theta$ and $n$ $${\protect{\underline{{\protect{\mathcal{R}}}}_{L}}}(e_n) \geq \rho(\theta_1,\theta_2) \left( \frac{1}{4} - \frac{n}{4} V(e(\theta_1),e(\theta_2)) \right).$$
To use lemma \[Replicated LeCam\], one defines $\theta_1 = \phi_1(n)$ and $\theta_2 = \phi_2(n)$ for $n \in [0,\infty)$, with the property $$\frac{1}{4} - \frac{n}{4} V(e(\theta_1),e(\theta_2)) \geq \frac{1}{8}$$ or equivalently $V(e(\theta_1), e(\theta_2)) \leq \frac{1}{2 n}$. This yields a lower bound of $${\protect{\underline{{\protect{\mathcal{R}}}}_{L}}}( e_n) \geq \frac{1}{8} \rho(\phi_1(n), \phi_2(n) ).$$ To obtain *tight* lower bounds, $\phi$ needs to be designed in a problem dependent fashion. However, as our goal here is to reason *relatively* we assume that $\phi$ is given.
### Other Methods for Obtaining Minimax Lower Bounds {#Other Methods}
There are many other techniques for lower bounds in terms of functions of pairwise $KL$ divergences [@Yu1997] (for example Assouad’s method) as well as functions of pairwise f-divergences [@Guntuboyina2011]. While such methods are often required to get tighter lower bounds, all of what follows can be applied to these more intricate lower bounding techniques. Therefore, for the sake of conceptual clarity, we proceed with Le Cam’s method.
Measuring the Amount of Corruption
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Rather than the experiment $e$, in corrupted learning we work with the corrupted experiment $\tilde{e}$. By the information processing theorem for $f$-divergences [@Reid2009b], states that $$V(T(P),T(Q)) \leq V(P,Q) ,\ \forall P, Q$$ Thus any lower bound achieved by Le Cam’s method for $e$ can be directly transferred to one for $\tilde{e}$. This is just a manifestation of theorems presented in [@Reid2009b; @Garca-Garca] and alluded to in section 2.4. However, this provides us with no means to rank different $T$. For some $T$, the information processing theorem can be *strengthened*, in the sense that one can find $\alpha(T) < 1$ such that $$\forall P,Q,\ V(T(P),T(Q)) \leq \alpha(T)V(P,Q).$$ The coefficient $\alpha(T)$ provides a means to measure the amount of corruption present in $T$. For example if $T$ is constant and maps all $P$ to the same distribution, then $\alpha(T)=0$. If $T$ is an invertible function, then $\alpha(T) = 1$. Together with lemma \[Replicated LeCam\] this strong information processing theorem [@Cohen1998] leads to meaningful lower bounds that allow the comparison of different corrupted experiments.
A Generic Strong Data Processing Theorem. {#sec:a-generic-strong-data-processing-theorem.}
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Following [@Cohen1998], we present a strong data processing theorem that works for all $f$-divergences.
Let $X$ be a set and $f : {\protect\mathbb{R}}_+ \rightarrow {\protect\mathbb{R}}$ a convex function with $f(1) = 0$. For all distributions $P,Q \in {\protect\mathbb{P}}(X)$ the *$f$-divergence* between $P$ and $Q$ is $$D_f(P,Q) = \int_X f\left(\frac{dQ}{dP}\right) dP.$$
Both the variational and KL divergence are examples of $f$ divergences. For fixed $T$ we seek an $\alpha(T)$ such that $$D_f(T(P),T(Q)) \leq \alpha(T) D_f(P,Q) \ \forall P, Q, f.$$ To do so we first relate the amount $T$ *contracts* $P$ and $Q$ to a certain deconstruction for Markov kernels before proving when such a deconstruction can occur.
\[Deconstruction and KL Lemma\] For all Markov kernels $T : X \rightsquigarrow Y$ and distributions $P,Q \in {\protect\mathbb{P}}(X)$, if there exists $F,G \in M(X,Y)$ and $\lambda \in [0,1]$ such that $T = \lambda F + (1-\lambda) G$ with $F(P) = F(Q)$ then ${\protect{D_{f}}}(T(P),T(Q)) \leq (1-\lambda) {\protect{D_{f}}}(P,Q)$.
Hence the amount $T$ contracts $P$ and $Q$ is related to the amount of $T$ that fixes $P$ and $Q$. We seek the largest $\lambda$ such that a decomposition $T = \lambda F + (1-\lambda) G$ is always possible, no matter what pair of distributions $F$ is required to fix.
\[Existence of Decontruction Lemma\] For all Markov kernels $T : X \rightsquigarrow Y$ define $\lambda(T) = \min_{i,j} \sum_k \min(T_{k,i}, T_{k,j})$. Then $\lambda \leq \lambda(T)$ if and only if for all pairs of distributions $P,Q$ there exists a decomposition $$T = \lambda F + (1-\lambda) G$$ with $F,G\in M(X,Y)$ and $F(P) = F(Q)$.
\[Strong Data Processing\] For all Markov kernels $T : X \rightsquigarrow Y$ define $\alpha(T) = 1 -\lambda(T)$. Then for all $P,Q,f$, $${\protect{D_{f}}}(T(P),T(Q))\leq \alpha(T) {\protect{D_{f}}}(P,Q).$$
The proof is a simple application of lemma \[Deconstruction and KL Lemma\] and lemma \[Existence of Decontruction Lemma\]. It is easy to see that $0 \leq \alpha(T) \leq 1$. Furthermore $\alpha(T) = 0$ if and only if all of the columns of $T$ are the same. While this $\alpha$ may not be the tightest for a given $f$, it is *generic* and as such can be applied in all lower bounding methods mentioned previously.
Relating $\alpha$ to Variational Divergence {#sec:Variational Alpha}
-------------------------------------------
It can be shown [@Cohen1998] that $\alpha(T) = \max_{x_1, x_2} V(T(x_1), T(x_2)) = \frac{1}{2} \max_{i,j} \sum_k|T_{ki} - T_{kj}|$, the maximum $L1$ distance between the columns of $A$ [@Reid2009b]. Furthermore $$\alpha(T) = \sup_{P, Q \in {\protect\mathbb{P}}(X)} \frac{V(T(P), T(Q))}{V(P, Q) } = \sup_{v \in S} \frac{\lVert T(v) \rVert_1}{\lVert v \rVert_1}$$ where $S = \{v : \sum v_i = 0, v \neq 0\}$. Hence $\alpha(T)$ is the operator 1-norm of T when restricted to $S$. The above also shows that $\alpha(T)$ provides the tightest strong data processing theorem possible when using variational divergence, and hence it gives the tightest generic strong data processing theorem. We also have the following compositional property of $\alpha$.
\[Alpha Composition\] For all Markov kernels $T_1 : X \rightsquigarrow Y$ and $T_2 : Y \rightsquigarrow Z$, $$\alpha(T_2 T_1) \leq \alpha(T_2) \alpha(T_1) \leq \min(\alpha(T_2), \alpha(T_1)).$$
Hence $T_2 T_1$ is at least as corrupt as either of the $T_i$.\
\
The first use of $\alpha(T)$ occurs in the work of [@Dobrushin1956] where it is called the coefficient of ergodicity and is used (much like in [@Boyen1998]) to prove rates of convergence of Markov chains to their stationary distribution.
Lower bounds Relative to the Amount of Corruption
-------------------------------------------------
For all experiments $e$, loss functions $L$, $\theta_1, \theta_2 \in \Theta$, $n$ and corruptions $T : {\protect\mathcal{O}}\rightsquigarrow \tilde{{\protect\mathcal{O}}}$ $${\protect{\underline{{\protect{\mathcal{R}}}}_{L}}}(\tilde{e}_n) \geq \rho(\theta_1,\theta_2) \left( \frac{1}{4} - \frac{\alpha(T) n}{4} V(e(\theta_1),e(\theta_2)) \right).$$
The proof is a simple application of lemma \[Replicated LeCam\] and the strong data processing. Suppose we have proceeded as in section \[Duchi Method\], defining $\theta_1 = \phi_1(n)$ and $\theta_2 = \phi_2(n)$ with $V(e(\theta_1), e(\theta_2) ) \leq \frac{1}{2 t}$. Letting $\tilde{\theta}_1 = \phi_1(\alpha(T) n)$ and $\tilde{\theta}_2 = \phi_2(\alpha(T) n)$ gives $V(e(\tilde{\theta}_1), e(\tilde{\theta}_2)) \leq \frac{1}{2 \alpha(T) n}$. Furthermore $${\protect{\underline{{\protect{\mathcal{R}}}}_{L}}}(\tilde{e}_n) \geq \frac{1}{8} \rho(\phi_1(\alpha(T) n), \phi_2(\alpha(T) n) ).$$ In words, if ever Le Cam’s method gives a lower bound of $f(n)$ for repetitions of the clean experiment, we obtain a lower bound of $f(\alpha(T) n)$ for repetitions of the corrupted experiment. Hence the *rate* is unaffected, only the constants. However, a penalty of factor $\alpha(T)$ is unavoidable no matter what learning algorithm is used, suggesting that $\alpha(T)$ is a valid way of measuring the amount of corruption. We summarize the results of this section in the following theorem.
\[Relative Lower Bound\] For all corruptions $T : {\protect\mathcal{O}}\rightsquigarrow \tilde{{\protect\mathcal{O}}}$ and experiments $e : \Theta \rightsquigarrow {\protect\mathcal{O}}$, if Le Cam’s method yields a lower bound ${\protect{\underline{{\protect{\mathcal{R}}}}_{L}}}(e_n) \geq f(n)$ then ${\protect{\underline{{\protect{\mathcal{R}}}}_{L}}}(\tilde{e}_n) \geq f(\alpha(T) n).$
In particular if one has a lower bound of $\frac{C}{\sqrt{n}}$ for the clean problem, as is usual for many machine learning problems, theorem \[Relative Lower Bound\] yields a lower bound of $\frac{C}{\sqrt{\alpha(T) n}}$ for the corrupted problem.
Lower Bounds for Combinations of Corrupted Data {#sec:lower-bounds-for-combinations-of-corrupted-data}
-----------------------------------------------
As in section \[sec:upper-bounds-for-combinations-of-corrupted-data\] we present lower bounds for combinations of corrupted data. For example in learning with noisy labels out of $n$ datum, there could be $n_1$ clean, $n_2$ slightly noisy and $n_3$ very noisy samples and so on.
\[Collection of Corrupted Lower Bound\] Let $T_i : {\protect\mathcal{O}}\rightsquigarrow \tilde{{\protect\mathcal{O}}}_i$, $i \in [1;k]$, be reconstructible Markov kernels. Let $T = \otimes_{i=i}^k T_i^{n_i}$ with $n = \sum_{i=i}^k n_k$. If Le Cam’s method yields a lower bound ${\protect{\underline{{\protect{\mathcal{R}}}}_{L}}}(e_n) \geq f(n)$ then\
${\protect{\underline{{\protect{\mathcal{R}}}}_{L}}}(T e_n) \geq f(K)$ where $K = \left( \sum\limits_{i=1}^{k}\alpha(T_i) n_i \right)$.
As in section \[sec:upper-bounds-for-combinations-of-corrupted-data\] this bound suggest means of choosing data sets, via the following integer program $$\max \sum\limits_{i=1}^{k}\alpha(T_i) n_i \ \ \text{subject to} \sum\limits_{i=1}^k c_i n_i \leq C$$ where $c_i$ is the cost of acquiring data corrupted by $T_i$ and $C$ is the maximum total cost. This is exactly the unbounded knapsack problem [@Dantzig1957] which admits the following near optimal greedy algorithm. First, choose data from the $T_i$ with highest $\frac{\alpha(T_i)}{c_i}$ until picking more violates the constraints. Then pick from the second highest and so on.
Measuring the Tightness of the Upper Bounds and Lower Bounds
============================================================
In the previous sections we have shown upper bounds that depend on $\lVert \tilde{\ell}\rVert_\infty$ as well as lower bounds that depend on $\alpha(T)$. Recall from theorem that \[MUBE\] $\tilde{\ell}_a = R^* \ell_a$, as such the worst case ratio $\frac{\lVert \tilde{\ell}\rVert_\infty}{\lVert \ell\rVert_\infty}$ is determined by the *operator norm* of $R^*$. For a linear map $R : {\protect\mathbb{R}}^X \rightarrow {\protect\mathbb{R}}^Y$ define $$\begin{aligned}
\lVert R \rVert_{1} := \sup_{v \in {\protect\mathbb{R}}^X} \frac{\lVert R v \rVert_{1}}{\lVert v\rVert_{1}} & \ , \lVert R \rVert_{\infty} := \sup_{v \in {\protect\mathbb{R}}^X} \frac{\lVert R v \rVert_{\infty}}{\lVert v\rVert_{\infty}}\end{aligned}$$ which are two operator norms of $R$. They are equal to the maximum absolute column and row sum of $R$ respectively [@Bernstein2009]. Hence $\lVert R \rVert_{1} = \lVert R^* \rVert_{\infty}$.
\[noisy loss norm\] For all losses $\ell$, $T : {\protect\mathcal{O}}\rightarrow \tilde{{\protect\mathcal{O}}}$ and reconstructions $R$, $\frac{\lVert \tilde{\ell}\rVert_\infty}{\lVert \ell\rVert_\infty} \leq \lVert R^* \rVert_{\infty}$.
\[adjoint inverses\] If $T : X \rightsquigarrow Y$ is reconstructible, with reconstruction $R$, then $$\frac{1}{\alpha(T)} \leq 1 / \left( \inf_{u \in {\protect\mathbb{R}}^X} \frac{\lVert T u\rVert_{1}}{\lVert u \rVert_{1}} \right) \leq \lVert R^* \rVert_{\infty}.$$
The intuition here is if $T$ contracts a particular $v \in {\protect\mathbb{R}}^X$ greatly, which would occur if $$\inf_{P, Q \in {\protect\mathbb{P}}(X)} \frac{\lVert T(P - Q) \rVert_1}{\lVert P - Q \rVert_1}$$ was small (here $v = P - Q$), then $R^*$ could greatly increase the norm of a loss $\ell$. However, it need not increase the norm of the particular loss of interest. Note that for lower bounds we look at the *best* case separation of columns of $T$, for upper bounds we essentially use the *worst*. We also get the following compositional theorem.
\[Loss norm Composition\] If $T_1 : X \rightsquigarrow Y$ and $T_2 : Y \rightsquigarrow Z$ are reconstructible, with reconstructions $R_1$ and $R_2$ then $T_2 T_1$ is reconstructible with reconstruction $R_1 R_2$. Furthermore $\frac{1}{\alpha(T_1)\alpha(T_2)} \leq \lVert R_1 R_2\rVert_{1} \leq \lVert R_1\rVert_{1} \lVert R_2\rVert_{1} $.
The first statement is obvious. For the first inequality simply use lemma \[adjoint inverses\] followed by lemma \[Alpha Composition\]. The second inequality is an easy to prove property of operator norms
Comparing Theorems \[Corrupted PAC Bayes\] and \[Relative Lower Bound\]
-----------------------------------------------------------------------
What we have shown is the following implication, for all reconstructible $T$ $$\begin{aligned}
\frac{C_1}{\sqrt{n}} \leq {\protect{\underline{{\protect{\mathcal{R}}}}_{L}}}(e_n) \leq \frac{C_2 \lVert \ell \rVert_\infty}{\sqrt{n}} \Rightarrow \frac{C_1}{\sqrt{\alpha(T) n}} \leq {\protect{\underline{{\protect{\mathcal{R}}}}_{L}}}(\tilde{e}_n) \leq \frac{C_2 \lVert \tilde{\ell} \rVert_\infty}{\sqrt{n}}.\end{aligned}$$ By lemma \[adjoint inverses\], in the worse case $\lVert \tilde{\ell} \rVert_\infty \geq \frac{\lVert \ell \rVert_\infty}{\alpha(T)}$, and in the “optimistic worst case" we arrive at bounds a factor of $\alpha(T)$ apart. We do not know if this is the fault of our upper or lower bounding techniques. However, when considering *specific* $\ell$ and $T$ this gap is no longer present (see section \[Examples\]).
Comparing Theorems \[Collection of Corrupted PAC Bayes\] and \[Collection of Corrupted Lower Bound\]
----------------------------------------------------------------------------------------------------
Assuming $c_T$ is the cost of acquiring data corrupted by $T$, theorem \[Collection of Corrupted Lower Bound\] the ranks the utility of different corruptions by $\frac{1}{\lVert \tilde{\ell} \rVert_\infty^2 c_T}$ where as theorem \[Collection of Corrupted Lower Bound\] ranks by $\frac{\alpha(T)}{c_T}$. By lemma \[adjoint inverses\], $\frac{1}{\alpha(T)}$ is a proxy for $\frac{\lVert \ell \rVert_\infty}{\lVert \tilde{\ell} \rVert_\infty}$ meaning both theorems are “doing the same thing". In theorems \[Collection of Corrupted Lower Bound\] and \[Collection of Corrupted PAC Bayes\] we have best case and a worst case loss specific method for choosing data sets. Theorem \[Collection of Corrupted PAC Bayes\] combined with 1emma \[noisy loss norm\] provides a worst case loss insensitive method for choosing data sets.
What if Clean Learning is Fast?
===============================
The preceding largly solves the problem of learning from corrupted data when learning from the clean distribution occurs at a slow ($\frac{1}{\sqrt{n}}$) rate. The reader is directed to section \[fast learning\] for some preliminary work on when corrupted learning also occurs at a fast rate.
Proper Losses and Convexity
===========================
A loss $\ell : {\protect\mathcal{O}}\times {\protect\mathbb{P}}({\protect\mathcal{O}}) \rightarrow {\protect\mathbb{R}}$ is *proper* if $$P \in \operatorname*{arg\,min}_{Q \in {\protect\mathbb{P}}({\protect\mathcal{O}})} {\protect\mathbb{E}}_{z{\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}P} \ell(z,Q).$$ It is *stricly proper* if $P$ is the *unique* minimizer.
Proper losses provide suitable surrogate losses for learning problems. All strictly proper losses can be *convexified* through the use of the *canonical* link function [@Reid2010; @Vernet2011]. Ultimately one works with a loss of the form $$\ell(z,v) = v(z) + \Psi(v) {\protect\bm{1}}$$ with $v \in {\protect\mathbb{R}}^{\protect\mathcal{O}}$, ${\protect\bm{1}}\in {\protect\mathbb{R}}^{\protect\mathcal{O}}$ the constant function $\bm{z} = 1$ and $\Psi : {\protect\mathbb{R}}^{\protect\mathcal{O}}\rightarrow {\protect\mathbb{R}}$ a convex function.
Let $v \in {\protect\mathbb{R}}^{\protect\mathcal{O}}$ and $\Psi : {\protect\mathbb{R}}^{\protect\mathcal{O}}\rightarrow {\protect\mathbb{R}}$ be a convex function. Define the loss $\ell(z,v) = v(z) + \Psi(v)$. Then $$\tilde{\ell}(\tilde{z}, v ) = R^*v(\tilde{z}) + \Psi(v).$$ Furthermore this loss is convex in $v$.
This was first noticed in [@Cid-Sueiro2014].
Uses in Supervised Learning
===========================
Recall in supervised learning ${\protect\mathcal{O}}= X \times Y$ and the goal is to find a function that predicts $Y$ from $X$ with low expected loss. Many supervised learning techniques proceed by minimizing a proper loss. Given a suitable function class ${\protect\mathcal{F}}\subseteq {\protect\mathbb{P}}(Y)^X$ and a strictly proper loss $\ell$, they attempt to find $$f^* = \operatorname*{arg\,min}_{f \in {\protect\mathcal{F}}} {\protect\mathbb{E}}_{(x,y) {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}P} \ell(y,f(x)).$$ Using the canonical link function and a careful chosen function class, leaves the learner with a convex problem. If we assume the labels have been corrupted by a corruption $T : Y \rightsquigarrow \tilde{Y}$, we can correct for the corruptions and solve for $$\operatorname*{arg\,min}_{f \in {\protect\mathcal{F}}} {\protect\mathbb{E}}_{(x,\tilde{y}) {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}\tilde{P}} \tilde{\ell}(\tilde{y},f(x)).$$ This objective is equivalent to the first and will also be convex.
Conclusion
==========
We have sought to solve the problem of how to rank different forms of corrupted data with the ultimate goal of making informed decisions regarding to the acquisition of data sets. To do so we have introduced a general framework in which many corrupted learning tasks can be expressed. Furthermore, we have derived general upper and lower bounds for the reconstructible subset of corrupted learning problems. Finally, we have shown that in some examples these bounds are tight enough to be of use and that they produce the quantities one would expect. These bounds facilitate the ranking of different corrupted data, either through the use of best case lower bounds or worst case upper bounds. We have shown both *loss specific* and *worst case as the loss is varied* bounds. Future work will attempt to further refine these methods as well as extend the framework to non reconstructible problems such as multiple instance learning and learning with label proportions. Theorems \[Collection of Corrupted PAC Bayes\] and \[Collection of Corrupted Lower Bound\] provide means of choosing between data sets that feature collections of different corrupted data.
Appendix
========
Examples {#Examples}
--------
We now show examples of common corrupted learning problems. Once again, our focus is corruption of the *labels* and not the *instances*. Thus we work directly with losses $\ell : Y \times A \rightarrow {\protect\mathbb{R}}$. In particular we work with classification problems. We present the worst case upper bound, $\lVert R^* \rVert_{\infty}$, as well as the upper bound relevant for $01$ loss, $\ell_{01}$.
### Noisy Labels
We consider the problem of learning from noisy binary labels [@Angluin1988; @Natarajan2013]. Here $\sigma_{i}$ is the probability that class $i$ is flipped. We have $$\begin{aligned}
T = {\left(\begin{array}{cc}
1-\sigma_{-1} & \sigma_1 \\
\sigma_{-1} & 1-\sigma_1
\end{array} \right)}& \ \ R^* = {\frac{1}{1 - \sigma_{-1} - \sigma_1} \left( \begin{array}{cc}
1-\sigma_1 & -\sigma_{-1} \\
-\sigma_{1} & 1-\sigma_{-1}
\end{array} \right) }.\end{aligned}$$ This yields $$\tilde{\ell}(y,a) = \frac{(1-\sigma_{-y}) \ell(y,a) - \sigma_y \ell(-y,a)}{1- \sigma_{-1} - \sigma_1}.$$ The above equation is lemma 1 in [@Natarajan2013] and is the original method of unbiased estimators. Interestingly, even if $\ell$ is positive, $\tilde{\ell}$ can be negative. If the noise is symmetric with $\sigma_{-1}= \sigma_{1} = \sigma$ and $\ell$ is $01$ loss then $$\tilde{\ell}(y,a) = \frac{\ell_{01}(y,a) - \sigma}{1 - 2 \sigma}$$ which is just a rescaled and shifted version of $01$ loss. If we work in the realizable setting, ie there is some $f \in \mathcal{F}$ with $${\protect\mathbb{E}}_{(x,y){\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}P} \ell_{01}(y, f(x)) = 0$$ then the above provides an interesting correspondence between learning with symmetric label noise and learning under distributions with large Tsybakov margin [@Audibert2007]. Taking $\sigma = \frac{1}{2} - h$ with $P$ *separable* in turn implies $\tilde{P}$ has Tsybakov margin $h$. This means bounds developed for this setting [@Massart2006] can be transferred to the setting of learning with symmetric label noise. Our lower bound reproduces the results of [@Massart2006]\
\
Below is a table of the relevant parameters for learning with noisy binary labels. These results directly extend those present in [@Kearns1998] that considered only the case of symmetric label noise.
[l||l]{}\
$T$ & ${\left(\begin{array}{cc}
1-\sigma_{-1} & \sigma_1 \\
\sigma_{-1} & 1-\sigma_1
\end{array} \right)}$\
$R^*$ & ${\frac{1}{1 - \sigma_{-1} - \sigma_1} \left( \begin{array}{cc}
1-\sigma_1 & -\sigma_{-1} \\
-\sigma_{1} & 1-\sigma_{-1}
\end{array} \right) }$\
$\alpha(T)$ & $|1-\sigma_{-1} - \sigma_1|$\
$\lVert R^* \rVert_{\infty}$ & $\frac{1}{|1-\sigma_{-1} - \sigma_1|}\max(1 - \sigma_{-1} + \sigma_1, 1-\sigma_1 + \sigma_{-1})$\
$\lVert \tilde{\ell}_{01} \rVert_{\infty}$ & $\frac{1}{|1-\sigma_{-1} - \sigma_1|}\max(1 - \sigma_{-1} , 1-\sigma_1 , \sigma_{-1} , \sigma_{1})$\
We see that as long as $\sigma_{-1} + \sigma_{1} \neq 1$ $T$ is reconstructible. The pattern we see in this table is quite common. $\lVert R^* \rVert_{\infty}$ tends to be marginally greater than $\frac{1}{\alpha(T)}$, with $\lVert \tilde{\ell}_{01} \rVert_{\infty}$ less than both. In the symmetric case our lower bound reproduces those of [@Aslam1996].
### Semi-Supervised Learning
We consider the problem of semi-supervised learning [@Chapelle2010]. Here $1-\sigma_i$ is the probability class $i$ has a missing label. We first consider the easier symmetric case where $\sigma_{-1}= \sigma_{1} = \sigma$.
[l||l]{}\
$T$ & ${\left(\begin{array}{cc}
\sigma & 0 \\
0 & \sigma \\
1-\sigma & 1-\sigma
\end{array} \right)}$\
$R^*$ & ${\left(\begin{array}{cc}
\frac{1 - 2 \sigma + 2 \sigma^2}{1- 3\sigma + 5 \sigma^2 - 3 \sigma^3} & \frac{-\sigma^2}{1- 3\sigma + 5 \sigma^2 - 3 \sigma^3} \\
\frac{-\sigma^2}{1- 3\sigma + 5 \sigma^2 - 3 \sigma^3} & \frac{1 - 2 \sigma + 2 \sigma^2}{1- 3\sigma + 5 \sigma^2 - 3 \sigma^3} \\
\frac{\sigma }{1 - 2 \sigma + 3 \sigma^2} & \frac{\sigma }{1 - 2 \sigma + 3 \sigma^2}
\end{array} \right)}$\
$\alpha(T)$ & $\sigma$\
$\lVert R^* \rVert_{\infty}$ & $\frac{1}{\sigma}$\
$\lVert \tilde{\ell}_{01} \rVert_{\infty}$ & $\frac{1 - 2 \sigma + 2 \sigma^2}{2\sigma + 3\sigma - 5 \sigma^2} $\
Once again $\lVert \tilde{\ell}_{01} \rVert_{\infty} \leq \frac{1}{\alpha(T)}$. As long as $\sigma \neq 0$. Our lower bound confirms that in general unlabelled data does not help [@Balcan2010]. Rather than using the method of unbiased estimators, one could simply throw away the unlabelled data leaving behind $\sigma n$ labelled instances on average.
[l||l]{}\
$T$ & ${\left(\begin{array}{cc}
\sigma_{-1} & 0 \\
0 & \sigma_{1} \\
1 - \sigma_{-1} & 1 - \sigma_{1}
\end{array} \right)}$\
$\alpha(T)$ & $\max_i \sigma_i$\
Other parameters for the more general case are omitted due to complexity (they involve the maximum of three 4th order rational equations). They are available in closed form.
### Three Class Symmetric Label Noise
In line with [@Kearns1998], here we present parameters for the three class variant of symmetric label noise. We have $\tilde{Y} = Y = \{1,2,3\}$ with $P(\tilde{Y} = \tilde{y} | Y =y ) = 1 -\sigma$, if $y = \tilde{y}$ and $\frac{\sigma}{2}$ otherwise.
[l||l]{}\
$T$ & ${\left(\begin{array}{ccc}
1-\sigma & \frac{\sigma}{2} & \frac{\sigma}{2} \\
\frac{\sigma}{2} & 1-\sigma & \frac{\sigma}{2} \\
\frac{\sigma}{2} & \frac{\sigma}{2} & 1-\sigma
\end{array} \right)}$\
$R^*$ & ${\left(\begin{array}{ccc}
\frac{2-\sigma}{2 - 3\sigma} & \frac{-\sigma}{2-3\sigma} & \frac{-\sigma}{2-3\sigma} \\
\frac{-\sigma}{2-3\sigma} & \frac{2-\sigma}{2 - 3\sigma} & \frac{-\sigma}{2-3\sigma} \\
\frac{-\sigma}{2-3\sigma} & \frac{-\sigma}{2-3\sigma} & \frac{2-\sigma}{2 - 3\sigma}
\end{array} \right)}$\
$\alpha(T)$ & $|1 - \frac{3}{2} \sigma |$\
$\lVert R^* \rVert_{\infty}$ & $\frac{2 + \sigma}{|2 - 3 \sigma |}$\
$\lVert \tilde{\ell}_{01} \rVert_{\infty}$ & $\frac{2}{| 2 - 3 \sigma |} \max(\sigma, 1- \sigma)$\
We see that as long as $\sigma \neq \frac{2}{3}$ $T$ is reconstructible. Once again $\lVert \tilde{\ell}_{01} \rVert_{\infty} \leq \frac{1}{\alpha(T)}$.
### Partial Labels
Here we follow [@Cour2011] with $Y = \{1,2,3\}$ and $\tilde{Y} = \{0,1\}^{Y}$ the set of partial labels. A partial label of $(0,1,1)$ indicates that the true label is either $2$ or $3$ but not $1$. We assume that a partial label always includes the true label as one of the possibilities and furthermore that spurious labels are added with probability $\sigma$.
[l||l]{}\
$T$ & $\left(
\begin{array}{ccc}
0 & 0 & (1-\sigma )^2 \\
0 & (1-\sigma )^2 & 0 \\
0 & (1-\sigma ) \sigma & (1-\sigma ) \sigma \\
(1-\sigma )^2 & 0 & 0 \\
(1-\sigma ) \sigma & 0 & (1-\sigma ) \sigma \\
(1-\sigma ) \sigma & (1-\sigma ) \sigma & 0 \\
\sigma ^2 & \sigma ^2 & \sigma ^2 \\
\end{array}
\right)$\
$\alpha(T)$ & $1 - \sigma$\
We see that as long as $\sigma \neq 1$ $T$ is reconstructible. In this case $\lVert \tilde{\ell}_{01} \rVert_{\infty}$ and $\lVert R^* \rVert_{\infty}$ are given by more complicated expressions (however they are both available in closed form). We display their interrelation in a graph in below. To the best of our knowledge, there are no upper and lower bounds are present in the literature for this problem.
{width="0.8\linewidth"}
PAC Bayesian Bounds {#PAC Bayes Appendix}
-------------------
PAC Bayesian bounds provide methods to assess the quality of any algorithm ${\protect\mathcal{A}}: {\protect\mathcal{O}}\rightsquigarrow A$. All of the bounds presented in this section appear in [@Zhang2006]. We use the shorthand $\ell(S,a) = \frac{1}{|S|}\sum_{z\in S} \ell(z,a)$.
\[PAC-Bayes\] For all sets ${\protect\mathcal{O}}$, $P \in {\protect\mathbb{P}}({\protect\mathcal{O}})$, priors $\pi \in {\protect\mathbb{P}}(A)$, algorithms ${\protect\mathcal{A}}: {\protect\mathcal{O}}\rightsquigarrow A$, functions $L : {\protect\mathcal{O}}\times A \rightarrow {\protect\mathbb{R}}$ and $\beta > 0$ $${\protect\mathbb{E}}_{z {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}P} {\protect\mathbb{E}}_{a {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}{\protect\mathcal{A}}(z)} - \frac{1}{\beta} \log ({\protect\mathbb{E}}_{z' {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}P} e^{- \beta L(z',a)} ) \leq {\protect\mathbb{E}}_{z {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}P}\left[ L(z,{\protect\mathcal{A}}(z)) +\frac{D_{KL}({\protect\mathcal{A}}(z),\pi) }{\beta} \right].$$ Furthermore with probability at least $1- \delta$ on a draw $x{\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}P$ with $\pi$, $\beta$ and ${\protect\mathcal{A}}$ fixed before the draw, $${\protect\mathbb{E}}_{a {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}{\protect\mathcal{A}}(z)} - \frac{1}{\beta} \log ({\protect\mathbb{E}}_{z' {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}P} e^{- \beta L(z',a)} ) \leq L(z,{\protect\mathcal{A}}(z)) +\frac{D_{KL}({\protect\mathcal{A}}(z),\pi) + {\log\left(\frac{1}{\delta}\right)}}{\beta}.$$
Combined with standard bounds of the cumulant generating function, theorem \[PAC-Bayes\] leads to useful generalization bounds.
\[cumulantlowerbound\] Let $\phi : {\protect\mathcal{O}}\rightarrow [-a,a]$, then for all $\beta >0$ and all $P$ $${\protect\mathbb{E}}_{P} \phi - \frac{a^2 \beta }{2} \leq -\frac{1}{\beta}\log({\protect\mathbb{E}}_{P} e^{-\beta \phi} )$$
See appendix A.1 of [@Cesa-Bianchi2006].
Proof of Theorem \[Collection of Corrupted PAC Bayes\]
------------------------------------------------------
Define $L(\tilde{S},a) = \sum_{i=1}^k \sum_{\tilde{z} \in \tilde{S}_i } \tilde{\ell}_i(\tilde{z}_i,a)$, the sum of the corrupted losses on the sample. We have by theorem \[PAC-Bayes\] $$\begin{aligned}
{\protect\mathbb{E}}_{\tilde{S} {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}Q} {\protect\mathbb{E}}_{a {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}{\protect\mathcal{A}}(\tilde{S})} - \frac{1}{\beta} \log ({\protect\mathbb{E}}_{\tilde{S}' {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}Q} e^{- \beta L(\tilde{S}',a)} ) &\leq {\protect\mathbb{E}}_{\tilde{S} {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}Q}\left[ L(\tilde{S},{\protect\mathcal{A}}(\tilde{S})) +\frac{D_{KL}({\protect\mathcal{A}}(\tilde{S}),\pi)}{\beta} \right] \\
\sum\limits_{i=1}^{k} n_i {\protect\mathbb{E}}_{\tilde{S} {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}Q} {\protect\mathbb{E}}_{a {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}{\protect\mathcal{A}}(\tilde{S})} - \frac{1}{\beta} \log ({\protect\mathbb{E}}_{\tilde{z} {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}\tilde{P}_i} e^{- \beta \tilde{\ell_i}(\tilde{z},a)} ) &\leq {\protect\mathbb{E}}_{\tilde{S} {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}Q}\left[ L(\tilde{S},{\protect\mathcal{A}}(\tilde{S})) +\frac{D_{KL}({\protect\mathcal{A}}(\tilde{S}),\pi)}{\beta} \right]\end{aligned}$$ where the first line follows from theorem \[PAC-Bayes\] and the second from properties of the cumulant generating function. Invoking lemma \[cumulantlowerbound\] yields $$\sum\limits_{i=1}^{k}n_i \left({\protect\mathbb{E}}_{\tilde{S} {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}Q} \tilde{\ell}_i(\tilde{P}_i,{\protect\mathcal{A}}(\tilde{S})) - \frac{\lVert \tilde{\ell}_i\rVert^2_\infty \beta}{2} \right) \leq {\protect\mathbb{E}}_{\tilde{S} {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}Q}\left[ L(\tilde{S},{\protect\mathcal{A}}(\tilde{S})) +\frac{D_{KL}({\protect\mathcal{A}}(\tilde{S}),\pi) }{\beta} \right].$$ As the $T_i$ are reconstructible, $${\protect\mathbb{E}}_{\tilde{S} {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}Q} \ell(P,{\protect\mathcal{A}}(\tilde{S})) \leq \frac{1}{n} {\protect\mathbb{E}}_{\tilde{S} {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}Q}\left[ L(\tilde{S},{\protect\mathcal{A}}(\tilde{S})) +\frac{D_{KL}({\protect\mathcal{A}}(\tilde{S}),\pi)}{\beta} \right] + \frac{\left(\sum\limits_{i=1}^{k} r_i \lVert \tilde{\ell}_i\rVert^2_\infty \right) \beta}{2}.$$ Optimizing over $\beta$ yields the desired result.
Le Cam’s Method and Minimax Lower Bounds {#sec:le-cam's-method-and-minimax-lower-bounds PROOF}
----------------------------------------
The development here closely follows [@Duchi2013] with some streamlining. We consider a general learning problem with unknowns $\Theta$, observation space ${\protect\mathcal{O}}$ and loss $L : \Theta \times A \rightarrow {\protect\mathbb{R}}$. For any learning algorithm ${\protect\mathcal{A}}: {\protect\mathcal{O}}\rightsquigarrow \Theta$, we wish to lower bound the max risk $$\sup_{\theta} {\protect\mathbb{E}}_{z {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}e(\theta)} L(\theta,{\protect\mathcal{A}}(z)).$$ The method proceeds by reducing a general decision problem to an easier binary classification problem. First one considers a supremum over a restricted set $\{\theta_1, \theta_2\}$. Using Markov’s inequality we then relate this to the minimum $01$ loss in a particular binary classification problem. Finally one finds a lower bound for this quantity. With $\theta {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}\{\theta_1, \theta_2\}$ meaning $\theta$ is drawn uniformly at random from the set $\{\theta_1, \theta_2\}$, we have $$\begin{aligned}
\sup_{\theta} {\protect\mathbb{E}}_{z {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}e(\theta)}{\protect\mathbb{E}}_{a {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}{\protect\mathcal{A}}(z)} {\protect{\Delta L}}(\theta,a) &\geq \sup_{\{\theta_1, \theta_2\}} {\protect\mathbb{E}}_{z {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}e(\theta)}{\protect\mathbb{E}}_{a {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}{\protect\mathcal{A}}(z)} {\protect{\Delta L}}(\theta,a) \nonumber \\
&\geq {\protect\mathbb{E}}_{\theta {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}\{\theta_1, \theta_2\}} {\protect\mathbb{E}}_{z {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}e(\theta)}{\protect\mathbb{E}}_{a {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}{\protect\mathcal{A}}(z)} {\protect{\Delta L}}(\theta,a) \nonumber \\
&\geq \delta {\protect\mathbb{E}}_{\theta {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}\{\theta_1, \theta_2\}} {\protect\mathbb{E}}_{z {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}e(\theta)}{\protect\mathbb{E}}_{a {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}{\protect\mathcal{A}}(z)} {[\![ {\protect{\Delta L}}(\theta,a) \geq \delta ]\!]}.\end{aligned}$$ Recall the *separation* $\rho : \Theta \times \Theta \rightarrow {\protect\mathbb{R}}$, $\rho(\theta_1,\theta_2) = \inf_a {\protect{\Delta L}}(\theta_1,a) + {\protect{\Delta L}}(\theta_2,a)$. The separation measures how hard it is to act well against both $\theta_1$ and $\theta_2$ simultaneously. We now assume $\rho(\theta_1,\theta_2) > 2 \delta$. Define $f : A \rightarrow \{\theta_1, \theta_2, \text{error}\}$ where $f(a) = \theta_i$ if ${\protect{\Delta L}}(\theta_i,a) < \delta$ and error otherwise. This function is well defined as if there exists an action $a$ with ${\protect{\Delta L}}(\theta_1,a) < \delta$ and ${\protect{\Delta L}}(\theta_2,a) < \delta$ then $\rho(\theta_1,\theta_2) < 2\delta$ a contradiction. Let $\hat{{\protect\mathcal{A}}}$ be the classifier that first draws $a{\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}{\protect\mathcal{A}}(z)$ and then outputs $f(a)$ we have $$\begin{aligned}
\sup_{\theta} {\protect\mathbb{E}}_{z {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}e(\theta)}{\protect\mathbb{E}}_{a {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}{\protect\mathcal{A}}(z)} {\protect{\Delta L}}(\theta,a) &\geq \delta {\protect\mathbb{E}}_{\theta {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}\{\theta_1, \theta_2\}} {\protect\mathbb{E}}_{z {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}e(\theta)}{\protect\mathbb{E}}_{\theta' {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}\hat{{\protect\mathcal{A}}}(z)} {[\![ \theta\neq \theta' ]\!]} \\
&\geq \delta \inf_{\hat{{\protect\mathcal{A}}} : {\protect\mathcal{O}}\rightsquigarrow \Theta} {\protect\mathbb{E}}_{\theta {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}\{\theta_1, \theta_2\}} {\protect\mathbb{E}}_{z {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}e(\theta)}{\protect\mathbb{E}}_{\theta' {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}\hat{{\protect\mathcal{A}}}(z)} {[\![ \theta\neq \theta' ]\!]} \\
&= \delta \left(\frac{1}{2} - \frac{1}{2} V(e(\theta_1), e(\theta_2)) \right)\end{aligned}$$ where the first line is a rewriting of (1) in terms of the classifier $\hat{{\protect\mathcal{A}}}$, the second takes an infimum over all classifiers and the final line is a standard result in theoretical statistics [@Reid2009b]. Taking $\delta = \frac{\rho(\theta_1,\theta_2)}{2}$ yields lemma \[Le Cam Lemma\].
Proof of Lemma \[Variational Divergence for product distribitions\]
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Firstly $V$ is a *metric* on ${\protect\mathbb{P}}(\times_{n=1}^k {\protect\mathcal{O}}_i)$ [@Reid2009b]. Thus $$\begin{aligned}
V(\otimes_{i=1}^k P_i, \otimes_{i=1}^k Q_i) &= V(P_1 \otimes (\otimes_{i=2}^k P_i), Q_1 \otimes (\otimes_{i=2}^k Q_i)) \\
&\leq V(P_1 \otimes (\otimes_{i=2}^k P_i) , Q_1 \otimes (\otimes_{i=2}^k P_i)) + V(Q_1 \otimes (\otimes_{i=2}^k P_i) , Q_1 \otimes (\otimes_{i=2}^k Q_i)) \\
&= V(P_1, Q_1) + V(\otimes_{i=2}^k P_i, \otimes_{i=2}^k Q_i)\end{aligned}$$ where the first line is by definition, the second as $V$ is a metric and the third is easily verified from the definition of $V$. To complete the proof proceed inductively.
Proof of Lemma \[Deconstruction and KL Lemma\]
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$$\begin{aligned}
{\protect{D_{f}}}(T(P),T(Q)) &= {\protect{D_{f}}}(\lambda F(P) + (1-\lambda) G(P),\lambda F(Q) + (1-\lambda) G(Q)) \\
&\leq \lambda {\protect{D_{f}}}(F(P),F(Q)) + (1 - \lambda) {\protect{D_{f}}}(G(P),G(Q)) \\
&= (1 - \lambda) {\protect{D_{f}}}(G(P),G(Q)) \\
&\leq (1 - \lambda) {\protect{D_{f}}}(P,Q)\end{aligned}$$
Where the first line follows from the definition, the second from the joint convexity of $f$-divergences [@Reid2009b], the third because $F(P)=F(Q)$ and $D_f(P,P) = 0$ and finally the fourth is from the standard data processing inequality [@Reid2009b].
Proof of Lemma \[Existence of Decontruction Lemma\]
---------------------------------------------------
The proof of the forward implication is lemma 2 of [@Boyen1998]. We prove the reverse implication.
As this decomposition works for all pairs of distributions we can take $P = \delta_{x_i} = e_i$ and $Q = \delta_{x_j} =e_j$. As $F(P) = F(Q)$ we must have $F_{ki} = F_{kj} = v_k$ for all $k$. As all of the entries of $(1-\lambda) G$ are positive, we have $\lambda v_k \leq T_{ki}$ and $\lambda v_k \leq T_{kj}$. Hence $\lambda v_k \leq \min(T_{ki},T_{kj})$. Summing over $k$ and remembering that $F$ is column stochastic gives $\lambda \leq \sum_k \min(T_{k,i}, T_{k,j})$. As $i$ and $j$ are arbitrary we have the desired result.
Proof of Theorem \[Collection of Corrupted Lower Bound\]
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Let $$T = \otimes_{i=i}^k T_i^{n_i} = \underbrace{T_1 \otimes \dots \otimes T_1}_{n_1 \ \text{times}} \otimes \underbrace{T_2 \otimes \dots \otimes T_2}_{n_2 \ \text{times}} \dots \otimes \underbrace{T_k \otimes \dots \otimes T_k}_{n_k \ \text{times}}.$$ One has $T(e_n(\theta)) = T_1(e(\theta))^{n_1}\otimes T_2(e(\theta))^{n_2} \otimes \dots \otimes T_k(e(\theta))^{n_k}$. By lemma \[Variational Divergence for product distribitions\], $$\begin{aligned}
V(T(e_n(\theta_1)), T(e_n(\theta_2)) &\leq \sum_{i=1}^k n_i V(T_i(e(\theta_1)), T_i(e(\theta_2)) ) \\
&\leq \left( \sum\limits_{i=1}^{k}\alpha(T_i) n_i \right) V(e(\theta_1),e(\theta_2)).\end{aligned}$$ Now proceed as in the proof of theorem \[Relative Lower Bound\].
Proof of Lemma \[Alpha Composition\]
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$$\begin{aligned}
\alpha(T_2 T_1) &= \sup_{P, Q \in {\protect\mathbb{P}}(X)} \frac{\lVert T_2 T_1(P) - T_2 T_1(Q) \rVert_1}{\lVert P - Q \rVert_1} \\
&= \sup_{P, Q \in {\protect\mathbb{P}}(X)} \frac{\lVert T_2 T_1(P) - T_2 T_1(Q) \rVert_1}{\lVert T_1(P) - T_2(Q) \rVert_1} \frac{\lVert T_1(P) - T_2(Q) \rVert_1}{\lVert P - Q \rVert_1} \\
&\leq \sup_{P, Q \in {\protect\mathbb{P}}(X)} \frac{\lVert T_2 T_1(P) - T_2 T_1(Q) \rVert_1}{\lVert T_1(P) - T_2(Q) \rVert_1} \sup_{P, Q \in {\protect\mathbb{P}}(X)} \frac{\lVert T_1(P) - T_2(Q) \rVert_1}{\lVert P - Q \rVert_1} \\
&\leq \sup_{P, Q \in {\protect\mathbb{P}}(Y)} \frac{\lVert T_2 (P) - T_2 (Q) \rVert_1}{\lVert P - Q \rVert_1} \sup_{P, Q \in {\protect\mathbb{P}}(X)} \frac{\lVert T_1(P) - T_2(Q) \rVert_1}{\lVert P - Q \rVert_1} \\
&= \alpha(T_2) \alpha(T_1)\end{aligned}$$
Where the first line follows from the definitions, the second follows if $T_1(P) \neq T_2(Q)$ and the rest are simple rearrangements. For the final inequality, remember that $\alpha(T) \leq 1$.
Proof of Lemma \[noisy loss norm\]
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By definition $\lVert \tilde{\ell} \rVert_\infty = \sup_{z,a} |\tilde{\ell}(z,a)| = \sup_a \lVert \tilde{\ell}_a \rVert_\infty$. Hence $$\begin{aligned}
\lVert \tilde{\ell} \rVert_\infty &= \sup_a \lVert \tilde{\ell}_a \rVert_\infty \\
&\leq \sup_a \lVert R^*\rVert_\infty \lVert \ell_a \rVert_\infty \\
&= \lVert R^*\rVert_\infty \lVert \ell \rVert_\infty\end{aligned}$$ where the second line follows from the definition of the operator norm $\lVert R^*\rVert_\infty$.
Proof of Lemma \[adjoint inverses\]
-----------------------------------
Firstly $\lVert R \rVert_{1} = \lVert R^* \rVert_{\infty}$ [@Bernstein2009]. From the definition of $\lVert R \rVert_{1}$ we have
$$\begin{aligned}
\lVert R \rVert_{1} &= \sup_{v \in {\protect\mathbb{R}}^Y} \frac{\lVert R v \rVert_{1}}{\lVert v\rVert_{1}} \\
&\geq \sup_{u \in {\protect\mathbb{R}}^X} \frac{\lVert R T u \rVert_{1}}{\lVert T u\rVert_{1}} \\
&= \sup_{u \in {\protect\mathbb{R}}^X} \frac{\lVert u \rVert_{1}}{\lVert T u\rVert_{1}} \\
&= 1 / \left( \inf_{u \in {\protect\mathbb{R}}^X} \frac{\lVert T u\rVert_{1}}{\lVert u \rVert_{1}} \right)\end{aligned}$$
this proves the first inequality. Recall one of the equivalent definitions of $\alpha(T)$ from section \[sec:Variational Alpha\] $$\alpha(T) = \sup_{v \in S} \frac{\lVert T(v) \rVert_1}{\lVert v \rVert_1}$$ where $S = \{v \in {\protect\mathbb{R}}^X : \sum v_i = 0, v \neq 0\}$. Hence trivially $\inf_{u \in {\protect\mathbb{R}}^X} \frac{\lVert T u \rVert_{1}}{\lVert u \rVert_{1}} \leq \alpha(T)$.
Corrupted Learning when Clean Learning is Fast {#fast learning}
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There are many conditions under which clean learning is fast, here we focus on the Bernstein condition presented in [@Erven2012].
Let $P\in {\protect\mathbb{P}}({\protect\mathcal{O}})$, $\ell$ a loss and $a_P = \operatorname*{arg\,min}_a {\protect\mathbb{E}}_{z {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}P} \ell(z,a)$. A pair $(\ell, P)$ satisfies the *Bernstein condition* with constant $K$ if for all $a \in A$ $${\protect\mathbb{E}}_{z {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}P} (\ell(z,a) - \ell(z,a_P))^2 \leq K \ {\protect\mathbb{E}}_{z {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}P} \ell(z,a) - \ell(z,a_P)$$
When $A$ is finite, such a condition leads to $\frac{1}{n}$ rates of convergence. From results in [@Zhang2006] we have the following theorem.
\[PAC Bayes Bernstein\] Let $\gamma = \frac{(e^\beta - 1 - \beta)}{\beta \lVert \ell \rVert_{\infty}}$. For all $P$, priors $\pi$, algorithms ${\protect\mathcal{A}}$, bounded losses $\ell$ and $\beta > 0$ $${\protect\mathbb{E}}_{S{\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}P^n} \left[ \ell(P,{\protect\mathcal{A}}(S)) - \gamma \ell^2(P,{\protect\mathcal{A}}(S)) \right]\leq {\protect\mathbb{E}}_{S{\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}P^n}\left[\ell(S,{\protect\mathcal{A}}(S)) + \lVert \ell \rVert_{\infty}\left(\frac{D_{KL}({\protect\mathcal{A}}(S),\pi)}{\beta n} \right)\right] .$$ Furthermore with probability at least $1-\delta$ on a draw $S {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}P^n$ with ${\protect\mathcal{A}}$, $\beta$ and $\pi$ chosen before the draw $$\ell(P,{\protect\mathcal{A}}(S)) - \gamma \ell^2(P,{\protect\mathcal{A}}(S)) \leq \left[\ell(S,{\protect\mathcal{A}}(S)) + \lVert \ell \rVert_{\infty}\left(\frac{D_{KL}({\protect\mathcal{A}}(S),\pi) + {\log\left(\frac{1}{\delta}\right)}}{\beta n} \right) \right].$$
We are now in a position to show that the Bernstein condition leads to fast rates for ERM.
(Fast Rates for ERM) Let ${\protect\mathcal{A}}$ be ERM with $A$ finite. If $(\ell,P)$ satisfies the Bernstein condition then for some constant $C$ $${\protect\mathbb{E}}_{S {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}P^n} \ell(P,{\protect\mathcal{A}}(S)) - \ell(P,a_P) \leq \frac{C \log(|A|)}{n}.$$ Furthermore with probability at least $1- \delta$ on a draw from $P^n$ one has $$\ell(P,{\protect\mathcal{A}}(S)) - \ell(P,a_P) \leq \frac{C\left(\log(|A|) + {\log\left(\frac{1}{\delta}\right)}\right)}{n}.$$
First, define $\ell_P(z,a) = \ell(z,a) - \ell(z,a_P)$. $l_P$ measures the loss relative to the best action for the distribution $P$. It is easy to verify that for bounded $\ell$, $\lVert \ell_P \rVert_\infty \leq 2 \lVert \ell \rVert_\infty$. We now utilize theorem \[PAC Bayes Bernstein\] with $\ell_P$ and $\pi$ uniform on $A$. This yields $${\protect\mathbb{E}}_{S{\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}P^n} \left[ \ell_P(P,{\protect\mathcal{A}}(S)) - \gamma \ell_P^2(P,{\protect\mathcal{A}}(S)) \right]\leq \frac{1}{n} {\protect\mathbb{E}}_{S{\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}P^n}\left[\ell_P(S,{\protect\mathcal{A}}(S)) + \lVert \ell_P \rVert_{\infty}\left(\frac{\log(|A|)}{\beta} \right)\right]$$ with $\gamma = \frac{(e^\beta - 1 - \beta)}{\beta \lVert \ell_P \rVert_{\infty}}$. Firstly ERM minimizes the right hand side of the bound meaning $$\frac{1}{n} {\protect\mathbb{E}}_{S{\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}P^n}\left[\ell_P(S,{\protect\mathcal{A}}(S)) + \lVert \ell_P \rVert_{\infty}\left(\frac{\log(|A|)}{\beta} \right)\right] \leq \frac{1}{n} \left[\lVert \ell_P \rVert_{\infty}\left(\frac{\log(|A|)}{\beta} \right)\right].$$ To see this consider the algorithm that always outputs $a_P$, this algorithm generalizes very well however it may be suboptimal on the sample. Secondly $(\ell,P)$ satisfies the Bernstein condition with constant $K$. Therefore $$(1 - \gamma K) {\protect\mathbb{E}}_{S{\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}P^n} \ell_P(P,{\protect\mathcal{A}}(S)) \leq \frac{1}{n} \left[\lVert \ell_P \rVert_{\infty}\left(\frac{\log(|A|)}{\beta} \right)\right].$$ Finally chose $\beta$ small enough so that $\gamma K \leq 1$. This can always be done as $\gamma \rightarrow 0$ as $\beta \rightarrow 0_+$. The high probability version proceeds in a similar way.
A natural question to ask is when does $(\tilde{\ell}, \tilde{P} )$ satisfy the Bernstein condition?
\[Noisy Bernstein\] If $(\tilde{\ell}, \tilde{P} )$ satisfies the Bernstein condition with constant $K$ then $(\ell,P)$ also satisfies the Bernstein condition with the same constant.
$$\begin{aligned}
K {\protect\mathbb{E}}_{z {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}P} \ell(z,a) - \ell(z,a_P) &= K {\protect\mathbb{E}}_{\tilde{z} {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}\tilde{P}} \tilde{\ell}(z,a) - \tilde{\ell}(z,a_P) &\\
&\geq {\protect\mathbb{E}}_{\tilde{z} {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}\tilde{P}} (\tilde{\ell}(\tilde{z},a) - \tilde{\ell}(\tilde{z},a_P))^2 \\
&= {\protect\mathbb{E}}_{z {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}P}{\protect\mathbb{E}}_{\tilde{z} {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}T(z)} (\tilde{\ell}(\tilde{z},a) - \tilde{\ell}(\tilde{z},a_P))^2 \\
&\geq{\protect\mathbb{E}}_{z {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}P} ({\protect\mathbb{E}}_{\tilde{z} {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}T(z)}\tilde{\ell}(\tilde{z},a) - {\protect\mathbb{E}}_{\tilde{z} {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}T(z)}\tilde{\ell}(\tilde{z},a_P))^2 \\
&= {\protect\mathbb{E}}_{z {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}P} (\ell(z,a) - \ell(z,a_P))^2\end{aligned}$$
where the first line follows from the definition of $\ell$ and because $a_{P} = a_{\tilde{P}}$, the second as $(\tilde{\ell}, \tilde{P} )$ satisfies the Bernstein condition and finally we have used the convexity of $f(x) = x^2$.
This theorem (almost) rules out pathological behaviour where ERM learns quickly from corrupted data and yet slowly for clean data. At present it is unknown if the converse to theorem \[Noisy Bernstein\] is true, with the same or possibly different constant. Here we present a partial converse.
Let $T : {\protect\mathcal{O}}\rightsquigarrow \tilde{{\protect\mathcal{O}}}$ be a Markov kernel and $\ell$ a loss. A pair $(\ell,T)$ are *$\eta$-compatible* if for all $z \in {\protect\mathcal{O}}$ and $a_1, a_2 \in A$ $${\protect\mathbb{E}}_{\tilde{z} {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}T(z)} (\tilde{\ell}(\tilde{z},a_1) - \tilde{\ell}(\tilde{z},a_2))^2 \leq \eta (\ell(z,a_1) - \ell(z,a_2))^2.$$
If the pair $(\ell,P)$ satisfies the Bernstein condition with constant $K$ and the pair $(\ell,T)$ are $\eta$-compatible then $(\tilde{l},\tilde{P})$ satisfies the Bernstein condition with constant $\eta K$.
$$\begin{aligned}
{\protect\mathbb{E}}_{\tilde{z} {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}\tilde{P}} (\tilde{\ell}(\tilde{z},a) - \tilde{\ell}(\tilde{z},a_P))^2 & = {\protect\mathbb{E}}_{z {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}P}{\protect\mathbb{E}}_{\tilde{z} {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}T(z)} (\tilde{\ell}(\tilde{z},a) - \tilde{\ell}(\tilde{z},a_P))^2 \\
&\leq \eta {\protect\mathbb{E}}_{z {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}P} (\ell(z,a) - \ell(z,a_P))^2 \\
&\leq \eta K {\protect\mathbb{E}}_{z {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}P} \ell(z,a) - \ell(z,a_P) \\
&= \eta K {\protect\mathbb{E}}_{\tilde{z} {\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}\tilde{P}} \tilde{\ell}(\tilde{z},a) - \tilde{\ell}(\tilde{z},a_P) \end{aligned}$$
where we have first used $\eta$-compatibility, then the fact that $(\ell,P)$ satisfies the Bernstein condition with constant $K$ and finally the definition of $\tilde{\ell}$.
While by no means the final line in fast corrupted learning, this theorem does allow one to prove interesting results in the binary classification setting.
\[Bernstein Label Noise\] Let $T$ be label noise, $T = {\left(\begin{array}{cc}
1-\sigma_{-1} & \sigma_1 \\
\sigma_{-1} & 1-\sigma_1
\end{array} \right)}$, then the pair $(\ell_{01}, T)$ is $\eta$-compatible with $\eta = \max(\left(\frac{1+\sigma_{-1} - \sigma_1}{1-\sigma_{-1} - \sigma_1}\right)^2, \left(\frac{1+\sigma_{1} - \sigma_{-1}}{1-\sigma_{-1} - \sigma_1}\right)^2)$.
Due to the symmetry of the left and right hand sides of the Bernstein condition, one only needs to check the case where $a_1 = 1$, $a_2 = -1$. Recall $$\begin{aligned}
\tilde{\ell}_{01}(\tilde{y},a) &= \frac{(1-\sigma_{-y}) \ell_{01}(\tilde{y},a) - \sigma_y \ell_{01}(-\tilde{y},a)}{1- \sigma_{-1} - \sigma_1} \\
&= \frac{(1-\sigma_{-y} + \sigma_y)\ell_{01}(\tilde{y},a) - \sigma_y}{1- \sigma_{-1} - \sigma_1}.\end{aligned}$$ For $y = 1$ it is easy to confirm $\left(\ell_{01}(1,1) - \ell_{01}(1,-1) \right)^2 = 1$. We have $$\begin{aligned}
\tilde{\ell}_{01}(\tilde{y},1) - \tilde{\ell}_{01}(\tilde{y},-1) &= \frac{(1-\sigma_{-y} + \sigma_y)(\ell_{01}(\tilde{y},1)- \ell_{01}(\tilde{y},-1))}{1- \sigma_{-1} - \sigma_1} \\
&= \frac{-\tilde{y}(1-\sigma_{-y} + \sigma_y)}{1- \sigma_{-1} - \sigma_1}.\end{aligned}$$ Squaring, taking maximums and finally expectations yields the desired result.
One very useful example of a pair $(P,\ell)$ satisfying the Bernstein condition with constant $1$ is when $P$ is separable, $\ell$ is $01$ loss and the Bayes optimal classifier is in the function class. Theorem \[Bernstein Label Noise\] guarantees that in such a setting one can learn at a fast rate from noisy examples.
|
---
author:
- 'J.–M. Grie[ß]{}meier'
- 'P. Zarka'
- 'H. Spreeuw'
date: Version of
title: 'Predicting low-frequency radio fluxes of known extrasolar planets[^1] '
---
Introduction
============
In the solar system, all strongly magnetised planets are known to be intense nonthermal radio emitters. For a certain class of extrasolar planets (the so-called Hot Jupiters), an analogous, but much more intense radio emission is expected. In the recent past, such exoplanetary radio emission has become an active field of research, with both theoretical studies and ongoing observation campaigns.
Recent theoretical studies have shown that a large variety of effects have to be considered, e.g. kinetic, magnetic and unipolar interaction between the star (or the stellar wind) and the planet, the influence of the stellar age, the potential role of stellar CMEs, and the influence of different stellar wind models. So far, there is no single publication in which all of these aspects are put together and where the different interaction models are compared extensively. We also discuss the escape of exoplanetary radio emission from its planetary system, which depends on the local stellar wind parameters. As will be shown, this is an additional constraint for detectability, making the emission from several planets impossible to observe.
The first observation attempts go back at least to @Yantis77. At the beginning, such observations were necessarily unguided ones, as exoplanets had not yet been discovered. Later observation campaigns concentrated on known exoplanetary systems. So far, no detection has been achieved. A list and a comparison of past observation attempts can be found elsewhere [@Griessmeier51PEG05]. Concerning ongoing and future observations, studies are performed or planned at the VLA [@Lazio04], GMRT [@Majid05; @Winterhalter06], UTR2 [@Ryabov04], and at LOFAR [@Farrell04]. To support these observations and increase their efficiency, it is important to identify the most promising targets.
The target selection for radio observations is based on theoretical estimates which aim at the prediction of the main characteristics of the exoplanetary radio emission. The two most important characteristics are the maximum frequency of the emission and the expected radio flux. The first predictive studies [e.g. @Zarka97; @Farrell99] concentrated on only a few exoplanets. A first catalog containing estimations for radio emission of a large number of exoplanets was presented by @Lazio04. This catalog included 118 planets (i.e. those known as of 2003, July 1) and considered radio emission energised by the kinetic energy of the stellar wind (i.e. the *kinetic model*, see below). Here, we present a much larger list of targets (i.e. 197 exoplanets found by radial velocity and/or transit searches as of 2007, January 13, taken from http://exoplanet.eu/), and compare the results obtained by all four currently existing interaction models, not all of which were known at the time of the previous overview study. As a byproduct of the radio flux calculation, we obtain estimates for the planetary magnetic dipole moment of all currently known extrasolar planets. These values will be useful for other studies as, e.g., star-planet interaction or atmospheric shielding.
To demonstrate which stellar and planetary parameters are required for the estimation of exoplanetary radio emission, some theoretical results are briefly reviewed (section \[sec:theory\]). Then, the sources for the different parameters (and their default values for the case where no measurements are available) are presented (section \[sec:modelling\]). In section \[sec:results\], we present our estimations for exoplanetary radio emission. This section also includes estimates for planetary magnetic dipole moments. Section \[sec:conclusion\] closes with a few concluding remarks.
Exoplanetary radio emission theory {#sec:theory}
==================================
Expected radio flux {#sec:flux}
-------------------
In principle, there are four different types of interaction between a planetary obstacle and the ambient stellar wind, as both the stellar wind and the planet can either be magnetised or unmagnetised. @Zarka06PSS [Table 1] show that for three of these four possible situations intense nonthermal radio emission is possible. Only in the case of an unmagnetised stellar wind interacting with an unmagnetised body no intense radio emission is possible.
In those cases where strong emission is possible, the expected radio flux depends on the source of available energy. In the last years, four different energy sources were suggested: a) In the first model, the input power $P_{\text{input}}$ into the magnetosphere is assumed to be proportional to the total *kinetic energy* flux of the solar wind protons impacting on the magnetopause [@Desch84; @Zarka97; @Farrell99; @Zarka01; @Farrell04; @Lazio04; @Stevens05; @Griessmeier05; @GriessmeierPREVI; @GriessmeierPSS06] b) Similarly, the input power $P_{\text{input}}$ into the magnetosphere can be assumed to be proportional to the *magnetic energy* flux or electromagnetic Poynting flux of the interplanetary magnetic field [@Zarka01; @Farrell04; @Zarka04a; @Zarka06PREVI; @Zarka06PSS]. From the data obtained in the solar system, it is not possible to distinguish which of these models is more appropriate [the constants of proportionality implied in the relations given below are not well known, see @Zarka01], so that both models have to be considered. c) For unmagnetised or weakly magnetised planets, one may apply the *unipolar interaction* model. In this model, the star-planet system can be seen as a giant analog to the Jupiter-Io system [@Zarka01; @Zarka04a; @Zarka06PREVI; @Zarka06PSS]. Technically, this model is very similar to the magnetic energy model, but the source location is very different: Whereas in the kinetic and in the magnetic model, the emission is generated near the planet, in the unipolar interaction case a large-scale current system is generated and the radio emission is generated in the stellar wind between the star and the planet. Thus, the emission can originate from a location close to the stellar surface, close to the planetary surface, or at any point between the two. This is possible in those cases where the solar wind speed is lower than the Alfvén velocity [i.e. for close-in planets, see e.g. @Preusse05]. Previous studied have indicated that this emission is unlikely to be detectable, except for stars with an extremely strong magnetic field [@Zarka01; @Zarka04; @Zarka06PREVI; @Zarka06PSS]. Nevertheless, we will check whether this type of emission is possible for the known exoplanets. d) The fourth possible energy source is based on the fact that close-in exoplanets are expected to be subject to frequent and violent stellar eruptions [@Khodachenko05] similar to solar coronal mass ejections (CMEs). As a variant to the kinetic energy model, the *CME* model assumes that the energy for the most intense planetary radio emission is provided by CMEs. During periods of such CME-driven radio activity, considerably higher radio flux levels can be achieved than during quiet stellar conditions [@GriessmeierPREVI; @GriessmeierPSS06]. For this reason, this model is treated separately.
For the *kinetic energy* case, the input power was first derived by @Desch84, who found that it is given by $$P_{\text{input,kin}} \propto n v{_{\text{eff}}}^3 R{_{\text{s}}}^2. \label{eq:Pin:kin}$$ In eq. (\[eq:Pin:kin\]), $n$ is the stellar wind density at the planetary orbit, $v{_{\text{eff}}}$ is the velocity of the stellar wind in the reference frame of the planet (i.e. including the aberration due to the orbital velocity of the planet, which is not negligible for close-in planets), and $R{_{\text{s}}}$ denotes the magnetospheric standoff distance.
The *magnetic energy* case was first discussed by @Zarka01. Here, the input power is given by $$P_{\text{input,mag}} \propto v{_{\text{eff}}}B_{\perp}^2 R{_{\text{s}}}^2 \label{eq:Pin:mag}$$ In eq. (\[eq:Pin:mag\]), $v{_{\text{eff}}}$ is the velocity of the stellar wind in the reference frame of the planet, $B_{\perp}$ if the component of the interplanetary magnetic field (IMF) perpendicular to the stellar wind flow in the reference frame of the planet, and $R{_{\text{s}}}$ denotes the magnetospheric standoff distance.
For the *unipolar interaction* case [@Zarka01], the input power is given by $$P_{\text{input,unipolar}} \propto v{_{\text{eff}}}B_{\perp}^2 R_{\text{ion}}^2 \label{eq:Pin:uni}$$ Eq. (\[eq:Pin:uni\]) is identical to eq. (\[eq:Pin:mag\]), except that the obstacle is not the planetary magnetosphere, but its ionosphere, so that $R{_{\text{s}}}$ is replaced by $R_{\text{ion}}$, the radius of the planetary ionosphere.
*CME*-driven radio emission was first calculated by @GriessmeierPREVI. In that case, the input power is given by $$P_{\text{input,kin,CME}} \propto n{_{\text{CME}}}v_{\text{eff,CME}}^3 R{_{\text{s}}}^2. \label{eq:Pin:kin:CME}$$ Eq. (\[eq:Pin:kin:CME\]) is identical to eq. (\[eq:Pin:kin\]), except that the stellar wind density and velocity are replaced by the corresponding values encountered by the planet during a CME.
A certain fraction $\epsilon$ of the input power $P_{\text{input}}$ given by eq. (\[eq:Pin:kin\]), (\[eq:Pin:mag\]), (\[eq:Pin:uni\]) or (\[eq:Pin:kin:CME\]) is thought to be dissipated within the magnetosphere: $$P_d=\epsilon P_{\text{input}}$$
Observational evidence suggests that the amount of power emitted by radio waves $P_{\text{rad}}$ is roughly proportional to the power input $P_{\text{input}}$ [see, e.g. @Zarka06PSS Figure 6]. This can be written as: $$P_{\text{radio}}=\eta_{\text{radio}} P_d=\eta_{\text{radio}}\epsilon P_{\text{input}}$$ As $P_d$ cannot be measured directly, one correlates the observed values of $P_{\text{radio}}$ with the calculated (model dependent) values of $P_{\text{input}}$. Thus, one replaces $P_{\text{input}}$ by $P_{\text{radio}}$ on the left-hand side of the proportionalities given by (\[eq:Pin:kin\]), (\[eq:Pin:mag\]), (\[eq:Pin:uni\]) and (\[eq:Pin:kin:CME\]). The proportionality constant is determined by comparison with Jupiter. The analysis of the jovian radio emission allows to define three values for the typical radio spectrum: (a) the power during *average conditions*, (b) the average power during periods of *high activity*, and (c) the *peak power* [@Zarka04]. In this work, we will use the average power during periods of high activity as a reference value for all four cases, with $P_{\text{radio,J}}=2.1 \cdot 10^{11}$ W.
The radio flux $\Phi$ seen by an observer at a distance $s$ from the emitter is related to the emitted radio power $P_{\text{radio}}$ by [@GriessmeierPSS06]: $$\Phi=\frac{P_{\text{radio}}}
{\Omega s^2 \Delta f}
=\frac{4 \pi^2 m_e R{_{\text{p}}}^3 P_{\text{radio}}} {e \mu_0 \Omega s^2 \mathcal{M}}. \label{eq:Phi:s}$$ Here, $\Omega$ is the solid angle of the beam of the emitted radiation [$\Omega= 1.6$ sr, see @Zarka04], and $\Delta f$ is the bandwidth of the emission. We use $\Delta f=f{_{\text{c}}}^\text{max}$ [@GriessmeierPSS06], where $f{_{\text{c}}}^\text{max}$ is the maximum cyclotron frequency. Depending on the model, $P_{\text{radio}}$ is given by eq. (\[eq:Pin:kin\]), (\[eq:Pin:mag\]), (\[eq:Pin:uni\]) or (\[eq:Pin:kin:CME\]). The maximum cyclotron frequency $f{_{\text{c}}}^\text{max}$ is determined by the maximum magnetic field strength $B{_{\text{p}}}^{\text{max}}$ close to the polar cloud tops [@Farrell99]: $$f{_{\text{c}}}^\text{max}= \frac{eB{_{\text{p}}}^\text{max}}{2\pi m_e}=\frac{e \mu_0 \mathcal{M}}{4\pi^2 m_e R{_{\text{p}}}^3}
\approx 24 \,\text{MHz} \, \frac{\widetilde{\mathcal{M}}}{\widetilde{R{_{\text{p}}}}^3}
.\label{eq:f}$$ Here, $m_e$ and $e$ are the electron mass and charge, $R{_{\text{p}}}$ is the planetary radius, $\mu_0$ is the magnetic permeability of the vacuum, and $\mathcal{M}$ is the planetary magnetic dipole moment. $\widetilde{\mathcal{M}}$ and $\widetilde{R{_{\text{p}}}}$ denote the planetary magnetic moment and its radius relative to the respective value for Jupiter, e.g. $\widetilde{\mathcal{M}}=\mathcal{M}/\mathcal{M}{_{\text{J}}}$, with $\mathcal{M}{_{\text{J}}}=1.56 \cdot 10^{27}$ Am$^2$ [@Cain95] and $R{_{\text{J}}}=71492$ km.
The radio flux expected for the four different models according to eqs. (\[eq:Pin:kin\]) to (\[eq:Phi:s\]) and the maximum emission frequency according to (\[eq:f\]) are calculated in section \[sec:results\] for all known exoplanets.
Escape of radio emission {#sec:escape}
------------------------
To allow an observation of exoplanetary radio emission, it is not sufficient to have a high enough emission power at the source and emission in an observable frequency range. As an additional requirement, it has to be checked that the emission can propagate from the source to the observer. This is not the case if the emission is absorbed or trapped (e.g. in the stellar wind in the vicinity of the radio-source), which happens whenever the plasma frequency $$f_{\text{plasma}}=\frac{1}{2\pi}\sqrt{\frac{ne^2}{\epsilon_0 m_e}} \label{eq:fplasma}$$ is higher than the emission frequency at any point between the source and the observer. Thus, the condition of observability is $$f_{\text{plasma}}^\text{max}<f{_{\text{c}}}^\text{max} \label{eq:escape},$$ where $f{_{\text{c}}}^\text{max} $ is taken at the radio source (e.g. the planet), whereas $f_{\text{plasma}}^\text{max}$ is evaluated along the line of sight. As the density of the electrons in the stellar wind $n$ decreases with the distance to the star, this condition is more restrictive at the orbital distance of the planet than further out. Thus, it is sufficient to check whether condition (\[eq:escape\]) is satisfied at the location of the radio-source (i.e. for $n=n(d)$, where $d$ is the distance from the star to the radio-source). In that case, the emission can escape from the planetary system and reach distant observers. In section \[sec:results\], condition (\[eq:escape\]) is checked for all known exoplanets at their orbital distance.
Note however that, depending on the line of sight, not all observers will be able to see the planetary emission at all times. For example, the observation of a secondary transit implies that the line of sight passes very close to the planetary host star, where the plasma density is much higher. For this reason, some parts of the orbit may be unobservable even for planets for which eq. (\[eq:escape\]) is satisfied.
Radiation emission in the unipolar interaction model {#sec:unipolar}
----------------------------------------------------
An additional constraint arises because certain conditions are necessary for the generation of radio emission. Planetary radio emission is caused by the cyclotron maser instability (CMI). This mechanism is only efficient in regions where the ratio between the electron plasma frequency and the electron cyclotron frequency is small enough. This condition can be written as $$\frac{f_{\text{plasma}}}{f{_{\text{c}}}} \lesssim 0.4 \label{eq:f:generation},$$ where the electron cyclotron frequency $f{_{\text{c}}}$ is defined by the local magnetic field $${f{_{\text{c}}}} = \frac{eB}{2\pi m_e}$$ and $f_{\text{plasma}}$ is given by eq. (\[eq:fplasma\]). Observations seem to favor a critical frequency ratio close to the $0.1$, while theoretical work supports a critical frequency ratio close to $0.4$ [@LeQueau85; @Hilgers92; @Zarka01cutoff]. Fundamental O mode or second harmonic O and X mode emission are possible also for larger frequency ratios, but are much less efficient [@Treumann00; @Zarka06PSS]. To avoid ruling out potential emission, we use the largest possible frequency ratio, i.e. $\frac{f_{\text{plasma}}}{f{_{\text{c}}}} \le 0.4$.
The condition imposed by eq. (\[eq:f:generation\]) has to be fulfilled for any of the four models presented in section \[sec:flux\]. For the three models where the radio emission is generated directly in the planetary magnetosphere, $n$ decreases much faster with distance to the planetary surface than $B$, so that eq. (\[eq:f:generation\]) can always be fulfilled. For the *unipolar interaction* model, the emission takes place in the stellar wind, and the electron density $n$ can be obtained from the model of the stellar wind. In this case, it is not a priori clear where radio emission is possible. It could be generated anywhere between the star and the planet. In section \[sec:results\], we will check separately for each planetary system whether unipolar interaction satisfying eq. (\[eq:f:generation\]) is possible at any location between the stellar surface and the planetary orbit.
Required parameters {#sec:modelling}
===================
In the previous section, it has been shown that the detectability of planetary radio emission depends on a few planetary parameters:
- the planetary radius $R{_{\text{p}}}$
- the planetary magnetic moment $\mathcal{M}$
- the size of the planetary magnetosphere $R{_{\text{s}}}$
- the size of the planetary ionosphere $R_{\text{ion}}$
- the stellar wind density $n$ and its velocity $v{_{\text{eff}}}$
- the stellar magnetic field (IMF) $B_\perp$ perpendicular to the stellar wind flow in the frame of the planet
- the distance of the stellar system (to an earth-based observer) $s$
- the solid angle of the beam of the emitted radiation $\Omega$
The models used to infer the missing stellar and planetary quantities require the knowledge of a few additional planetary parameters. These are the following:
- the planetary mass $M{_{\text{p}}}$
- the planetary radius $R{_{\text{p}}}$
- its orbital distance $d$
- the planetary rotation rate $\omega$
- the stellar magnetic field (IMF) components $B_r, B_\phi$
- the stellar mass $M_\star$
- the stellar radius $R_\star$
- the stellar age $t_\star$
In this section, we briefly describe how these each of these quantities can be obtained.
Basic planetary parameters
--------------------------
As a first step, basic planetary characteristics have to be evaluated:
- $d, \omega_{\text{orbit}}$ and $s$ are directly taken from the Extrasolar Planets Encyclopaedia at [http://exoplanet.eu]{}, as well as the observed mass $M_{\text{obs}}$ and the orbital eccentricity $e$. Note that in most cases the observed mass is the “projected mass” of the planet, i.e. $M_{\text{obs}}= M{_{\text{p}}}\sin i$, where $i$ is the angle of inclination of the planetary orbit with respect to the observer.
- For eccentric planets, we calculate the periastron from the semi-major axis $d$ and the orbital eccentricity $e$: $d_{\text{min}}=d/(1-e)$. Thus, the results for planetary radio emission apply to the periastron.
- For most exoplanets, the planetary mass is not precisely known. Instead, usually only the projected mass $M{_{\text{p}}}\sin i$ is accessible to measurements, where $i$ is the inclination of the planetary orbit with respect to the observer. This projected mass is taken from [http://exoplanet.eu]{} and converted to the median value of the mass: $\text{median}(M{_{\text{p}}})=M_{\text{obs}} \cdot \text{median}(\frac{1}{\sin i})
= \sqrt{4/3} \cdot M_{\text{obs}}
\approx 1.15 \, M_{\text{obs}}$.
- For transiting planets, $R{_{\text{p}}}$ is taken from original publications. For non-transiting planets, the planetary radius is $R{_{\text{p}}}$ not known. In this case, we estimate the planetary radius based on its mass $M{_{\text{p}}}$ and orbital distance $d$, as explained in appendix \[sec:appendix\]. The radius of a “cold” planet of mass $M{_{\text{p}}}$ is given by $$R{_{\text{p}}}(d=\infty)=
\frac{ \left( \alpha M{_{\text{p}}}\right)^{1/3}}{1+\left( \frac{M{_{\text{p}}}}{M_{\text{max}}} \right)^{2/3}}
\approx 1.47 R{_{\text{J}}}\,\frac{ \widetilde{M{_{\text{p}}}} ^{1/3}}{1+\left( \frac{\widetilde{M{_{\text{p}}}}}{\widetilde{M_{\text{max}}}} \right)^{2/3}}$$ with $\alpha=6.1 \cdot 10^{-4}$ m$^3$ kg$^{-1}$ (for a planet with the same composition as Jupiter) and $M_{\text{max}}=3.16 \, M{_{\text{J}}}$. Again, $\widetilde{M{_{\text{p}}}}$ and $\widetilde{M_{\text{max}}}$ denote values relative to the respective value for Jupiter (using ${M}{_{\text{J}}}=1.9 \cdot 10^{27}$ kg). The radius of an irradiated planet is then given by $$\frac{R{_{\text{p}}}(d)}{R{_{\text{p}}}(d=\infty)}=\frac{\widetilde{R{_{\text{p}}}}(d)}{\widetilde{R{_{\text{p}}}}(d=\infty)}=
\cdot \left[ 1+0.05\left(\frac{T_{\text{eq}}}{T_0}\right)^{\gamma} \right]
\label{eq:temp}$$ where $T_{\text{eq}}$ is the equilibrium temperature of the planetary surface. The coefficients $T_0$ and $\gamma$ depend on the planetary mass (see appendix \[sec:appendix\]).
Stellar wind model {#sec:stellarwind}
------------------
The stellar wind density $n$ and velocity $v{_{\text{eff}}}$ encountered by a planet are key parameters defining the size of the magnetosphere and thus the energy flux available to create planetary radio emission. As these stellar wind parameters strongly depend on the stellar age, the expected radio flux is a function of the estimated age of the exoplanetary host star [@Stevens05; @Griessmeier05]. At the same time it is known that at close distances the stellar wind velocity has not yet reached the value it has at larger orbital distances. For this reason, a distance-dependent stellar wind models has to be used to avoid overestimating the expected planetary radio flux [@GriessmeierPHD06; @GriessmeierPSS06].
It was shown [@GriessmeierPSS06] that for stellar ages $>0.7$ Gyr, the radial dependence of the stellar wind properties can be described by the stellar wind model of @Parker58, and that the more complex model of @Weber67 is not required. In the Parker model, the interplay between stellar gravitation and pressure gradients leads to a supersonic gas flow for sufficiently large substellar distances $d$. The free parameters are the coronal temperature and the stellar mass loss. They are indirectly chosen by setting the stellar wind conditions at 1 AU. More details on the model can be found elsewhere [e.g. @Mann99; @PreussePHD05; @GriessmeierPHD06].
The dependence of the stellar wind density $n$ and velocity and $v{_{\text{eff}}}$ on the age of the stellar system is based on observations of astrospheric absorption features of stars with different ages. In the region between the astropause and the astrospheric bow shock (analogs to the heliopause and the heliospheric bow shock of the solar system), the partially ionized local interstellar medium (LISM) is heated and compressed. Through charge exchange processes, a population of neutral hydrogen atoms with high temperature is created. The characteristic Ly$\alpha$ absorption (at 1216 $\text{\AA}$) of this population was detectable with the high-resolution observations obtained by the Hubble Space Telescope (HST). The amount of absorption depends on the size of the astrosphere, which is a function of the stellar wind characteristics. Comparing the measured absorption to that calculated by hydrodynamic codes, these measurements allowed the first empirical estimation of the evolution of the stellar mass loss rate as a function of stellar age [@Wood02; @Wood04; @Wood05]. It should be noted, however, that the resulting estimates are only valid for stellar ages $\ge0.7$ Gyr [@Wood05]. From these observations, [@Wood05] calculate the age-dependent density of the stellar wind under the assumption of an age-independent stellar wind velocity. This leads to strongly overestimated stellar wind densities, especially for young stars [@Griessmeier05; @Holzwarth07]. For this reason, we combine these results with the model for the age-dependence of the stellar wind velocity of @Newkirk80. One obtains [@GriessmeierPSS06]: $$v(1 \text{AU}, t)= v_0 \left( 1+\frac{t}{\tau}\right)^{-0.43}.
\label{eq:scaleV}$$ The particle density can be determined to be $$n(1 \text{AU}, t)= n_0 \left( 1+\frac{t}{\tau}\right)^{-1.86\pm0.6}.
\label{eq:scaleN}$$ with $v_0=3971$ km/s, $n_0=1.04\cdot10^{11}\text{ m}^{-3}$ and $\tau=2.56\cdot10^7 \text{yr}$.
For planets at small orbital distances, the keplerian velocity of the planet moving around its star becomes comparable to the radial stellar wind velocity. Thus, the interaction of the stellar wind with the planetary magnetosphere should be calculated using the effective velocity of the stellar wind plasma relative to the planet, which takes into account this “aberration effect” [@Zarka01]. For the small orbital distances relevant for Hot Jupiters, the planetary orbits are circular because of tidal dissipation [@Goldreich66; @DobbsDixon04; @Halbwachs05]. For circular orbits, the orbital velocity $v{_{\text{orbit}}}$ is perpendicular to the stellar wind velocity $v$, and its value is given by Kepler’s law. In the reference frame of the planet, the stellar wind velocity then is given by $$v{_{\text{eff}}}=\sqrt{v^2_{\text{orbit}}+v^2}. \label{eq:veff}$$
Finally, for the magnetic energy case, the interplanetary magnetic field ($ B_r, B_\phi$) is required. At 1 AU, the average field strength of the interplanetary magnetic field is $B_{\text{imf}}\approx 3.5\,\text{nT}$ [@Mariani90; @Proelss04]. According to the Parker stellar wind model [@Parker58], the radial component of the interplanetary magnetic field decreases as $$B_{\text{imf},r}(d)=B_{r,0} \left(\frac{d}{d_0}\right)^{-2}. \label{eq:Bimfr}$$ This was later confirmed by Helios measurements. One finds $B_{r,0}\approx 2.6$ nT and $d_0=1$ AU [@Mariani90; @Proelss04]. At the same time, the azimuthal component $B_{\text{imf},\varphi}$ behaves as $$B_{\text{imf},\varphi}(d)=B_{\varphi,0} \left(\frac{d}{d_0}\right)^{-1},
\label{eq:Bimfphi}$$ with $B_{\varphi,0}\approx2.4$ nT [@Mariani90; @Proelss04]. The average value of $B_{\text{imf},\theta}$ vanishes ($B_{\text{imf},\theta}\approx 0$). From $B_{\text{imf},r}(d)$ and $B_{\text{imf},\varphi}(d)$, the stellar magnetic field (IMF) $B_\perp(d)$ perpendicular to the stellar wind flow in the frame of the planet can be calculated [@Zarka06PSS]: $$B_\perp = \sqrt{B_{\text{imf},r}^2+B_{\text{imf},\varphi}^2} \left| \sin\left(\alpha-\beta \right) \right|$$ with $$\alpha = \arctan \left( \frac{B_{\text{imf},\varphi}}{B_{\text{imf},r}} \right)$$ and $$\beta = \arctan \left( \frac{v_{\text{orbit}}}{v} \right).$$
We obtain the stellar magnetic field $B{_{\star}}$ relative to the solar magnetic field $B{_\odot}$ under the assumption that it is inversely proportional to the rotation period $P{_{\star}}$ [@CollierCameron94; @GriessmeierPSS06]: $$\frac{B{_{\star}}}{B{_\odot}} = \frac{P{_\odot}}{P{_{\star}}},$$ where we use $P{_\odot}=25.5$ d and take $B_\odot=1.435\cdot 10^{-4}$ T as the reference magnetic field strength at the solar surface [@Preusse05]. The stellar rotation period $P{_{\star}}$ is calculated from the stellar age $t$ [@Newkirk80]: $$P{_{\star}}\propto \left(1+\frac{t}{\tau}\right)^{0.7},$$ where the time constant $\tau$ is given by $\tau=2.56\cdot10^7$ yr [calculated from @Newkirk80].
Note that first measurements of stellar magnetic fields for planet-hosting stars are just becoming available from the spectropolarimeter ESPaDOnS [@Catala07]. This will lead to an improved understanding of stellar magnetic fields, making more accurate models possible in the future.
Stellar CME model
-----------------
For the CME-driven radio emission, the stellar wind parameters $n$ and $v{_{\text{eff}}}$ are effectively replaced by the corresponding CME parameters $n{_{\text{CME}}}$ and $v_\text{eff,CME}$, potentially leading to much more intense radio emission than those driven by the kinetic energy of the steady stellar wind [@GriessmeierPHD06; @GriessmeierPREVI; @GriessmeierPSS06].
These CME parameters are estimated by @Khodachenko05, who combine in-situ measurements near the sun (e.g. by Helios) with remote solar observation by SoHO. Two interpolated limiting cases are given, denoted as [*weak*]{} and [*strong*]{} CMEs, respectively. These two classes have a different dependence of the average density on the distance to the star $d$. In the following, these quantities will be labeled $n{_{\text{CME}}}^w(d)$ and $n{_{\text{CME}}}^s(d)$, respectively.
For weak CMEs, the density $n{_{\text{CME}}}^w(d)$ behaves as $$n{_{\text{CME}}}^w(d) = n_{\text{CME},0}^w \left( d/d_0 \right)^{-2.3} \label{eq:sme:nw}$$ where the density at $d_0=1$ AU is given by $n_{\text{CME},0}^w=n{_{\text{CME}}}^w(d=d_0)=4.9 \cdot 10^6$ m$^{-3}$.
For strong CMEs, @Khodachenko05 find $$n{_{\text{CME}}}^s(d) = n_{\text{CME},0}^s \left( d/d_0 \right)^{-3.0} \label{eq:sme:ns}$$ with $n_{\text{CME},0}^s=n{_{\text{CME}}}^s(d=d_0)=7.1 \cdot 10^6$ m$^{-3}$, and $d_0=1$ AU.
As far as the CME velocity is concerned, one has to note that individual CMEs have very different velocities. However, the [*average*]{} CME velocity $v$ is approximately independent of the subsolar distance, and is similar for both types of CMEs: $$v{_{\text{CME}}}^w=v{_{\text{CME}}}^s=v{_{\text{CME}}}\approx 500 \, \text{km/s}. \label{eq:sme:v}$$ Similarly to the steady stellar wind the CME velocity given by eq. (\[eq:sme:v\]) has to be corrected for the orbital motion of the planet: $$v_{\text{eff,CME}}=\sqrt{\frac{M_\star G}{d}+v{_{\text{CME}}}^2}. \label{eq:veff:CME}$$
In addition to the density and the velocity, the temperature of the plasma in a coronal mass ejection is required for the calculation of the size of the magnetosphere. According to @Khodachenko05 [@Khodachenko06PSS], the front region of a CME consists of hot, coronal material ($T \approx 2$ MK). This region may either be followed by relatively cool prominence material ($T \approx 8000$ K), or by hot flare material ($T \approx 10$ MK). In the following, the temperature of the leading region of the CME will be used, i.e. $T{_{\text{CME}}}=2$ MK.
Planetary magnetic moment and magnetosphere {#sec:protation}
-------------------------------------------
For each planet, the value of the planetary magnetic moment $\mathcal{M}$ is estimated by taking the geometrical mean of the maximum and minimum result obtained by different scaling laws. The associated uncertainty was discussed by @GriessmeierPSS06. The different scaling laws are compared, e.g., by @Farrell99[^2], @Griessmeier04 and @GriessmeierPHD06. In order to be able to apply these scaling laws, some assumptions on the planetary size and structure are required. The variables required in the scaling laws are $r{_{\text{c}}}$ (the radius of the dynamo region within the planet), $\rho{_{\text{c}}}$ (the density within this region), $\sigma$ (the conductivity within this region) and $\omega$ (the planetary rotation rate).
#### The size of the planetary core $r{_{\text{c}}}$ and its density $\rho{_{\text{c}}}$
The density profile within the planet $\rho(r)$ is obtained by describing the planet as a polytropic gas sphere, using the solution of the Lane-Emden equation [@Chandrasekhar57; @Sanchez04]: $$\rho(r)=\left( \frac{\pi M{_{\text{p}}}}{4 R{_{\text{p}}}^3} \right) \frac{\sin\left(\pi \frac{r}{R{_{\text{p}}}}\right)}{\left(\pi \frac{r}{R{_{\text{p}}}}\right)}.$$ The size of the planetary core $r{_{\text{c}}}$ is found by searching for the value of $r$ where the density $\rho(r)$ becomes large enough for the transition to the liquid-metallic phase [@Sanchez04; @Griessmeier05; @GriessmeierPHD06]. The transition was assumed to occur at a density of 700 kg/m$^3$, which is consistent with the range of parameters given by @Sanchez04. For Jupiter, we obtain $r{_{\text{c}}}=0.85\, R{_{\text{J}}}$.
The average density in the dynamo region $\rho{_{\text{c}}}$ is then obtained by averaging the density $\rho(r)$ over the range $0\le r\le r{_{\text{c}}}$. For Jupiter, we obtain $r{_{\text{c}}}\approx 1800$ kg m$^{-3}$.
#### The planetary rotation rate $\omega$
Depending on the orbital distance of the planet and the timescale for synchronous rotation ${\tau_{\text{sync}}}$ (which is derived in appendix \[sec:tlocking\]), three cases can be distinguished:
1. For planets at small enough distances for which the timescale for tidal locking is small (i.e. ${\tau_{\text{sync}}}\le 100$ Myr), the rotation period is taken to be synchronised with the orbital period ($\omega=\omega_\text{f}\approx\omega_{\text{orbit}}$), which is known from measurements. This case will be denoted by “TL”. Typically, this results in smaller rotation rates for tidally locked planets than for freely rotating planets.
2. Planets with distances resulting in $100 \ \text{Myr} \le {\tau_{\text{sync}}}\le 10\ \text{Gyr}$ may or may not be subject to tidal locking. This will, for example, depend on the exact age of the planetary system, which is typically in the order of a few Gyr. For this reason, we calculate the expected characteristics of these “potentially locked” planets *twice*: once with tidal locking (denoted by “(TL)”) and once without tidal locking (denoted by “(FR)”).
3. For planets far away from the central star, the timescale for tidal locking is very large. For planets with ${\tau_{\text{sync}}}\ge 10$ Gyr, the effect of tidal interaction can be neglected. In this case, the planetary rotation rate can be assumed to be equal to the initial rotation rate $\omega_\text{i}$, which is assumed to be equal to the current rotation rate of Jupiter, i.e. $\omega=\omega{_{\text{J}}}$ with $\omega{_{\text{J}}}=1.77\cdot10^{-4}$ s$^{-1}$. This case will be denoted by “FR”.
Note that tidal interaction does not perfectly synchronise the planetary rotation to its orbit. Thermal atmospheric tides resulting from stellar heating can drive planets away from synchronous rotation [@Showman02; @Correia03; @Laskar04]. According to @Showman02, the corresponding error for $\omega$ could be as large as a factor of two. On the basis of the example of $\tau$ Bootis b, @GriessmeierPSS06 show that the effect of imperfect tidal locking (in combination with the spread of the results found by different scaling laws) can lead to magnetic moments and thus emission frequencies up to a factor 2.5 higher than for the nominal case. Keeping this “error bar” in mind, we will nevertheless consider only the reference case in this work.
#### The conductivity in the planetary core $\sigma{_{\text{c}}}$
Finally, the conductivity in the dynamo region of extrasolar planets remains to be evaluated. According to @Nellis00, the electrical conductivity remains constant throughout the metallic region. For this reason, it is not necessary to average over the volume of the conducting region. As the magnetic moment scaling is applied relative to Jupiter, only the relative value of the conductivity, i.e. $\sigma/\sigma{_{\text{J}}}$ is required. In this work, the conductivity is assumed to be the same for extrasolar gas giants as for Jupiter, i.e. $\sigma/\sigma{_{\text{J}}}=1$.
#### The size of the magnetosphere
The size of the planetary magnetosphere $R{_{\text{s}}}$ is calculated with the parameters determined above for the stellar wind and the planetary magnetic moment. For a given planetary orbital distance $d$, of the different pressure contributions only the magnetospheric magnetic pressure is a function of the distance to the planet. Thus, the standoff distance $R_s$ is found from the pressure equilibrium [@Griessmeier05]: $$R_s(d) =
\left[ \frac{\mu_0f_0^2\mathcal{M}^2}
{8\pi^2 \left(m n(d) v{_{\text{eff}}}(d)^2+2 \, n(d)k_BT\right)} \right]^{1/6}.
\label{eq:Rs}$$ Here, $m$ is the proton’s mass, and $f_0$ is the form factor of the magnetosphere. It describes the magnetic field created by the magnetopause currents. For a realistic magnetopause shape, a factor $f_0 = 1.16$ is used [@Voigt95]. In term of units normalized to Jupiter’s units, this is equivalent to $$R_s(d) \approx 40 R{_{\text{J}}}\left[ \frac{\widetilde{\mathcal{M}}^2}
{ \tilde{n}(d) \widetilde{v{_{\text{eff}}}}(d)^2+\frac{2 \, \tilde{n}(d)k_BT}{m {v_\text{eff,J}}^2}} \right]^{1/6},$$ with $\tilde{n}(d)=n(d)/n{_{\text{J}}}$, $\widetilde{v{_{\text{eff}}}}(d)=v{_{\text{eff}}}(d)/v_\text{eff,J}$, $n{_{\text{J}}}=2.0 \cdot10^5$ m$^{-3}$, and $v_\text{eff,J}=520$ km/s.
Note that in a few cases, especially for planets with very weak magnetic moments and/or subject to dense and fast stellar winds of young stars, eq. (\[eq:Rs\]) yields standoff distances $R_s<R{_{\text{p}}}$, where $R{_{\text{p}}}$ is the planetary radius. Because the magnetosphere cannot be compressed to sizes smaller than the planetary radius, we set $R_s=R{_{\text{p}}}$ in those cases.
Additional parameters
---------------------
The other required parameters are obtained from the following sources:
- The stellar ages $t_\star$ are taken from @Saffe05 [who gave ages for 112 exoplanet host stars]. @Saffe05 compare stellar age estimations based on five different methods (and for one of them they use two different calibrations), some of which give more reliable results than others. Because not all of these methods are applicable to all stars, we use the age estimations in the following order or preference: chromospheric age for ages below 5.6 Gyr (using the D93 calibration), isochrone age, chromospheric age for ages above 5.6 Gyr (using the D93 calibration), metallicity age. Note that error estimates for the two most reliable methods, namely isochrone and chromospheric ages, are already relatively large (30%-50%). In those cases where the age is not known, a default value of 5.2 Gyr is used [the median chromospheric age found by @Saffe05]. This relatively high average age is due to a selection effect (planet detection by radial velocity method is easier to achieve for older, more slowly rotating stars). As the uncertainty of the radio flux estimation becomes very large for low stellar ages [@Griessmeier05], we use a minimum stellar age of 0.5 Gyr.
- For the solid angle of the beam, we assume the emission to be analogous to the dominating contributions of Jupiter’s radio emission and use $\Omega= 1.6$ sr [@Zarka04].
Expected radio flux for know exoplanets {#sec:results}
=======================================
The list of known exoplanets {#sec:table}
----------------------------
Table 1 [^3] shows what radio emission we expect from the presently known exoplanets (13.1.2007). It contains the maximum emission frequency according to eq. (\[eq:f\]), the plasma frequency in the stellar wind at the planetary location according to eq. (\[eq:fplasma\]) and the expected radio flux according to the magnetic model (\[eq:Pin:kin\]), the kinetic model (\[eq:Pin:mag\]), and the kinetic CME model (\[eq:Pin:kin:CME\]). The unipolar interaction model is discussed in the text below. Table 1 also contains values for the expected planetary mass $M{_{\text{p}}}$, its radius $R{_{\text{p}}}$ and its planetary magnetic dipole moment $\mathcal{M}$. For each planet, we note whether tidal locking should be expected.
Note that [http://exoplanet.eu]{} contains a few more planets than table 1, because for some planets, essential data required for the radio flux estimation are not available (typically $s$, the distance to the observer).
The numbers given in table 1 are not accurate results, but should be regarded as refined estimations intended to guide observations. Still, the errors and uncertainties involved in these estimations can be considerable. As was shown in @GriessmeierPSS06, the uncertainty on the radio flux at Earth, $\Phi$, is dominated by the uncertainty in the stellar age $t{_{\star}}$ [for which the error is estimated as $\approx 50$% by @Saffe05]. For the maximum emission frequency, $f{_{\text{c}}}^{\text{max}}$, the error is determined by the uncertainty in the planetary magnetic moment $\mathcal{M}$, which is uncertain by a factor of two. For the planet $\tau$ Bootes b, these effects translate into an uncertainty of almost one order of magnitude for the flux (the error is smaller for planets around stars of solar age), and an uncertainty of a factor of 2-3 for the maximum emission frequency. This error estimate is derived and discussed in more detail by @GriessmeierPSS06.
The results given in table 1 cover the following range:
- The maximum emission frequency found to lie beween 0 to almost 200 MHz. However, all planets with $ f_{\text{c}}^{\text{max}} > 70$ MHz have negligible flux. Also note that any emission with $ f_{\text{c}} \le 5$ to $10$ MHz will not be detectable on Earth because it cannot propagate through the Earth’s ionosphere (“ionospheric cutoff”). For this reason, the most appropriate frequency window for radio observations seems to be between 10 and 70 MHz.
- The radio flux according to the *magnetic energy* model, $ \Phi_{\text{sw,mag}} $ lies between 0 and 5 Jy (for GJ 436 b). For 15 candidates, $ \Phi_{\text{sw,mag}} $ is larger than 100 mJy, and for 37 candidates it is above 10 mJy.
- The flux prediction according to the *kinetic energy* model, $ \Phi_{\text{sw,kin}} $ is much lower than the flux according to the magnetic energy model. Only in one case it exceeds the value of 10 mJy.
- The increased stellar wind density and velocity during a *CME* leads to a strong increase of the radio flux when compared to the *kinetic energy* model. Correspondingly, $ \Phi_{\text{CME,kin}} $ exceeds 100 mJy in 3 cases and 10 mJy in 11 cases.
- Table 1 does not contain flux estimations for the *unipolar interaction* model. The reason is that the condition given by eq. (\[eq:f:generation\]) is not satisfied in *any* of the studied cases. This is consistent with the result of @Zarka06PREVI [@Zarka06PSS], who found that stars 100 times as strongly magnetised as the sun are required for this type of emission. The approach taken in section \[sec:stellarwind\] for the estimation stellar magnetic fields does not yield such strong magnetic fields for stars with ages $>0.5$ Gyr. Stronger magnetic fields are possible (e.g. for younger stars). Strongly magnetised stars (even those without known planets) could be defined as specific targets to test this model.
- The plasma frequency in the stellar wind, $ f_{\text{p,sw}} $, is negligibly small in most cases. For a few planets, however, it is of the same order of magnitude as the maximum emission frequency. In these cases, the condition given by eq. (\[eq:escape\]) makes the escape of the radio emission from its source towards the observer impossible. Of the 197 planets of the current census, eq. (\[eq:escape\]) is violated in 8 cases. If one takes into account the uncertainty of the stellar age [30-50%, see @Saffe05], an uncertainty of similar size is introduced for the plasma frequency: In the example of $\tau$ Bootis, the age uncertainty translates into a variation of up to 50% for the plasma frequency. With such error bars, between 6 and 14 planets are affected by eq. (\[eq:escape\]). None of the best targets are affected.
- The expected planetary magnetic dipole moments lie between 0 and 5.5 times the magnetic moment of Jupiter. However, the highest values are found only for very massive planets: Planets with masses $M\le2M{_{\text{J}}}$ have magnetic moments $\mathcal{M}\le2\mathcal{M}{_{\text{J}}}$. For planets with masses of $M\le M{_{\text{J}}}$, the models predict magnetic moments $\mathcal{M}\le1.1\mathcal{M}{_{\text{J}}}$.
The results of table 1 confirm that the different models for planetary radio emission lead to very different results. The largest fluxes are found for the *magnetic energy* model, followed by the *CME* model and the *kinetic energy* model. This is consistent with previous expectations [@Zarka01; @Zarka06PREVI; @GriessmeierPREVI; @Zarka06PSS; @GriessmeierPSS06]. The *unipolar interaction* model does not lead to observable emission for the presently known exoplanets. Furthermore, the impact of tidal locking is clearly visible in the results. As it is currently not clear which of these models best describes the auroral radio emission, it is not sufficient to restrict oneself to one scaling law (e.g. the one yielding the largest radio flux). For this reason, all possible models have to be considered. Once exoplanetary radio emission is detected, observations will be used to constrain and improve the models.
Table 1 also shows that planets subject to tidal locking have a smaller magnetic moment and thus a lower maximum emission frequency than freely rotating planets. The reduced bandwidth of the emission can lead to an increase of the radio flux, but frequently emission is limited to frequencies not observable on earth (i.e. below the ionospheric cutoff).
The results of table 1 are visualized in figure \[fig:radiopredictionSWmag\] (for the *magnetic energy* model), figure \[fig:radiopredictionCMEkin\] (for the *CME* model) and figure \[fig:radiopredictionSWkin\] (for the *kinetic energy* model). The predicted planetary radio emission is denoted by open triangles (two for each “potentially locked” planet, otherwise one per planet). The typical uncertainties (approx. one order of magnitude for the flux, and a factor of 2-3 for the maximum emission frequency) are indicated by the arrows in the upper right corner. The sensitivity limit of previous observation attempts are shown as filled symbols and as solid lines [a more detailed comparison of these observations can be found in @Zarka04a; @Griessmeier51PEG05; @GriessmeierPHD06]. The expected sensitivity of new and future detectors (for 1 hour integration and 4 MHz bandwidth, or any equivalent combination) is shown for comparison. Dashed line: upgraded UTR-2, dash-dotted lines: low band and high band of LOFAR, left dotted line: LWA, right dotted line: SKA. The instruments’ sensitivities are defined by the radio sky background. For a given instrument, a planet is observable if it is located either above the instrument’s symbol or above and to its right. Again, large differences in expected flux densities are apparent between the different models. On average, the *magnetic energy* model yields the largest flux densities, and the *kinetic energy* model yields the lowest values. Depending on the model, between one and three planets are likely to be observable using the upgraded system of UTR-2. Somewhat higher numbers are found for LOFAR. Considering the uncertainties mentioned above, these numbers should not be taken literally, but should be seen as an indicator that while observation seem feasible, the number of suitable candidates is rather low. It can be seen that the maximum emission frequency of many planets lies below the ionospheric cutoff frequency, making earth-based observation of these planets impossible. A moon-based radio telescope however would give access to radio emission with frequencies of a few MHz [@Zarka06PSS]. As can be seen in figures \[fig:radiopredictionSWmag\], \[fig:radiopredictionCMEkin\] and \[fig:radiopredictionSWkin\], this frequency range includes a significant number of potential target planets with relatively high flux densities.
Figures \[fig:radiopredictionSWmag\], \[fig:radiopredictionCMEkin\] and \[fig:radiopredictionSWkin\] also show that the relatively high frequencies of the LOFAR high band and of the SKA telescope are probably not very well suited for the search for exoplanetary radio emission. These instruments could, however, be used to search for radio emission generated by *unipolar interaction* between planets and strongly magnetised stars.
![ Maximum emission frequency and expected radio flux for known extrasolar planets according to the *magnetic energy* model, compared to the limits of past and planned observation attempts. Open triangles: Predictions for planets. Solid lines and filled circles: Previous observation attempts at the UTR-2 (solid lines), at Clark Lake (filled triangle), at the VLA (filled circles), and at the GMRT (filled rectangle). For comparison, the expected sensitivity of new detectors is shown: upgraded UTR-2 (dashed line), LOFAR (dash-dotted lines, one for the low band and one for the high band antenna), LWA (left dotted line) and SKA (right dotted line). Frequencies below $~10$ MHz are not observable from the ground (ionospheric cutoff). Typical uncertainties are indicated by the arrows in the upper right corner. []{data-label="fig:radiopredictionSWmag"}](7397fig1.ps){width="1.0\linewidth"}
![Maximum emission frequency and expected radio flux for known extrasolar planets according to the *CME* model, compared to the limits of past and planned observation attempts. Open triangles: Predictions for planets. All other lines and symbols are as defined in figure \[fig:radiopredictionSWmag\].[]{data-label="fig:radiopredictionCMEkin"}](7397fig2.ps){width="1.0\linewidth"}
![Maximum emission frequency and expected radio flux for known extrasolar planets according to the *kinetic energy* model, compared to the limits of past and planned observation attempts. Open triangles: Predictions for planets. All other lines and symbols are as defined in figure \[fig:radiopredictionSWmag\].[]{data-label="fig:radiopredictionSWkin"}](7397fig3.ps){width="1.0\linewidth"}
A few selected cases {#sec:cases}
--------------------
According to our analysis, the best candidates are:
- HD 41004 B b, which is the best case in the *magnetic energy* model with emission above 1 MHz. Note that the mass of this object is higher than the upper limit for planets ($\approx 13 M{_{\text{J}}}$), so that it probably is a brown dwarf and not a planet.
- Epsilon Eridani b, which is the best case in the *kinetic energy* model.
- Tau Boo b, which is the best case in the *magnetic energy* model with emission above the ionospheric cutoff (10 MHz).
- HD 189733 b, which is the best case in both the *magnetic energy* model and in the *CME* model which has emission above 1 MHz.
- Gliese 876 c, which is the best case in the *CME* model with emission above the ionospheric cutoff (10 MHz).
- HD 73256 b, which has emission above 100 mJy in the *magnetic energy* model and which is the second best planet in the *kinetic energy* model.
- GJ 3021 b, which is the third best planet in the *kinetic energy* model.
To this list, one should add the planets around Ups And (b, c and d) and of HD 179949 b, whose parent stars exhibit an increase of the chromospheric emission of about 1-2% [@Shkolnik03; @Shkolnik04; @Shkolnik05]. The observations indicate one maximum per planetary orbit, a “Hot Spot” in the stellar chromosphere which is in phase with the planetary orbit. The lead angles observed by @Shkolnik03 and @Shkolnik05 were recently explained with an Alfvén-wing model using realistic stellar wind parameters obtained from the stellar wind model by Weber and Davis [@PreussePHD05; @Preusse06]. This indicates that a magnetised planet is not required to describe the present data. The presence of a planetary magnetic field could, however, be proven by the existence of planetary radio emission. Although our model does not predict high radio fluxes from these planets (see table 1), the high chromospheric flux shows that a strong interaction is taking place. As a possible solution of this problem, an intense stellar magnetic field was suggested [@Zarka06PREVI; @Zarka06PSS]. In that case, table 1 underestimates the radio emission of Ups And b, c, d and HD 179949 b, making these planets interesting candidates for radio observations (eg. through the *magnetic energy* model or the *unipolar interaction* model). For this reason, it would be especially interesting to obtain measurements of the stellar magnetic field for these two planet-hosting stars [e.g. by the method of @Catala07].
Considering the uncertainties mentioned above, it is important not to limit observations attempts to these best cases. The estimated radio characteristics should only be used as a guide (e.g. for the target selection, or for statistical analysis), but individual results should not be regarded as precise values.
Statistical discussion {#sec:statistics}
----------------------
It may seem surprising that so few good candidates are found among the 197 examined exoplanets. However, when one checks the list of criteria for “good” candidates [e.g. @Griessmeier51PEG05], it is easily seen that only a few good targets can be expected: (a) the planet should be close to the Earth (otherwise the received flux is too weak). About 70% of the known exoplanets are located within 50 pc, so that this is not a strong restriction. (b) A strongly magnetised system is required (especially for frequencies above the ionospheric cutoff). For this reason, the planet should be massive (as seen above, we find magnetic moments $\mathcal{M}\ge2\mathcal{M}{_{\text{J}}}$ only for planets with masses $M\ge2M{_{\text{J}}}$). About 60% of the known exoplanets are at least as massive as Jupiter (but only 40% have $M{_{\text{p}}}\ge 2.0 M{_{\text{J}}}$). (c) The planet should be located close to its host star to allow for strong interaction (dense stellar wind, strong stellar magnetic field). Only 25% of the known exoplanets are located within 0.1 AU of their host star.
By multiplying these probabilities, one finds that close ($s \le 50$ pc), heavy ($M{_{\text{p}}}\ge 2.0 M{_{\text{J}}}$), close-in ($d \le 0.1$ AU) planets would represent 8% ($\approx$ 15 of 197 planets) of the current total if the probabilities for the three conditions were independent. However, this is not the case. In the current census of exoplanets, a correlation between planetary mass and orbital distance is clearly evident, with a lack of close-in massive planets [see e.g. @Udry03]. This is not a selection effect, as massive close-in planets should be easier to detect than low-mass planets. This correlation was explained by the stronger tidal interaction effects for massive planets, leading to a faster decrease of the planetary orbital radius until the planet reaches the stellar Roche limit and is effectively destroyed [@Paetzold02; @Jiang03]. Because of this mass-orbit correlation, the fraction of good candidates is somewhat lower (approx. 2%, namely 3 of 197 planets: HD 41004 B b, Tau Boo b, and HD 162020 b).
Comparison to previous results {#sec:comparison}
------------------------------
A first comparative study of expected exoplanetary radio emission from a large number of planets was performed by @Lazio04, who compared expected radio fluxes of 118 planets (i.e. those known as of 2003, July 1). Their results differ considerably from those given in table 1:
- As far as the maximum emission frequency $f_\text{c}^{\text{max}}$ is concerned, our results are considerably lower than the frequencies given by @Lazio04. For Tau Bootes, their maximum emission frequency is six times larger than our result. For planets heavier than Tau Bootes, the discrepancy is even larger, reaching more than one order of magnitude for the very heavy cases (e.g. HD 168433c, for which they predict radio emission with frequencies up to 2670 MHz). These differences have several reasons: Firstly, @Farrell99 and @Lazio04 assume that $R{_{\text{p}}}=R{_{\text{J}}}$. Also, these works rely on the magnetic moment scaling law of @Blackett47, which has a large exponent in $r{_{\text{c}}}$. This scaling law should not be used, as it was experimentally disproven [@Blackett52]. Thirdly, these works make uses $r{_{\text{c}}}\propto M{_{\text{p}}}^{1/3}$. Especially for planets with large masses like $\tau$ Bootes, this yields unrealistically large core radii (even $r{_{\text{c}}}>R{_{\text{p}}}$ in some cases), magnetic moments, and emission frequencies. Note that a good estimation of the emission frequency is particularly important, because a difference of a factor of a few can make the difference between radiosignals above and below the Earth’s ionospheric cutoff frequency.
- The anticipated radio flux obtained with the *kinetic energy* model $\Phi_{\text{sw,kin}}$ is much lower than the estimates of @Lazio04. Typically, the difference is approximately two orders of magnitude, but this varies strongly from case to case. For example, for Tau Boo b, our result is smaller by a factor of 30 (where the difference is partially compensated by the low stellar age which increases our estimation), for Ups And b, the results differ by a factor of 220, and for Gliese 876 c, the difference is as large as a factor 6300 (which is partially due to the fact we take into account the small stellar radius and the high stellar age).
- As was mentioned above, the analysis of the jovian radio emission allows to define three terms for the typical radio spectrum: (a) the power during *average conditions*, (b) the average power during periods of *high activity*, and (c) the *peak power* [@Zarka04]. When comparing the results of our table 1 to those of @Lazio04, one has to note that their table I gives the *peak power*, while the results in our table 1 were obtained using the average power during periods of *high activity*. Similarly to @Farrell99, @Lazio04 assume that the peak power caused by variations of the stellar wind velocity is two orders of magnitude higher than the average power. However, the values @Farrell99 use for average conditions correspond to periods of high activity, which are less than one order of magnitude below the peak power [@Zarka04]. During periods of peak emission, the value given in our table 1 would be increased by the same amount [approximately a factor of 5, see @Zarka04]. For this reason, the peak radio flux is considerably overestimated in these studies.
- Estimated radio fluxes according to the *magnetic energy* model and the *CME* model have not yet been published for large numbers of planets. This is the first time the results from these models are compared for a large number of planets.
Conclusions {#sec:conclusion}
===========
Predictions concerning the radio emission from all presently known extrasolar planets were presented. The main parameters related to such an emission were analyzed, namely the planetary magnetic dipole moments, the maximum frequency of the radio emission, the radio flux densities, and the possible escape of the radiation towards a remote observer.
We compared the results obtained with various theoretical models. Our results confirm that the four different models for planetary radio emission lead to very different results. As expected, the largest fluxes are found for the *magnetic* energy model, followed by the *CME* model and the *kinetic* energy model. The results obtained by the latter model are found to be less optimistic than by previous studies. The *unipolar interaction* model does not lead to observable emission for any of the currently known planets. As it is currently not clear which of these models best describes the auroral radio emission, it is not sufficient to restrict oneself to one scaling law (e.g. the one yielding the largest radio flux). Once exoplanetary radio emission is detected, observations will be used to constrain and improve the model.
These results will be particularly useful for the target selection of current and future radio observation campaigns (e.g. with the VLA, GMRT, UTR-2 and with LOFAR). We have shown that observation seem feasible, but that the number of suitable candidates is relatively low. The best candidates appear to be HD 41004 B b, Epsilon Eridani b, Tau Boo b, HD 189733 b, Gliese 876 c, HD 73256 b, and GJ 3021 b. The observation of some of these candidates is in progress.
We thank J. Schneider for providing data via “The extrasolar planet encyclopedia” (http://exoplanet.eu/), I. Baraffe and C. Vocks for helpful discussions concerning planetary radii. We would also like to thank the anonymous referee for his helpful comments. This study was jointly performed within the ANR project “La détection directe des exoplanètes en ondes radio” and within the LOFAR transients key project (TKP). J.-M. G. was supported by the french national research agency (ANR) within the project with the contract number NT05-1\_42530 and partially by Europlanet (N3 activity). P.Z. acknowledges support from the International Space Science Institute (ISSI) within the ISSI team “Search for Radio Emissions from Extra-Solar Planets”.
An empirical mass-radius relation {#sec:appendix}
=================================
For the selection of targets for the search for radio emission from extrasolar planets, an estimation of the expected radio flux $\Phi$ and of the maximum emission frequency $f{_{\text{c}}}^{\text{max}}$ is required. For the calculation of these values, both the planetary mass and the planetary radius are required [see, e.g. @Farrell99; @GriessmeierPSS06; @Zarka06PSS]. However, only for a few planets (i.e. the 16 presently known transiting planets) both mass and radius are known.
In the absence of observational data, it is in principle possible to obtain planetary radii from numerical simulation, e.g. similar to those of @Bodenheimer03 or @Baraffe03 [@Baraffe05], requiring one numerical run per planet. We chose instead to derive a simplified analytical fit to such numerical results.
The accuracy of the fit {#sec-precision}
-----------------------
The description presented here is necessarily only preliminary, as (a) numerical models are steadily further developed and improved, and as (b) more transit observations (e.g. by the COROT satellite, which was launched recently) will provide a much better database in the future. This will considerably improve our understanding of the dependence of the planetary radius on various parameters as, e.g. planetary mass, orbital distance, or stellar metallicity [as suggested by @Guillot06].
### What accuracy can we accept?
Within the frame of the models presented in section \[sec:theory\], an increase in $R{_{\text{p}}}$ by 40% increases the expected radio flux by a factor of 2, and the estimated maximum emission frequency decreases by 40%. More generally, for a fixed planetary mass, $\Phi$ is roughly proportional to $R{_{\text{p}}}^{7/3}$ and $f{_{\text{c}}}^{\text{max}}$ is approximately proportional to $R{_{\text{p}}}^{-1}$. Thus, it appears that the assumption of a single standard radius for all planets leads to a relatively large error. Comparing this to the other uncertainties involved in the estimation of radio characteristics [these are discussed in @GriessmeierPSS06], it seems sufficient to estimate $R{_{\text{p}}}$ with 20% accuracy.
### What accuracy can we expect?
Several effects limit the precision in planetary radius we can hope to achieve:
- The definition of the radius: The “transit radius” measured for transiting planets is not exactly identical to the standard 1 bar radius. The differences are of the order of about 5-10%, but depend on the mass of the planet [@Burrows03; @Burrows04]. Because we compare modelled radii and observed radii without correcting for this effect, this limits the maximum precision we can potentially obtain.
- The abundance of heavy elements: The transiting planet around HD 149026 is substantially enriched in heavy elements [@Sato05]. Models [@Bodenheimer03] yield a smaller radius for planets with a heavy core than for coreless planets of the same mass (more than 10% difference for small planets). For a planet with unknown radius, a strong enrichment in heavy elements cannot be ruled out, as this case cannot be distinguished observationally from a pure hydrogen giant (i.e. one without heavy elements).
For these reasons, we conclude that an analytical description which agrees with the (numerical) data within $\sim$20% seems sufficient. For such a description, the error introduced by the fit will not be the dominant one. To get a better result, it is not sufficient to improve the approximation for the radius estimation, but the more fundamental problems mentioned above have to be addressed.
An analytical mass-radius relation {#sec-model}
----------------------------------
### The influence of the planetary mass {#sec-model-mass}
A simple mass-radius relation, valid within a vast mass range, has been proposed by @LyndenBell01preprint and @LyndenBell01: $$R{_{\text{p}}}=
\frac{1}{\left( \frac{4}{3} \pi \rho_0 \right)^{1/3}}
\frac{M{_{\text{p}}}^{1/3}}{1+\left( \frac{M{_{\text{p}}}}{M_{\text{max}}} \right)^{2/3}}
\label{eq:mass-radius}$$ The density $\rho_0$ is depends on the planetary atomic composition. Eq. (\[eq:mass-radius\]) has a maximum in $R{_{\text{p}}}$ when $M{_{\text{p}}}=M_{\text{max}}$ (corresponding to the planet of maximum radius). Fitting Jupiter, Saturn and the planet of maximum radius [$R_{\text{max}}=1.16 R{_{\text{J}}}$, see @Hubbard84], we obtain $M_{\text{max}}=3.16 M{_{\text{J}}}$ and $\rho_0=394$ kg m$^{-3}$ for a Jupiter-like mixture of hydrogen and helium (75% and 25% by mass, respectively). For a pure hydrogen planet, $\rho_0=345$ kg m$^{-3}$. With there parameters, eq. (\[eq:mass-radius\]) fits well the results for cold planets of @Bodenheimer03. In this work, this estimation is used for the radii of non-irradiated (cold) planets.
### The influence of the planetary age {#sec-model-age}
It is known that the radius of a planet with a given mass depends on its age. According to models for the radii of isolated planets [@Baraffe03], the assumption of a time-independent planetary radius leads to an error $\le$11% for planets with ages above 0.5 Gyr and with masses above $0.5 M{_{\text{J}}}$. In view of the uncertainties discussed above, this error seems acceptable for a first approximation, and we use $$R_{\text{p}}(M{_{\text{p}}},t)\approx R_{\text{p}}(M{_{\text{p}}}) \label{eq-Riso}.$$
### The influence of the planetary orbital distance {#sec-model-distance}
It is commonly expected that planets subjected to strong stellar radiation have a larger planetary radius than isolated, but otherwise identical planets. This situation is typical for “Hot Jupiters”, where strong stellar irradiation is supposed to delay the planetary contraction [@Burrows00; @Burrows03; @Burrows04]. Clearly, this effect depends on the planetary distance to its star, $d$, and on the stellar luminosity $L{_{\star}}$. In the following, we denote the radius increase by $r$, which we define as $$r=\frac{R{_{\text{p}}}(M{_{\text{p}}},d)}{R{_{\text{p}}}(M{_{\text{p}}},d=\infty)}.$$ Herein, $R{_{\text{p}}}(M{_{\text{p}}},t,d)$ denotes the radius of a planet under the irradiation by its host star, and $R{_{\text{p}}}(M{_{\text{p}}},t,d=\infty)$ is the radius of a non-irraditated, but otherwise identical planet.
Different exoplanets have vastly different host stars. A difference of a factor of two in stellar mass can result in a difference of more than order of magnitude in stellar luminosity. For this reason, it is not sufficient to take the orbital distance as the only parameter determining the radius increase by irradiation.
Here, we select the equilibrium temperature of the planetary surface as the basic parameter. It is defined as [@Bodenheimer03] $$T_{\text{eq}}=\left[
\frac{(1-A)L{_{\star}}}{16 \pi \sigma_{\text{SB}} d^2 (1+e^2/2)^2}
\right]^{1/4},$$ where the planetary albedo is set as $A=0.4$ in the following. The stellar luminosity $L{_{\star}}$ is calculated from the stellar mass according to the analytical fit given by @Tout96 for the zero-age main sequence. In this, we assumed solar metallicity for all stars. Finally, $\sigma_{\text{SB}}$ denotes the Stefan-Boltzmann constant.
Using the data of @Bodenheimer03, we use the following fit for the radius increase $r$: $$r=1+0.05\left(\frac{T_{\text{eq}}}{T_0}\right)^{\gamma}. \label{eq:rratio}$$ This form was selected because it has the correct qualitative behaviour: It yields a monotonous decrease of $r$ with decreasing equilibrium temperature $T_{\text{eq}}$ (increasing orbital distance $d$). In the limit $T_{\text{eq}}\to0$ ($d\to\infty$) we find $r\to 1$. As an alternative to a continuous fit like eq. (\[eq:rratio\]), @Lazio04 use a step-function, i.e. assume an increased radius of $r=1.25$ for planets closer than a certain distance $d_1=0.1$ AU only. However, this treatment does not reduce the number of fit-constants. Also, for planets with orbital distances close to $d_1$, the results obtained with a step-function strongly depend on the somehow arbitrary choice of $d_1$. A continuous transition from “irradiated” to “isolated” planets is less prone to this effect.
The numerical results of @Bodenheimer03 show that the ratio $r$ also depends on the mass of the planet: for small planets, $r=1.4$ for an equilibrium temperature of 2000 K, whereas for large planets, $r\le 1.10$. For this reason, we allow to coefficients $T_0$ and $\gamma$ to vary with $M{_{\text{p}}}$: $$T_0=c_{\text{t,1}}\cdot \left( \frac{M{_{\text{p}}}}{M{_{\text{J}}}}\right)^{c_{\text{t,2}}}\label{eq:T0}$$ and $$\gamma=1.15+0.05 \cdot \left( \frac{c_{\gamma,1}} {M{_{\text{p}}}} \right)^{c_{\gamma,2}}.\label{eq:gamma}$$ With the set of coefficient $c_{\text{t,1}}=764$ K , $c_{\text{t,2}}=0.28$, $c_{\gamma,1}=0.59 M{_{\text{J}}}$ and $c_{\gamma,2}=1.03$, we obtain an analytical fit to the numerical results of @Bodenheimer03. The maximum deviation from the numerical results is below 10% (cf. Fig. \[fig:rofTeff\]).
For comparison, Fig. \[fig:rofTeff\] also shows the value of $r$ for the transiting exoplanets OGLE-TR-10b, OGLE-TR-56b, OGLE-TR-111b, OGLE-TR-113b, OGLE-TR-132b, XO-1b, HD 189733b, HD 209458b, TrES-1b and TrES-2b as small crosses. Here, $r$ is calculated as the ratio of the observed value of $R{_{\text{p}}}(M{_{\text{p}}},d)$ and the value for $R{_{\text{p}}}(M{_{\text{p}}},d=\infty)$ calculated according to eq. (\[eq:mass-radius\]). As the mass of all transiting planets lies between $0.11M{_{\text{J}}}\le M{_{\text{p}}}\le 3.0 M{_{\text{J}}}$, one should expect to find all crosses between the two limiting curves. For most planets, this is indeed the case: only for HD 209458b, $r$ is considerably outside the area delimited by the two curves. Different explanations have been put forward for the anomalously large radius of this planet, but so far no conclusive answer to this question has been found [see e.g. @Guillot06 and references therein]. As numerical models cannot reproduce the observed radius of this planet, one cannot expect our approach (which is based on a fit to numerical results) to reproduce it either. For the other planets, Fig. \[fig:rofTeff\] shows that our approach is a valid approximation.
![Radius increase $r$ as a function of the equilibrium temperature $T_{\text{eq}}$ according to eq. (\[eq:rratio\]), (\[eq:T0\]) and (\[eq:gamma\]). Upper line: radius increase $r$ due to irradiation for planets of mass $M{_{\text{p}}}=0.11M{_{\text{J}}}$. Lower line: Upper line: radius increase $r$ due to irradiation for planets of mass $M{_{\text{p}}}=3.0M{_{\text{J}}}$. Open symbols: results of the numerical calculation of @Bodenheimer03 for planetary masses $M{_{\text{p}}}=0.11M{_{\text{J}}}$ (diamonds) and $M{_{\text{p}}}=3.0M{_{\text{J}}}$ (triangles). Crosses: radius increase for observed transiting planets (see text). []{data-label="fig:rofTeff"}](7397fig4.ps){width="1.0\linewidth"}
Tidal locking? {#sec:tlocking}
==============
For the estimation of the planetary magnetic dipole moment in section \[sec:protation\], we require the planetary rotation rate. This rotation rate rotation greatly depends on whether the planet can be considered as tidally locked, as freely rotating, or as potentially locked. In this appendix, we discuss how we evaluate the tidal locking timescale ${\tau_{\text{sync}}}$, which decides to which of these categories a planet belongs.
The value of the planetary rotation $\omega$ depends on its distance from the central star. For close-in planets, the planetary rotation rate is reduced by tidal dissipation. In this case, tidal interaction gradually slows down the planetary rotation from its initial value $\omega_\text{i}$ until it reaches the final value $\omega_\text{f}$ after tidal locking is completed.
It should be noted that for planets in eccentric orbits, tidal interaction does not lead to the synchronisation of the planetary rotation with the orbital period. Instead, the rotation period also depends on the orbital eccentricity. At the same time, the timescale to reach this equilibrium rotation rate is reduced [@Laskar04]. Similarly, for planets in an oblique orbit, the equilibrium rotation period is modified [@Levrard07]. However, the locking of a hot Jupiter in a non-synchronous spin-orbit resonance appears to be unlikely for distances $\leq$ 0.1 AU [@Levrard07]. For this reason, we will calculate the timescale for tidal locking under the assumptions of circular orbits and zero obliquity. In the following, the tidal locking timescale for reaching $\omega_\text{f}$ is calculated under the following simplifying assumptions: prograde orbit, spin parallel to orbit (i.e. zero obliquity), and zero eccentricity [@Murray99 Chapter 4].
The rate of change of the planetary rotation velocity $\omega$ for a planet with a mass of $M{_{\text{p}}}$ and radius of $R{_{\text{p}}}$ around a star of mass $M{_{\star}}$ is given by [@Goldreich66; @Murray99]: $$\frac{d\omega}{dt}= \frac{9}{4} \frac{1}{\alpha \, Q{_{\text{p}}}'} \left( \frac{GM{_{\text{p}}}}{R{_{\text{p}}}^3} \right)
\left( \frac{M{_{\star}}}{M{_{\text{p}}}} \right)^2
\left( \frac{R{_{\text{p}}}}{d} \right)^6
,\label{eq:omegadot}$$ where the constant $\alpha$ depends on the internal mass distribution within the planet. It is defined by $\alpha=I/(M{_{\text{p}}}R{_{\text{p}}}^2)$, where $I$ is the planetary moment of inertia. For a sphere of homogeneous density, $\alpha$ is equal to 2/5. For planets, generally $\alpha\le 2/5$. $Q{_{\text{p}}}'$ is the modified $Q$-value of the planet. It can be expressed as [@Murray99] $$Q{_{\text{p}}}'=\frac{3Q{_{\text{p}}}}{2k_{2,\text{p}}}, \label{eq:k_2}$$ where $k_{2,\text{p}}$ is the Love number of the planet. $Q{_{\text{p}}}$ is the planetary tidal dissipation factor (the larger it is, the smaller is the tidal dissipation), defined by @MacDonald64 and @Goldreich66. The time scale for tidal locking is obtained by a comparison of the planetary angular velocity and its rate of change: $${\tau_{\text{sync}}}=\frac{\omega_\text{i}-\omega_\text{f}}{\dot{\omega}}\,. \label{eq:tau}$$ A planet with angular velocity $\omega_\text{i}$ at $t=0$ (i.e. after formation) will gradually lose angular momentum, until the angular velocity reaches $\omega_\text{f}$ at $t={\tau_{\text{sync}}}$. Insertion of eq. (\[eq:omegadot\]) into eq. (\[eq:tau\]) yields the following expression for ${\tau_{\text{sync}}}$: $${\tau_{\text{sync}}}\approx \frac{4}{9} \alpha \, Q{_{\text{p}}}' \left( \frac{R{_{\text{p}}}^3}{GM{_{\text{p}}}} \right)
\left( \omega_\text{i}-\omega_\text{f} \right)
\left( \frac{M{_{\text{p}}}}{M{_{\star}}} \right)^2
\left( \frac{d}{R{_{\text{p}}}} \right)^6
\label{eq:locking}$$
The importance of this effect strongly depends on the distance (${\tau_{\text{sync}}}\propto d^{6}$). Thus, a planet in a close-in orbit ($d\lesssim 0.1$ AU) around its central star is subject to strong tidal interaction, leading to gravitational locking on a very short timescale.
In the following, we briefly describe the parameters ($\alpha$, $Q{_{\text{p}}}'$, $\omega_\text{i}$ and $\omega_\text{f}$) required to calculate the timescale for tidal locking of Hot Jupiters.
#### Structure parameter $\alpha$
For large gaseous planets, the equation of state can be approximated by a polytrope of index $\kappa=1$. In that case, the structure parameter $\alpha$ (defined by $\alpha=I/M{_{\text{p}}}R{_{\text{p}}}^2$) is given by $\alpha=0.26$ [@Gu03].
#### Tidal dissipation factor $Q_\mathrm{p}'$ {#sec:Q':gas}
For planets with masses of the order of one Jupiter mass, one finds that $k_{2,\text{p}}$ has a value of $k_{2,\text{p}}\approx 0.5$ for Jupiter [@Murray99; @Laskar04] and $k_{2,\text{p}}\approx 0.3$ for Saturn [@Peale99; @Laskar04]. The value of $k_{2,\text{p}}=0.5$ will be used in this work. With eq. (\[eq:k\_2\]), this results in $Q{_{\text{p}}}'\approx 3 Q{_{\text{p}}}$.
For Jupiter, one finds the following range of allowed values: $6.6\cdot10^4\lesssim Q{_{\text{p}}}\lesssim2.0\cdot10^6$ [@Peale99]. Several estimations of the turbulent dissipation within Jupiter yield $Q{_{\text{p}}}$-values larger than this upper limit, while other theories predict values consistent with this upper limit [@Marcy97; @Peale99 and references therein]. This demonstrates that the origin of the value of $Q{_{\text{p}}}$ is not well understood even for Jupiter [@Marcy97].
Extrasolar giant planets are subject to strongly different conditions, and it is difficult to constrain $Q{_{\text{p}}}$. Typically, Hot Jupiters are assumed to behave similarly to Jupiter, and values in the range $1.0\cdot10^5 \le Q{_{\text{p}}}' \le 1.0\cdot10^6$ are used. In the following, we will distinguish three different regimes: close planets (which are tidally locked), distant planets (which are freely rotating), and planets at intermediate distances (which are potentially tidally locked). The borders between the “tidally locked” and the “potentially locked” regime is calculated by setting ${\tau_{\text{sync}}}=100$ Myr and $Q{_{\text{p}}}'=10^6$. The border between the “potentially locked” and the “freely rotating” regime is calculated by setting ${\tau_{\text{sync}}}=10$ Gyr and $Q{_{\text{p}}}'=10^5$. Thus, the area of “potentially locked” planets is increased.
#### Initial rotation rate $\omega_\mathrm{i}$
The initial rotation rate $\omega_\text{i}$ is not well constrained by planetary formation theories. The relation between the planetary angular momentum density and planetary mass observed in the solar system [@MacDonald64] suggests a primordial rotation period of the order of 10 hours [@Hubbard84 Chapter 4]. We assume the initial rotation rate to be equal to the current rotation rate of Jupiter (i.e. $\omega=\omega_\text{i}=\omega{_{\text{J}}}$) with $\omega{_{\text{J}}}=1.77\cdot10^{-4}$ s$^{-1}$.
#### Final rotation rate $\omega_\mathrm{f}$
As far as eq. (\[eq:locking\]) is concerned, $\omega_\mathrm{f}$ can be neglected [@GriessmeierPHD06].
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[^1]: Table 1 is only available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/
[^2]: containing a typo in the sixth equation of the appendix.
[^3]: Table 1 is only available in electronic form at the CDS.
|
---
abstract: 'We study the inverse proximity effect in a bilayer consisting of a thin [$s$- or $d$-wave]{} superconductor (S) and a topological insulator (TI). Integrating out the topological fermions of the TI, we find that spin-orbit coupling is induced in the S, which leads to spin-triplet $p$-wave ($f$-wave) correlations in the anomalous Green’s function [for an $s$-wave ($d$-wave) superconductor]{}. Solving the self-consistency equation for the superconducting order parameter, we [find that the inverse proximity effect can be strong for parameters for which the Fermi momenta of the S and TI coincide]{}. [The suppression of the gap is approximately proportional to $e^{-1/\lambda}$, where $\lambda$ is the dimensionless superconducting coupling constant. This is consistent with the fact that a higher $\lambda$]{} gives a more robust superconducting state. For an $s$-wave S, the interval of TI chemical potentials for which the suppression of the gap is strong is centered at ${\mu_{\mathrm{TI}}}= \pm\sqrt{2m{v_{\mathrm{F}}}^2\mu}$, and increases quadratically with [the]{} hopping parameter $t$. Since the S chemical potential $\mu$ typically is high for conventional superconductors, the inverse proximity effect is negligible except for $t$ above a critical value. For sufficiently low $t$, however, the inverse proximity effect is negligible, in agreement with what has thus far been assumed in most works studying the proximity effect in S-TI structures. [In superconductors with low Fermi energies, such as high-$T_c$ cuprates with $d$-wave symmetry, we again find a suppression of the order parameter. However, since $\mu$ [is]{} much smaller in this case, a strong inverse proximity effect can occur at ${\mu_{\mathrm{TI}}}=0$ for much lower values of $t$. Moreover, the onset of a strong inverse proximity effect is preceded by an increase in the order parameter, allowing the gap to be tuned by several orders of magnitude by small variations in ${\mu_{\mathrm{TI}}}$.]{}'
author:
- 'Henning G. Hugdal'
- Morten Amundsen
- Jacob Linder
- 'Asle Sudb[ø]{}'
title: 'Inverse proximity effect in s-wave and d-wave superconductors coupled to topological insulators'
---
Introduction
============
Topological insulators are insulating in the bulk, but host metallic surface states protected by the topology of the material.[@Hasan2010; @Qi2011; @Wehling2014] For three-dimensional topological insulators, the two-dimensional (2D) surface states can be described by a massless analog of the relativistic Dirac equation, having linear dispersions and spin-momentum locking. Many interesting phenomena are predicted to occur by coupling the TI to a superconductor, thus inducing a superconducting gap in the TI.[@Alicea2012] For instance, such systems have been predicted to host Majorana bound states,[@Fu2008] which could be used for topological quantum computing. Moreover, the Dirac-like Hamiltonian ${\bm{{\sigma}}}\cdot{{\bf{k}}}$ has consequences for the response to exchange fields, allowing the phase difference in a Josephson junction to be tuned by [an]{} in-plane magnetization to values other than $0$ and $\pi$,[@Tanaka2009] and inducing vortexes by an in-plane magnetic field.[@Zyuzin2016; @Amundsen2018]
Numerous papers have studied the interesting phenomena [that have been discovered]{} in topological insulators with proximity-induced superconductivity.[@Fu2009; @Akhmerov2009; @Linder2010a; @Linder2010b; @Zhang2011; @Cook2011; @Qu2012; @Cook2012; @Sochnikov2013; @Koren2013; @Galletti2014; @Sochnikov2015; @Li2015; @Kim2016] To our knowledge, however, much less attention has been paid to the inverse superconducting, or topological,[@Shoman2015] proximity effect, i.e. the effect that the topological insulator has on the superconductor order parameter. There have been indications that superconductivity might be suppressed,[@Sochnikov2013] while other studies have found no suppression.[@Sochnikov2015] One recent study demonstrated that the proximity to the TI induces spin-orbit coupling in the S, possibly making a Fulde-Ferrel[@Fulde1964] superconducting state energetically more favorable near the interface of a magnetically doped TI.[@Park2017] Another study showed that the TI surface states can leak into the superconductor, resulting in a Dirac cone in the density of states.[@Sedlmayr2018] In this paper, we focus on the superconducting gap itself and study under what circumstances the inverse proximity effect is negligible, as is often assumed in theoretical works.
Using a field-theoretical approach, we study an atomically thin Bardeen-Cooper-Schrieffer (BCS) $s$-wave superconductor and $d$-wave superconductor coupled to a TI. While this is an approximation for most conventional and high-$T_c$ superconductors such as e.g. Nb, Al and YBa$_2$Cu$_3$O$_7$ (YBCO), superconductivity has been observed in e.g. single-layer NbSe$_2$[@Ugeda2016] and FeSe.[@Wang2012; @Liu2012; @He2013] Integrating out the TI fermions, we obtain an effective action for the S electrons. Due to the induced spin-orbit coupling, spin-triplet $p$-wave ($f$-wave) correlations are induced in the $s$-wave ($d$-wave) superconductor.
Solving the mean-field equations, using parameters valid for both conventional $s$-wave superconductors and high-$T_c$ $d$-wave superconductors, we find that in both cases a strong suppression of the superconducting gap is possible. For conventional superconductors, where the Fermi energy $\mu$ is high compared to the cut-off frequency, the coupling between the S and the TI has to be quite large in order for the inverse proximity effect to be strong for relevant TI chemical potentials ${\mu_{\mathrm{TI}}}$. This can explain the lack of any inverse proximity effect in experiments.[@Sochnikov2015] [In high-$T_c$ $d$-wave superconductors, on the other hand, where the Fermi energy is much smaller, we find a strong gap suppression at much lower coupling strengths, which might therefore be experimentally observable.]{} For these systems, we also find an increase in the gap for ${\mu_{\mathrm{TI}}}$ just outside the region of strong inverse proximity effect.
The remainder of the article is organized as follows: The model system is presented in Sec. \[sec:model\], and the effective action for the S fermions and order parameter is derived in Sec. \[sec:eff\_S\]. In Sec. \[sec:mean\_field\] we derive the mean field gap equations for the order parameter. Numerical results for the superconducting gap are presented and discussed in Sec. \[sec:results\], and summarized in Sec. \[sec:summary\]. Further details on the calculation of the criteria for strong proximity effect, the Nambu space field integral, the zero-temperature, non-interacting gap solutions, [and the numerical methods used,]{} are presented in the Appendices.
Model {#sec:model}
=====
We model the bilayer consisting of a thin superconductor (S) coupled to a TI by the action $$S = S_{\mathrm{S}} + S_{\mathrm{TI}} + S_t.$$ In [Matsubara and reciprocal space]{}, the superconductor is described by $$\begin{aligned}
S_{\mathrm{S}} ={}&\frac{1}{\beta V} \sum_k c^\dagger(k)\left(-i{\omega_n}+\frac{{{\bf{k}}}^2}{2m} -\mu\right)c(k) \nonumber\\
&- \sum_{k,k',q} \frac{V_{{{\bf{k}}}',{{\bf{k}}}}}{(\beta V)^3}c_{\uparrow}^\dagger(k') c_{\downarrow}^\dagger(-k'+q) c_{\downarrow}(-k+q) c_{\uparrow}(k),\end{aligned}$$ where $c(k) = [c_{\uparrow}(k) ~ c_{\downarrow}(k)]^T$ with $c_{{\uparrow}({\downarrow})}(k)$ denoting the annihilation operator for spin-up (spin-down) electrons, $m$ is the electron mass, $\mu$ is the chemical potential in the S. [$\beta=1/k_B T$ and $V=L_xL_y$ are the inverse temperature and system area respectively. We have used the notation $k=({\omega_n},{{\bf{k}}})$ ($q=({{\Omega_n}},{{\bf{q}}})$), where ${\omega_n}$ (${{\Omega_n}}$) is a fermionic (bosonic) Matsubara frequency, and ${{\bf{k}}}$ (${{\bf{q}}}$) the fermionic (bosonic) in-plane wavevector. $V_{{{\bf{k}}},{{\bf{k}}}'}$ is the pairing potential, which can be written[@Fossheim2004] $$V_{{{\bf{k}}},{{\bf{k}}}'} = g v({{\bf{k}}})v({{\bf{k}}}'),$$ where $v({{\bf{k}}}) = 1$ for $s$-wave pairing, and $v({{\bf{k}}}) = \sqrt{2}\cos(2\phi_{{\bf{k}}})$ for $d_{x^2-y^2}$-wave pairing, [where $\phi_{{{\bf{k}}}}$ is the angle of ${{\bf{k}}}$ relative to the $k_x$ axis]{}. The coupling constant $g$]{} is assumed to be non-zero only when $-\omega_-<{{\bf{k}}}^2/2m - \mu<\omega_+$, where $\pm \omega_\pm$ is the upper (lower) cut-off frequency. For conventional $s$-wave superconductors this is typically taken to be the characteristic frequency $\omega_D$ of the phonons, while the cut-off frequencies in high-$T_c$ superconductors are of the order of the characteristic energy of the anti-ferromagnetic fluctuations [present in these materials]{}.[@Moriya1990; @Monthoux1992; @Pines1993; @Moriya1994] We will set $\hbar = 1$ throughout the paper. For the TI we use the Dirac action $$S_{\mathrm{TI}} = \frac{1}{\beta V}\sum_k\Psi^\dagger(k)(-i{\omega_n}+ {v_{\mathrm{F}}}{{\bf{k}}}\cdot{\bm{{\sigma}}}- {\mu_{\mathrm{TI}}})\Psi(k),$$ where $\Psi({{\bf{r}}}) = [\psi_{\uparrow}({{\bf{r}}}) ~ \psi_{\downarrow}({{\bf{r}}})]^T$ describes the TI fermions, ${v_{\mathrm{F}}}$ is the Fermi velocity, and ${\mu_{\mathrm{TI}}}$ is the TI chemical potential. The S and TI layers are coupled by a hopping term[@BlackSchaffer2013; @Takane2014; @Park2017; @Sedlmayr2018] $$S_t = -\frac{1}{\beta V}\sum_k t[c^\dagger(k)\Psi(k) + \Psi^\dagger(k)c(k)]. \label{eq:hopping}$$ Similar models were recently used in Refs. when studying similar systems with an $s$-wave S. [The full partition function of the system is therefore $$\begin{aligned}
Z = \int {\mathcal{D}}[c^\dagger, c] e^{-S_{\mathrm{S}}} \left(\int {\mathcal{D}}[\Psi^\dagger, \Psi] e^{-S_{\mathrm{TI}} - S_t}\right).\end{aligned}$$]{}
Effective action {#sec:eff_S}
================
As we are interested in the inverse proximity effect in the S and its consequences for the superconducting gap, we integrate out the TI fermions by performing the functional integral $Z_{{\mathrm{TI}},t} = \int {\mathcal{D}}[\Psi^\dagger,\Psi] e^{-S_{{\mathrm{TI}},t}}$, where $$\begin{aligned}
S_{{\mathrm{TI}},t} ={}& \frac{1}{\beta V}{\sum_k} \Big\{\Psi^\dagger(k)(-G^{-1}_{\mathrm{TI}})\Psi(k)\nonumber\\
&- t[c^\dagger(k)\Psi(k) + \Psi^\dagger(k)c(k)]\Big\}.
\end{aligned}$$ Here, we have defined the matrix $G^{-1}_{\mathrm{TI}} = i{\omega_n}- {v_{\mathrm{F}}}{{\bf{k}}}\cdot{\bm{{\sigma}}}+{\mu_{\mathrm{TI}}}$. Performing the functional integration leads to an additional term in the S action, $$\delta S_{\mathrm{S}} = \frac{t^2}{\beta V} \sum_k c^\dagger(k) G_{\mathrm{TI}} c(k),$$ with the TI Green’s function $$G_{\mathrm{TI}} = \frac{i{\omega_n}+{\mu_{\mathrm{TI}}}+{v_{\mathrm{F}}}{{\bf{k}}}\cdot{\bm{{\sigma}}}}{(i{\omega_n}+ {\mu_{\mathrm{TI}}})^2-{v_{\mathrm{F}}}^2{{\bf{k}}}^2}.\label{eq:G_TI}$$ The effective S action thus reads $$\begin{aligned}
S_{\mathrm{S}}^{\mathrm{eff}} = {} &-\frac{1}{\beta V}\sum_k c^\dagger(k)G_0^{-1}c(k) \nonumber\\
&- \sum_{k,k',q} \frac{V_{{{\bf{k}}}',{{\bf{k}}}}}{(\beta V)^3} c^\dagger_{\uparrow}(k')c^\dagger_{\downarrow}(-k'+q) c_{\downarrow}(-k+q)c_{\uparrow}(k),\end{aligned}$$ where we have defined the inverse non-interacting Green’s function $$G_0^{-1} = i{\omega_n}-\frac{{{\bf{k}}}^2}{2m} + \mu - t^2G_{\mathrm{TI}}.$$ From this we see that the coupling to $G_{\mathrm{TI}}$ in Eq. (\[eq:G\_TI\]) leads to an induced spin-orbit coupling $\sim {{\bf{k}}}\cdot{\bm{{\sigma}}}$ in the S, in agreement with Ref. .
Performing a Hubbard-Stratonovich decoupling,[@Altland2010] the 4-fermion term in the S action can be rewritten in terms of bosonic fields $\varphi(q)$ and $\varphi^\dagger(q)$, $$\begin{aligned}
&- \sum_{k,k',q}\frac{V_{{{\bf{k}}}',{{\bf{k}}}}}{(\beta V)^3} c^\dagger_{\uparrow}(k')c^\dagger_{\downarrow}(-k'+q) c_{\downarrow}(-k+q)c_{\uparrow}(k)\nonumber\\
&\quad\rightarrow -\frac{1}{\beta V}\sum_{k,q}\left[\varphi(q)v({{\bf{k}}})c_{\uparrow}^\dagger(k)c_{\downarrow}^\dagger(-k+q) + {\mathrm{h.c.}}\right].\end{aligned}$$ This also leads to an additional term in the total system action $$S_\varphi^0 = \frac{\beta V}{g}\sum_q\varphi^\dagger(q)\varphi(q),\label{eq:phi0action}$$ and a functional integration of the bosonic fields in the partition function. Note that the decoupling is performed such that the bosonic fields have units of energy.
[By]{} defining the Nambu spinor $${\mathcal{C}}(k) = [c_{\uparrow}(k) ~ c_{\downarrow}(k) ~ c^\dagger_{\uparrow}(-k) ~ c^\dagger_{\downarrow}(-k)]^T,\label{eq:nambu}$$ the effective S action can be written $$S_{\mathrm{S}}^{\mathrm{eff}} = -\frac{1}{2\beta V} \sum_{k,k'} {\mathcal{C}}^\dagger(k) {\mathcal{G}}^{-1}(k,k') {\mathcal{C}}(k'),\label{eq:Seff_nambu}$$ where $$\begin{aligned}
{\mathcal{G}}^{-1}(k,&k') =\nonumber\\
& \begin{pmatrix}
G_0^{-1}(k)\delta_{k,k'} & \varphi(k-k')v({{\bf{k}}})i{\sigma}_y\\
-\varphi^\dagger(-k+k')v({{\bf{k}}})i{\sigma}_y & -[G_0^{-1}(-k)]^T\delta_{k,k'}
\end{pmatrix}.\label{eq:Ginv}\end{aligned}$$ Performing the functional integration over the fermionic fields, we arrive at the effective action for the bosonic fields $$S_\varphi = \frac{\beta V}{g} \sum_q \varphi^\dagger(q) \varphi(q) - \frac{1}{2}\operatorname{Tr}\ln (-{\mathcal{G}}^{-1}).$$ The additional factor $1/2$ in front of the trace is due to the change in integration measure when changing to the Nambu spinor notation, see Appendix \[sec:Nambu\_funcint\] and [[e.g.]{} ]{}Ref. for details.
Mean field theory {#sec:mean_field}
=================
Since $G_0^{-1}(i{\omega_n}, {{\bf{k}}})$ is still inversion symmetric in the diagonal basis (see below), we assume that the bosonic field $\varphi(q)$ is temporally and spatially homogeneous as in the regular BCS case. However, a recent study has shown that introducing in-plane magnetic fields in the TI breaks this symmetry and can make a Fulde-Ferrel[@Fulde1964] order parameter energetically more favorable in an $s$-wave S.[@Park2017] Calculating the matrix ${\mathcal{G}}(k)$ assuming a spatially homogeneous bosonic field $\phi(q) = \delta_{q,0}\Delta$, and defining the superconducting order parameter $\Delta({{\bf{k}}}) = \Delta \cdot v({{\bf{k}}})$, we get $$\mathcal{G}(k) = \begin{pmatrix}
G(k) & F(k)\\
F^\dagger(k) & -G^T(-k)
\end{pmatrix},$$ where to leading order in $t$
$$\begin{aligned}
G(k) = {}& -\frac{{\epsilon}_{{\bf{k}}}+ i{\omega_n}}{\xi_{{\bf{k}}}^2+{\omega_n}^2} - t^2\frac{({\epsilon}_{{\bf{k}}}+i {\omega_n})^2[(i{\omega_n}+{\mu_{\mathrm{TI}}})+{v_{\mathrm{F}}}{{\bf{k}}}\cdot{\bm{{\sigma}}}]}{(\xi_{{\bf{k}}}^2+{\omega_n}^2)^2[{v_{\mathrm{F}}}^2{{\bf{k}}}^2-(i{\omega_n}+{\mu_{\mathrm{TI}}})^2]} - t^2\frac{|\Delta({{\bf{k}}})|^2[(i{\omega_n}-{\mu_{\mathrm{TI}}})-{v_{\mathrm{F}}}{{\bf{k}}}\cdot{\bm{{\sigma}}}]}{(\xi_{{\bf{k}}}^2+{\omega_n}^2)^2[{v_{\mathrm{F}}}^2{{\bf{k}}}^2-(i{\omega_n}-{\mu_{\mathrm{TI}}})^2]},\\
F(k) ={}& \frac{\Delta({{\bf{k}}})}{\xi_{{\bf{k}}}^2 + {\omega_n}^2}\bigg\{1+2t^2\frac{({v_{\mathrm{F}}}^2{{\bf{k}}}^2-{\mu_{\mathrm{TI}}}^2-{\omega_n}^2){\epsilon}_{{\bf{k}}}{\mu_{\mathrm{TI}}}- {\omega_n}^2({v_{\mathrm{F}}}^2{{\bf{k}}}^2+{\mu_{\mathrm{TI}}}^2+{\omega_n}^2)}{(\xi_{{\bf{k}}}^2 + {\omega_n}^2)[({v_{\mathrm{F}}}|{{\bf{k}}}|-{\mu_{\mathrm{TI}}})^2+{\omega_n}^2][({v_{\mathrm{F}}}|{{\bf{k}}}|+{\mu_{\mathrm{TI}}})^2+{\omega_n}^2]}\nonumber\\
&+2t^2\frac{({v_{\mathrm{F}}}^2{{\bf{k}}}^2-{\mu_{\mathrm{TI}}}^2+{\omega_n}^2){\epsilon}_{{\bf{k}}}- 2{\omega_n}^2{\mu_{\mathrm{TI}}}}{(\xi_{{\bf{k}}}^2 + {\omega_n}^2)[({v_{\mathrm{F}}}|{{\bf{k}}}|-{\mu_{\mathrm{TI}}})^2+{\omega_n}^2][({v_{\mathrm{F}}}|{{\bf{k}}}|+{\mu_{\mathrm{TI}}})^2+{\omega_n}^2]}{v_{\mathrm{F}}}{{\bf{k}}}\cdot{\bm{{\sigma}}}\bigg\}i{\sigma}_y,\label{Fk}\end{aligned}$$
with ${\epsilon}_{{\bf{k}}}= {{\bf{k}}}^2/2m - \mu$ and $\xi_{{\bf{k}}}= \sqrt{{\epsilon}_{{\bf{k}}}^2 + |\Delta({{\bf{k}}})|^2}$. As mentioned above, the proximity-induced spin-orbit coupling leads to non-diagonal terms in $G(k)$. Moreover, $F(k)$ now has diagonal terms $\propto {{\bf{k}}}\cdot{\bm{{\sigma}}}i{\sigma}_y$, signaling that [$p$-wave ($f$-wave) triplet superconducting correlations are induced in the $s$-wave ($d$-wave) superconductor. This has been shown to be the case in $s$-wave superconductors]{} when the spin-degeneracy is lifted by spin-orbit coupling.[@Gorkov2001] A similar expression was found for the anomalous Green’s function on the TI side of an S-TI bilayer in Ref. . [The results in Ref. also suggest that odd-frequency triplet pairing could be induced in the S by including a magnetic exchange term ${{\bf{m}}}\cdot{\bm{{\sigma}}}$ in the TI Lagrangian.]{}
Gap equation
------------
While the above Green’s functions contain information about the correlations in the superconductor, the superconducting gap must be determined self-consistently. We first change to the basis which diagonalizes the non-superconducting normal inverse Green’s function $G_0^{-1}$. We find $G_{d,0}^{-1}(k) = P(k)G_{0}^{-1}(k)P^\dagger(k)$, where $G_{d,0}^{-1}(k) = \operatorname{diag}(G_{+,0}^{-1}(k), G_{-,0}^{-1}(k))$, with $$G_{\pm,0}^{-1}(k) = i{\omega_n}-{\epsilon}_{{\bf{k}}}- \frac{t^2}{i{\omega_n}+ {\mu_{\mathrm{TI}}}\mp {v_{\mathrm{F}}}|{{\bf{k}}}|}$$ and $$P(k) = \frac{1}{\sqrt{2}}\begin{pmatrix}
1 & e^{-i\phi_{{\bf{k}}}}\\
1 & -e^{-i\phi_{{\bf{k}}}}
\end{pmatrix}.$$ Here $\phi_{{\bf{k}}}$ is the angle of ${{\bf{k}}}$ relative the $k_x$ axis. $+$ $(-)$ here denotes the Green’s function for positive (negative) chirality states. Inverting $G_{d,0}^{-1}$ we find the Green’s functions $$G_{\pm,0}(k) = \frac{i{\omega_n}\mp{v_{\mathrm{F}}}|{{\bf{k}}}| +{\mu_{\mathrm{TI}}}}{[i{\omega_n}-{\epsilon}_\pm^+({{\bf{k}}})][i{\omega_n}-{\epsilon}_\pm^-({{\bf{k}}})]},$$ where $$\begin{aligned}
{\epsilon}_\alpha^\gamma({{\bf{k}}}) ={}& \frac{1}{2}\big[{\epsilon}_{{\bf{k}}}+ \alpha {v_{\mathrm{F}}}|{{\bf{k}}}| - {\mu_{\mathrm{TI}}}\nonumber\\*
&+ \gamma \sqrt{({\epsilon}_{{\bf{k}}}-\alpha{v_{\mathrm{F}}}|{{\bf{k}}}|+{\mu_{\mathrm{TI}}})^2 + 4t^2}\big],\label{eq:nonsc_bands}\end{aligned}$$ with $\alpha,\gamma = \pm 1$. The Green’s function has residues $$\begin{aligned}
w_\alpha^\gamma({{\bf{k}}}) = \frac{1}{2}+\frac{{\epsilon}_{{\bf{k}}}-\alpha {v_{\mathrm{F}}}|{{\bf{k}}}|+{\mu_{\mathrm{TI}}}}{2\gamma\sqrt{({\epsilon}_{{\bf{k}}}-\alpha{v_{\mathrm{F}}}|{{\bf{k}}}|+{\mu_{\mathrm{TI}}})^2+4t^2}}.\label{eq:residues}\end{aligned}$$
We next transform the entire inverse Green’s function ${\mathcal{G}}$ using ${\mathcal{G}}_d^{-1}(k) = \mathcal{P}(k){\mathcal{G}}^{-1}(k)\mathcal{P}^\dagger(k),$ where $$\mathcal{P}(k) = \begin{pmatrix}
P(k) & 0 \\
0 & P^*(-k)
\end{pmatrix},$$ which yields $${\mathcal{G}}_d^{-1}(k) = \begin{pmatrix}
G_{d,0}^{-1}(k) & -\Delta({{\bf{k}}}) e^{-i\phi_{{\bf{k}}}}{\sigma}_z \\
-\Delta^\dagger({{\bf{k}}}) e^{i\phi_{{\bf{k}}}}{\sigma}_z & -G_{d,0}^{-1}(-k)
\end{pmatrix}.$$ Hence the full Green’s function matrix for the superconductor is $${\mathcal{G}}_d(k) = \begin{pmatrix}
G_d(k) & F_d(k) \\
F_d^\dagger(k) & -G_d(-k)
\end{pmatrix},$$ where we have defined the $2\times2$ matrices $G_d(k) = \operatorname{diag}(G_+(k), G_-(k))$ and $F_d(k) = \operatorname{diag}(F_+(k), F_-(k))$, and Green’s functions
$$\begin{aligned}
G_\pm(k) &= \frac{[i{\omega_n}+ {\epsilon}_\pm^+({{\bf{k}}})][i{\omega_n}+ {\epsilon}_\pm^-({{\bf{k}}})][i{\omega_n}\mp {v_{\mathrm{F}}}|{{\bf{k}}}|+{\mu_{\mathrm{TI}}}]}{[i{\omega_n}-\xi_\pm^+({{\bf{k}}})][i{\omega_n}+\xi_\pm^+({{\bf{k}}})][i{\omega_n}-\xi_\pm^-({{\bf{k}}})][i{\omega_n}+\xi_\pm^-({{\bf{k}}})]}\label{Gdiag}\\
F_\pm(k) &= \pm\frac{\Delta({{\bf{k}}}) e^{-i\phi_{{\bf{k}}}}[(i{\omega_n})^2- (\pm {v_{\mathrm{F}}}|{{\bf{k}}}|-{\mu_{\mathrm{TI}}})^2]}{[i{\omega_n}-\xi_\pm^+({{\bf{k}}})][i{\omega_n}+\xi_\pm^+({{\bf{k}}})][i{\omega_n}-\xi_\pm^-({{\bf{k}}})][i{\omega_n}+\xi_\pm^-({{\bf{k}}})]}.\label{Fdiag}\end{aligned}$$
The eigenenergies of the system are now given by the poles in the above equation, where $$\begin{aligned}
\xi_\alpha^\gamma({{\bf{k}}}) = \frac{1}{\sqrt{2}}\Big\{\xi_{{\bf{k}}}^2 + (\alpha {v_{\mathrm{F}}}|{{\bf{k}}}| - {\mu_{\mathrm{TI}}})^2 + 2t^2 + \gamma \sqrt{[\xi_{{\bf{k}}}^2 - (\alpha {v_{\mathrm{F}}}|{{\bf{k}}}| - {\mu_{\mathrm{TI}}})^2]^2 + 4t^2[({\epsilon}_{{\bf{k}}}+ \alpha {v_{\mathrm{F}}}|{{\bf{k}}}| - {\mu_{\mathrm{TI}}})^2 + |\Delta({{\bf{k}}})|^2]}\Big\}^{1/2}.\end{aligned}$$
The gap equation for the amplitude $\Delta$ is found by requiring [$\frac{\delta S_\varphi}{\delta \Delta} = 0$,]{}[@Altland2010] which yields $$\Delta^\dagger =
-\frac{g}{2\beta V}\sum_k \operatorname{tr}F_d^\dagger(k)v({{\bf{k}}}){\sigma}_z e^{-i\phi_{{\bf{k}}}}.$$ Inserting the hermitian conjugate of Eq. (\[Fdiag\]) and performing the sum over Matsubara frequencies, we get the gap equation, $$\begin{aligned}
1 = {}& \frac{g}{4V} \sum_{{{\bf{k}}}} v({{\bf{k}}})^2\Big\{\frac{\xi_+^+({{\bf{k}}})^2-({v_{\mathrm{F}}}|{{\bf{k}}}|-{\mu_{\mathrm{TI}}})^2}{\xi_+^+({{\bf{k}}})[\xi_+^+({{\bf{k}}})^2-\xi_+^-({{\bf{k}}})^2]}\tanh\frac{\beta \xi_+^+({{\bf{k}}})}{2}\nonumber\\
&-\frac{\xi_+^-({{\bf{k}}})^2-({v_{\mathrm{F}}}|{{\bf{k}}}|-{\mu_{\mathrm{TI}}})^2}{\xi_+^-({{\bf{k}}})[\xi_+^+({{\bf{k}}})^2-\xi_+^-({{\bf{k}}})^2]}\tanh\frac{\beta \xi_+^-({{\bf{k}}})}{2}\nonumber\\
&+\frac{\xi_-^+({{\bf{k}}})^2-({v_{\mathrm{F}}}|{{\bf{k}}}|+{\mu_{\mathrm{TI}}})^2}{\xi_-^+({{\bf{k}}})[\xi_-^+({{\bf{k}}})^2-\xi_-^-({{\bf{k}}})^2]}\tanh\frac{\beta \xi_-^+({{\bf{k}}})}{2}\nonumber\\
&-\frac{\xi_-^-({{\bf{k}}})^2-({v_{\mathrm{F}}}|{{\bf{k}}}|+{\mu_{\mathrm{TI}}})^2}{\xi_-^-({{\bf{k}}})[\xi_-^+({{\bf{k}}})^2-\xi_-^-({{\bf{k}}})^2]}\tanh\frac{\beta \xi_-^-({{\bf{k}}})}{2}
\Big\}.\label{eq:gapeq}\end{aligned}$$ Setting $t=0$ simply yields the regular BCS gap equation, which results in a gap $\Delta_0 = 2\omega_D e^{-1/\lambda}$ in the $s$-wave case,[@Bardeen1957] where $\lambda = g D_0/V$ is a dimensionless coupling constant, and $D_0$ is the density of states at the Fermi level. $d$-wave pairing results in a slightly smaller gap for the same values for $\lambda$ and the cut-off frequencies, see Appendix \[sec:zerotempgap\] for details. For $t\neq 0$, the above equation can be expressed in terms of an energy integral over ${\epsilon}_{{\bf{k}}}$ using ${v_{\mathrm{F}}}|{{\bf{k}}}| = {v_{\mathrm{F}}}\sqrt{2m({\epsilon}_{{\bf{k}}}+ \mu)}$.
{width="\columnwidth"} {width="50.00000%"}
Results and discussion {#sec:results}
======================
From the expressions for the system eigenenergies in the non-superconducting case, Eq. (\[eq:nonsc\_bands\]) we see that the S and TI bands have hybridized, leading to avoided crossings. The effect of this hybridization is largest when the chemical potential of both the S and TI is tuned such that the Fermi momenta coincide, i.e. for ${\mu_{\mathrm{TI}}}= \pm\sqrt{2m{v_{\mathrm{F}}}^2\mu}$. [A possibly strong proximity effect should therefore be expected to occur in a region close to these values of ${\mu_{\mathrm{TI}}}$, the size of which increases with increased hopping $t$]{}. In the following we numerically solve the gap equations for both $s$- and $d$-wave superconductors for relevant parameter values.
s-wave pairing
--------------
Using numerical values $\mu \sim \SI{5}{\electronvolt}$, a cut-off corresponding to the Debye frequency, $\hbar\omega_\pm = \hbar\omega_D\sim\SI{25}{\milli \electronvolt}$[@Ashcroft1976], $\hbar^2/2m\sim \SI{40}{\milli \electronvolt \cdot \nano\meter \squared}$, $\hbar {v_{\mathrm{F}}}\sim \SI{300}{\milli \electronvolt\cdot \nano\meter}$,[@Brune2011; @Sochnikov2015] and $\lambda = 0.2$, we solve the gap equation in Eq. (\[eq:gapeq\]) for different values of $t$ and ${\mu_{\mathrm{TI}}}$ at $T=0$ for an $s$-wave superconductor. The results in Fig. \[fig:gap0\](a) show that the absolute value of the gap is not changed significantly due to the inverse proximity effect for small $t$, except for ${\mu_{\mathrm{TI}}}$ close to $\sqrt{2m{v_{\mathrm{F}}}^2\mu}$. Both for ${\mu_{\mathrm{TI}}}$ above and below this region, the inverse proximity effect is small, signifying that the disappearing gap in the region where the inverse proximity effect is strong cannot be simply related to the increasing density of states in the TI. [For increasing $t$, the region where superconductivity is suppressed increases quadratically with $t$, eventually leading to suppressed superconductivity also at ${\mu_{\mathrm{TI}}}=0$.]{}
The strong suppression of the order parameter can be understood from the fact that the pairing potential is attractive only when $|{{\bf{k}}}^2/2m -\mu|\le \omega_D$, [corresponding to wavevectors between $k_\pm \equiv \sqrt{2m(\mu \pm \omega_\pm)}$. This means that the Fermi wavevectors $k_F$ of the bands in Eq. (\[eq:nonsc\_bands\]), the value of $|{{\bf{k}}}|$ for which ${\epsilon}^\gamma_\alpha({{\bf{k}}}) = 0$, have to satisfy $k_-< k_F < k_+$]{} in order to contribute significantly to the integral in the gap equations and thus give a finite gap. This can be seen by comparing the left panels in Fig. \[fig:gap0\](b), where the upper left panel shows the integrand of the gap equation, Eq. (\[eq:gapeq\]), and the lower left panel plots [$k_F$ for]{} the bands in Eq. (\[eq:nonsc\_bands\]) as a function of ${\mu_{\mathrm{TI}}}$. The main contribution to the gap equation clearly comes from [the values $|{{\bf{k}}}| = k_F$.]{} From Fig. \[fig:gap0\](b) we also see that as ${\mu_{\mathrm{TI}}}$ approaches $\pm\sqrt{2m{v_{\mathrm{F}}}^2\mu}$, the value where the Fermi wavevectors for the bare the S and TI bands, [$k_F^S$ and $k_F^\mathrm{TI}({\mu_{\mathrm{TI}}})$]{} cross, the [wavevector]{} of one of the bands exceeds $k_+$ and thus does not contribute to the gap equation. Now there is only one non-degenerate band inside the relevant region, meaning that the density of states and thus $\lambda$ is halved compared to the $t=0$ case, where the band is doubly degenerate. Hence the resulting gap is suppressed to [$\Delta_0 e^{-1/\lambda} = 2\omega_De^{-2/\lambda}$]{}, in good agreement with the numerical results, as shown by the dashed line in the inset in Fig. \[fig:gap0\](a). This also means that the suppression is less severe for higher $\lambda$, which we have confirmed by numerical simulations.
For positive ${\mu_{\mathrm{TI}}}$, [the Fermi wavevector]{} in one band exits the integration [interval $[k_-,k_+]$]{} at ${\mu_{\mathrm{TI}}}={\mu_{\mathrm{TI}}}^{+,-}$, while a new band enters this region at ${\mu_{\mathrm{TI}}}={\mu_{\mathrm{TI}}}^{+,+}$, where we have defined $$\begin{aligned}
{\mu_{\mathrm{TI}}}^{\alpha,\pm}(t) = \alpha\sqrt{2m{v_{\mathrm{F}}}^2(\mu \mp \omega_D)} \pm \frac{t^2}{\omega_D}, \label{eq:muti_crit}\end{aligned}$$ see appendix \[sec:proximity\_criterion\] for details. A similar argument holds for negative ${\mu_{\mathrm{TI}}}$, and hence superconductivity is strongly suppressed for $$\begin{aligned}
{\mu_{\mathrm{TI}}}^{\alpha,-} < {\mu_{\mathrm{TI}}}< {\mu_{\mathrm{TI}}}^{\alpha,+},\label{eq:prox_crit}\end{aligned}$$ indicated by the dashed and dotted lines in Fig. \[fig:gap0\]. If the hopping parameter is large enough, $t^2 > \omega_D \sqrt{2m{v_{\mathrm{F}}}^2(\mu\mp\omega_D)}\equiv (t_\mp)^2$, ${\mu_{\mathrm{TI}}}^{-,+}$ and ${\mu_{\mathrm{TI}}}^{+,-}$ change sign. Hence, for $|t|>|t_+|>|t_-|$ and ${\mu_{\mathrm{TI}}}^{+,-}<{\mu_{\mathrm{TI}}}<{\mu_{\mathrm{TI}}}^{-,+}$, no bands have a Fermi [wavevector between $k_-$ and $k_+$]{}, resulting in $\Delta=0$, as seen for $t\approx\SI{0.3}{eV}$ and low ${\mu_{\mathrm{TI}}}$ in Fig. \[fig:gap0\]. Since $\mu \gg \omega_D$, all results are close to symmetric about ${\mu_{\mathrm{TI}}}=0$, as seen in Fig. \[fig:gap0\](b).
In order for strong suppression to occur for some value of ${\mu_{\mathrm{TI}}}$, we must require ${\mu_{\mathrm{TI}}}^{\alpha,-} < {\mu_{\mathrm{TI}}}^{\alpha,+}$. For $\alpha=-1$ this always holds, while for $\alpha = +1$ we get a lower limit for $t^2$, $$\begin{aligned}
t^2 > \omega_D\left[\sqrt{2m{v_{\mathrm{F}}}^2(\mu+\omega_D)} - \sqrt{2m{v_{\mathrm{F}}}^2(\mu-\omega_D)}\right].\end{aligned}$$ For conventional $s$-wave superconductors $\mu \gg \omega_D$, meaning strong suppression can occur even at low values of $t$, though for TI chemical potentials close to $\pm\sqrt{2m{v_{\mathrm{F}}}^2\mu}$.
While this result is strictly only valid in the limit of an atomically thin superconductor, we expect that this effect in principle could reduce the zero temperature gap and thus also reduce the critical temperature in superconducting thin films. [However, f]{}or typical parameter values in TIs and $s$-wave superconductors, the values of ${\mu_{\mathrm{TI}}}$ where superconductivity vanishes is inaccessible, tuning ${\mu_{\mathrm{TI}}}$ by several eV would place the Fermi level inside the bulk bands of the TI, where our model is no longer valid. The only exception from this is when $|t| \gtrsim |t_-|$, when superconductivity is suppressed even at ${\mu_{\mathrm{TI}}}=0$. The fact that no strong inverse proximity effect has been observed, e.g. in Ref. , might indicate that the coupling constant $t$ is below this limit, meaning that an unphysical high chemical potential is needed in the TI to observe the vanishing of superconductivity. Since conventional $s$-wave superconductors have high Fermi energies, it might not be possible to reach the parameter regions where superconductivity vanishes, unless [the chemical potential in the S can be lowered significantly]{}, the Fermi velocity of the TI is [lowered by renormalization]{}, as was proposed in Ref. , or the coupling between the layers can be increased beyond $t_-$. However, [as we show below,]{} similar effects [are]{} present also for unconventional, high-$T_c$ superconductors, for which the Fermi energy is lower. Examples of such superconductors would be the high-$T_c$ cuprates and the heavy-fermion superconductors.[^1]
{width="\columnwidth"} {width="50.00000%"}
d-wave pairing
--------------
Using a much lower chemical potential in the S, $\mu \sim \SI{35}{\milli \electronvolt}$,[@Gerbstein1989] and an upper cut-off frequency comparable to the spin fluctuation energy in the high-$T_c$ cuprates, $\omega_+ \sim$ ,[@Moriya1990; @Monthoux1992; @Nagaosa1997] $\omega_-=\mu$, and parameters otherwise as for the $s$-wave case, we solve the gap equations for a $d$-wave superconductor. First of all, the effect of the $d$-wave gap structure, compared to an $s$-wave gap, is an overall change in scaling, just as is the case for $\Delta_0$ (see Appendix \[sec:zerotempgap\]). Hence, the results for $\Delta^{\textrm{$s$-wave}}/\Delta_0^{\textrm{$s$-wave}}$ are identical to $\Delta^{\textrm{$d$-wave}}/\Delta_0^{\textrm{$d$-wave}}$ when using the same parameters, and we have therefore solved the numerically more efficient $s$-wave gap equations with parameters valid for high-$T_c$ superconductors.
Fig. \[fig:gap0\_dwave\](a) shows the numerical results for the normalized gap as a function of ${\mu_{\mathrm{TI}}}$ and $t$. The most prominent difference compared to the results in Fig. \[fig:gap0\] is that the results are no longer symmetric about ${\mu_{\mathrm{TI}}}=0$, which can be understood from the fact that $\omega_\pm$ is of the same order of magnitude or larger than $\mu$. Due to the anti-crossing of the Fermi [wavevectors]{} at negative ${\mu_{\mathrm{TI}}}$, there is only one Fermi [wavevector between $k_-$ and $k_+$ for]{} ${\mu_{\mathrm{TI}}}^- < {\mu_{\mathrm{TI}}}< {\mu_{\mathrm{TI}}}^+$ (dashed lines in Fig. \[fig:gap0\_dwave\](a)), leading to strong suppression for negative ${\mu_{\mathrm{TI}}}$. This is illustrated in Fig. \[fig:gap0\_dwave\](b), where we plot the Fermi [wavevectors]{} of the bands together with the normalized gap as a function of ${\mu_{\mathrm{TI}}}$ for different values of $t$. The figure also shows how the regions of strong mixing between the bands increases with increasing $t$. Interestingly, the suppression of the gap is preceded by an increased $\Delta$ at ${\mu_{\mathrm{TI}}}^\pm$, due to the bending of the Fermi [wavevectors]{} away from the crossing point of [$k_F^S$ and $k_F^\mathrm{TI}({\mu_{\mathrm{TI}}})$]{},[ which leads to an increase in the density of states at the Fermi level. This is illustrated in Fig. \[fig:bands\](b), where for TI chemical potentials ${\mu_{\mathrm{TI}}}^\pm$ the bands have a minimum (maximum) at the Fermi level, resulting in high densities of states. The difference in the gap enhancement between ${\mu_{\mathrm{TI}}}^+$ and ${\mu_{\mathrm{TI}}}^-$ is due to the combined effects of different spectral weights, indicated by the line widths in Fig. \[fig:bands\](b), and the size of the Fermi surface, leading to a net larger increase in $|\Delta|$ at ${\mu_{\mathrm{TI}}}^-$.]{} In the small $t$ limit, we find the approximate expressions $$\begin{aligned}
{\mu_{\mathrm{TI}}}^\pm = - \sqrt{2m{v_{\mathrm{F}}}^2\mu} \pm 2\left(\frac{m{v_{\mathrm{F}}}^2}{2\mu}\right)^{1/4}t + \frac{1}{4\mu}t^2. \label{eq:mupm}\end{aligned}$$ These lines are plotted in Fig. \[fig:gap0\_dwave\](a) (dotted lines) together with the exact numerical solutions (dashed lines), see Appendix \[sec:proximity\_criterion\] for details. [This increase in $|\Delta|$ is not due to the the $d$-wave symmetry, and should therefore be present for ${\mu_{\mathrm{TI}}}= {\mu_{\mathrm{TI}}}^\pm$ whenever the interval $[k_-, k_+]$ includes either of the points $k_F^S \pm |\delta k_F|$, where $\delta k_F$ is defined in Eq. (\[eq:delta\_k\]).]{}
For positive ${\mu_{\mathrm{TI}}}$ there is a small reduction in $\Delta$ close to ${\mu_{\mathrm{TI}}}= \sqrt{2m{v_{\mathrm{F}}}^2\mu}$, even though there are three bands with [$k_F \in [k_-, k_+]$]{}. However, since the numerator of each term in the gap equation Eq. (\[eq:gapeq\]) can be written $\xi^\pm_\alpha({{\bf{k}}})^2 - (\alpha{v_{\mathrm{F}}}|{{\bf{k}}}|-{\mu_{\mathrm{TI}}})^2$, regions where $\xi^\pm_\alpha({{\bf{k}}})$ are similar to the bare TI bands contribute little to the gap equations, resulting in a small decrease of $\Delta$.
The effect of using a lower upper cut-off in the solution of the gap equations is also shown in Fig. \[fig:gap0\_dwave\]. Comparing the $\omega_+ = \SI{0.15}{eV}$ and $\SI{0.04}{eV}$ lines, we see that for high $t$, the mixing of the S and TI bands is still significant at [$k_F=k_+$,]{} leading to abrupt changes in $\Delta$. For the negative ${\mu_{\mathrm{TI}}}$ the main effect of lowering the upper cut-off $\omega_+$ is a further increase of $\Delta$ at ${\mu_{\mathrm{TI}}}^\pm$.
{width="\textwidth"}
From the above results, it is clear that a strong suppression of the gap is more probable in S-TI bilayers consisting of a high-$T_c$ S, where both the chemical potential [$-\sqrt{2m{v_{\mathrm{F}}}^2\mu}$ corresponding to $k_F^S = k_F^\mathrm{TI}({\mu_{\mathrm{TI}}})$]{}, and the hopping strength needed for strong suppression at ${\mu_{\mathrm{TI}}}=0$ is much lower. Hence, we may expect a strong inverse proximity effect in such systems, with a strength determined by $\lambda$, as illustrated in Fig. \[fig:lambda\_dep\] for both the $s$- and $d$-wave case. Increasing $\lambda$ leads to a reduced suppression of the gap, consistent with the fact that the superconducting state is more robust for higher $\lambda$. [For the $s$-wave case, the suppression is proportional to $e^{-1/\lambda}$. This holds only approximately for the $d$-wave case due to other factors than Fermi level crossings affecting the suppression, such as changes in the spectral densities at the Fermi level and changes in the size of the Fermi surface (see Fig. \[fig:bands\]), effects which are small in the $s$-wave case.]{} From the results in Fig. \[fig:gap0\_dwave\] we also see that it should be possible to change $\Delta$ by several orders of magnitude by small changes in ${\mu_{\mathrm{TI}}}$, again depending on the value of $\lambda$ as illustrated in Fig. \[fig:lambda\_dep\].
![\[fig:lambda\_dep\] The figure shows how the dimensionless coupling constant $\lambda$ affects the suppression of the superconducting gap for $s$-wave S with $t=\SI{0.2}{eV}$ (top) and $d$-wave S with $t=\SI{0.05}{eV}$ and $\omega_+ = \SI{0.15}{eV}$ (bottom). Increasing $\lambda$ makes the superconducting state more robust, reducing both the suppression of $\Delta$, and also the increase in $\Delta$ at ${\mu_{\mathrm{TI}}}^\pm$ in the $d$-wave case.](L_dependence.pdf){width="\columnwidth"}
Summary {#sec:summary}
=======
We have theoretically studied the inverse superconducting proximity effect between a thin $s$-wave or $d$-wave superconductor and a topological insulator. Using a field-theoretical approach, we have found that in both cases there are regions in parameter space where the inverse proximity effect is strong, leading to a strong suppression of the gap approximately proportional to $e^{-1/\lambda}$. The suppression can be related to the hybridization of the TI and S bands, and the large degree of mixing which occurs when the Fermi wavevectors of the S and TI coincide for chemical potential ${\mu_{\mathrm{TI}}}=\pm\sqrt{2m{v_{\mathrm{F}}}^2\mu}$. A larger value of $\lambda$ results in a more robust superconducting state, and hence less suppression.
For parameter values relevant for $s$-wave superconductors, the interval of suppression grows quadratically with the hopping $t$, and eventually leads to strong suppression even at ${\mu_{\mathrm{TI}}}= 0$. However, since there have been no experimental indications of a strong inverse proximity effect, we must conclude that the hopping is too weak to lead to suppression for experimentally accessible values of ${\mu_{\mathrm{TI}}}$. Neglecting the inverse proximity effect regarding the stability of the superconducting order therefore seems to be a good approximation for conventional $s$-wave superconductors.
A similar effect of suppressed superconductivity is also present for $d$-wave superconductors with parameter values relevant for the high-$T_c$ superconductors. In this case the strong suppression is found for TI chemical potentials close to $-\sqrt{2m{v_{\mathrm{F}}}^2\mu}$, where the interval of strong suppression of the gap grows approximately linearly with $t$. Since the Fermi energy $\mu$ is much lower for high-$T_c$ superconductors, both the magnitude of the chemical potential $-\sqrt{2m{v_{\mathrm{F}}}^2\mu}$, and the hopping strength needed for strong suppression at ${\mu_{\mathrm{TI}}}=0$ is much lower, making a strong inverse proximity effect more probable in such systems. In contrast to the $s$-wave case, the region of strong suppression was preceded by an increase in $\Delta$ above $\Delta_0$. [This is, however, not a consequence of the pairing symmetry, but rather the difference in system parameters. For large enough cut-off frequencies, the integration region will include a band minimum/maximum just touching the Fermi level, leading to a large increase in the density of states, and thus increased gap.]{} We also find that the spin-triplet $p$-wave ($f$-wave) superconducting correlations are induced in the $s$-wave ($d$-wave) S due to the proximity-induced spin-orbit coupling. Possible further work could include breaking the translation symmetry in the $x$ or $y$ direction and probing the density of states normal to the z-axis, possibly revealing signatures of $p$-wave or $f$-wave pairing. Moreover, it could be interesting to study the spatial variation of the order parameter in a superconductor with finite thickness.
J. L. and A. S. acknowledge funding from the Research Council of Norway Center of Excellence Grant Number 262633, Center for Quantum Spintronics. A. S. and H. G. H. also acknowledge funding from the Research Council of Norway Grant Number 250985. J. L. acknowledges funding from Research Council of Norway Grant No. 240806. J. L. and M. A. also acknowledge funding from the NV-faculty at the Norwegian University of Science and Technology. H. G. H. thanks F. N. Krohg for useful discussions.
Criteria for strong proximity effect {#sec:proximity_criterion}
====================================
For superconductivity to occur, the Fermi wavevector of at least one of the bands has to lie within the interval of attractive pairing, which for $s$-wave superconductors is $\sqrt{2m(\mu-\omega_D)} < |{{\bf{k}}}| < \sqrt{2m(\mu + \omega_D)}$. We find the Fermi wavevector of the energy bands by setting ${\epsilon}_\alpha^\gamma({{\bf{k}}}) = 0$, which yields the equation $$\begin{aligned}
\left[\alpha{v_{\mathrm{F}}}|{{\bf{k}}}|-{\mu_{\mathrm{TI}}}\right]{\epsilon}_{{\bf{k}}}- t^2 = 0.\label{eq:fermi_lvl_condition}\end{aligned}$$ Inserting $|{{\bf{k}}}| = k_ \pm$ we get the value of ${\mu_{\mathrm{TI}}}$ for which the Fermi wavevector of a band enters or leaves the interval of attractive pairing, $$\begin{aligned}
{\mu_{\mathrm{TI}}}^{\alpha,\pm}(t) = \alpha\sqrt{2m{v_{\mathrm{F}}}^2(\mu \mp \omega_D)} \pm \frac{t^2}{\omega_D}.\end{aligned}$$ The Fermi wavevectors of the bands ${\epsilon}_\alpha^-({{\bf{k}}})$ exceed $k_+$ at ${\mu_{\mathrm{TI}}}^{\alpha,-}$, while the Fermi wavevectors of ${\epsilon}_\alpha^+({{\bf{k}}})$ enter the interval $[k_-, k_+]$ at ${\mu_{\mathrm{TI}}}^{\alpha,+}$. ${\mu_{\mathrm{TI}}}^{+, +}$ (${\mu_{\mathrm{TI}}}^{-, -}$) is always positive (negative), while ${\mu_{\mathrm{TI}}}^{+, -}$ and ${\mu_{\mathrm{TI}}}^{-, +}$ change sign when $t^2 > \omega_D\sqrt{2m{v_{\mathrm{F}}}^2(\mu +\omega_D)}\equiv (t_0^+)^2$ and $t^2> \omega_D \sqrt{2m{v_{\mathrm{F}}}^2(\mu-\omega_D)}\equiv (t_0^-)^2$ respectively, where $|t_0^+| > |t_0^-|$.
Hence we have strong suppression when $$\begin{aligned}
{\mu_{\mathrm{TI}}}^{\alpha,-}<{\mu_{\mathrm{TI}}}<{\mu_{\mathrm{TI}}}^{\alpha,+},\end{aligned}$$ which for $\alpha=+1$ requires $$t^2 > \omega\left[\sqrt{2m{v_{\mathrm{F}}}^2(\mu+\omega_D)} - \sqrt{2m{v_{\mathrm{F}}}^2(\mu-\omega)}\right].$$ Moreover, for $|t|>|t_+|$ and ${\mu_{\mathrm{TI}}}^{+,-}<{\mu_{\mathrm{TI}}}<{\mu_{\mathrm{TI}}}^{-,+}$ no bands have a Fermi [wavevector]{} inside the relevant interval, and the gap is zero.
For the $d$-wave S we find an increase in the gap function for certain values of ${\mu_{\mathrm{TI}}}$. An increase in the gap would occur in regions where the Fermi wavevectors of two bands approach each other and finally coincide as a function of ${\mu_{\mathrm{TI}}}$, resulting in a [region of closely spaced Fermi wavevectors]{}. This can be seen to happen in Fig. \[fig:gap0\_dwave\](b). To find the value of ${\mu_{\mathrm{TI}}}$ corresponding to the increase in $\Delta$ we find the local minima of [ $$\begin{aligned}
{\mu_{\mathrm{TI}}}(k_F) = \alpha{v_{\mathrm{F}}}k_F - \frac{t^2}{{\epsilon}_{k_F}}\label{eq:muti_eF}\end{aligned}$$ by requiring ${\partial}_{k_F} {\mu_{\mathrm{TI}}}(k_F) = 0$, from which we get the equation for ${k_F}$ $$\begin{aligned}
\alpha {v_{\mathrm{F}}}+ \frac{t^2 k_F}{m {\epsilon}_{k_F}^2} = 0.\end{aligned}$$ Solving this equation numerically with $\alpha=-1$ and inserting the results into Eq. (\[eq:muti\_eF\]) yields the dashed lines in Fig. \[fig:gap0\_dwave\], in good agreement with the numerical results of the gap equation. To get an approximate analytical expression, we assume that $k_F = k_F^S + \delta k_F$, where $\delta k_F \ll k_F^S$, which is valid for sufficiently small $t$. Neglecting terms of $\mathcal{O}(\delta k_F^3)$ and higher, we get $$\begin{aligned}
\delta k_F^2 + \frac{t^2m}{\alpha {v_{\mathrm{F}}}k_F^S} + \frac{t^2m}{\alpha {v_{\mathrm{F}}}(k_F^S)^2}\delta k_F = 0.\end{aligned}$$ Neglecting the last term yields, effectively keeping terms up to $\mathcal{O}(t^2)$, results in $$\begin{aligned}
\delta k_F = {}& \pm \sqrt{-\frac{1}{\alpha}}\left(\frac{m}{2{v_{\mathrm{F}}}^2\mu}\right)^{1/4}t,\label{eq:delta_k}\end{aligned}$$ from which it is clear that we only have solutions for $\alpha=-1$. Inserting this expression into Eq. (\[eq:muti\_eF\]), we get to $\mathcal{O}(t^2)$ $$\begin{aligned}
{\mu_{\mathrm{TI}}}^\pm \approx - \sqrt{2m{v_{\mathrm{F}}}^2\mu} \pm 2\left(\frac{m{v_{\mathrm{F}}}^2}{2\mu}\right)^{1/4}t + \frac{1}{4\mu}t^2.\end{aligned}$$ ]{} This result is plotted as dotted lines in Fig. \[fig:gap0\_dwave\](a), and is in good agreement with the exact numerical results for small $t$. For ${\mu_{\mathrm{TI}}}^- < {\mu_{\mathrm{TI}}}< {\mu_{\mathrm{TI}}}^+$, there is only one Fermi [wavevector]{} in the integration region, leading to a suppressed gap.
Functional integral in Nambu spinor notation {#sec:Nambu_funcint}
============================================
We begin by considering the Gaussian integral over Grassmann variables,[@Wegner2016] $$\begin{aligned}
I &= \left(\prod_i\int {\mathrm{d}}a_i\right) ~ e^{-\frac{1}{2}\sum_{i,j} a_iM_{ij}a_j} \nonumber\\
&= \left(\prod_i\int{\mathrm{d}}a_i\right)\prod_{i,j}(1 - \frac{1}{2}a_iM_{ij}a_j) = \operatorname{Pf}\left(\frac{M-M^T}{2}\right)\label{eq:Pfaffian},\end{aligned}$$ where $\operatorname{Pf}((M-M^T)/2)$ is the Pfaffian of the antisymmetric part of $M$, where $\operatorname{Pf}(A)^2 = \det(A)$. As an example we consider only two variables, $a_1$ and $a_2$. In this case, terms containing $M_{ii}$ disappear, since $a_i^2 = 0$, as do second order terms in $M$. For the above integral we therefore get $$\begin{aligned}
I &= \int {\mathrm{d}}a_1 {\mathrm{d}}a_2 ~ \frac{1}{2}(-a_1 M_{12} a_2 - a_2 M_{21} a_1) = \frac{M_{12} - M_{21}}{2} \nonumber\\
&= \sqrt{\det \frac{M - M^T}{2}} = \sqrt{\det M^A} = \operatorname{Pf}(M^A).\end{aligned}$$ Here, $M^A$ is the anti-symmetric part of $M$.
Applying this to the problem of integrating $\exp(-S_{\mathrm{S}}^{\mathrm{eff}})$, we first write the action in terms of the Nambu spinor ${\mathcal{C}}$:
$$\begin{aligned}
S_{\mathrm{S}}^{\mathrm{eff}} &= -\frac{1}{\beta V}\sum_{k,k'} {\mathcal{C}}^T(-k) \begin{pmatrix}
\varphi^\dagger(k'-k)\frac{{\sigma}_x-i{\sigma}_y}{2} & 0\\
{\mathcal{G}}_0^{-1}(k)\delta_{k,k'} & \varphi(k-k')\frac{{\sigma}_x+i{\sigma}_y}{2}
\end{pmatrix}{\mathcal{C}}(k') \equiv -\frac{1}{2\beta V}\sum_{k,k'} {\mathcal{C}}^T(-k) A(k,k') {\mathcal{C}}(k') \nonumber\\
&=-\frac{1}{\beta V}\sum_{k,k'} {\mathcal{C}}^T(k) \begin{pmatrix}
-\varphi^\dagger(k-k')\frac{{\sigma}_x+i{\sigma}_y}{2} & -[{\mathcal{G}}_0^{-1}(k)]^T\delta_{k,k'}\\
0 & -\varphi(k'-k)\frac{{\sigma}_x-i{\sigma}_y}{2}
\end{pmatrix}{\mathcal{C}}(-k')\equiv -\frac{1}{2\beta V}\sum_{k,k'} {\mathcal{C}}^T(k) [-A(k',k)]^T {\mathcal{C}}(-k').\nonumber\end{aligned}$$
Combining these two expressions, we get $$\begin{aligned}
S_{\mathrm{S}}^{\mathrm{eff}} &= -\frac{1}{2\beta V}\sum_{k,k'} {\mathcal{C}}^T(-k) \begin{pmatrix}
-\varphi^\dagger(k'-k)i{\sigma}_y & -[{\mathcal{G}}_0^{-1}(-k)]^T\delta_{k,k'}\\
{\mathcal{G}}_0^{-1}(k)\delta_{k,k'} & \varphi(k-k')i{\sigma}_y
\end{pmatrix}{\mathcal{C}}(k') \nonumber\\
&= -\frac{1}{2\beta V} \sum_{k,k'} {\mathcal{C}}^T(-k) \frac{A(k,k') - A^T(-k',-k)}{2} {\mathcal{C}}(k') = -\frac{1}{2\beta V} \sum_{k,k'} {\mathcal{C}}^T(-k) A^A(k,k') {\mathcal{C}}(k'),\label{eq:Seff_CT}\end{aligned}$$ where $A^A(k,k')$ denotes the anti-symmetric part of $A$. [This is exactly equal to Eq. (\[eq:Seff\_nambu\]), as can be seen by the following manipulations. For notational simplicity we use the $2$-vector notation]{} $${\mathcal{C}}(k) = \begin{pmatrix}
c(k)\\
c^*(-k)
\end{pmatrix},$$ [i.e $[{\mathcal{C}}(k)]_1 = c(k),~[{\mathcal{C}}(k)]_2 = c^*(-k)$. Hence the matrix multiplication in Eq. (\[eq:Seff\_CT\]) can be written]{} $$\begin{aligned}
\sum_{ij} [{\mathcal{C}}^T(-k)]_i [A^A(k,k')]_{ij} [{\mathcal{C}}(k')]_j =
{}&-[{\mathcal{C}}^T(-k)]_1 \varphi^\dagger(k'-k)i{\sigma}_y [{\mathcal{C}}(k')]_1
- [{\mathcal{C}}^T(-k)]_1 [{\mathcal{G}}_0^{-1}(-k)\delta_{k,k'}]^T [{\mathcal{C}}(k')]_2\nonumber\\
&+[{\mathcal{C}}^T(-k)]_2 {\mathcal{G}}_0^{-1}(-k)\delta_{k,k'} [{\mathcal{C}}(k')]_1
+[{\mathcal{C}}^T(-k)]_2 \varphi(k-k')i{\sigma}_y [{\mathcal{C}}(k')]_2.\end{aligned}$$ [We use the fact that $[{\mathcal{C}}^\dagger(k)]_1 = [{\mathcal{C}}^T(-k)]_2$ and $[{\mathcal{C}}^\dagger(k)]_2 = [{\mathcal{C}}^T(-k)]_1$, and relate the remaining factors to the elements of ${\mathcal{G}}^{-1}(k,k')$ in Eq. (\[eq:Ginv\]),]{} $$\begin{aligned}
C^T(-k) A^A(k,k') C(k') = {}&[{\mathcal{C}}^\dagger(k)]_2[{\mathcal{G}}^{-1}(k,k')]_{21}[{\mathcal{C}}(k')]_1 + [{\mathcal{C}}^\dagger(k)]_2[{\mathcal{G}}^{-1}(k,k')]_{22}[{\mathcal{C}}(k')]_2 \nonumber\\
&+[{\mathcal{C}}^\dagger(k)]_1[{\mathcal{G}}^{-1}(k,k')]_{11}[{\mathcal{C}}(k')]_1 + [{\mathcal{C}}^\dagger(k)]_1[{\mathcal{G}}^{-1}(k,k')]_{12}[{\mathcal{C}}(k')]_2\nonumber\\
={} & {\mathcal{C}}^\dagger(k) {\mathcal{G}}^{-1}(k,k'){\mathcal{C}}(k'),\end{aligned}$$
which shows that Eq. (\[eq:Seff\_CT\]) and Eq. (\[eq:Seff\_nambu\]) are equivalent. Using Eq. (\[eq:Pfaffian\]), the functional integral of the action in Eq. (\[eq:Seff\_CT\]) results in $$Z = \int {\mathcal{D}}c^\dagger {\mathcal{D}}c ~ e^{-S_{\mathrm{S}}^{\mathrm{eff}}} = \sqrt{\det\left[-A^A\right]},$$ where we have neglected various numerical constants. By interchanging an even number of rows, it can be shown that $A^A(k,k') \rightarrow {\mathcal{G}}^{-1}(k,k')$, and since the determinant is invariant under an even number interchanges, we find[@Krohg2018] $$Z = e^{\frac{1}{2}\operatorname{Tr}\ln (-{\mathcal{G}}^{-1})}.$$
Zero temperature gap for t=0 {#sec:zerotempgap}
============================
When $t=0$, the gap equation, Eq. (\[eq:gapeq\]), reduces to $$\begin{aligned}
1 = \frac{g}{2V}\sum_{{\bf{k}}}\frac{v^2({{\bf{k}}})}{\sqrt{{\epsilon}_{{\bf{k}}}+ |\Delta_0({{\bf{k}}})|^2}}\end{aligned}$$ in the zero temperature limit. Transforming this to an integration over $\phi_{{\bf{k}}}$ and energy, we get $$\begin{aligned}
1 &= \frac{\lambda}{2} \int_{-\omega_-}^{\omega_+}{\mathrm{d}}{\epsilon}\int_0^{2\pi}\frac{{\mathrm{d}}\phi_{{\bf{k}}}}{2\pi} \frac{v^2(\phi_{{\bf{k}}})}{\sqrt{{\epsilon}+ |\Delta_0(\phi_{{\bf{k}}})|^2}},\end{aligned}$$ where $\omega_\pm$ are positive. Performing the energy integral we get $$\begin{aligned}
1 &= \frac{\lambda}{2} \int_0^{2\pi}\frac{{\mathrm{d}}\phi_{{\bf{k}}}}{2\pi}v^2(\phi_{{\bf{k}}}) \ln\frac{\sqrt{|\Delta_0(\phi_{{\bf{k}}})|^2 + \omega_+^2} + \omega_+}{\sqrt{|\Delta_0(\phi_{{\bf{k}}})|^2 + \omega_-^2}-\omega_-^2}\nonumber\\
&\approx \frac{\lambda}{2} \int_0^{2\pi}\frac{{\mathrm{d}}\phi_{{\bf{k}}}}{2\pi}v^2(\phi_{{\bf{k}}}) \left[\ln\frac{4\omega_-\omega_+}{\Delta_0^2} - 2\ln|v(\phi_{{\bf{k}}})|\right],\end{aligned}$$ where we in the last line have assumed that the gap is small compared to the cut-off energy. For an $s$-wave superconductor $v(\phi_{{\bf{k}}}) = 1$, and we get simply $\Delta_0 = 2\sqrt{\omega_-\omega_+} e^{-1/\lambda}$. For $d$-wave pairing we can instead write the gap as $$\begin{aligned}
\Delta_0 = 2\sqrt{\omega_-\omega_+} e^{-\frac{1}{\lambda} - I},\end{aligned}$$ where we have defined the integral $$\begin{aligned}
I = \int_0^{2\pi} \frac{{\mathrm{d}}\phi_{{\bf{k}}}}{2\pi} v^2(\phi_{{\bf{k}}})\ln|v(\phi_{{\bf{k}}})| = \frac{1-\ln2}{2} \approx 0.153426.\end{aligned}$$ Hence, the maximum $d$-wave gap-amplitude is marginally smaller than the $s$-wave gap for the same values of $\lambda$ and $\omega_\pm$.
Numerical integration procedures
================================
When solving the gap equation numerically, the ${{\bf{k}}}$ sum is rewritten in terms of an energy integral over ${\epsilon}_{{\bf{k}}}$ and an integral over $\phi_{{\bf{k}}}$, which in the $s$-wave case is simply equal to $2\pi$. In the $s$-wave case we therefore only have to perform the energy integral for energies in the interval $[-\omega_-, \omega_+]$, in our case using Python and the implementation `trapz` of the trapezoidal method in the `scipy` library. In the $d$-wave case, we use the `quadpy` library’s implementation of the numerical integration method in Ref. when calculating the 2D integral in the ${\epsilon}_{{\bf{k}}}- \phi_{{\bf{k}}}$ plane.
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|
---
author:
- 'M.Hitschfeld'
- 'M.Aravena'
- 'C.Kramer'
- 'F.Bertoldi'
- 'J.Stutzki'
- 'F.Bensch'
- 'L.Bronfman'
- 'M.Cubick'
- 'M.Fujishita'
- 'Y.Fukui'
- 'U.U.Graf'
- 'N.Honingh'
- 'S.Ito'
- 'H.Jakob'
- 'K.Jacobs'
- 'U.Klein'
- 'B.-C.Koo'
- 'J.May'
- 'M.Miller'
- 'Y.Miyamoto'
- 'N.Mizuno'
- 'T.Onishi'
- 'Y.-S.Park'
- 'J.L.Pineda'
- 'D.Rabanus'
- 'M.Röllig'
- 'H.Sasago'
- 'R.Schieder'
- 'R.Simon'
- 'K.Sun'
- 'N.Volgenau'
- 'H.Yamamoto'
- 'Y.Yonekura'
bibliography:
- 'p\_nanten\_galaxies.bib'
date: 'Received / Accepted '
title: '$^{12}$CO 4–3 and $[$CI$]$ 1–0 at the centers of NGC4945 and Circinus'
---
[ Studying molecular gas in the central regions of the star burst galaxies NGC4945 and Circinus enables us to characterize the physical conditions and compare them to previous local and high-z studies.]{} [We estimate temperature, molecular density and column densities of CO and atomic carbon. Using model predictions we give a range of estimated CO/C abundance ratios.]{} [Using the new NANTEN2 4m sub-millimeter telescope in Pampa La Bola, Chile, we observed for the first time CO 4–3 and [\[\]]{} $^3P_1-^3P_0$ at the centers of both galaxies at linear scale of 682 pc and 732 pc respectively. We compute the cooling curves of $^{12}$CO and $^{13}$CO using radiative transfer models and estimate the physical conditions of CO and $[$CI$]$.]{} [The centers of NGC4945 and Circinus are very [\[\]]{} bright objects, exhibiting [\[\]]{} $^3P_1-^3P_0$ luminosities of 91 and 67Kkms$^{-1}$kpc$^{2}$, respectively. The [\[\]]{} $^3P_1-^3P_0$/CO 4–3 ratio of integrated intensities are large at 1.2 in NGC4945 and 2.8 in Circinus. Combining previous CO $J$= 1–0 , 2–1 and 3–2 and $^{13}$CO $J$= 1–0 , 2–1 studies with our new observations, the radiative transfer calculations give a range of densities, $n(\rm H_{2})=10^{3}-3 \times 10^{4}$cm$^{-3}$, and a wide range of kinetic temperatures, $T_{\rm kin}= 20-100$K, depending on the density. To discuss the degeneracy in density and temperature, we study two representative solutions. In both galaxies the estimated total $[$CI$]$ cooling intensity is stronger by factors of $\sim 1-3$ compared to the total CO cooling intensity. The CO/C abundance ratios are 0.2-2, similar to values found in Galactic translucent clouds. ]{} [Our new observations enable us to further constrain the excitation conditions and estimate the line emission of higher–$J$ CO– and the upper $[$CI$]$–lines. For the first time we give estimates for the CO/C abundance ratio in the center regions of these galaxies. Future CO $J$= 7–6 and $[$CI$]$ 2–1 observations will be important to resolve the ambiguity in the physical conditions and confirm the model predictions.]{}
Introduction
============
The spiral galaxies NGC4945 and Circinus at distances of $\sim 3.7$ and $\sim 4$Mpc belong to the nearest and infrared brightest galaxies in the sky. Their strong central star burst activity is fed by large amounts of molecular material and this has been studied extensively at millimeter wavelengths [e.g. @Curran2001; @Wang2004]. However, sub-millimeter observations are largely missing. The important rotational transitions of CO with $J\ge4$ and the fine structure transitions of atomic carbon have not yet been observed. These transitions often contribute significantly to the thermal budget of the interstellar gas in galactic nuclei and are therefore important tracers of the physical conditions of the warm and dense gas.
Emission of CO, [\[\]]{} (and [\[\]]{}) traces the bulk of carbon bearing species in molecular clouds that play an important role in their chemical network. The CO 4–3 transition in particular is a sensitive diagnostic of the dense and warm gas while the CO 1–0 transition traces the total molecular mass. The [\[\]]{} $^3P_1-^3P_0$ (henceforth 1–0) line has, to date, been detected in about 30 galactic nuclei [@Gerin2000; @IsraelBaas2002; @Israel2004] and appears to trace the surface regions of clumps illuminated by the FUV field of newborn, massive stars [e.g. @Kramer2004; @Kramer2005]. The use of CI as an accurate tracer of the cloud mass has been discussed with controversy [@Frerking1989; @Papadopoulos2004; @Mookerjea2006].
The strong cooling emission is balanced by equally strong heating caused by the vigorous star formation activity in the galaxy centers. The variation of CO cooling intensities with rotational number, i.e. the peak of the CO cooling curve, reflects the star forming activity [@bayet2006] and, possibly, also the underlying heating mechanisms. Several mechanisms have been proposed to explain the heating of the ISM in galactic nuclei and it is currently rather unclear which of these dominates in individual sources [e.g. @Wang2004]. Heating by X-rays from the active galactic nuclei (AGN) may lead to strongly enhanced intensities of high-$J$ CO transitions [@Meijerink2007], possibly allowing one to discriminate this heating mechanism from e.g. stellar ultraviolet heating via photon dominated regions (PDRs) [e.g. @bayet2006]. The greatly enhanced supernova rate by several orders of magnitude relative to the solar system value leads to an enhanced cosmic ray flux, providing another source of gas heating in the centers [@Farquhar1994; @bradford2003]. Another mechanism for heating is provided by shocks. The most important, large-scale shocks are produced by density wave instabilities which induce gravitational torques (in spiral arms and/or bars) and make the gas fall into the nucleus [@Usero2006]. A non-negligible contribution is also given by shocks produced by supernovae explosions. On smaller scales, bipolar outflows from young stellar objecs (YSO) can also contribute to this heating, although to a lesser extent [e.g. @garcia-burillo2001].
NGC4945
-------
NGC4945, a member of the Centaurus group of galaxies, is seen nearly edge-on (Table\[tab\_properties\]) with an optical diameter of $\sim 20'$ [@devaucouleurs1991]. kinematics indicate a galaxy mass of $1.4\,10^{11}$[M$_{\odot}$]{} within a radius of $6.3'$ with molecular and neutral atomic gas contributing $\sim2\%$ respectively [@Ott2001].
With a dynamical mass of $\sim3\,10^9\,$[M$_{\odot}$]{} in the central 600pc [@Mauersberger1996], it is one of the strongest IRAS point sources with almost all the far infrared luminosity coming from the nucleus [@Brock1998]. Observations of the X-ray spectrum are consistent with a Seyfert nucleus [@Iwasawa1993], and further analysis of optical imaging and infrared spectra [@Moorwood1994] suggests that this object is in a late stage on the transition from starburst to a Seyfert galaxy. Its nucleus was the first source in which a powerful H$_2$O mega maser was detected [@DosSantos1979].
Studies in [@Ables1987] and low-$J$ CO transitions [@Whiteoak1990; @Dahlem1993; @Ott1995; @Mauersberger1996] suggest the presence of a face-on circumnuclear molecular ring. Millimeter molecular multiple transition studies [@Wang2004; @Cunningham2005] are consistent with this result. The bright infrared and radio emission in the nucleus [@Ghosh1992], and the evidence that large amounts of gas seem to coexist in the central $30''$ [@Henkel1994] make it particularly suited for studying the high density environment in the center of this galaxy.
Circinus
--------
The nearby starburst spiral Circinus has a small optical angular diameter of $\sim 7$compared to its hydrogen diameter $D_{\rm
H} = 36'$ defined by the $1\, 10^{20}$atoms cm$^{-2}$ contour in @Freeman1977.
Circinus has a dynamical mass of $\sim 3\ 10^9\ $ M$_{\odot}$ within the inner 560pc [@Curran1998] matching the value obtained for NGC4945. Large amounts of molecular gas have been found by studies of low-$J$ CO observations [@Johansson1991; @Aalto1995; @Elmouttie1998; @Curran1998] including C$^{17}$O, C$^{18}$O and HCN [@Curran2001]. The strong H$_{2}$O maser emission found [@Gardner1982] traces a thin accretion disk of 0.8 pc radius, with a significant population of masers lying away from this disk, possibly in an outflow [@Greenhill1998]. Its obscured nucleus is classified as an X-ray Compton thick Seyfert 2. It shows a circumnuclear star burst on scales of 100 - 200pc [@Maiolino1998] with a complex structure of as seen in Hubble Space Telescope (HST) H$\alpha$ images [@Wilson2000]. Adaptive optics studies find there has been a recent star burst ($\sim100$ Myr old) in the central 8pc accounting for 2% of the total luminosity [@MuellerS]. The strong FIR emission, the large molecular gas reservoir, the existence of a molecular ring associated with star burst activity [@Curran1998] and the similarity with NGC4945 make them ideal objects for comparative studies of the dense and warm ISM in their nuclei.
Observations
============
[lrrrrr]{} & Circinus & NGC4945 & NGC253\
RA(2000) & 14:13:09.9 & 13:05:27.4 & 00:47:33.1\
DEC(2000) & -65:20:21 & -49:28:05 & -25:17:18\
Type & SA(s)b$^{a}$ & SB(s)cd$^{a}$& SAB(s)c$^{a}$\
Distance \[Mpc\] &$4.0^{d}$ &$3.7^{b}$& 3.5$^{e}$\
$38''$ correspond to & 732pc & 682pc & 646pc\
LSR velocity \[kms$^{-1}$\] & 434 & 555 & 243\
Inclination \[deg\] & 65$^{c}$ & $78^{b}$ & $78^{a}$ &\
$L_{\rm IR}$ \[$10^{10}$ [L$_{\odot}$]{}\] & 1.41$^{g}$ & $1.39^{g}$ & 2.67$^{g}$ &\
$S_{100} [{\rm Jy}]$ & $3.16\,10^2$ & $6.86\,10^2$ & $10.4\,10^2$\
We used the new NANTEN2 sub-millimeter telescope [e.g. @Kramer2007] on Pampa la Bola at an elevation of 4900m together with a dual channel 490/810 GHz receiver (operated jointly by the Nagoya University radioastronomy group and the KOSMA group from Universität zu Köln) to observe the centers of NGC4945, Circinus, and NGC253 (Table \[tab\_properties\]) in CO 4–3 and [\[\]]{} 1–0. The observations were performed from September to October 2006 using the position-switch mode with 20sec On- and 20sec Off-time. In October 2007 we reobserved $^{12}$CO 4–3 in the centers of NGC4945 and Circinus with the newly available chopping tertiary in double beam-switch mode yielding significantly improved baselines and removing atmospheric features in the spectra. In NGC4945 the total integration times ON-source are 17min and 10min for $^{12}$CO 4–3 and [\[\]]{} 1–0 respectively and in Circinus integration times ON-source are 10min and 34min for $^{12}$CO 4–3 and [\[\]]{} 1–0 respectively at source elevations of 40-60[$^\circ$]{}. The relative calibration uncertainty derived from repeated pointings on the nuclei is about 15%.
Position-switching was conducted by moving the telescope $10'$ in azimuth, i.e. out of the galaxy. Using double beam-switch (dbs) mode, the chopper throw is fixed at $162''$ in azimuth with a chopping frequency of 1 Hz. Typical double-sideband receiver temperatures of the dual-channel 460/810GHz receiver were $\sim$250K at 460GHz and 492GHz. The system temperature varied between $\sim$850 and $\sim$1200K. As backends we used two acusto optical spectrometers (AOS) with a bandwidth of 1GHz and a channel resolution of 0.37kms$^{-1}$ at 460GHz and 0.21kms$^{-1}$ at 806GHz. The pointing was regularly checked and pointing accuracy was stable with corrections of $\sim10''$. The half power beam width (HPBW) at 460GHz and 492GHz is 38$''$ with a beam efficiency $B_{\rm eff}=0.50$ and a forward efficiency $F_{\rm eff}=0.86$ [@Simon2007]. The calibrated data on $T_{A}^{*}$-scale were converted to $T_{\rm mb}$ by multiplying by the ratio $F_{\rm eff}$/$B_{\rm eff}$. The standard calibration procedure derives the atmospheric transmission (averaged over the bandpass) from the observed difference spectrum of hot load and blank sky, the latter taken at a reference position.Next, the model atmosphere [atm]{} is used to derive the atmospheric opacity taking into account the sideband imbalance. This is an important correction, especially when observing the CO 4-3 and \[CI\] 1-0 lines. This pipeline produces the standard spectra on the antenna temperature scale ($T_A^*$). In addition, we removed baselines up to first order.
The 810GHz channel was not used for these observations due to insufficient baseline stability.
All data presented in this paper are on the $T_{\rm mb}$ scale.
$^{12}$CO 4–3 in NGC253
-----------------------
To check the NANTEN2 calibration scheme, we retrieved the $^{12}$CO 4–3 map of NGC253 taken at APEX [@Guesten2006] and smoothed it to the NANTEN2 HPBW of $38''$ using a Gaussian kernel. The resulting spectra at the center position are shown in Fig.\[fig\_ngc253\]. Line temperatures and shapes show very good agreement.
Spectra of CO 4–3 and [\[\]]{} 1–0
==================================
Figure\[fig\_spectra\] shows the $^{12}$CO 4–3 and [\[\]]{} 1–0 spectra of NGC4945 and Circinus obtained with the NANTEN2 telescope. CO 4–3 spectra peak at 700mK in NGC4945 and 250mK in Circinus. Outside the velocity ranges of 350–800kms$^{-1}$ and 200–600kms$^{-1}$ respectively, the baseline rms values are 11mK and 25mK, respectively, at the velocity resolution of 15kms$^{-1}$. [\[\]]{} 1–0 spectra peak at 900mK in both galaxies while the rms values are 110mK and 140mK, respectively. The [\[\]]{} 1–0 area-integrated luminosities are 91Kkms$^{-1}$kpc$^{2}$ and 67Kkms$^{-1}$kpc$^{2}$ in NGC4945 and Circinus. @Curran2001 [@Curran1998] and @Mauersberger1996 mapped the low-$J$ 2–1, and 3–2 transitions of CO in the centers of both galaxies with SEST. We smoothed these data to the resolution of the NANTEN2 data, i.e. to $38''$. Only the $^{12}$CO 3–2 spectrum in Circinus is shown at its original resolution of $15''$ because a map of the central region could not be retrieved. Calibration of the CO 3–2 spectrum in NGC4945 was confirmed recently at APEX [@Risacher2006]. The $^{12}$CO 1–0, $^{13}$CO 1–0 and $^{13}$CO 2–1 spectra of the central region of NGC4945 and Circinus are presented in @Curran2001. We list integrated intensities in Table\[tab\_intint\_circinus\].
NGC4945 shows broad emission between 350 and 800kms$^{-1}$. The CO 4–3 line shape resembles the line shapes of 1–0 and 2–1.The line shape of [\[\]]{} 1–0 is similar to that of the CO transitions with a slighlty higher peak temperature than CO 4–3. In CO 1–0 and 2–1, Circinus shows broad emission between 200 and about 600kms$^{-1}$. The velocity component at $\sim550$kms$^{-1}$ becomes weaker with rising rotational number. Emission of [\[\]]{} 1–0 is restricted to 200 and $\sim500$kms$^{-1}$ only. In Circinus, the [\[\]]{} peak temperature is a factor $\sim3$ stronger than CO 4–3.
[lrrrc ]{} & Circinus & NGC4945 &\
line transition & $I_{\rm int}$ & $I_{\rm int}$ & FWHM\
& \[Kkms$^{-1}$\] & \[Kkms$^{-1}$\] & \[$''$\]\
CO 1–0$^{a}$ & 180 & 510 & 45\
CO 2–1$^{a}$ & 177 & 390 & 38\
CO 3–2$^{b}$ & - & 330 & 38\
CO 3–2$^{b}$ & 230 & - & 15\
CO 4–3 & 58 & 212 & 38 &\
$^{13}$CO 1–0$^{a}$ & 12 & 30 & 45\
$^{13}$CO 2–1$^{a}$ & 19 & 45 & 38\
$[$CI$]$ 1–0 & 163 & 248 & 38 &\
Physical conditions
===================
LTE
---
In the optically thin limit, the integrated intensities of [\[\]]{} and $^{13}$CO listed in Table\[tab\_intint\_circinus\] are proportional to the total column densities. LTE column densities of carbon are rather independent of the assumed excitation temperatures [e.g. @Frerking1989]. We find $N_{\rm C}=3.4-3.9\, 10^{18}$cm$^{-2}$ in NGC4945 and $N_{\rm C}=2.2-2.5\,10^{18}$cm$^{-2}$ in Circinus for a temperature range of $T_{\rm ex}= 20 - 150$K.
We used the $^{13}$CO $J$=1–0 and $J$=2–1 integrated intensities to derive total CO column densities, assuming LTE, optically thin $^{13}$CO emission, a CO/$^{13}$CO abundance ratio of 40 [@Curran2001], and $T_{\rm ex}=
20$K. We find a total CO column density of $N_{\rm CO}$= 1.0-1.710$^{18}$cm$^{-2}$ and $N_{\rm CO}$= 4.1-6.710$^{17}$cm$^{-2}$ in NGC4945 and Circinus respectively depending on which transition is used, 1–0 or 2–1. Varying the temperature to $T_{\rm ex}=150$K the column density of CO slightly increases up to $N_{\rm CO}$= 3.210$^{18}$cm$^{-2}$ and $N_{\rm CO}$= 1.410$^{18}$cm$^{-2}$ for NGC4945 and Circinus.
Another method uses the Galactic CO 1–0 to H$_{\rm 2}$ conversion factor $X_{\rm MW}=2.3\,10^{20}$cm$^{-2}$(Kkms$^{-1})^{-1}$ [@Strong1988; @strong_mattox1996] and the canonical CO to H$_{\rm 2}$ abundance of $8.5\, 10^{-5}$ [@frerking1982] to derive $N_{\rm CO}$= 3.510$^{18}$cm$^{-2}$ for Circinus, i.e. a factor of $\sim$ 10 larger than the LTE estimate indicating that the X-factor is only 1/10 Galactic. As the abundance of CO maybe different in NGC4945 and Circinus this result has to be taken with caution.
For NGC4945 @Wang2004 derive an X-factor 7 times smaller than the Galactic value, which leads to $N_{\rm CO}$= 1.410$^{18}$cm$^{-2}$, in good agreement with the LTE approximation from $^{13}$CO. The LTE column density derived from $^{13}$CO in Circinus also indicates an X-factor around 10 times smaller than the Galactic value.
The CO/C abundance ratio is 0.29-0.50 in NGC4945 and 0.19-0.27 in Circinus using the LTE column densities.
Radiative transfer analysis of CO and $^{13}$CO
-----------------------------------------------
We modeled the $^{12}$CO and $^{13}$CO emission lines using an escape probability radiative transfer model for spherical clumps [@Stutzki1985] using the CO collision rates of @schinke1985. This non-LTE model assumes a uniform density and temperature in a homogenous clump. The physical parameters kinetic temperature $T_{\rm kin}$, molecular density $n(\rm H_{2})$, and column density $N_{\rm CO}$ determine the excitation conditions in this model (Table\[tab\_results\]).
In NGC4945 and Circinus, we used the ratios of the observed integrated intensities of CO 1–0 to 4–3 and the $^{13}$CO 1–0 and 2–1 transitions [@Curran2001] to obtain column densities, density and kinetic temperature, assuming a constant $^{12}$CO/$^{13}$CO abundance ratio. We used an abundance ratio of 40 in both sources in accordance with @Curran2001. The escape probility code uses an internal clump line width $\Delta v_{\rm
mod}$ which hardly effects the outcome of the model in the reasonable range of 1-20kms$^{-1}$.
In a simultaneous fit a $\chi^{2}$-fitting routine then compared the $J$ line ratios of the model output $R^{j}_{\rm mod}$ to the ratios of the observed integrated intensities $R^{j}_{\rm obs}$ and determined the model with the minimal $\chi^{2}$. We compute the normalized $\chi^{2}$ with the degrees of freedom $d=J-p$ and $J$ being the number of independent ratios and $p$ the number of parameters, in our case $T_{\rm kin}$, $n(\rm H_{2})$ and $N_{\rm CO}$, to be determined :
$$\chi^2 = \frac{1}{d} \sum_{j=1}^{J}(R^j_{\rm mod} - R^j_{\rm obs})/\sigma_{j}.$$
The errors $\sigma_{j}$ due to calibration uncertainties are estimated to be 20$\%$.
In Circinus the $^{12}$CO 3–2 line is missing, leaving one degree of freedom compared to two in NGC4945.
$T_{\rm kin}$ and $n(\rm H_{2})$ are determined with this step. To compare the modeled integrated intensities $I_{\rm mod}$ to the absolute observed intensities $I_{\rm obs}$ we have to account for the velocity filling, due to the velocity width $\Delta$v$_{\rm mod}$ of an individual clump to the width of the galaxy spectrum $\Delta$v$_{\rm obs}$ and the beam dilution, due to the size of the modeled clump A$_{\rm cl}$ compared to the beam area A$_{\rm beam}$.
The large velocity width of the observed spectra implies several clumps in the beam $N_{\rm cl}$=$n\,\Delta$v$_{\rm obs}$/$\Delta$v$_{\rm mod}$ with $n\geq
1$ (Table\[tab\_results\]). The beam dilution is determined by the fraction of modeled clump area to the beam size which we express in terms of an area filling factor per clump $\phi_{\rm
A,cl}$= A$_{\rm cl}$/A$_{\rm beam}$. The total area filling factor is $\phi_{\rm A}$= $N_{cl}\,\phi_{\rm A,cl}$. The size of the clump with radius $R$, A$_{\rm cl}=\pi R^2$, can be inferred via the mass $M$ and density $n(\rm H_{2})$ of the clump: $R=(3/(4\pi)
M/n)^{(1/3)}$. In summary, the modeled intensities of the individual clumps $I_{\rm mod}$ are converted to the intensities of a clump ensemble $I_{\rm ens}$ which can then be compared with the observed intensities:
$$I_{\rm ens} = I_{\rm mod} \times N_{\rm cl}\times \phi_{\rm A,cl}=I_{\rm mod} \times \phi_{\rm A}.$$
[lcccc]{} & &\
$\chi^{2}$ & 2.0 & 9.6 & 4.8 & 12.4\
$n(\rm H_{2})_{\rm loc}$\[cm$^{-3}$\] & $ 10^{4}$& $10^{3}$ & $3\,10^{4}$ & $10^{3}$\
$T_{\rm kin}$ \[K\] & 20 & 100 & 20 & 100\
$N_{\rm CO}$ \[10$^{16}$cm$^{-2}]$ & 35 & 50 & 76 & 63\
$N_{\rm C}$ \[10$^{16}$cm$^{-2}]$ & 230 & 30 & 330 & 98\
$N_{\rm H_{2}}$ \[10$^{20}$cm$^{-2}]$ & 37 & 46.5 & 89 & 74\
$M$ \[10$^{6}$[M$_{\odot}$]{}\] & 630 & 792 & 1385 & 1114\
$\Delta v_{\rm mod}$ \[kms$^{-1}$\] & 5 & 5 & 10 & 10\
$\Delta v_{\rm obs}$ \[kms$^{-1}$\] & 186 & 186 & 188 & 188\
$N_{\rm cl}$ & 50 & 38 & 35 & 40\
$\phi_{\rm A}$ & 2.0 & 6.3 & 1.5 & 4.0\
$^{12}$CO/$^{13}$CO abundance ratio & 40 & 40 & 40 & 40\
CO/C abundance ratio & 0.15 & 1.67 & 0.23 & 0.64\
[\[\]]{} cooling intensity & 4.1 & 7.88 & 7.2 & 11.8\
CO cooling intensity & 2.1 & 2.8 & 6.6 & 5.7\
[\[\]]{} /CO cooling intensity ratio & 2.0 & 2.8 & 1.1 & 2.1\
### NGC4945
Figure\[fig\_seds\]b shows the observed intensities of CO, $^{13}$CO, and [\[\]]{} together with two representative solutions of the radiative transfer calculations (see also Table\[tab\_results\]).
#### CO.
Fitting the CO and $^{13}$CO lines, we find a degeneracy in the n(H$_{2})$-T$_{\rm kin}$ plane of the solutions for a rather constant pressure n(H$_{2}) \times \, T_{\rm kin}$ $\sim 10^{5}$Kcm$^{-3}$. The best fits are achieved for a $^{12}$CO/$^{13}$CO abundance ratio of 40, similar to the value found by @Curran2001. Low $\chi^{2}$-values (see Table\[tab\_results\]) constrain the densities to $n(\rm H_{2})=10^{3}-10^{5}$cm$^{-3}$ and temperatures to a wide range of $T_{\rm kin}$=20-180K with higher temperature solutions corresponding to lower densities. The best fitting solution is $n(\rm H_{2})=3\,10^4$cm$^{-3}$ and $T_{\rm
kin}$=20K. However, this solution is not significantly better than e.g. $n(\rm H_{2})=10^3$cm$^{-3}$ and $T_{\rm kin}$=100K (Fig.\[fig\_seds\]b, Table\[tab\_results\]).
The fitted column density $N_{\rm CO}$ agrees with the LTE approximation for the CO-column density within a factor of 2–3. The peak of the modeled CO cooling curve at $J$=4 contains 35.6% of the total $^{12}$CO cooling intensity of $6.6\,10^{-5}$ ergs$^{-1}$cm$^{-2}$sr$^{-1}$ computed by summing the cooling intensity of the $^{12}$CO transitions from $J$=1 to 20 for the $T_{\rm
kin}$=20K solution.
We use the radiative transfer model to predict the $^{12}$CO 7–6 intensity. It is rather weak, depending strongly on $T_{\rm kin}$. It varies from k 1.3 Kkms$^{-1}$ for the $T_{\rm
kin}$=20K solution to 4.1 Kkms$^{-1}$ for the $T_{\rm kin}$=100K fit.
@Curran2001 find in this source $n(\rm H_{2})$=3$\times
10^{3}$cm$^{-3}$ and $T_{\rm kin}$=100K from $^{12}$CO observations of the 3 lowest transitions and $^{13}$CO data of the two lowest transitions. In their multi-transition study @Wang2004 estimate a density of $n(\rm H_{2})$= $10^{3}$cm$^{-3}$ for an assumed temperature $T_{\rm kin}$=50K from CO transitions up to $J=3$. In contrast, the observed CN and CH$_{3}$OH lines indicate local densities around $10^4$cm$^{-3}$. The solutions for CO and $^{13}$CO found in the literature are in good agreement with our parameter space of solutions showing that the additional CO 4–3 line does not help significantly to improve the fits.
#### Atomic carbon.
Assuming the same density, kinetic temperature, velocity filling, and beam dilution for carbon as for CO, we use the observed [\[\]]{} intensity and the radiative transfer model to estimate the carbon column density and hence the CO/C abundance ratio. The CO/C abundance ratio varies between 0.23 and 0.64 for the two solutions listed in Table\[tab\_results\].The corresponding carbon column densities are $N_{\rm C}=3.3\,10^{18}$cm$^{-2}$ and $N_{\rm C}=9.8\,10^{17}$cm$^{-2}$ for the high and the low density solution, respectivly. The column density for the latter solution is a factor of 3 lower compared to the LTE carbon column density. The assumption of optically thin emission in the LTE might be overestimating the column density compared to the escape probability modelling. We also use the model to predict the [\[\]]{} 2–1 intensity. It is 16 Kkms$^{-1}$ for the $T_{\rm kin}$=20K/$n(\rm H_{2})=3\,10^4$cm$^{-3}$ solution. However, this result critically depends on the kinetic temperature. The $T_{\rm kin}$=100K/$n(\rm H_{2})=10^3$cm$^{-3}$ solution yields 182Kkms$^{-1}$. Also, the [\[\]]{} 2–1/[\[\]]{} 1–0 line ratios change from 0.07 to 0.75, depending on the solutio.This shows that the high-lying transitions can be used to break the degeneracy.
The total cooling intensity of both [\[\]]{} lines is listed in Table\[tab\_results\] for both presented solutions. The cooling intensity of the two [\[\]]{} lines is thus of the order of the total cooling intensity of CO with the C/CO cooling intensity ratio varying between 1.1-2.1 for the discussed solutions.
### Circinus
#### CO.
The modeled CO cooling curves are shown in Figure\[fig\_seds\]a and the fit results are summarized in Table\[tab\_results\].The CO line ratios are given for a $\Delta\,v_{mod} $=5kms$^{-1}$ so there are about 40 clouds in the beam to account for the observed velocity width of $\Delta\,v_{obs}$=186kms$^{-1}$. The assumed $^{12}$CO/$^{13}$CO abundance ratio of 40 is slightly lower than the values of $\sim$ 60, found by @Curran2001.
Good fits corresponding to low $\chi^{2}$ (see Table\[tab\_results\]) can be found for densities of $n(\rm H_{2})=10^{3}-10^{4.5}$cm$^{-3}$ and a large range of temperatures of $T_{\rm kin}$=20-160K while the product of $n(\rm H_{2}) \times \, T_{\rm kin}$ stays approximately constant at $\sim
10^{5}$Kcm$^{-3}$. Again, a number of solutions provide consistent CO cooling curves.
The lowest $\chi^{2}$ is obtained for $n(\rm H_{2})=10^{4}$cm$^{-3}$, $T_{\rm kin}$=20K and a column density of $N_{\rm
CO}=3.5\,10^{17}$cm$^{-2}$ assuming 50 modeled clumps in the beam. The column density $N_{\rm CO}$ is well determined showing a steep gradient of $\chi^{2}$-values for varying densities and temperatures. This is in reasonable agreement with the LTE-approximation from $^{13}$CO. A second solution at $n=10^{3}$cm$^{-3}$ and $T_{\rm kin}$=100K also lies within the 1$\sigma$-contour of the $\chi^2$ distribution (cf. Fig.\[fig\_seds\] and Tab.\[tab\_results\]).
The results agree well with the solutions found by @Curran2001. They find $T_{\rm kin}$=50-80K and $n(\rm H_{2})=2\,10^{3}$cm$^{-3}$ from observations of the 3 lowest $^{12}$CO transitions and the 2 lowest $^{13}$CO transitions.
The modeled CO cooling curve peaks at $J$=4 containing 35% of the total $^{12}$CO cooling intensity of $2.1\,10^{-5}$ ergs$^{-1}$cm$^{-2}$sr$^{-1}$ for the $T_{\rm kin}$=20K fit.
The predicted integrated intensity of $^{12}$CO 7–6 is very weak, varying strongly from 0.1 Kkms$^{-1}$ for the $T_{\rm kin}$=20K solution to 1.9 Kkms$^{-1}$ for $T_{\rm kin}=100$ K.
#### Atomic carbon.
Assuming the same density and temperature for carbon as for CO, we use the best fitting CO model to derive the [\[\]]{} 1–0 intensity, carbon column densities, and CO/C abundance ratios. The predicted CO/C abundance ratio is 0.15, again consistent with the optically thin LTE result. The predicted [\[\]]{} 2–1 integrated intensity is 6.1 Kkms$^{-1}$.
As in NGC4945, changing the temperature has a large effect on the [\[\]]{}2–1 intensity. The $T_{\rm kin}$=100K solution yields 111 Kkms$^{-1}$ and a higher CO/C abundance ratio of 1.67. Thus the [\[\]]{}2–1/[\[\]]{}1–0 line ratios changes from 0.04 to 0.74.
The corresponding carbon column densities for the two presented solutions are $N_{\rm C}=2.3\,10^{18}$cm$^{-2}$ and $N_{\rm C}=3.0\,10^{17}$cm$^{-2}$, repectivly. For the latter solution the column density is about a factor of 10 lower compared to the LTE carbon column density. The optically thin assumption for the LTE modelling is obviously not appropriate in the high temperature and low density scenario.
The total cooling intensity ratio of [\[\]]{}CO varies from 2.1 to 2.8 for the presented solutions.Carbon is a stronger coolant than CO by a factor of 2-3.
Discussion
==========
[\[\]]{} 1–0 luminosities
-------------------------
The [\[\]]{} 1–0 luminosities for the centers of the Seyfert galaxies NGC4945 and Circinus are 91 and 67Kkms$^{-1}$kpc$^{2}$ (Fig.\[fig\_israel\]). To date, about 30 galactic nuclei have been studied in the 1–0 line of atomic carbon, most of which are presented in @Gerin2000 and @IsraelBaas2002. The [\[\]]{} luminosity of a source is an important property since it gives the amount of energy emitted per time which is proportional to the number of emitting atoms, i.e. proportional to the [\[\]]{} column density in the limit of optically thin emission. The NANTEN2 38 beam achieves $\sim$700pc resolution in Circinus and NGC4945 which both lie at $\sim$4Mpc. To achieve the same spatial resolution for galaxies at $\sim$12Mpc distance, e.g. for NGC278, NGC660, NGC1068, NGC3079 and NGC7331 listed in @IsraelBaas2002, one would need an angular resolution of $\sim13''$, comparable to the $10''$ JCMT beam at 492GHz. The luminosities studied in these 7 sources thus all sample the innermost $\sim$700pc. Area integrated [\[\]]{}luminosities are found to vary strongly between $\sim$1 and $\sim160$Kkms$^{-1}$kpc$^{2}$ in these 7 galaxies (Fig.\[fig\_israel\]) [@Israel2005; @IsraelBaas2002]. Quiescent centers show modest luminosities $1\le L$([\[\]]{})$\le5$Kkms$^{-1}$kpc$^{2}$, while starburst nuclei in general show higher luminosities. The largest luminosities are found in the active nuclei of NGC1068 and NGC3079 which show 50 and 160Kkms$^{-1}$kpc$^{2}$ [@IsraelBaas2002]. NGC4945 and Circinus also fall in this category (Fig.\[fig\_israel\]).
[\[\]]{} 1–0/CO 4–3 line ratios
-------------------------------
The [\[\]]{} 1–0/CO 4–3 ratio of integrated intensities is 1.2 in NGC4945 and 2.8 in Circinus. For Circinus the ratio is larger than any ratios previously observed in other galactic nuclei or in the Milky Way. The [\[\]]{}1-0/CO 4–3 ratio is shown in Figure\[fig\_israel\] versus the area integrated [\[\]]{} luminosity. As also discussed in @IsraelBaas2002, we see no functional dependence. Galactic sources (not shown in this figure) would lie in the lower left corner. @Israel2005 studied 13 galactic nuclei and found that the [\[\]]{}1–0 line is in general weaker than the CO 4–3 line, but not by much. Ratios vary over one order of magnitude between 0.1 in Maffei2 and 1.2 NGC4826. Galactic star forming regions like W3Main or the Carina clouds show much lower values, between about 0.1 and 0.5, which are consistent with emission from photon dominated regions (PDRs) [@Kramer2004; @Jakob2007; @Kramer2007]. @Fixsen1999 find 0.22 in the Galactic center and 0.31 in the Inner Galaxy. The variation seen in the various Galactic and extragalactic sources appears to be intrinsic and not due to observations of different angular resolutions. This is because the frequencies of the two lines are very close and angular resolutions are therefore similar if the same telescope is used.
[\[\]]{} 1–0/ $^{13}$CO 2–1 line ratios
---------------------------------------
The [\[\]]{} 1–0/ $^{13}$CO 2–1 line ratios in NGC4945 and Circinus are 5.51 and 8.57, respectively (Fig.\[fig\_israel\]).In NGC4945, the observed ratio is consistent with results found in previous studies [@IsraelBaas2002; @Gerin2000] ranging up to ratios of 5. For Circinus, the ratio is again higher than any previous measurement.
Two thirds of the sample of galaxies studied by @IsraelBaas2002 show [\[\]]{} 1–0/ $^{13}$CO 2–1 line ratios well above unity. The sample consists of quiescent, star burst and active nuclei. The highest [\[\]]{} 1–0/ $^{13}$CO 2–1 ratios are found in star burst and active nuclei, consistent with our observations. @Gerin2000 find a similar result with two thirds of the galaxies in their sample exceeding a ratio of 2. High ratios can be qualitatively understood in low column density enviroments with mild UV radiation fields. In these regimes most CO will be dissociated and the gas-phase carbon will be neutral atomic [@IsraelBaas2002]. This implies that dense, star forming molecular cloud cores are not the major emission source in galaxy centers.
In general the studied centers of external galaxies show stronger [\[\]]{}emission than one would expect from Galactic observations, which show typical ratios of 0.2-1.1 [@Mookerjea2006]. In Galactic sources high ratios are found in low gas column densitis and medium UV radiation enviroments where $^{13}$CO will be dissociated and atomic carbon remains neutral in the gas phase i.e. in translucent clouds and at cloud edges [@Israel2005]. He concludes that the dominant emission from galaxy centers does not stem from PDRs.
@Meijerink2007 studied irradiated dense gas in galaxy nuclei using a grid of XDR and PDR models. For the same density the predicted [\[\]]{} 1–0/ $^{13}$CO 2–1 line ratios are significantly higher for the XDR- compared to PDR-models (Fig. 10 in Meijerink et al. 2007). We observed ratios of 61 and 95 in NGC4945 and Circinus respectivly, on the erg-scale. These ratios can be explained by XDR-models at high densities $n(\rm
H_{2})=2\,10^{3}-10^{5}$cm$^{-3}$. PDR-models explain the observed ratios in low density regimes with $n(\rm H_{2})=2\,10^{2}-6\,10^{2}$cm$^{-3}$. The high [\[\]]{}1–0/ $^{13}$CO 2–1 line ratios observed in NGC4945 and Circinus may hint at a significant role of X-ray heating in these galaxy nuclei as our predicted densities of $n(\rm H_{2})=10^{3}-10^{4}$cm$^{-3}$ agree with the high-density XDR-models in @Meijerink2007.\
Total CO and [\[\]]{} cooling intensity
---------------------------------------
In all sources studied by @bayet2006 but NGC6946, the CO cooling intensity exceeds that of atomic carbon. They find the [\[\]]{} to CO cooling ratio to vary between $\sim0.3$ for M83 to $\sim 2$ for NGC6946. NGC4945 and Circinus show similarly high values as the latter galaxy: In NGC4945 this ratio is $\sim 1-2$ and in Circinus we find $\sim 2-3$. These values are around 2 orders of magnitude higher than the typical values found in Galactic star forming regions [@Jakob2007; @Kramer2007].
Shape of the CO cooling curve
-----------------------------
The modeled CO cooling curve of NGC4945 and Circinus peaks at $J=4$. Observations of the higher lying CO lines i.e. the CO 6-5 and CO 7-6 lines will however be important to verify our model predictions and the importance of CO cooling relative to C.
The shape and maximum of the cooling curve of $^{12}$CO has been studied in a number of nearby and high-z galaxies. @Fixsen1999 find a peak at $J$=5 in the central part of the Milky Way using FIRAS/COBE data and rotational transitions up to 8–7. @bayet2006 observed 13 nuclei in mid-$J$ CO lines up to 7–6 and find that the peak of the cooling curves vary with nuclear activity. While normal nuclei exhibit peaks near $J_{\rm up}=4$ or 5, active nuclei show a peak near $J_{\rm up}=6$ or 7. NGC253 was observed at APEX in CO upto $J_{\rm up}=7$ and shows a maximum at $J=6$ [@Guesten2006]. On the other hand, studies of high redshift galaxies show cooling curves peaking as low as $J$=4 for SMM16359 [@WeissA2005], as high as $J$=9 in the case of the QSO APM08279 [@WeissA2007], and peaking at $J$=7 for the very high redshift ($z=6.4$) QSO J1148 [@WalterF2003].
Pressure of the molecular gas
-----------------------------
A wide range of temperatures $T_{\rm kin}= 25-150$K and densities $n(\rm H_{2})=5\,10^{2}- 7\,10^{5}$cm$^{-3}$ has been found in similar studies of external nuclei, including ULIRGs, normal spirals, star burst galaxies, and interacting galaxies [e.g. @bayet2006; @IsraelBaas2002]. In the irregular galaxy IC10 rather high densities $n(\rm H_{2})\sim 10^6$cm$^{-3}$ and low temperatures of $T_{\rm kin}= 25$K are found while the molecular gas in the center of the spiral galaxy NGC6946 is found to be less dense, $n(\rm H_{2})=10^{3}$cm$^{-3}$, but much hotter, $T_{\rm
kin}=130$K. @Guesten2006 studied NGC253 to obtain $n(\rm H_{2})=10^{3.9}$cm$^{-3}$ and $T_{\rm kin}=60$K while @bradford2003 investigated the same source and derived a higher $T_{\rm kin}=120$K and density $n(\rm H_{2})=4.5 \, 10^{4}$cm$^{-3}$. Both studies used $^{12}$CO 7–6 observations. However, the temperature/density degeneracy cannot be resolved. The solutions we present in our study of NGC4945 and Circinus show densities and temperatures of $n(\rm H_{2})=10^{3}-10^{4}$cm$^{-3}$ with a less well constrained temperature $T_{\rm kin}=20-100$K; depending on the density, as discussed before.
[\[\]]{} 2–1/1–0 line ratio
---------------------------
The modeled [\[\]]{} 2–1/1–0 line ratios change from 0.07 to 0.75 in NGC4945 and from 0.04 to 0.74 in Circinus for the presented escape probability solutions. Observed ratios vary from 0.48 in G333.0-0.4 [@Tieftrunk2001] to 2.9 in W3 main for galactic and extragalactic sources [@Kramer2004]. For M82, @stutzki1998 found a ratio of 0.96. @bayet2006 observed ratios ranging between 1.2 (in NGC253) to 3.2 (in IC342). The [\[\]]{} 2–1/1–0 line ratio for the low temperature solution predicted for NGC4945 and Circinus is significantly lower than previous results in the literature.
CO/C abundance ratio
--------------------
Compared to an abundance ratio CO/C of 3-5 in NGC253 [@bayet2004] and an average value of 2 in the nucleus of M83 [@White1994; @IsraelBaas2001], NGC4945 and Circinus have a higher fraction of atomic carbon in their nuclei which resembles the low CO/C abundance ratios found in Galactic translucent clouds. We find values of 0.23-0.64 in NGC4945 and 0.15-1.67 in Circinus. Due to the degeneracy of $n(\rm H_{2})$,T the abundance ratios cannot be determined more accurately. In galactic molecular clouds values range between 0.16-100, low values are found in translucent clouds [@Stark1994] while massive star forming regions show high CO/C abundances [@Mookerjea2006].
Future observations
-------------------
Future observations of CO 7–6 and [\[\]]{} 2–1 will be important to better understand the the kinetic temperature and density to resolve their degeneracy, and for an understanding of the dominating heating mechanism i.e. X-ray or UV heating. We estimate detection of the [\[\]]{} 2–1 line with NANTEN2 within less than 10 minutes total observation time under average weather conditions . The CO 7-6 will be harder to detect, depending on the excitation conditions. At 810 GHz the 1GHz bandwidth of the receiver translates into a velocity range of 370kms$^{-1}$. We therefore plan to stack future observations to be able to cover the broad line widths of Circinus and NGC4945.
We thank S.Curran for providing us with the SEST-data for Circinus, R. Mauersberger and G. Rydberg for SEST-data for NGC4945, and R.Güsten and S. Philipp for the APEX-data of NGC253. The NANTEN2 project (southern submillimeter observatory consisting of a 4-meter telescope) is based on a mutual agreement between Nagoya University and The University of Chile and includes member universities from six countries, Australia, Republic of Chile, Federal Republic of Germany, Japan, Republic of Korea, and Swiss Confederation. We acknowledge that this project could be realized by financial contributions of many Japanese donators and companies. This work is financially supported in part by a Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology of Japan (No.15071203) and from JSPS (No. 14102003 and No.18684003), and by the JSPS core-to-core program (No.17004). This work is also financially supported in part by the grant SFB494 of the Deutsche Forschungsgemeinschaft, the Ministerium für Innovation, Wissenschaft, Forschung und Technologie des Landes Nordrhein-Westfalen and through special grants of the Universität zu Köln and Universität Bonn. L. B. and J. M. acknowledge support from the Chilean Center for Astrophysics FONDAP 15010003.
|
---
author:
- Sarabjeet Singh
- 'David J. Schneider'
- 'Christopher R. Myers'
title: 'The structure of infectious disease outbreaks across the animal–human interface'
---
The authors would like to thank Jason Hindes, Oleg Kogan, Marshall Hayes, Drew Dolgert and Jamie Lloyd-Smith for helpful discussions and comments on the manuscript. This work was supported by the Science & Technology Directorate, Department of Homeland Security via interagency agreement no. HSHQDC-10-X-00138.
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abstract: |
We use the considered axial deformed relativistic mean field theory to perform systematical calculations for $Z=112$ and 104 isotopic chains with force parameters NL3, NL-SH and NL-Z2 sets. Three deformed chains (oblate, moderate prolate and super-deformed chain) are found for $Z=112$ and 104 isotopic chains. It is found that there is a chain of super-deformed nuclei which can increase the stability of superheavy nuclei in the $Z=112$ isotopic chain. Shape coexistence is found for $Z=112, 104$ isotopic chain and the position is defined. For moderate prolate deformed chains of $Z=112$ and 104, there is shell closure at $N=184$ for moderate prolate deformed chain. For oblate deformed chain of $Z=112$, the shell closure appears around at $N=176$. For super-deformed chains of $Z=112$ and 104, the position of shell closure have strong parameter dependence. There is shell anomalism for oblate or superdeformed nuclei.
Key words: relativistic mean field, superheavy element, deformed configuration, shape coexistence, superdeformed chain
author:
- |
X.H. Zhong, [^1] L. Li, [^2] P.Z. Ning [^3]\
*Department of Physics, Nankai University, Tianjin 300071, P. R. China*
title: '[^4]'
---
cite\#1\#2[^\[[\#1@tempswa\ ,\ \#2]{}\]^]{} 25.0cm -2.5cm -.5cm -0.5cm epsf
Introduction
============
Since the possible existence of superheavy elements was predicted in 1960s by nuclear theoreticians, the search of superheavy elements in nature has become a hot topic for scientists. Empirically, many heavy elements were identified by nuclear synthesis. At first, the elements $Z=105\sim108$ were successfully produced and both physicists and chemists agree on the existence of these elements although their half-lives are not very long. During $1995\sim1996$, because of the participating of more and more large laboratories in researches of new elements, the elements $Z=110\sim112$ were produced by Hofmann et al. at GSI in Germany[@1; @2; @3]. $Z=114$ was produced by Oganessian et al. at Dubna in Russia in 1999[@4; @5]. One year later it was again reported that $Z=116$ was synthesized at Dubna[@pp]. Recently, $^{288}115$ and $^{287}115$ were synthesized at FLNR, JINR[@pp1]. The achievement in producing new elements speeds up the researches on superheavy nuclei not only in experiment but also in theory. Theoretically, several models have been used and many aspects such as the collisions, structure, and stability have been investigated [@p1; @p2; @p3; @p4; @p5; @30; @31; @32; @33; @34; @35] for heavy and superheavy elements.
Recently, there are systematically calculations for superheavy elements within the framework of deformed relativistic mean-field (RMF) theory [@d1; @d2; @d3]. These calculations predicted that there are shape coexistence and super-deformation in the ground state of superheavy nuclei and deformation can be an important cause for the stability of superheavy nuclei based on a constraint RMF calculation. The conclusion changes the usual conception that the existence of superheavy elements is due to their spherical shell structure and makes people recognize that deformed configurations are as important as the spherical one for stability of superheavy elements in theory.
However, the constraint RMF calculation consumes much time, thus the calculation is very limited. Although there are many systematical calculations for superheavy nuclei which find there are a moderate prolate solution, an oblate solution, and a superdeformed prolate solution for the same element[@d2; @d3], there is no calculation which can give out deformed configurations of a whole isotopic chain. No information tells us that whether there are two or more deformed configurations for most of the nuclei in a whole isotopic chain or only exist in a few nuclei in the isotopic chain. Our aim is to study these properties in a long isotopic chain. Thus, we select $Z=104$ and $Z=112$ isotopic chains and carry out systematical calculation within the framework of deformed relativistic mean-field (RMF) theory. The binding energy, quadrupole deformation, root mean square radii, shape coexistence and shell closure are the investigative projects.
The formalism of the relativistic mean-field theory
===================================================
The relativistic mean-field theory has been widely used to describe finite nuclei, in the RMF method, the local Lagrangian density is given as[@c1; @c2] $$\begin{aligned}
{\mathcal{L}}= \bar{\psi} (
i\gamma^{\mu}\partial_{\mu}- M)\psi- g_{\sigma} \bar{\psi}
\sigma\psi-
g_{\omega}\bar{\psi}\gamma^{\mu}\omega_{\mu}\psi- g_{\rho}
\bar{\psi }\gamma^{\mu}\rho^{a}_{\mu}\tau^{a}\psi\cr+
\frac{1}{2}\partial^{\mu}\sigma\partial_{\mu}\sigma-\frac{1}{2}m
_{\sigma}^{2}\sigma^{2}- \frac{1}{3}g
_{2}^{2}\sigma^{3}-\frac{1}{4} g _{3}^{2}\sigma^{4}\cr-
\frac{1}{4}\Omega^{\mu\nu}\Omega_{\mu\nu} +\frac{1}{2}
m_{\omega}^{2}\omega^{\mu}\omega_{\mu} -\frac{1}{4} R ^{a \mu
\nu} R _{\mu\nu}^{a}+
\frac{1}{2} m_{\rho}^{2} {\rho^{a\mu}} {\rho
^{a}_{\mu}}
\cr-\frac{1}{4} F^{\mu\nu}F_{\mu\nu}-e \bar{\psi }\gamma^{\mu}
A^{\mu}\frac{1}{2}(1-\tau^{3})\psi.\end{aligned}$$ The meson fields included are the isoscalar $\sigma$ meson, the isoscalar-vector $\omega$ meson and the isovector-vector $\rho$ meson. $M$, $m_{\sigma}$, $m_{ \omega}$ and $m_{\rho}$ are the nucleon-, the $\sigma$-, the $\omega$- and the $\rho$-meson masses, respectively, while $g_{\sigma}$, $g_{\omega}$, $g_{\rho}$ and $e^{2}/4\pi=1/137$ are the corresponding coupling constants for the mesons and the photon. The isospin Pauli matrices are written as $\tau^{a}, \tau^{3}$ being the third component of $\tau^{a}$. The field tensors of the vector mesons and of the electromagnetic fields take the following form: $$\begin{aligned}
\Omega^{\mu\nu}&=&\partial^{\mu}\omega^{\nu}-\partial^{\nu}\omega^{\mu},\cr
R ^{a\mu\nu}&=&\partial^{\mu}\rho^{a\nu}-\partial^{\nu}\rho^{a\mu},\cr
F ^{\mu\nu}&=&\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}.\end{aligned}$$
The variational principle gives the equations of motion. For the static case the meson fields and photon field operators are assumed to be classical fields and they are time independent. They are replaced by their expectation values. The symmetries of the system simplify the calculations considerably. In all the systems considered in this work, there exists time reversal symmetry, so there are no currents in the nucleus and therefore the spatial vector components of $\omega^{\mu}$, $\rho^{a\mu}$ and $A^{\mu}$ vanish. This leaves only the time-like components, $\omega^{0}$, $\rho^{a0}$ and $A^{0}$. Charge conservation guarantees that only the 3-component of the isovector $\rho^{00}$ survives. Finally we have the following Dirac equation for the nucleon:
$$\{-i \alpha \nabla+V(r)+\beta[M+S(r)]\}\psi_{i}
=\varepsilon_{i}\psi_{i},$$
where $V(r)$ is the vector potential $$V(r)=g_{\omega}\omega^{0}(r)+g_{\rho}\tau^{3}\rho^{00}+e\frac{1+\tau^{3}}{2}A^{0}(r),$$ and $S(r)$ is the scalar potential $$S(r)=g_{\sigma}\sigma(r).$$ The Klein-Gordon equations for the mesons and the electromagnetic fields with the densities as sources are $$\begin{aligned}
\{-\Delta+m_{\sigma}^{2}\} \sigma(r)&=&-\:g_{\sigma}\rho_{s}(r)-g_{2}\sigma^{2}(r)-g_{3}\sigma^{3}(r),\\
\{-\Delta+m_{\omega}^{2}\} \omega_{0}(r)&=&g_{\omega}\rho_{v}(r),\\
\{-\Delta+m_{\rho}^{2}\} \rho_{00}(r)&=&g_{\rho}\rho_{3}(r),\\
-\Delta A^{0}(r)&=&e\rho_{c}(r).
\end{aligned}$$ The corresponding densities are $$\begin{aligned}
\rho_{s}(r)&=&\sum_{i=1} ^{A}\overline{\psi}_{i}(r)\psi_{i}(r) ,\\
\rho_{v}(r)&=&\sum_{i=1} ^{A} \psi^{\dagger} _{i}(r)\psi_{i}(r) ,\\
\rho_{3}(r)&=&\sum_{i=1} ^{A} \psi^{\dagger} _{i}(r)\tau^{3}\psi_{i}(r) ,\\
\rho_{c}(r)&=&\sum_{i=1} ^{A}
\psi^{\dagger}_{i}(r)((1-\tau^{3})/2)\psi_{i}(r) .\end{aligned}$$
Now we have a set of coupled equations for mesons and nucleons and they will be solved consistently by iterations.
Numerical calculation and analysis
==================================
The validity of deformed relativistic mean-field (RMF) theory in the calculation for superheavy nuclei is tested in previous papers[@d1; @d2; @d3], so we do not test the validity any more in our work. In the process calculation, three typical sets of force parameters NL3[@q1], NL-SH[@q2], and NL-Z2[@p3] in RMF model are chosen. The method of harmonic basis expansions is used in solving the coupled RMF equations. The number of bases is chosen as $N_{f}=12, N_{b}=20$. Pairing has been included using the BCS formalism. In the BCS calculations we have used constant pairing gaps $\Delta_{n}=\Delta_{p}=11.2/\sqrt{A}$ MeV[@f1]. This input of pairing gaps is used in nuclear physics for many years. Although the BCS model may fail for light neutron-rich nuclei, the nuclei studied here are not light neutron-rich nuclei and the RMF results with BCS treatment should be reliable. The different inputs of $\beta_{0}$ lead to different iteration numbers of the self-consistent calculation and different computational time, but physical quantities such as the binding energy and the deformation do not change much, which is tested in Ref.[@d2]. Thus, when we carry out calculation, we only choose a proper initial $\beta_{0}$ and neglect its effect on our results.
*Quadrupole deformation*
-------------------------
The quadrupole deformation of isotopic chain $Z=112$, $160\leq N
\leq 200$ with the force parameters NL3, NL-SH and NL-Z2 are listed in figure 1. There are three deformed chains 1, 2 and 3, denoted with circle, triangle and star respectively, which can be seen for all the force parameters NL3, NL-SH and NL-Z2 in figure 1. 1 is a oblate chain, the quantities of quadrupole deformation $\beta_{2}\leq-0.3$. 2 is a moderate or light prolate deformed chain except that there is light oblate deformation in the NL-Z2 calculation when $N\geq186$. It is very interesting that 3 is a super-deformed chain with $\beta_{2}\geq 0.4$ except $\beta_{2}$=0.34, 0.33, 0.32 for $N$=196, 198, 200 in the NL-SH calculation, but they are still around $\beta_{2}$=0.4. The phenomenon of several deformed configurations for a single superheavy element is predicted in Refs.[@d2; @d3] with constraint RMF calculation. In our work, it is the first time to obtain several deformed chains in a isotopic chain without using any constraint RMF calculations at all.
Is it a general phenomenon that there are several deformed chains for a isotopic chain of superheavy elements? To answer this question, we also perform RMF calculation for $Z=104$, $152\leq N
\leq 198 $ isotopic chain. The results are shown in figure 1 as well. All the results of the force parameters (NL3, NL-SH, and NL-Z2) show three deformed chains (oblate, moderate prolate and super-deformed prolate deformation ) for the $Z=104$ isotopic chain as well as $Z=112$. Although there are three deformed bands for the $Z=104$ isotopic chain, the super-deformed configurations appear with neutron number $N\geq 168$ (for NL3 and NL-Z2 calculation) or $N\geq 170$ (for NL-SH calculation). The oblate deformation appears in the region of $N \leq 168$ for NL3 calculation, $N \leq 170$ for NL-SH calculation, however, it appears in the whole region of $Z=104$ isotopic chain for the NL-Z2 calculation. Why oblate deformation does not appear in the whole region for NL3 and NL-SH calculations? Oblate deformed configurations which are predicted by NL-Z2 set but not by NL3 and NL-SH sets in the region. Are they physical solutions? In figure 1, we see there is a sudden change at $N=176$ in the oblate chain for the NL-Z2 calculation, when $N\geq 178$ the oblate deformation changes gradually with the variation of the neutron number. It indicates that the results of NL-Z2 calculation in the region $N\geq 178$ are physical solutions. The NL-Z2 set may be better than the other parameters in describing the deformation in larger neutron region. From the above mentioned analysis, we conclude that it is a general phenomenon that there are several deformed configurations for a isotopic chain of superheavy elements, and that the super-deformed nuclei can exist in the superheavy elements. Since there are several deformed configurations for the superheavy elements, is there shape coexistence in a superheavy element? This phenomenon will be discussed in the subsection 3.3 in detail.
From figure 1, we can see there are minima or maxima in the deformed chains. In super-deformed chain of $Z=112$, the minimum appears at $N=172$ for NL3, $N=168$ for NL-SH and $N=176$ for NL-Z2, and the maximum appears at $N=182$ for all the force parameters. And in super-deformed chain of $Z=104$, there is no minima for NL3; for NL-SH, the minimum is at $N=188$; for NL-Z2, sudden changes occur at $N=182,184$, which is the very minimum. In the moderate deformed chain of both $Z=112$ and $Z=104$, the minimum appears at around $N=184$ for all the three parameters. In oblate chain of $Z=112$, there is a maximum at $N=176$ for all the force parameters. There is no obvious kink for NL3 and NL-SH calculation in oblate chain of $Z=104$, but there is a large peak at $N=176$ and a small peak at $N=190$ for NL-Z2 calculation. Generally the minimum of the prolate deformed chain and the maximum or peak of oblate deformed chain correspond to shell closure. In the moderate deformed chain, the minimum ($\beta_{2}\approx 0$) appears at $N\simeq184$, which agrees with the prediction in refs.[@g1; @g2; @g3; @g4; @g5; @p3] that spherical neutron shell closures occur at $N=184$. For superdeformed heavy nuclei the shell closures depend on the force parameters. There is strong trend that shell closure occurs at $N=176$ for $Z=112$ in the oblate deformed chain, for all the parameters giving a peak at $N=176$. If the minimum or maximum is the sign of shell closure, the above analysis indicates that the positions of shell closure for the super-deformed and oblate deformed nuclei are different from that of spherical nuclei and have strong parameter dependence. We believe that there may be shell anomalism in oblate and super-deformed prolate superheavy nuclei. In fact, the phenomenon of shell anomalism is predicted in some Refs[@k1; @k2; @k3].
*Root mean square radii (rms)*
------------------------------
The root mean square (rms) radii are the project of investigation, for they contain a lot of important information of ground state properties. In figure 2, the rms radii of oblate, moderate prolate and super-deformed prolate configurations with NL3, NL-SH and NL-Z2 for the isotopic chains of $Z=112, 104$ are listed. The solid symbol stands for neutron and empty symbol stands for proton radii. The circles, up triangles and down triangles denote the oblate, moderate prolate and super- deformed configuration respectively.
In figure 2, it is seen that there is a gap $0.15\sim 0.38 fm$ for $Z=112$ and $0.17\sim 0.45 fm$ for $Z=104$ isotopic chain between neutron rms radii and proton rms radii of the same deformed configuration for all the force parameters. The gap become larger and larger with the increasing of the neutron number. For the three parameters NL3, NL-SH and NL-Z2, if we select the rms radii of NL3 as the standard, the NL-SH calculation underestimates the rms radii about $0.05 fm $ and the NL-Z2 calculation overestimates the rms radii about $0.1 fm$ in the same deformed configuration for both $Z=112$ and $Z=104$ isotopes. We also can see three obvious chains of rms radii for the three deformed configurations respectively in figure 2.
For $Z=112$ ($160\leq N\leq 200$) isotopic chain, NL3, NL-SH and NL-Z2 calculations show that the rms radii of moderate prolate deformed configuration are the smallest among the three deformed ones, and the rms radii of super-deformed one are the largest in the region $N \leq 190$, namely, nuclei with large absolute values $\beta_{2}$ tend to have larger rms radii in this region, however, when $N \geq 192$ the rms radii of oblate deformation are the largest. The gap of rms radii between the different deformed configurations is about $0.1\sim 0.2 fm$ in general. For moderate prolate deformed configuration, there are obvious kinks in the proton rms radii at $N=184$, which agrees with the results in subsection 3.1 that there is shell closure at N=184. For oblate deformed one, the calculation of NL3, NL-SH and NL-Z2 shows kink in the proton rms radii at $N=176$, but the NL-Z2 calculation is not very obvious. Together with the prediction of subsection 3.1, we are convinced that there is shell closure at $N=176$ for oblate deformed configuration. Finally, for super-deformed configuration, the NL3 and NL-SH calculations show kinks at $N \simeq 168, 184$ in the proton rms radii, and the kink appears at $N \simeq 190$ for NL-Z2, there also are obvious kinks at $N\simeq 186$ for NL3 and NL-SH at $N\simeq 192$ in proton rms radii. In summary, the characters of shell closure for super-deformed configuration are not very obvious in the region $N<184$, and have strong parameter dependence. There are some sudden changes at $N=198,200$ for moderate prolate deformed configuration, at $N=196, 198, 200$ for super-deformed one with NL-SH calculation and at $N=198,200$ for oblate deformed configuration with NL-Z2 calculation. The anomalism maybe come from the validity of the force parameter in these regions.
For $Z=104$ ($152\leq N\leq 198$) isotopic chain, figure 2 shows that the rms radii of moderate prolate deformed configuration are larger than those of oblate one for both NL3 and NL-SH calculations. The gap between them is about $0.03\sim 0.07 fm$. For NL-Z2 calculation, it scarcely shows any gap in the rms radii between moderate prolate and oblate deformed configuration when $152\leq N \leq 176$ in figure 2. All the calculations of NL3, NL-SH and NL-Z2 indicate that the radii of super-deformed configuration are larger than that of the moderate. The radii of oblate deformed configuration for NL-Z2 in the region $N\geq 178$ lie between the radii of super-deformed configuration and the moderate one. When $N\geq 186$, it is very close to the super-deformed configuration’s. As a whole, nuclei with large absolute values $\beta_{2}$ tend to have larger rms radii. In figure 2, obvious kinks can be seen at $N=184$ in the proton rms radii for moderate deformed configuration of all the parameters, which consists with the prediction in subsection 3.1 that $N=184$ is a magic number and shell closure exists there. Then we see the results for superdeformed configuration in figure 2. The rms radii suddenly become small at $N=172$ with NL3 calculation. $N=172$ is a magic number for spherical nuclei in many Refs.[@p1; @p3]. Is $N=172$ still a magic number for superdeformed configuration, and does the sudden change come from the shell closure at $N=172$ as well? It is very difficult to answer the question, because the other two force parameters NL-SH and NL-Z2 can not reproduce that phenomenon. There is also sudden change at $N=194$ with NL3 calculation. The NL-SH calculation shows that the rms radii change gradually with the variation of neutron number and there is a kink at $N=176$, from which the radii increases much more slowly with the increasing of neutron. It is seen that the rms radii suddenly become small at $N=182$ and 184 with NL-Z2 calculation. Is it caused by the shell closure at around $N=184$? However, it is not predicted by NL3 and NL-SH calculations. Finally, let’s see the results of oblate deformed configuration. We only list the results in the region $152\leq N \leq 168$ for NL3 and $152\leq N \leq
170$ for NL-SH, because when we perform calculation with NL3 at $N=170$ and with NL-SH at $N=172$, the results suddenly change to moderate deformed configuration’s. The calculation with NL-Z2 set shows there is sudden change at N=176. In summary, it is a very strange region in $N=168\sim 178$, there maybe exist complicate shell structure.
From the analysis for the rms radii of $Z=112$ and 104 isotopes, we find that there is strong parameter dependence in predicting the position of shell closure for oblate and super-deformed configuration. Shell anomalism maybe occur in the oblate and super-deformed configuration.
---------- -------------- --------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- --
[$N$ ]{} $B_{1}$(MeV) $B_{2}$ (MeV) $B_{3}$(MeV) $B_{1}$(MeV) $B_{2}$(MeV) $B_{3}$(MeV) $B_{1}$(MeV) $B_{2}$(MeV) $B_{3}$(MeV)
160 1946.55 1958.84 1953.41 1948.74 1962.90 1955.63 1946.00 1955.60 1952.48
162 1961.41 1973.53 1968.65 1963.73 1977.47 1970.63 1960.83 1970.81 1967.61
164 1975.69 1987.53 1982.56 1978.47 1991.75 1984.57 1975.06 1984.51 1981.87
166 1989.47 2001.13 1995.92 1992.56 2005.42 1998.00 1988.91 1997.96 1995.55
168 2002.88 2014.09 2009.13 2005.83 2017.59 2011.23 2002.49 2011.19 2008.79
170 2015.93 2026.18 2022.19 2018.61 2028.06 2024.43 2015.78 2023.79 2021.84
172 2028.60 2037.76 2035.05 2031.03 2040.07 2037.43 2028.69 2036.09 2034.51
174 2040.84 2049.03 2047.34 2043.28 2051.35 2049.84 2041.13 2047.98 2046.75
176 2052.75 2059.90 2058.66 2054.83 2062.30 2060.72 2053.22 2058.94 2058.62
178 2064.00 2070.54 2069.42 2065.97 2072.74 2070.93 2065.05 2070.14 2070.03
180 2074.84 2080.39 2079.65 2076.35 2082.60 2080.87 2076.62 2080.34 2080.93
182 2085.21 2090.59 2089.50 2086.11 2092.27 2090.59 2087.72 2090.29 2091.18
184 2094.79 2100.68 2099.03 2095.33 2101.54 2099.97 2097.96 2100.84 2100.90
186 2103.90 2108.92 2108.07 2104.13 2109.80 2108.89 2107.68 2109.50 2110.45
188 2112.76 2116.72 2116.79 2112.56 2118.12 2117.82 2117.09 2117.75 2119.70
190 2121.26 2124.88 2125.48 2120.50 2126.36 2126.67 2126.22 2125.67 2128.76
192 2129.42 2132.67 2134.31 2128.10 2134.23 2135.18 2135.07 2133.36 2137.73
194 2137.31 2139.97 2142.95 2135.52 2141.62 2143.35 2143.66 2141.01 2146.74
196 2144.84 2146.65 2151.09 2142.76 2148.55 2151.00 2151.76 2148.87 2155.49
198 2151.98 2152.87 2158.68 2149.67 2152.87 2158.97 2159.33 2156.53 2163.79
200 2158.78 2165.98 2165.99 2156.29 2161.52 2166.40 2167.00 2163.21 2171.58
---------- -------------- --------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- --
: Binding energies for $Z=112$ isotopic chains with calculation of NL3, NL-SH and NL-Z2 sets. $B_{1},
B_{2}$ and $B_{3}$ denote the binding energies of oblate, moderate prolate and super-deformed prolate configurations. $N$ is the neutron number. []{data-label=""}
---------- -------------- --------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- --
[$N$ ]{} $B_{1}$(MeV) $B_{2}$ (MeV) $B_{3}$(MeV) $B_{1}$(MeV) $B_{2}$(MeV) $B_{3}$(MeV) $B_{1}$(MeV) $B_{2}$(MeV) $B_{3}$(MeV)
152 1876.43 1889.25 1878.83 1892.57 1875.29 1887.03
154 1890.14 1902.90 1892.08 1906.31 1889.64 1900.68
156 1903.29 1915.89 1904.97 1919.03 1902.92 1913.95
158 1915.93 1928.35 1917.46 1931.35 1915.50 1926.77
160 1928.13 1940.28 1929.56 1943.22 1927.78 1938.71
162 1939.99 1951.40 1941.38 1953.99 1940.00 1950.02
164 1951.44 1961.15 1952.84 1964.21 1952.00 1960.09
166 1962.32 1970.77 1963.94 1974.32 1963.79 1969.74
168 1972.60 1980.32 1976.98 1974.68 1984.01 1978.43 1975.48 1979.02 1978.00
170 1989.86 1986.94 1985.01 1993.13 1988.48 1986.82 1988.72 1988.07
172 1999.24 1997.96 2003.14 1997.96 1997.78 1998.55 1997.77
174 2008.53 2005.61 2012.32 2006.88 2008.14 2008.02 2007.25
176 2017.68 2014.23 2021.20 2015.58 2018.38 2017.68 2016.55
178 2026.58 2022.61 2029.66 2023.68 2024.41 2027.31 2025.90
180 2034.82 2030.82 2037.56 2031.36 2033.09 2035.74 2034.88
182 2042.81 2038.60 2045.37 2038.75 2041.30 2044.36 2040.45
184 2050.54 2045.89 2052.65 2045.97 2049.34 2052.73 2048.21
186 2056.40 2052.80 2058.30 2051.94 2056.83 2059.16 2056.42
188 2061.92 2059.48 2063.67 2058.63 2063.97 2065.98 2063.95
190 2069.40 2066.02 2069.59 2065.06 2071.20 2073.86 2071.44
192 2075.42 2072.37 2075.05 2071.31 2078.14 2080.90 2078.79
194 2081.42 2079.81 2079.84 2077.46 2084.85 2087.56 2085.65
196 2087.61 2085.75 2086.46 2083.17 2091.27 2093.73 2092.12
198 2093.75 2091.22 2092.22 2098.35 2097.51 2100.33 2098.32
---------- -------------- --------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- --
: Binding energies for $Z=104$ isotopic chains with calculation of NL3, NL-SH and NL-Z2 sets. $B_{1},
B_{2}$ and $B_{3}$ denote the binding energies of oblate, moderate prolate and super-deformed prolate configurations. $N$ is the neutron number. []{data-label=""}
*Binding energy*
----------------
In the above subsections, we have discussed the deformation and root mean square radii of $Z=112, 104$ isotopic chains, and find that there are three deformed chains for each isotopic chain. Since there are multi-deformed configurations for the superheavy nuclei, is there shape coexistence in them? In this subsection, we will discuss it in detail. The binding energies of the three deformed chains with the force parameters NL3, NL-SH and NL-Z2 are listed in table 1 and table 2 for $Z=112,104$ respectively. $B_{1}, B_{2}$ and $B_{3}$ denote the binding energies of oblate, moderate prolate and super-deformed prolate configurations respectively. The binding energies with NL3, NL-SH and NL-Z2 in the same deformed chain are very close, the difference between them is not larger than 0.05 percent (about 10 MeV). One of the important conditions for the shape coexistence is that the deference of binding energies between two deformed configurations is very small, usually less than 1 MeV. If it is larger than 1 MeV, the translation between two different configurations is very difficult and the probability of shape coexistence is very little. To find the sign of shape coexistence, we compare the binding energies of different deformed chain with each other, and list the results in figure 3. The circle denotes the values of $B_{1}-B_{2}$, triangle is for $B_{2}-B_{3}$ and star is for $B_{1}-B_{3}$. In figure 3, it is seen that all the values of $B_{1}-B_{2}$, $B_{2}-B_{3}$ and $B_{1}-B_{3}$ are in the region $-14.5\sim 8$ MeV. It is obvious that the abstract values of them decrease with the increasing of the neutron number until to a certain region ($N\sim 184$) and the trend changes when $N > 184$ for $Z=112$ isotopic chain. For $Z=104$ isotopic chain, the phenomenon is less clear than that of $Z=112$. It is interesting that these values are very close to the fission barriers as high as $8\sim12$ MeV around the double-shell closure $Z =114,
N=184$[@t0; @t1; @t2]. This means the deformation is important for the stability of superheavy nuclei. From figure 3, we can also clearly find in which region the shape coexistence exists and what kind of shape coexistence is in the long isotopic chain.
For $Z=112$ isotopic chain, from table 1 and figure 3, we can see $B_{2}$ is always the largest one and $B_{1}$ is the smallest among $B_{1}, B_{2}$ and $B_{3}$ in the region $N\leq 186$ for NL3, $N\leq 188$ for NL-SH and $N\leq 178$ for NL-Z2 calculation respectively. It indicates that the moderate prolate deformed configuration is the ground state of nuclei $N\leq 186$ for NL3, $N\leq 188$ for NL-SH and $N\leq 178$ for NL-Z2 calculation and the oblate or superdeformed configuration is the exciting state of them. The most distinct character is that $B_{3}$ is the largest one when $N\geq 188$ for NL3, $N\geq 190$ for NL-SH and $N\geq
180$ for NL-Z2 calculation, namely, there is a chain of super-deformed configurations which become the ground state and are more stable than the other deformed configurations. Although there are some differences with the different force parameters in predicting the region of super-deformation, all the parameters’ calculation show there is a more stable super-deformed region in the $Z=112$ isotopic chain. It agrees with the Bohr and Mottelson’ suggestion[@t] that deformation can increase the stability of the heavy nuclei. But it should be noted that it is not for all heavy nuclei that deformation can increase the stability. This phenomenon appears only in special region of superdeformed chain. Figure 3 also shows that the results with NL3, NL-SH and NL-Z2 set are different in some details, but the total trend is consistent. In figure 3, the star is below and far from the dotted line $\delta E$=-1 MeV, it means that it is impossible for oblate and super-deformed prolate deformation to coexist in $Z=112$ isotopes . There is no shape coexistence in the region $N\leq 172$, for no symbols in the region 1 MeV $\leq\delta E\leq-1$ MeV as shown in figure 3. The NL3 results show there maybe exist moderate- and super-deformed shape coexistence in the region $176\leq N\leq 190$ of $Z=112$ isotopes, for the value of $B_{2}-B_{3}$ (triangles) is around 1 MeV or within 1 MeV. Although the NL-SH results for $B_{2}-B_{3}$ (triangles) are larger than 1 MeV in the region $174\leq N\leq 184$, they are very close to 1 MeV, together with the values within 1 MeV, we believe that moderate- and super-deformed shape coexistence maybe occur in $174\leq N\leq
192$. The NL-Z2 results predict the moderate- and super-deformed shape coexistence should occur in $174\leq N\leq 186$. It is interesting that all the regions of shape coexistence predicted by the parameters NL3, NL-SH and NL-Z2 contain the magic number N=184. It maybe have some trends that shape coexistence occur usually at around the magic number. NL3 calculation also shows moderate prolate and oblate deformation coexist at $N=198, Z=112$, and NL-Z2 shows moderate prolate and oblate deformation coexist at $N=188,190$. The calculation of Ref.[@d3] predicts $N=172,Z=112$ exists shape coexistence, but our result does not indicate there is any shape coexistence.
On the other hand, for $Z=104$ isotopic chain, from table 2 and figure 3, we can find there is no super-deformed configuration until $ N\geq 168$ and no oblate deformed configuration when $
N\geq 168$ for NL3, $ N\geq 170$ for NL-SH calculation. $B_{2}$ is always larger than $B_{3}$ except $N=198$ for NL-SH calculation. It may be an anomalism or un-physical solution. Thus, for $Z=104$ isotopic chain, the moderate prolate deformed configuration may be the ground state, and the deformation can not increase the stability in this case at all. In figure 3, the results of NL3 and NL-SH are very close, the triangles, circles or stars are not in the region -1 MeV$\leq \delta E \leq$ 1 MeV. It indicates there is no shape coexistence for $Z=104$ isotopes with NL3 and NL-SH calculations. But the results that NL-Z2 parameter shows are very different from those of NL3 and NL-SH sets. It predicts moderate prolate and oblate deformation maybe coexist at $N=172,174$ and 176, moderate prolate and super-deformation maybe coexist in $168\leq N \leq 180$ and oblate and super-deformation maybe coexist at $N=172, 174 $ and in the region $182\leq N \leq 198$. The shape coexistence also occurs around the magic number $N=174$ and 184.
Summary
=======
We use the considered axial deformed relativistic mean field theory to perform systematical calculations for $Z=112$ and 104 isotopic chains with force parameters NL3, NL-SH and NL-Z2 sets. First, three deformed chains (oblate, moderate prolate and super-deformed chain) are found for $Z=112$ and 104 isotopic chains. Second, a super-deformed chain can be the ground states for $Z=112$ isotopic chain when $N\geq 188$ with NL3, $N\geq 190$ with NL-SH and $N\geq 180$ with NL-Z2 calculation. This confirms that the deformation can increase the stability of some superheavy nuclei. Third, although shape coexistence is a general phenomenon for superheavy nuclei, it only appears in some special regions of an isotopic chain, and only some special kinds of shape coexistence can exist. For $Z=112$ isotopic chain, it is predicted that moderate prolate and super-deformation coexist in the region $174\leq N\leq 184$ for NL3 set, $174\leq N\leq 192$ for NL-SH set, $174\leq N\leq 186$ for NL-Z2 set. There is no other kind of shape coexistence for $Z=112$ isotopic chain except moderate prolate and oblate deformation at $N=198 $ for NL3 set, moderate prolate and oblate deformation at $N=188,190$ for NL-Z2 set. For $Z=104$ isotopic chain, there is no shape coexistence with NL3 and NL-SH calculations. However, the NL-Z2 calculation predicts that moderate prolate and oblate deformation maybe coexist at $N=172,174$ and 176, moderate prolate and super-deformation maybe coexist in $168\leq N \leq 180$ and oblate and super-deformation maybe coexist at $N=172, 174 $ and in the region $182\leq N \leq
198$. We predict that shape coexistence maybe occur around the magic number. Finally, it is found that there is shell closure at $N=184$ ($\beta_{2}\approx 0$) for moderate prolate deformed chains of $Z=112$ and 104, at $N=176$ for oblate deformed chain of $Z=112$, the position of shell closure have strong parameter dependence for super-deformed chains of $Z=112$ and 104. It is confirmed the prediction that there is shell anomalism for oblate or superdeformed nuclei.
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[^1]: E-mail: [email protected]
[^2]: E-mail: [email protected]
[^3]: E-mail: [email protected]
[^4]: Supported by National Natural Science Foundation of China (10275037) and Specialized Research Fund for the Doctoral Program of Higher Education of China (20010055012)
|
---
abstract: 'Natural language offers an intuitive and flexible means for humans to communicate with the robots that we will increasingly work alongside in our homes and workplaces. Recent advancements have given rise to robots that are able to interpret natural language manipulation and navigation commands, but these methods require a prior map of the robot’s environment. In this paper, we propose a novel learning framework that enables robots to successfully follow natural language route directions without any previous knowledge of the environment. The algorithm utilizes spatial and semantic information that the human conveys through the command to learn a distribution over the metric and semantic properties of spatially extended environments. Our method uses this distribution in place of the latent world model and interprets the natural language instruction as a distribution over the intended behavior. A novel belief space planner reasons directly over the map and behavior distributions to solve for a policy using imitation learning. We evaluate our framework on a voice-commandable wheelchair. The results demonstrate that by learning and performing inference over a latent environment model, the algorithm is able to successfully follow natural language route directions within novel, extended environments.'
author:
- |
Sachithra Hemachandra$^*$ Felix Duvallet$^*$ Thomas M. Howard\
Nicholas Roy Anthony Stentz Matthew R. Walter[^1][^2][^3][^4] [^5]
bibliography:
- 'references.bib'
title: |
**Learning Models for Following Natural Language Directions\
in Unknown Environments**
---
=1
Introduction {#sec:introduction}
============
Over the past decade, robots have moved out of controlled isolation and into our homes and workplaces, where they coexist with people in domains that include healthcare and manufacturing. One long-standing challenge to realizing robots that behave effectively as our partners is to develop command and control mechanisms that are both intuitive and efficient. Natural language offers a flexible medium through which people can communicate with robots, without requiring specialized interfaces or significant prior training. For example, a voice-commandable wheelchair [@hemachandra11] allows the mobility-impaired to independently and safely navigate their surroundings simply by speaking to the chair, without the need for traditional head-actuated switches or sip-and-puff arrays. Recognizing these advantages, much attention has been paid of late to developing algorithms that enable robots to interpret natural language expressions that provide route directions [@macmahon06; @kollar10; @chen11; @matuszek12], that command manipulation [@tellex11; @howard14a], and that convey environment knowledge [@walter13; @hemachandra14].
![Our goal is to enable robots to autonomously follow natural language commands without any prior knowledge of their environment.[]{data-label="fig:go_to_kitchen_down_hallway"}](./graphics/wheelchair-commanding2.pdf){width="1.0\linewidth"}
Natural language interpretation becomes particularly challenging when the expression references areas in the environment unknown to the robot. Consider an example in which a user directs the voice-commandable to “go to the kitchen that is down the hallway,” when the wheelchair is in an unknown environment and the hallway and kitchen are outside the field-of-view of its sensors (Fig. \[fig:go\_to\_kitchen\_down\_hallway\]). Unable to associate the hallway and kitchen with specific locations, most existing solutions to language understanding would result in the robot exploring until it happens upon a kitchen. By reasoning over the spatial and semantic environment information that the command conveys, however, the robot would be able to follow the spoken directions more efficiently.
In this paper, we propose a framework that follows natural language route directions within unknown environments by exploiting spatial and semantic knowledge implicit in the commands. There are three algorithmic contributions that are integral to our approach. The first is a learned language understanding model that efficiently infers environment annotations and desired behaviors from the user’s command. The second is an estimation-theoretic algorithm that learns a distribution over hypothesized world models by treating the inferred annotations as observations of the environment and fusing them as observations from the robot’s sensor streams (Fig. \[fig:belief\_world\]). The third is a belief space policy learned from human demonstrations that reasons directly over the world model distribution to identify suitable navigation actions.
This paper generalizes previous work by the authors [@duvallet14], which was limited to object-relative navigation within small, open environments. The novel contributions of this work enable robots to follow natural language route directions in large, complex environments. They include: a hierarchical framework that learns a compact probabilistic graphical model for language understanding; a semantic map inference algorithm that hypothesizes the existence and location of regions in spatially extended environments; and a belief space policy learned from human demonstrations that considers spatial relationships with respect to a hypothesized map distribution. We demonstrate these advantages through simulations and experiments with a voice-commandable wheelchair in an office-like environment.
Related Work {#sec:related_work}
============
Recent advancements in language understanding have enabled robots to understand free-form commands that instruct them to manipulate objects [@tellex11; @howard14a] or navigate through environments using route directions [@macmahon06; @kollar10; @chen11; @howard14a; @matuszek12a]. With few exceptions, most of these techniques require a priori knowledge of location, geometry, colloquial name, and type of all objects and regions within the environment [@kollar10; @howard14a; @tellex11]. Without known world models, however, interpreting free-form commands becomes much more difficult. Existing methods have dealt with this by learning a parser that maps the natural language command directly to plans [@macmahon06; @chen11; @matuszek12a]. Alternatively, Duvallet et al. [@duvallet2013] use imitation learning to train a policy that reasons about uncertainty in the grounding and that is able to backtrack as necessary. However, none of these approaches explicitly utilize the knowledge that the instruction conveys to influence their models of the environment, nor do they reason about its uncertainty. Instead, our framework treats language as an additional, albeit noisy, sensor that we use to learn a distribution over hypothesized world models, by taking advantage of information implicitly contained in a given command.
Related to our algorithm’s ability to learn world models, state-of-the-art semantic mapping frameworks exist that focus on using the robot’s sensor observations to update its representation of the world [@zender08; @pronobis10]. Some methods additionally incorporate natural language descriptions in order to improve the learned world models [@walter13; @hemachandra14]. These techniques, however, only use language to update regions of the environment that the robot has observed and are not able to extend the maps based on natural language. Our approach treats natural language as another sensor and uses it to extend the spatial representation by adding both topological and metric information regarding hypothesized regions in the environment, which is then used for planning. Williams et al. [@Williams2013] use a cognitive architecture to add unvisited locations to a partial map. However, they only reason about topological relationships to unknown places, do not maintain multiple hypotheses, and make strong assumptions about the environment that limit the applicability to real systems. In contrast, our approach reasons both topologically and metrically about regions, and can deal with ambiguity, which allows us to operate in challenging environments.
Approach Overview {#sec:overview}
=================
We define natural language direction following as one of inferring the robot’s trajectory $x_{t+1:T}$ that is most likely for a given command $\Lambda^t$: $${\ensuremath{\underset{x_{t+1:T} \, \in \, \Re^{n}}{\arg \! \max}\;}}
p\left(x_{t+1:T} | \Lambda^t, z^t, u^t \right),
\label{eqn:problem-statement}$$ where $z^t$ and $u^t$ are the history of sensor observations and odometry data, respectively. Traditionally, this problem has been solved by also conditioning the distribution over a known world model. Without any a priori knowledge of the environment, we treat this world model as a latent variable $S_t$. We then interpret the natural language command in terms of the latent world model, which results in a distribution over behaviors $\beta_t$. We then solve the inference problem by marginalizing over the latent world model and behaviors: $$\begin{split}
{\ensuremath{\underset{x_{t+1:T} \, \in \, \Re^{n}}{\arg \! \max}\;}}
\int\displaylimits_{\beta_t} \int\displaylimits_{S_{t}} p(x_{t+1:T}
&\vert \beta_t, S_{t}, \Lambda^t) \cdot
p(\beta_t \vert S_t, \Lambda^t)\\
&\cdot p(S_{t} \vert \Lambda^t) \, dS_{t} \, d\beta_t,
\end{split}\label{eqn:marginalization}$$ where we have omitted the measurement $z^t$ and odometry $u^t$ histories for lack of space.
By structuring the problem in this way, we are able to treat inference as three coupled learning problems. The framework (Fig. \[fig:framework\]) first converts the natural language direction into a set of environment annotations using learned language grounding models. It then treats these annotations as observations of the environment (i.e., the existence, name, and relative location of rooms) that it uses together with data from the robot’s onboard sensors to learn a distribution over possible world models (third factor in Eqn. \[eqn:marginalization\]). Our framework then infers a distribution over behaviors conditioned upon the world model and the command (second factor). We then solve for the navigation actions that are consistent with this behavior distribution (first factor) using a learned belief space policy that commands a single action to the robot. As the robot executes this action, we update the world model distribution based upon new utterances and sensor observations, and subsequently select an updated action according to the policy. This process repeats as the robot navigates.
The rest of this paper details each of these components in turn. We then demonstrate our approach to following natural language directions through large unstructured indoor environments on the robot shown in \[fig:go\_to\_kitchen\_down\_hallway\] as well as simulated experiments. We additionally evaluate our approach to learning belief space policies on a corpus of natural language directions through one floor of an indoor building.
(annotation-inference) at (0,0) [annotation inference]{}; (semantic-mapping) at (0,-2) [semantic mapping]{}; (behavior-inference) at (3.5,0) [behavior inference]{}; (policy-planner) at (3.5,-2) [policy planner]{}; (robot) at (-3.0,-2.0) [![Outline of the framework.[]{data-label="fig:framework"}](graphics/robot.png "fig:"){width="1.5cm"}]{}; (annotation-inference) – (semantic-mapping); (semantic-mapping) –(1.75,-2) – (1.75,0) – (behavior-inference); (semantic-mapping) – (policy-planner); (behavior-inference) – (policy-planner); (observationlabel) at (2.325,-1.0) [map distribution]{}; (annotationdistributionlabel) at (0.625,-1.0) [annotation distribution]{}; (behaviordistributionlabel) at (4.125,-1.0) [behavior distribution]{}; (speechlabel) at (-3.0,0) [“go to the kitchen that is down the hallway”]{}; (robot) – (speechlabel); (robot) – (semantic-mapping); (speechlabel) – (-3.0,1.0) – (0.0,1.0) – (annotation-inference); (speechlabel) – (-3.0,1.0) – (3.5,1.0) – (behavior-inference); (policy-planner) – (3.5,-3.5) – (-3.0,-3.5) – (robot); (observationslabel) at (-1.5,-1.75) [observations]{}; (parsetreelabel) at (0,1.25) [parse tree(s)]{}; (actionlabel) at (0,-3.25) [action]{};
Natural Language Understanding {#sec:nlu}
==============================
Our framework relies on learned models to identify the existence of annotations and behaviors conveyed by free-form language and to convert these into a form suitable for semantic mapping and the belief space planner. This is a challenge because of the diversity of natural language directions, annotations, and behaviors. We perform this translation using the Hierarchical Distributed Correspondence Graph (HDCG) model [@howard14b], which is a more efficient extension of the Distributed Correspondence Graph (DCG) [@howard14a]. The DCG exploits the grammatical structure of language to formulate a probabilistic graphical model that expresses the correspondence $\phi \in \Phi$ between linguistic elements from the command and their corresponding constituents (*groundings*) $\gamma \in \Gamma$. The factors $f$ in the DCG are represented by log-linear models with feature weights that are learned from a training corpus. The task of grounding a given expression then becomes a problem of inference on the DCG model.
The HDCG model employs DCG models in a hierarchical fashion, by inferring rules $\mathtt{R}$ to construct the space of groundings for lower levels in the hierarchy. At any one level, the algorithm constructs the space of groundings based upon a distribution over the rules from the previous level: $$\Gamma \rightarrow \Gamma\left(\mathtt{R}\right).$$ The HDCG model treats these rules and, in turn, the structure of the graph, as latent variables. Language understanding then proceeds by performing inference on the marginalized models: $$\begin{aligned}
&\operatorname*{arg\,max}_\Phi \int_{\mathtt{R}} p\left(\Phi\vert\mathtt{R},\Gamma\left(\mathtt{R}\right),\Lambda,\Psi\right) p\left(\mathtt{R} \vert \Gamma\left(\mathtt{R}\right),\Lambda,\Psi\right)\\
&\operatorname*{arg\,max}_\Phi \int_{\mathtt{R}} \prod_{i} \prod_{j} f\left(\Phi_{i_{j}},\Gamma_{i_{j}}\left(\mathtt{R}\right),\Lambda_{i},\Psi,\mathtt{R}\right)\times\\
&\qquad \qquad \qquad \prod_{i} \prod_{j} f\left(\mathtt{R},\Lambda_{i},\Psi,\Gamma_{i_{j}}\left(\mathtt{R}\right)\right).\nonumber\end{aligned}$$ We now describe how the HDCG model infers annotations (representing our knowledge of the environment inferred from the language) and behaviors (representing the intent of the command) to understand the natural language command given by the user.
Annotation Inference
--------------------
An annotation is a set of object types and subspaces. A subspace is defined here as a spatial relationship (e.g., down, left, right) with respect to an object type. In the experiments described in Section \[sec:results\] we assume 17 object types and 12 spatial relationships. We also permit object types to express a spatial relationship with another object type. We denote object types by their physical type (e.g., kitchen, hallway), subspaces as the relationship type with an object type argument (e.g., down(kitchen), left(hallway)), and object types with spatial relationships as an object type with a subspace argument (e.g., kitchen(down(hallway))). Since the number of possible combinations of annotations is equal to the power set of the number of symbols, $2^{3,485}$ annotations can be expressed by an instruction.[^6] The HDCG model infers a distribution of graphical models to efficiently generate annotations by assuming conditional independence of constituents and eliminating symbols that are learned to be irrelevant to the utterance. For example, Figure \[fig:annotation-inference\] illustrates the model for the direction “go to the kitchen that is down the hall.” In this example only 4 of the 3,485 symbols (two object types, one subspace, and one object type with a spatial relationship) are active in this model. Note that all factors with inactive correspondence variables are not illustrated in Figures \[fig:annotation-inference\] and \[fig:behavior-inference\]. At the root of the sentence the symbols for an object type (kitchen) and an object type with a spatial relationship (kitchen(down(hallway))) are sent to the semantic map to fuse with other observations.
Behavior Inference
------------------
A behavior is a set of objects, subspaces, actions, objectives, and constraints. Behavior inference differs from annotation inference by considering objects from the semantic map and subspaces defined with respect to objects from the semantic map instead of only object types. We denote actions by their type and an object or subspace argument (e.g., navigate(hallway)), objectives by their type (e.g., quickly, safely), and constraints as objects with spatial relationship from the semantic map (e.g., $o_{4}$(down($o_{3}$))). In the experiments presented in Section \[sec:results\] we assume 4 action types, 3 objectives, and 12 spatial relations. Just as with annotation inference, the HDCG model eliminates irrelevant action types, objective types, objects, and spatial relationships to efficiently infer behaviors. Figure \[fig:behavior-inference\] illustrates the model for the direction “go to the kitchen that is down the hall” in the context of an inferred map. In this example a *navigate* action with a goal relative to $o_{1}$ would be inferred as the most likely behavior for the policy planner.
Semantic Mapping
================
We represent the world model as a modified *semantic map* [@walter13] $S_t = \{G_t, X_t\}$, a hybrid metric and topological representation of the environment. The topology $G_t$ consists of nodes $n_i$ that denote locations in the environment, edges that denote inter-node connections, and non-overlapping regions $R_\alpha = \{n_1, n_2, \ldots, n_m\}$ that represent spatially coherent areas compatible with a human’s decomposition of space (e.g., rooms and hallways). We associate a pose $x_i$ with each node $n_i$, the vector of which constitutes the metric map $X_t$. Each region is also labeled according to its type (e.g., kitchen, hallway). An edge connects two regions that the robot has transitioned between or for which language indicates the existence of an inter-region spatial relation (e.g., that the kitchen is “down” the hallway).
Annotations extracted from a given command provide information regarding the existence, relative location, and type of regions[^7] in the environment. We learn a distribution over world models consistent with these annotations by treating them as observations $\alpha_t$ in a filtering framework. We combine these observations with those from other sensors onboard the robot (LIDAR and region appearance observations) $z_t$ to maintain a distribution over the semantic map:
$$\begin{aligned}
p(S_t \vert \Lambda^t, z^t, u^t)\! &\approx p(S_t \vert
\alpha^t, z^t, u^t)\\
\!&= p(G_t, X_t, \vert \alpha^t, z^t, u^t)\\
\!&= p(X_t \vert G_t, \alpha^t, z^t, u^t) p(G_t \vert \alpha^t, z^t, u^t),
\end{aligned}$$
where we assume that an utterance $\Lambda^t$ provides a set of annotations $\alpha_t$. The factorization within the last line models the metric map induced by the topology, as with pose graph representations [@kaess08]. We maintain this distribution over time using a Rao-Blackwellized particle filter (RBPF) [@doucet00], with a sample-based approximation of the distribution over the topology, and a Gaussian distribution over metric poses.
The robot observes transitions between environment regions and the semantic label of its current region. As scene understanding is not the focus of this work, we use AprilTag fiducials [@Olson2011] placed in each region that denotes its label. Unlike our earlier work [@hemachandra14] in which we segment regions based only on their spatial coherence using spatial clustering, here we additionally use the presence of conflicting spatial appearance tags to also segment the region. As such, we assume that we are aware of the segmentation of the space immediately, which is not possible with a purely spectral clustering based approach, allowing us to immediately evaluate each particle’s likelihood based on the observation of region appearance. In turn, we can down-weight particles that are inconsistent with the actual layout of the world sooner, reducing the number of actions the robot must take to satisfy the command.
We maintain each particle through the three steps of the RPBF. First, we propagate the topology by sampling modifications to the graph when the robot receives new sensor observations or annotations. Second, we perform a Bayesian update to the pose distribution based upon the sampled modifications to the underlying graph. Third, we update the weight of each particle based on the likelihood of generating the given observations, and resample as needed to avoid particle depletion. We now outline this process in more detail.
During the proposal step, we first add an additional node $n_t$ and edge to each particle’s topology that model the robot’s motion $u_t$, yielding a new topology $S_t^{(i)-}$. We then sample modifications to the topology $\Delta_t^{(i)} = \{\Delta_{\alpha_t}^{(i)}, \Delta_{z_t}^{(i)}\}$ based on the most recent annotations $\alpha_t$ and sensor observations $z_t$: $$\begin{gathered}
\label{eq:sample_graph}
p(S_t^{(i)} | S_{t-1}^{(i)}, \alpha_t, z_t, u_t)=p(\Delta_{\alpha_t}^{(i)} | S_{t}^{(i)-}, \alpha_t)\, \\
p(\Delta_{z_t}^{(i)} | S_{t}^{(i)-}, z_t)\,
p(S_t^{(i)-} | S_{t-1}^{(i)}, u_t).\end{gathered}$$ This updates the proposed graph topology $S_{t}^{(i)-}$ with the graph modifications $\Delta_t^{(i)}$ to yield the new semantic map $S_t^{(i)}$. The updates can include the addition and deletion of nodes and regions from the graph that represent newly hypothesized or observed regions, and edges that express express spatial relations inferred from observations or annotations.
We sample graph modifications from two independent proposal distributions for annotations $\alpha_t$ and robot observations $z_t$. This is done by sampling a grounding for each observation and modifying the graph according to the implied grounding.
Graph modifications based on natural language
---------------------------------------------
Given a set of annotations , we sample modifications to the graph for each particle. An annotation $\alpha_{t,j}$ contains a spatial relation and figure when the language describes one region (e.g., “go to the elevator lobby”), and an additional landmark when the language describes the relation between two regions (e.g., “go to the lobby through the hallway”). We use a likelihood model over the spatial relation to sample landmark and figure pairs for the grounding. This model employs a Dirichlet process prior that accounts for the fact that the annotation may refer to regions that exist in the map or to unknown regions. If either the landmark or the figure are sampled as new regions, we add them to the graph and create an edge between them. We also sample the metric constraint associated with this edge based on the spatial relation. The spatial relation models employ features that describe the locations of the regions, their boundaries, and robot’s location at the time of the utterance, and are trained based upon a natural language corpus [@tellex11].
Graph modifications based on robot observations
-----------------------------------------------
If the robot does not observe a region transition (i.e. the robot is in the same region as before), the algorithm adds the new node $n_t$ to the current region and modifies its spatial extent. If there are any edges denoting spatial relations to hypothesized regions, the algorithm resamples their constraint if its likelihood changes significantly due to the modified spatial extent of the current region. Alternatively, if the robot observes a region transition, the new node $n_t$ is assigned to a new or existing region as follows. First, the algorithm checks if the robot is in a previously visited region, based on spatial proximity, in which case it will add $n_t$ to that region. Otherwise, it will create a new region and check whether it matches a region that was previously hypothesized based on an annotation (for example, a newly-visited kitchen can be the same as a hypothesized kitchen described with language). We do so by sampling a grounding to any unobserved regions in the topology using a Dirichlet process prior. If this process results in a grounding to an existing hypothesized region, we remove the hypothesized region and adjust the topology accordingly, resampling any edges to yet-unobserved regions. For example, if an annotation suggested the existence of a “kitchen down the hallway,” and we grounded the robot’s current region to the hypothesized hallway, we would reevaluate the “down” relation for the hypothesized kitchen with respect to this detected hallway.
Re-weighting particles and resampling
-------------------------------------
After modifying each particle’s topology, we perform a Bayesian update to its Gaussian distribution. We then re-weight each particle according to the likelihood of generating language annotations and region appearance observations: $$\label{eq:weight_update_all}
w_t^{(i)}\!\!=\! p(z_t, \alpha_t | S_{t-1}^{(i)}) w_{t-1}^{(i)} \!\!=\! p(\alpha_t| S_{t-1}^{(i)}) p(z_t| S_{t-1}^{(i)}) w_{t-1}^{(i)}.$$ When calculating the likelihood of each region appearance observation, we consider the current node’s region type and calculate the likelihood of generating this observation given the topology. In effect, this down-weights any particle with a sampled region of a particular type existing on top of a known traversed region of a different type. We use a likelihood model that describes the observation of a region’s type, with a latent binary variable $v$ that denotes whether or not the observation is valid. We marginalize over $v$ to arrive at the likelihood of generating the given observation, where $R_u$ is the set of unobserved regions in particle $S_{t-1}^{(i)}$: $$\label{eq:likelihood_appearance}
p(z^t\vert{S_{t-1}^{(i)}}) = \prod_{R_i\in {R_u}}\left(\sum\limits_{v\in {1,0}}{p(z^t\vert{v,R_i}) \times p(v\vert{R_i})}\right).$$ For annotations, we use the language grounding likelihood under the map at the previous time step. As such, a particle with an existing pair of regions conforming to a specified language constraint will be weighted higher than one without. When the particle weights fall below a threshold, we resample particles to avoid particle depletion [@doucet00].
Reasoning and Learning in Belief Space {#sec:planning}
======================================
Searching for the complete trajectory that is optimal in the distribution of maps would be intractable. Instead, we treat direction following as sequential decision making under uncertainty, where a policy $\pi$ minimizes a single step of the cost function $c$ over the available actions $a \in A_t$ from state $x$: $$\label{eqnPolicy}
\pi {\left( x, {S_t}\right)} = {\ensuremath{\underset{a \in A_t}{\arg \! \min}\;}} c{\left( x, a, {S_t}\right)}.$$ After executing the action and updating the map distribution, we repeat this process until the policy declares it has completed following the direction using a separate stop action.
As the robot travels in the environment, it keeps track of the nodes in the topological graph $G_t$ it has visited (${\mathcal{V}}$) and frontiers (${\mathcal{F}}$) that lie at the edge of explored space. The action set $A_t$ consists of paths to nodes in the graph. An additional action ${a_\textrm{stop}}$ declares that the policy has completed following the direction. Intuitively, an action represents a single step along the path that takes the robot towards its destination. Each action may explore new parts of the environment (for example continuing to travel down a hallway) or backtrack if the policy has made a mistake (for example, traveling to a room in a different part of the environment). The following sections explain how the policy reasons in belief space, and the novel imitation learning formulation to train the policy from demonstrations of correct behavior.
Belief Space Reasoning using Distribution Embedding
---------------------------------------------------
The semantic map ${S_t}$ provides a distribution over the possible locations of the landmarks relevant to the command the robot is following. As such, the policy $\pi$ must reason about a *distribution* of action features when computing the cost of any action $a$. We accomplish this by embedding the action feature distribution in a Reproducing Kernel Hilbert Space (RKHS), using the mean feature map [@smola2007] consisting of the first $K$ moments of the features computed with respect to each map sample ${S_t^{{\left( i \right)}}}$ (and its likelihood): $$\begin{aligned}
\label{eqnPolicyDecomposition}
{\hat{\Phi}_1}{\left( x, a, {S_t}\right)} & = \sum_{{S_t^{{\left( i \right)}}}} p({S_t^{{\left( i \right)}}}) \: {\phi {\left( x, a, {S_t^{{\left( i \right)}}}\right)}}\\
{\hat{\Phi}_2}{\left( x, a, {S_t}\right)} & = \sum_{{S_t^{{\left( i \right)}}}} p({S_t^{{\left( i \right)}}}) \: {\left( {\phi {\left( x, a, {S_t^{{\left( i \right)}}}\right)}}- {\hat{\Phi}_1}\right)}^2\\
& \ldots \nonumber \\
{\hat{\Phi}_k}{\left( x, a, {S_t}\right)} & = \sum_{{S_t^{{\left( i \right)}}}} p({S_t^{{\left( i \right)}}}) \: {\left( {\phi {\left( x, a, {S_t^{{\left( i \right)}}}\right)}}- {\hat{\Phi}_1}\right)}^k\end{aligned}$$ Intuitively, this formulation computes features for the action and all hypothesized landmarks individually, aggregates these feature vectors, and then computes moments of the feature vector distribution (mean, variance, and higher order statistics). A simplified illustration, shown in , shows how our approach computes belief space features for two actions with a hypothesized kitchen (with two possible locations).
The cost function in can now be rewritten as a weighted sum of the first $K$ moments of the feature distribution: $$\label{eqnBeliefPolicySumK}
{c{\left( x, a, {S_t}\right)}}= \sum_{i=1}^K {w_i^T}\; {\hat{\Phi}_i}{{\left( x, a, {S_t}\right)}}.$$ By concatenating the weights and moments into respective column vectors $W := [w_1; \ldots; w_k]$ and $F := [{\hat{\Phi}_1}; \ldots; {\hat{\Phi}_k}]$, we can rewrite the policy in as minimizing a weighted sum of the feature moments $F_a$ for action $a$ : $$\label{eqnBeliefPolicyVectorK}
\pi {\left( x, {S_t}\right)} = {\ensuremath{\underset{a \in A_t}{\arg \! \min}\;}} W^T F_a.
$$
The vector $\phi (x, a, {S_t^{{\left( i \right)}}})$ are features of the action and a *single* landmark in ${S_t^{{\left( i \right)}}}$. It contains geometric features describing the shape of the action (e.g., the cumulative change in angle), the geometry of the landmark (e.g., the area of the landmark), and the relationship between the action and landmark (e.g., the difference between the ending and starting distances to the landmark). See [@duvallet2013] for more details.
Imitation Learning Formulation
------------------------------
We use imitation learning to train the policy by treating action prediction as a multi-class classification problem: given an expert demonstration, we wish to correctly predict their action among all possible actions for the same state. Although prior work introduced imitation learning for training a direction following policy, it operated in partially known environments [@duvallet2013]. Instead, we train a belief space policy that reasons in a *distribution* of hypothesized maps.
We assume the expert’s policy ${\pi^*}$ minimizes the unknown immediate cost $C(x, a^*, {S_t})$ of performing the demonstrated action $a^*$ from state $x$, under the map distribution ${S_t}$. However, since we cannot directly observe the true costs of the expert’s policy, we must instead minimize a surrogate loss that penalizes disagreements between the expert’s action $a^*$ and the policy’s action $a$, using the multi-class hinge loss [@Crammer2002]: $$\label{eqnSVMloss}
\ell {\left( x, {{a^{*}}}\!, c, {S_t}\right)} \! = \! \max \! {\left( \! 0, 1 \! + \! {c{\left( x, {{a^{*}}}\negmedspace, {S_t}\right)}}\! - \! \min_{a \ne {{a^{*}}}} { \left[ {c{\left( x, a, {S_t}\right)}}\right] } \! \right)}.$$ The minimum of this loss occurs when the cost of the expert’s action is lower than the cost of all other actions, with a margin of one. This loss can be re-written and combined with to yield: $$\label{eqLossAugmentation}
\ell {\left( x, {{a^{*}}}, W, {S_t}\right)} = {W^T}F_{{a^{*}}}- \min_a { \left[ {W^T}F_a - l_{xa} \right] },$$ where the margin $l_{xa} = 0$ if $a={{a^{*}}}$ and $1$ otherwise. This ensures that the expert’s action is better than all other actions by a margin [@Ratliff2006]. Adding a regularization term $\lambda$ to yields our complete optimization loss: $$\label{eqnOptimizationLoss}
\ell {\left( x, {{a^{*}}}, W, {S_t}\right)} \! = \! \frac{\lambda}{2} \Vert W \Vert^2 + {W^T}F_{{a^{*}}}- \min_{a} \left[ {W^T}F_a - l_{xa} \right].$$
Although this loss function is convex, it is not differentiable. However, we can optimize it efficiently by taking the subgradient of and computing action predictions for the loss-augmented policy [@Ratliff2006]: $$\begin{aligned}
{\frac{\partial \ell}{\partial W}}& = \lambda W + F_{{a^{*}}}- F_{a'} \\
a' & = {\ensuremath{\underset{a}{\arg \! \min}\;}} \left[ {W^T}F_a - l_{xa} \right].\end{aligned}$$ Note that $a'$ (the best loss-augmented action) is simply the solution to our policy using a loss-augmented cost. This leads to the update rule for the weights $W$: $$\label{eqnUpdateRule}
W_{t+1} \gets W_t - \alpha \; {\frac{\partial \ell}{\partial W}}$$ with a learning rate $\alpha \propto 1/t^\gamma$. Intuitively, if the current policy disagrees with the expert’s demonstration, decreases the weight (and thus the cost) for the features of the demonstrated action $F_{{a^{*}}}$, and increases the weight for the features of the planned action $F_{a'}$. If the policy produces actions that agree with the expert’s demonstration, the update will only be for the regularization term. As in our prior work, we train the policy using the [<span style="font-variant:small-caps;">DAgger</span>]{} (Dataset Aggregation) algorithm [@Ross2011], which learns a policy by iterating between collecting data (using the current policy) and applying expert corrections on all states visited by the policy (using the expert’s demonstrated policy).
Treating direction following in the space of possible semantic maps as a problem of sequential decision making under uncertainty provides an efficient approximate solution to the belief space planning problem. By using a kernel embedding of the distribution of features for a given action, our approach can learn a policy that reasons about the distribution of semantic maps.
Results {#sec:results}
=======
We implemented the algorithm on our voice-commandable wheelchair (Fig. \[fig:go\_to\_kitchen\_down\_hallway\]), which is equipped with three forward-facing cameras with a collective field-of-view of 120 degrees, and forward- and rearward-facing LIDARs. We set up an experiment in which the wheelchair was placed in a lobby within MIT’s Stata Center, with several hallways, offices, and lab spaces, as well as a kitchen on the same floor. As scene understanding is not the focus of this paper, we placed AprilTag fiducials [@Olson2011] to identify the existence and semantic type of regions in the environment. We trained the HDCG models from a parallel corpus of 54 fully-labeled examples. We then directed the wheelchair to execute the novel instruction “go to the kitchen that is down the hallway.”
[0.9]{}[Scccc]{} & &\
Algorithm & Mean & Std Dev & Mean & Std Dev\
Known Map & 13.10 & 0.67 & 62.48 & 16.61\
With Language & 12.62 & 0.62 & 122.14 & 32.48\
Without Language & 24.91 & 13.55 & 210.35 & 97.73\
\[tab:results\_robot\]
[0.9]{}[Scccc]{} & &\
Algorithm & Mean & Std Dev & Mean & Std Dev\
Known Map & 12.88 & 0.06 & 18.32 & 3.54\
With Language & 16.64 & 6.84 & 82.78 & 10.56\
Without Language & 25.28 & 12.99 & 85.57 & 17.80\
\[tab:results\_sim\]
We compare our framework against two other methods. The first emulates the previous state-of-the-art and uses a known map of the environment in order to infer the actions consistent with the route direction. The second assumes no prior knowledge of the environment (as with ours) and opportunistically grounds the command in the map, but does not use language to modify the map. We performed six experiments with our algorithm, three with the known map method, and five with the method that does not use language, all of which were successful (the robot reached the kitchen). compares the total distance traveled and execution time for the three methods. Our algorithm resulted in paths with lengths close to those of the known map, and significantly outperformed the method that did not use language. Our framework did require significantly more time to follow the directions than the known map case, due to the fact that it repeats the three steps of the algorithm when new sensor data arrives. shows a visualization of the semantic maps over several time steps for one successful run on the robot.
We performed a similar evaluation in a simulated environment comprised of an office, hallway, and kitchen. With the robot starting in the office, we ran ten simulations of each method. As with the physical experiment, our method resulted in an average length closer to that of the known map case, but with a longer average run time ().
To evaluate the performance of the learned belief space policy in isolation on a larger corpus of natural language directions (with more verbs, spatial relations, and landmarks), we performed cross-validation trials of the policy operating in a simplified simulated map. We evaluated the policy using a corpus of 55 multi-step natural language directions, some of which refer to navigation landmarks (for example, the direction shown in \[figFullDirectionAmbiguousEight\]). These directions are similar to those in our prior work [@duvallet2013]. For this cross-validation evaluation, we trained the policy on 28 randomly-sampled directions then evaluated the learned policy on the remaining 27 directions (measuring the average ending distance error across the held out directions). The results of this experiment, shown in \[figAmbiguousDistanceBoxPlots\], demonstrate the benefit of using the additional information available in the direction to infer a distribution of possible environment models. By contrast, our prior approach (without belief space reasoning) ignores this information which results in larger ending distance errors.
Conclusions
===========
Robots that can understand and follow natural language directions in unknown environments are one step towards intuitive human-robot interaction. Reasoning about parts of the environment that have not yet been detected would help enable seamless coordination in human-robot teams.
We have generalized our prior work to move beyond object-relative navigation in small, open environments. The primary contributions of this work include:
- a hierarchical framework that learns a compact probabilistic graphical model for language understanding;
- a semantic map inference algorithm that hypothesizes the existence and location of spatially coherent regions in large environments; and
- a belief space policy that reasons directly over the hypothesized map distribution and is trained based on expert demonstrations.
Together, these algorithms are integral to efficiently interpreting and following natural language route directions in unknown, spatially extended, and complex environments. We evaluated our algorithm through a series of simulations as well as demonstrations on a voice-commandable autonomous wheelchair tasked with following natural language route instructions in an office-like environment.
In the future, we plan to carry out experiments on a more diverse set of commands. Other future work will focus on handling sequences of commands, as well as streams of command that are given *during* execution to change the behavior of the robot.
Acknowledgments {#acknowledgments .unnumbered}
===============
[^1]: $^*$The first two authors contributed equally to this paper.
[^2]: S. Hemachandra and N. Roy are with the Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology, Cambridge, MA USA [ {sachih,tmhoward,nickroy}@csail.mit.edu]{}
[^3]: F. Duvallet and A. Stentz are with the Robotics Institute, Carnegie Mellon University, Pittsburgh, PA USA [ {felixd,tony}@cmu.edu]{}
[^4]: T.M. Howard is with the University of Rochester, Rochester, NY USA [[email protected]]{}
[^5]: M.R. Walter is with the Toyota Technological Institute at Chicago, Chicago, IL USA [[email protected]]{}
[^6]: 3,485 symbols = 17 object types, 204 subspaces, and 3,264 object types with spatial relationships (we exclude object types with spatial relationships to the same object type)
[^7]: Regions as defined by the mapping framework are also considered as objects for the purpose of natural language understanding.
|
---
abstract: 'We prove the standard conjectures for complex projective varieties that are deformations of the Hilbert scheme of points on a K3 surface. The proof involves Verbitsky’s theory of hyperholomorphic sheaves and a study of the cohomology algebra of Hilbert schemes of K3 surfaces.'
address:
- 'Département de Mathématiques et Applications, École Normale Supérieure, 45, rue d’Ulm, 75005 Paris, France'
- 'Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003, USA'
author:
- François Charles and Eyal Markman
title: The Standard Conjectures for holomorphic symplectic varieties deformation equivalent to Hilbert schemes of $K3$ surfaces
---
Introduction
============
An [*irreducible holomorphic symplectic manifold*]{} is a simply connected compact Kähler manifold $X$, such that $H^0(X,\Omega^2_X)$ is generated by an everywhere non-degenerate holomorphic two-form (see [@beauville; @huybrects-basic-results]). Let $S$ be a smooth compact Kähler $K3$ surface and $S^{[n]}$ the Hilbert scheme (or Douady space) of length $n$ zero dimensional subschemes of $S$. Beauville proved in [@beauville] that $S^{[n]}$ is an irreducible holomorphic symplectic manifold of dimension $2n$. If $X$ is a smooth compact Kähler manifold deformation equivalent to $S^{[n]}$, for some $K3$ surface $S$, then we say that $X$ is of [*$K3^{[n]}$-type*]{}. The variety $X$ is then an irreducible holomorphic symplectic manifold. The odd Betti numbers of $X$ vanish [@gottsche].
The moduli space of Kähler manifolds of $K3^{[n]}$-type is smooth and $21$-dimensional, if $n\geq 2$, while that of $K3$ surfaces is $20$-dimensional [@beauville]. It follows that if $S$ is a $K3$ surface, a general Kähler deformation of $S^{[n]}$ is not of the form $S'^{[n]}$ for a K3 surface $S'$. The same goes for projective deformations. Indeed, a general projective deformation of $S^{[n]}$ has Picard number $1$, whereas for a projective $S$, the Picard number of $S^{[n]}$ is at least $2$.
In this note, we prove the standard conjectures for projective varieties of $K3^{[n]}$-type. Let us recall general facts about the standard conjectures.
In the paper [@Gr69] of 1968, Grothendieck states those conjectures concerning the existence of some algebraic cycles on smooth projective algebraic varieties over an algebraically closed ground field. Here we work over $\mathbb C$. The Lefschetz standard conjecture predicts the existence of algebraic self-correspondences on a given smooth projective variety $X$ of dimension $d$ that give an inverse to the operations $$H^{i}(X) {{\rightarrow}}H^{2d-i}(X)$$ given by the cup-product $d-i$ times with a hyperplane section, for all $i\leq d$. Above and throughout the rest of the paper, the notation $H^i(X)$ stands for singular cohomology with rational coefficients.
Over the complex numbers, the Lefschetz standard conjecture implies all the standard conjectures. If it holds for a variety $X$, it implies that numerical and homological equivalence coincide for algebraic cycles on $X$, and that the Künneth components of the diagonal of $X\times X$ are algebraic. We refer to [@Kl68] for a detailed discussion.
Though the motivic picture has tremendously developed since Grothendieck’s statement of the standard conjectures, very little progress has been made in their direction. The Lefschetz standard conjecture is known for abelian varieties, and in degree $1$, where it reduces to the Hodge conjecture for divisors. The Lefschetz standard conjecture is also known for varieties $X$, for which $H^*(X)$ is isomorphic to the Chow ring $A^*(X)$, see [@Kl94]. Varieties with the latter property include flag varieties, and smooth projective moduli spaces of sheaves on rational Poisson surfaces [@ES; @markman-integral-generators].
In the paper [@arapura], Arapura proves that the Lefschetz standard conjecture holds for uniruled threefolds, unirational fourfolds, the moduli space of stable vector bundles over a smooth projective curve, and for the Hilbert scheme $S^{[n]}$ of every smooth projective surface ([@arapura], Corollaries 4.3, 7.2 and 7.5). He also proves that if $S$ is a $K3$ or abelian surface, $H$ an ample line-bundle on $S$, and ${{\mathcal M}}$ a smooth and compact moduli space of Gieseker-Maruyama-Simpson $H$-stable sheaves on $S$, then the Lefschetz standard conjecture holds for ${{\mathcal M}}$ ([@arapura], Corollary 7.9). Those results are obtained by showing that the motive of those varieties is very close, in a certain sense, to that of a curve or a surface. Aside from those examples and ones obtained by specific constructions from them (e.g. hyperplane sections, products, projective bundles, etc.), very few cases of the Lefschetz standard conjecture seem to be known.
The main result of this note is the following statement.
\[thm-lefschetz\] The Lefschetz standard conjecture holds for every smooth projective variety of $K3^{[n]}$-type.
Since the Lefschetz standard conjecture is the strongest standard conjecture in characteristic zero, we get the following corollary.
The standard conjectures hold for any smooth projective variety of $K3^{[n]}$-type.
Note that by the remarks above, Theorem \[thm-lefschetz\] does not seem to follow from Arapura’s results, as a general variety of $K3^{[n]}$-type is not a moduli space of sheaves on any K3 surface.
Theorem \[thm-lefschetz\] is proven in section \[sec-proof-of-main-thm\]. The degree $2$ case of the Lefschetz standard conjecture, for projective varieties of $K3^{[n]}$-type, has already been proven in [@markman-2010] as a consequence of results of [@charles]. Section 2 gives general results on the Lefschetz standard conjecture. Sections 3 to 5 introduce the algebraic cycles we need for the proof, while sections 6 and 7 contain results on the cohomology algebra of the Hilbert scheme of K3 surfaces.
Preliminary results on the Lefschetz standard conjecture
========================================================
Let $X$ be a smooth projective variety of dimension $d$. Let $\xi \in H^2(X)$ be the cohomology class of a hyperplane section of $X$. According to the hard Lefschetz theorem, for all $i\in\{0, \ldots, d\}$, cup-product with $\xi^{d-i}$ induces an isomorphism $$L^{[d-i]}:=\cup \xi^{d-i} : H^{i}(X){{\rightarrow}}H^{2d-i}(X).$$ The Lefschetz standard conjecture was first stated in [@Gr69], conjecture $B(X)$. It is the following.
Let $X$ and $\xi$ be as above. Then for all $i\in\{0, \ldots, d\}$, there exists an algebraic cycle $Z$ of codimension $i$ in the product $X\times X$ such that the correspondence $$[Z]_* : H^{2d-i}(X){{\rightarrow}}H^{i}(X)$$ is the inverse of $\cup \xi^{d-i}$.
If this conjecture holds for some specific $i$ on $X$, we will say that [*the Lefschetz conjecture holds in degree $i$ for the variety $X$.*]{}
We will derive Theorem \[thm-lefschetz\] as a consequence of Theorem \[thm-deformability\] and Corollary \[cor-comparison-between-kappa-and-f\] below. In this section, we prove some general results we will need. The reader can consult [@arapura], sections 1 and 4 for related arguments, and [@andre] for a more general use of polarizations and semi-simplicity. Let us first state an easy lemma.
\[devissage\] Let $X$ be a smooth projective variety of dimension $d$. Let $i\leq d$ be an integer.
1. Assume $i=2j$ is even, and let $\alpha\in H^{2j}(X)$ be the cohomology class of a codimension $j$ algebraic cycle in $X$. Then there exists a cycle $Z$ of codimension $i=2j$ in $X\times X$ such that the image of the correspondence $$[Z]_* : H^{2d-2j}(X){{\rightarrow}}H^{2j}(X)$$ contains $\alpha$.
2. Assume that $X$ satisfies the Lefschetz standard conjecture in degrees up to $i-1$. Then $X\times X$ satisfies the Lefschetz standard conjecture in degree up to $i-1$.
Let $j$ and $k$ be two positive integers with $i=j+k$. Then there exists a cycle $Z$ of codimension $i$ in $(X\times X)\times X$ such that the image of the correspondence $$[Z]_* : H^{4d-i}(X\times X){{\rightarrow}}H^{i}(X)$$ contains the image of $H^j(X)\otimes H^k(X)$ in $H^{j+k}(X)=H^i(X)$ by cup-product.
Let $\alpha\in H^{2j}(X)$ be the cohomology class of a codimension $j$ algebraic cycle $T$ in $X$. Let $Z$ be the codimension $i=2j$ algebraic cycle $T\times T$ in $X\times X$. Since the image in $H^i(X)\otimes H^i(X)$ of the cohomology class of $Z$ in $H^{2i}(X\times X)$ is $\alpha\otimes \alpha$, the image of the correspondence $$[Z]_* : H^{2d-i}(X){{\rightarrow}}H^{i}(X)$$ is the line generated by $\alpha$. This proves $(1)$.
Let us prove the first part of $(2)$. We repeat some of Kleiman’s arguments in [@Kl68]. Assume that $X$ satisfies the Lefschetz standard conjecture in degree up to $i-1$. We want to prove that $X\times X$ satisfies the Lefschetz standard conjecture in degree up to $i-1$. By induction, we only have to prove that $X\times X$ satisfies the Lefschetz standard conjecture in degree $i-1$.
For any $j$ between $0$ and $i-1$, there exists a codimension $j$ algebraic cycle $Z_j$ in $X\times X$ such that the correspondence $$[Z_j]_* : H^{2d-j}(X){{\rightarrow}}H^j(X)$$ is an isomorphism. For $k$ between $0$ and $2d$, let $\pi^k\in H^{2d-k}(X)\otimes H^k(X)
\subset H^{2d}(X\times X)$ be the $k$-th Künneth component of the diagonal. By [@Kl68], Lemma 2.4, the assumption on $X$ implies that the elements $\pi^0, \ldots, \pi^{i-1}, \pi^{2d-i+1},
\ldots, \pi^{2d}$ are algebraic. Identifying the $\pi^j$ with the correspondence they induce, this implies that for all $j$ between $0$ and $i-1$, the projections $$\pi^j : H^*(X){{\rightarrow}}H^j(X)\hookrightarrow H^*(X)$$ and $$\pi^{2d-j} : H^*(X){{\rightarrow}}H^{2d-j}(X)\hookrightarrow H^*(X)$$ are given by algebraic correspondences. Replacing the correspondence $[Z_j]_*$ by $[Z_j]_* \circ \pi_{2d-j}$, which is still algebraic, we can thus assume that the morphism $$[Z_j]_* : H^{2d-k}(X){{\rightarrow}}H^{2j-k}(X)$$ induced by $[Z_j]$ is zero unless $k=j$.
Now consider the codimension $i-1$ cycle $Z$ in $(X\times X)\times(X\times X)$ defined by $$Z=\sum_{j=0}^{i-1} Z_j \times Z_{i-1-j}.$$ We claim that the correspondence $$[Z]_* : H^{4d-i+1}(X\times X){{\rightarrow}}H^{i-1}(X\times X)$$ is an isomorphism.
Fix $j$ between $0$ and $i-1$. The hypothesis on the cycles $Z_j$ imply that the correspondence $$[Z_j\times Z_{i-1-j}]_* : H^{4d-i+1}(X\times X){{\rightarrow}}H^{i-1}(X\times X)$$ maps the subspace $H^{2d-k}(X)\otimes H^{2d-i+1+k}(X)$ of $H^{4d-i+1}(X\times X)$ to zero unless $k=j$, and it maps $H^{2d-j}(X)\otimes H^{2d-i+1+j}(X)$ isomorphically onto $H^{j}(X)\otimes H^{i-1-j}(X)$. The claim follows, as does the first part of $(2)$.
For the second statement, let $j$ and $k$ be as in the hypothesis. Since $j$ (resp. $k$) is smaller than or equal to $i-1$, $X$ satisfies the Lefschetz standard conjecture in degree $j$ (resp. $k$). As a consequence, there exists a cycle $T$ (resp. $T'$) in $X\times X$ such that the morphism $$[T]_* : H^{2d-j}(X){{\rightarrow}}H^{j}(X)$$ (resp. $[T']_* : H^{2d-k}(X){{\rightarrow}}H^{k}(X)$) is an isomorphism. Consider now the projections $p_{13}$ and $p_{23}$ from $X\times X\times X$ to $X\times X$ forgetting the second and first factor respectively, and let $Z$ in $CH^{i}(X\times X\times X)$ be the intersection of $p_{13}^*T$ and $p_{23}^*T'$. Since the cohomology class of $Z$ is just the cup-product of that of $p_{13}^*T$ and $p_{23}^*T'$, it follows that the image of the correspondence $$[Z]_* : H^{4d-i}(X\times X){{\rightarrow}}H^{i}(X)$$ contains the image of $H^j(X)\otimes H^k(X)$ in $H^{j+k}(X)=H^i(X)$ by cup-product.
The following result appears in [@charles], Proposition 8. Using it with the previous lemma, it will allow us to finish the proof of Theorem \[thm-lefschetz\].
\[Lef-criterion\] Let $X$ be a smooth projective variety of dimension $d$, and let $i\leq d$ be an integer. Then the Lefschetz conjecture is true in degree $i$ for $X$ if and only if there exists a disjoint union $S$ of smooth projective varieties of dimension $l\geq i$ satisfying the Lefschetz conjecture in degrees up to $i-2$ and a codimension $i$ cycle $Z$ in $X\times S$ such that the morphism $$[Z]_* : H^{2l-i}(S){{\rightarrow}}H^i(X)$$ induced by the correspondence $Z$ is surjective.
\[recurrence\] Let $X$ be a smooth projective variety of dimension $d$, and let $i\leq d$ be an integer. Suppose that $X$ satisfies the Lefschetz standard conjecture in degrees up to $i-1$.
Let $A^i(X)\subset H^i(X)$ be the subspace of classes, which belong to the subring generated by classes of degree $< i$, and let $Alg^i(X)\subset
H^i(X)$ be the subspace of $H^i(X)$ generated by the cohomology classes of algebraic cycles[^1].
Assume that there is a cycle $Z$ of codimension $i$ in $X\times X$ such that the image of the morphism $$[Z]_* : H^{2d-i}(X){{\rightarrow}}H^{i}(X)$$ maps surjectively onto the quotient space $H^{i}(X)/\left[Alg^i(X)+A^i(X)\right]$. Then $X$ satisfies the Lefschetz standard conjecture in degree $i$.
We use Lemma \[devissage\]. Let $\alpha_1, \ldots, \alpha_r$ be a basis for $Alg^i(X)$. We can find codimension $i$ cycles $Z_1, \ldots, Z_r$ in $X\times X$ and $(Z_{j,k})_{j, k>0, j+k=i}$ in $(X\times X)\times X$, such that the image of the correspondence $$[Z_l]_* : H^{2d-i}(X){{\rightarrow}}H^{i}(X)$$ contains $\alpha_l$ for $1\leq l \leq r$, and such that the image of the correspondence $$[Z_{j,k}]_* : H^{4d-i}(X\times X){{\rightarrow}}H^{i}(X)$$ contains the image of $H^j(X)\otimes H^k(X)$ in $H^{j+k}(X)=H^i(X)$, for $j+k=i$.
We proved that $X\times X$ satisfies the Lefschetz standard conjecture in degree up to $i-1$. The disjoint union of the cycles $Z\times X, (Z_l\times X)_{1\leq
l\leq r}$ and $(Z_{j,k})_{j, k>0, j+k=i}$ in a disjoint union of copies of $(X\times X)\times X$ thus satisfies the hypothesis of Theorem \[Lef-criterion\] (we took products with $X$ in order to work with equidimensional varieties). Indeed, the space generated by the images in $H^{i}(X)$ of the correspondences $[Z_{j,k}]_*$ contains $A^{i}(X)$ by definition. Adding the images in $H^{i}(X)$ of the $[Z_l\times X]_*$, which generate a space containing $Alg^i(X)$, and the image in $H^{i}(X)$ of $[Z\times
X]_*$, which maps surjectively onto $H^{i}(X)/\left[A^{i}(X)+Alg^i(X)\right]$, we get the whole space $H^{i}(X)$.
This ends the proof, and shows that $X$ satisfies the Lefschetz standard conjecture in degree $i$.
Let $X$ be a smooth projective variety with cohomology algebra generated in degree less than $i$, and assume that $X$ satisfies the Lefschetz standard conjecture in degree up to $i$. Then $X$ satisfies the standard conjectures.
Using induction and taking $Z=0$, the previous corollary shows that $X$ satisfies the Lefschetz standard conjecture, hence all the standard conjectures, since we work in characteristic zero.
Note that the Lefschetz conjecture is true in degree $1$, as it is a consequence of the Lefschetz theorem on $(1,1)$-classes. The preceding corollary hence allows us to recover the Lefschetz conjecture for abelian varieties which was proved in [@lieberman].
Moduli spaces of sheaves on a $K3$ surface {#sec-moduli-spaces-of-sheaves}
==========================================
Let $S$ be a projective $K3$ surface. Denote by $K(S,{{\mathbb Z}})$ the topological $K$ group of $S$, generated by topological complex vector bundles. The $K$-group of a point is ${{\mathbb Z}}$ and we let $\chi:K(S,{{\mathbb Z}})\rightarrow {{\mathbb Z}}$ be the Gysin homomorphism associated to the morphism from $S$ to a point. The group $K(S,{{\mathbb Z}})$, endowed with the [*Mukai pairing*]{} $$(v,w) \ \ := \ \ -\chi(v^\vee\otimes w),$$ is called the [*Mukai lattice*]{} and denoted by $\Lambda(S)$. Mukai identifies the group $K(S,{{\mathbb Z}})$ with $H^*(S,{{\mathbb Z}})$, via the isomorphism sending a class $F$ to its [*Mukai vector*]{} $ch(F)\sqrt{td_S}$. Using the grading of $H^*(S,{{\mathbb Z}})$, the Mukai vector of $F$ is $$\label{eq-Mukai-vector}
({{\rm rank}}(F),c_1(F),\chi(F)-{{\rm rank}}(F)),$$ where the rank is considered in $H^0$ and $\chi(F)-{{\rm rank}}(F)$ in $H^4$ via multiplication by the orientation class of $S$. The homomorphism $ch(\bullet)\sqrt{td_S}:\Lambda(S)\rightarrow H^*(S,{{\mathbb Z}})$ is an isometry with respect to the Mukai pairing on $\Lambda(S)$ and the pairing $$\left((r',c',s'),(r'',c'',s'')\right) \ \ = \ \
\int_{S}c'\cup c'' -r'\cup s''-s'\cup r''$$ on $H^*(S,{{\mathbb Z}})$ (by the Hirzebruch-Riemann-Roch Theorem). Mukai defines a weight $2$ Hodge structure on the Mukai lattice $H^*(S,{{\mathbb Z}})$, and hence on $\Lambda(S)$, by extending that of $H^2(S,{{\mathbb Z}})$, so that the direct summands $H^0(S,{{\mathbb Z}})$ and $H^4(S,{{\mathbb Z}})$ are of type $(1,1)$ [@mukai-hodge].
Let $v\in \Lambda(S)$ be a primitive class with $c_1(v)$ of Hodge-type $(1,1)$. There is a system of hyperplanes in the ample cone of $S$, called $v$-walls, that is countable but locally finite [@huybrechts-lehn-book], Ch. 4C. An ample class is called [*$v$-generic*]{}, if it does not belong to any $v$-wall. Choose a $v$-generic ample class $H$. Let ${{\mathcal M}}_H(v)$ be the moduli space of $H$-stable sheaves on the $K3$ surface $S$ with class $v$. When non-empty, the moduli space ${{\mathcal M}}_H(v)$ is a smooth projective irreducible holomorphic symplectic variety of $K3^{[n]}$ type, with $n=\frac{(v,v)+2}{2}$. This result is due to several people, including Huybrechts, Mukai, O’Grady, and Yoshioka. It can be found in its final form in [@yoshioka-abelian-surface].
Over $S\times {{\mathcal M}}_H(v)$ there exists a universal sheaf ${{\mathcal F}}$, possibly twisted with respect to a non-trivial Brauer class pulled-back from ${{\mathcal M}}_H(v)$. Associated to ${{\mathcal F}}$ is a class $[{{\mathcal F}}]$ in $K(S\times {{\mathcal M}}_H(v),{{\mathbb Z}})$ ([@markman-integral-generators], Definition 26). Let $\pi_i$ be the projection from $S\times {{\mathcal M}}_H(v)$ onto the $i$-th factor. Assume that $(v,v)>0$. The second integral cohomology $H^2({{\mathcal M}}_H(v),{{\mathbb Z}})$, its Hodge structure, and its Beauville-Bogomolov pairing [@beauville], are all described by Mukai’s Hodge-isometry $$\label{eq-Mukai-isomorphism}
\theta \ : \ v^\perp \ \ \ \longrightarrow \ \ \ H^2({{\mathcal M}}_H(v),{{\mathbb Z}}),$$ given by $\theta(x):=c_1\left(\pi_{2_!}\{\pi_1^!(x^\vee)\otimes [{{\mathcal F}}]\}\right)$ (see [@yoshioka-abelian-surface]). Above, $\pi_{2_!}$ and $\pi_1^!$ are the Gysin and pull-back homomorphisms in $K$-theory.
An algebraic cycle
==================
Let ${{\mathcal M}}:={{\mathcal M}}_H(v)$ be a moduli space of stable sheaves on the $K3$ surface $S$ as in section \[sec-moduli-spaces-of-sheaves\], so that ${{\mathcal M}}$ is of $K3^{[n]}$-type, $n\geq 2$. Assume that there exists an untwisted universal sheaf ${{\mathcal F}}$ over $S\times {{\mathcal M}}$. Denote by $\pi_{ij}$ the projection from ${{\mathcal M}}\times S\times {{\mathcal M}}$ onto the product of the $i$-th and $j$-th factors. Denote by $E^i$ the relative extension sheaf $$\label{eq-E}
{{\mathcal E}xt}^i_{\pi_{13}}\left(\pi_{12}^*{{\mathcal F}},\pi_{23}^*{{\mathcal F}}\right).$$ Let $\Delta\subset {{\mathcal M}}\times {{\mathcal M}}$ be the diagonal. Then $E^1$ is a reflexive coherent ${{\mathcal O}_{{{\mathcal M}}\times{{\mathcal M}}}}$-module of rank $2n-2$, which is locally free away from $\Delta$, by [@markman-2010], Proposition 4.5. The sheaf $E^0$ vanishes, while $E^2$ is isomorphic to ${{\mathcal O}_{\Delta}}$. Let $[E^i]\in K({{\mathcal M}}\times {{\mathcal M}},{{\mathbb Z}})$ be the class of $E^i$, $i=1,2$, and set $$\label{eq-K-class-E}
[E]:=[E^2]-[E^1].$$ Set $\kappa(E^1):=ch(E^1)\exp\left[-c_1(E^1)/(2n-2)\right]$. Then $\kappa(E^1)$ is independent of the choice of a universal sheaf ${{\mathcal F}}$. Let $\kappa_i(E^1)$ be the summand in $H^{2i}({{\mathcal M}})$. Then $\kappa_1(E^1)=0$. There exists a suitable choice of ${{\mathcal M}}_H(v)$, one for each $n$, so that the sheaf $E^1$ over ${{\mathcal M}}_H(v)\times {{\mathcal M}}_H(v)$ can be deformed, as a [*twisted*]{} coherent sheaf, to a sheaf $\widetilde{E}^1$ over $X\times X$, for every $X$ of $K3^{[n]}$-type [@markman-2010]. See [@caldararu-thesis] for the definition of a family of twisted sheaves. We note here only that such a deformation is equivalent to a flat deformations of ${{\mathcal E}nd}(E^1)$, as a reflexive coherent sheaf, together with a deformation of its associative algebra structure. The characteristic class $\kappa_i(\widetilde{E}^1)$ is well defined for twisted sheaves [@markman-2010]. Furthermore, $\kappa_i(\widetilde{E}^1)$ is a rational class of weight $(i,i)$, which is algebraic, whenever $X$ is projective. The construction is summarized in the following statement.
\[thm-deformability\] ([@markman-2010], Theorem 1.7) Let $X$ be a smooth projective variety of $K3^{[n]}$-type. Then there exists a smooth and proper family $\pi:{{\mathcal X}}\rightarrow C$ of irreducible holomorphic symplectic varieties, over a simply connected reduced (possibly reducible) projective curve $C$, points $t_0, t_1\in C$, isomorphisms ${{\mathcal M}}\cong \pi^{-1}(t_0)$ and $X\cong \pi^{-1}(t_1)$, with the following property. Let $p:{{\mathcal X}}\times_C{{\mathcal X}}\rightarrow C$ be the natural morphism. The flat section of the local system $R_{p_*}^*{{\mathbb Q}}$ through the class $\kappa([E^1])$ in $H^*({{\mathcal M}}\times {{\mathcal M}})$ is algebraic in $H^*(X_t\times
X_t)$, for every projective fiber $X_t$, $t\in C$, of $\pi$.
Verbitsky’s theory of hyperholomorphic sheaves plays a central role in the proof of the above theorem, see [@kaledin-verbitski-book].
A self-adjoint algebraic correspondence
=======================================
Let $K({{\mathcal M}})$ be the topological $K$-group with ${{\mathbb Q}}$ coefficients. Define the [*Mukai pairing*]{} on $K({{\mathcal M}})$ by $(x,y):=-\chi(x^\vee\otimes y)$. Let $D_{{\mathcal M}}:H^*({{\mathcal M}})\rightarrow H^*({{\mathcal M}})$ be the automorphism acting on $H^{2i}({{\mathcal M}})$ via multiplication by $(-1)^i$. Define the [*Mukai pairing*]{} on $H^*({{\mathcal M}})$ by $$(\alpha,\beta) \ \ \ := \ \ \ -\int_{{{\mathcal M}}}D_{{\mathcal M}}(\alpha)\beta.$$ Define $$\label{eq-mu}
\mu \ : \ K({{\mathcal M}}) \ \ \ \longrightarrow \ \ \ H^*({{\mathcal M}})$$ by $\mu(x):=ch(x)\sqrt{td_{{\mathcal M}}}$. Then $\mu$ is an isometry, by the Hirzebruch-Riemann-Roch theorem.
Note that the graded direct summands $H^i({{\mathcal M}})$ of the cohomology ring $H^*({{\mathcal M}})$ satisfy the usual orthogonality relation with respect to the Mukai pairing: $H^i({{\mathcal M}})$ is orthogonal to $H^j({{\mathcal M}})$, if $i+j\neq 4n$.
This is the main reason for the above definition of the Mukai pairing. The Chern character $ch:K({{\mathcal M}})\rightarrow H^*({{\mathcal M}})$ is an isometry with respect to another pairing $(x,y):=-\int_{{{\mathcal M}}}D_{{\mathcal M}}(x)\cdot y \cdot td_{{\mathcal M}}$. However, the graded summands $H^i({{\mathcal M}})$ need not satisfy the orthogonality relations above.
Given two ${{\mathbb Q}}$-vector spaces $V_1$ and $V_2$, each endowed with a non-degenerate symmetric bilinear pairing $(\bullet,\bullet)_{V_i}$, and a homomorphism $h:V_1\rightarrow V_2$, we denote by $h^\dagger$ the adjoint operator, defined by the equation $$(x,h^\dagger(y))_{V_1} \ \ := \ \ (h(x),y)_{V_2},$$ for all $x\in V_1$ and $y\in V_2$. Set $K(S):=K(S,{{\mathbb Z}})\otimes_{{\mathbb Z}}{{\mathbb Q}}$. We consider $H^*({{\mathcal M}})$, $K({{\mathcal M}})$, and $K(S)$, all as vector spaces over ${{\mathbb Q}}$ endowed with the Mukai pairing.
Let $\pi_i$ be the projection from ${{\mathcal M}}\times {{\mathcal M}}$ onto the $i$-th factor, $i=1,2$. Define $$\tilde{f}' \ : \ K({{\mathcal M}}) \ \ \ \rightarrow \ \ \ K({{\mathcal M}}),$$ by $\tilde{f}'(x):=\pi_{2_!}(\pi_1^!(x)\otimes [E])$, where $[E]$ is the class given in (\[eq-K-class-E\]), and $\pi_{2_!}$ and $\pi_1^!$ are the Gysin and pull-back homomorphisms in $K$-theory. Let $p_i$ be the projection from $S\times{{\mathcal M}}$ onto the $i$-th factor. Define $$\phi' \ : \ K(S) \ \ \ \rightarrow \ \ \ K({{\mathcal M}})$$ by $\phi'(\lambda):=p_{2_!}(p_1^!(\lambda)\otimes [{{\mathcal F}}])$. Define $$\psi' \ : \ K({{\mathcal M}}) \ \ \ \rightarrow \ \ \ K(S)$$ by $\psi'(x):=p_{1_!}(p_2^!(x)\otimes [{{\mathcal F}}^\vee])$, where ${{\mathcal F}}^\vee$ is the dual class. We then have the following identities $$\begin{aligned}
\label{eq-psi-is-adjoint-of-phi}
\psi' & = & (\phi')^\dagger,
\\
\label{eq-self-duality}
\tilde{f}' & = & \phi'\circ\psi'.\end{aligned}$$ Equality (\[eq-psi-is-adjoint-of-phi\]) is a $K$-theoretic analogue of the following well known fact in algebraic geometry. Let $\Phi:D^b(S)\rightarrow D^b({{\mathcal M}})$ be the Fourier-Mukai functor with kernel ${{\mathcal F}}$. Set ${{\mathcal F}}_R:={{\mathcal F}}^\vee\otimes p_1^*\omega_S[2]$ and let $\Psi:D^b({{\mathcal M}})\rightarrow D^b(S)$ be the Fourier-Mukai functor with kernel ${{\mathcal F}}_R$. Then $\Psi$ is the right adjoint functor of $\Phi$ ([@mukai-duality] or [@huybrechts-book-FM], Proposition 5.9). The classes of ${{\mathcal F}}^\vee$ and ${{\mathcal F}}_R$ in $K(S\times {{\mathcal M}})$ are equal, since $\omega_S$ is trivial. The equality (\[eq-psi-is-adjoint-of-phi\]) is proven using the same argument as its derived-category analogue. Equality (\[eq-self-duality\]) expresses the fact that the class $[E]$ is the convolution of the classes of ${{\mathcal F}}^\vee$ and ${{\mathcal F}}$. We conclude that $\tilde{f}'$ is [*self adjoint.*]{} Set $$f' \ \ := \ \ \mu \circ \tilde{f}' \circ \mu^\dagger.$$ Then $f'$ is the self adjoint endomorphism given by the algebraic class $$\left(\pi_1^*\sqrt{td_{{\mathcal M}}}\right)ch([E])\left(\pi_2^*\sqrt{td_{{\mathcal M}}}\right)$$ in $H^*({{\mathcal M}}\times{{\mathcal M}})$.
We normalize next the endomorphism $f'$ to an endomorphism $f$. The latter will be shown to have a monodromy-equivariance property in section \[sec-monodromy\]. Let $\alpha\in K({{\mathcal M}})$ be the class satisfying $ch(\alpha)=\exp\left(\frac{-c_1(\phi'(v^\vee))}{2n-2}\right)$. Note that $\alpha$ is the class of a ${{\mathbb Q}}$-line-bundle. Let $\tau_\alpha:K({{\mathcal M}})\rightarrow K({{\mathcal M}})$ be tensorization with $\alpha$, i.e., $\tau_\alpha(x):=x\otimes\alpha$. Then $\tau_\alpha$ is an isometry. Hence, $\tau_\alpha^\dagger=\tau_\alpha^{-1}$. Set $$\begin{aligned}
\phi & := & \tau_\alpha\circ \phi',
\\
\psi & := & \psi'\circ \tau_\alpha^{-1},
\\
\tilde{f} & := & \phi\circ \psi,
\\
f & := & \mu \circ \tilde{f}\circ \mu^\dagger.\end{aligned}$$ Then $f$ is the self adjoint endomorphism given by the algebraic class $$\label{eq-normalized-class-E-and-two-square-roots}
\pi_1^*\exp\left(\frac{c_1(\phi'(v^\vee))}{2n-2}\right)
\left(\pi_1^*\sqrt{td_{{\mathcal M}}}\right)ch([E])\left(\pi_2^*\sqrt{td_{{\mathcal M}}}\right)
\pi_2^*\exp\left(\frac{-c_1(\phi'(v^\vee))}{2n-2}\right).$$
Let $h_i:H^*({{\mathcal M}})\rightarrow H^{2i}({{\mathcal M}})$ be the projection, and $e_i:H^{2i}({{\mathcal M}})\rightarrow H^*({{\mathcal M}})$ the inclusion. Set $$f_i \ \ \ := \ \ \ h_i\circ f\circ e_{2n-i}.$$ Note that $f_i$ is induced by the graded summand in $H^{4i}({{\mathcal M}}\times{{\mathcal M}})$ of the class given in equation (\[eq-normalized-class-E-and-two-square-roots\]).
Generators for the cohomology ring and the image of $f_i$
=========================================================
Let $A^{2i}\subset H^{2i}({{\mathcal M}})$ be the subspace of classes, which belong to the subring generated by classes of degree $< 2i$. Set $$\overline{H}^{2i}({{\mathcal M}}) \ \ \ := \ \ \
H^{2i}({{\mathcal M}})/\left[A^{2i}+{{\mathbb Q}}\cdot c_i(TX)\right].$$
\[prop-surjectivity\] The composition $$\label{eq-bar-f-i}
H^{4n-2i}({{\mathcal M}}) \ {\stackrel{f_i}{\longrightarrow}} \ H^{2i}({{\mathcal M}}) \ \rightarrow \
\overline{H}^{2i}({{\mathcal M}})$$ is surjective, for $i\geq 2$.
The proposition is proven below after Claim \[claim-image-of-f-i\]. Let $g_i:H^{4n-2i}({{\mathcal M}})\rightarrow H^{2i}({{\mathcal M}})$ be the homomorphism induced by the graded summand of degree $4i$ of the cycle $$\label{eq-deformable-class}
-\left(\pi_1^*\sqrt{td_{{\mathcal M}}}\right)\kappa(E^1)\left(\pi_2^*\sqrt{td_{{\mathcal M}}}\right).$$ Denote by $\bar{f}_i:H^{4n-2i}({{\mathcal M}})\rightarrow\overline{H}^{2i}({{\mathcal M}})$ the homomorphism given in (\[eq-bar-f-i\]) and define $\bar{g}_i:H^{4n-2i}({{\mathcal M}})\rightarrow\overline{H}^{2i}({{\mathcal M}})$ similarly in terms of $g_i$.
\[cor-comparison-between-kappa-and-f\] $\bar{g}_i=\bar{f}_i$, for $i\geq 2$. In particular, $\bar{g}_i$ is surjective, for $i\geq 2$.
The equality $
c_1([E])=\pi_1^*c_1(\phi'(v^\vee))-\pi_2^*c_1(\phi'(v^\vee))
$ is proven in [@markman-2010], Lemma 4.3. Hence, the difference between the two classes (\[eq-normalized-class-E-and-two-square-roots\]) and (\[eq-deformable-class\]) is $ch({{\mathcal O}_{\Delta}})\pi_1^*td_{{{\mathcal M}}}$. Now $ch_j({{\mathcal O}_{\Delta}})=0$, for $0\leq j < 2n$. Hence, $f_i=g_i$, for $0\leq i \leq n-1$. The quotient group $\overline{H}^{2i}({{\mathcal M}})$ vanishes, for $i>n-1$, by [@markman-generators], Lemma 10, part 4. Consequently, $\bar{f}_i=0=\bar{g}_i$, for $i\geq n$.
Set $\eta:=\mu\circ \phi$, where $\mu$ is given in equation (\[eq-mu\]). Then $\eta^\dagger=\psi\circ \mu^\dagger$, and we have $$f \ \ = \ \ \eta\circ \eta^\dagger.$$ Set $\eta_i:=h_i\circ \eta$. We abuse notation and identify $h_i$ with the endomorphism $e_i\circ h_i$ of $H^*({{\mathcal M}})$. Similarly, we identify $e_{2n-i}$ with the endomorphism $e_{2n-i}\circ h_{2n-i}$ of $H^*({{\mathcal M}})$. With this notation we have $$h_i^\dagger=e_{2n-i}.$$
We get the following commutative diagram. $$\label{eq-diagram}
\xymatrix{
H^{4n-2i}({{\mathcal M}}) \ar[rr]^{f_i} \ar[d]^{e_{2n-i}}
\ar@/_7pc/[dddr] _{\eta_i^\dagger} & &
H^{2i}({{\mathcal M}})
\\
H^*({{\mathcal M}}) \ar[d]^{\mu^\dagger} \ar[rr]^{f}
& &
H^*({{\mathcal M}}) \ar[u]^{h_i}
\\
K({{\mathcal M}}) \ar[rr]^{\tilde{f}} \ar[dr]_{\psi} & &
K({{\mathcal M}}) \ar[u]^{\mu}_{\cong}
\\
& K(S) \ar[ur]_{\phi}
\ar@/_7pc/[uuur] _{\eta_i}
}$$
The two main ingredients in the proof of Proposition \[prop-surjectivity\] are the following Theorem and the monodromy equivariance of Diagram (\[eq-diagram\]) reviewed in section \[sec-monodromy\].
\[thm-generators\] The composite homomorphism $$K(S)\ {\stackrel{\eta_i}{\longrightarrow}} \ H^{2i}({{\mathcal M}}) \
\rightarrow \ \overline{H}^{2i}({{\mathcal M}})$$ is surjective, for all $i\geq 1$.
The subspaces $ch_i(\phi'(K(S))$, $i\geq 1$, generate the cohomology ring $H^*({{\mathcal M}})$, by ([@markman-generators], Corollary 2). When ${{\mathcal M}}=S^{[n]}$, this was proven independently in [@lqw]. The same statement holds for the subspaces $ch_i(\phi(K(S))$. Indeed, $ch_1(\phi(K(S))=ch_1(\phi'(K(S))=
H^2({{\mathcal M}})$, since $\phi'(\lambda^\vee)$ is a class of rank $0$, for $\lambda\in
v^\perp$, and so $c_1(\phi'(\lambda^\vee))=c_1(\phi(\lambda^\vee))$, for $\lambda\in v^\perp$. Now $ch_1(\phi'([v^\perp]^\vee))=H^2({{\mathcal M}})$, since Mukai’s isometry given in (\[eq-Mukai-isomorphism\]) is surjective. For $i>1$, the subspaces $ch_i(\phi'(K(S))$ and $ch_i(\phi(K(S))$ are equal modulo the subring generated by $H^2({{\mathcal M}})$. The surjectivity of the composite homomorphism follows.
If $\eta_i$ is injective, then $Im(f_i)=Im(\eta_i)$.
The assumption implies that $\eta_i^\dagger$ is surjective. Furthermore, we have $
f_i=
\eta_i\circ\eta_i^\dagger.
$ The equality $Im(f_i)=Im(\eta_i)$ follows.
In the next section we will prove an analogue of the above claim, without the assumption that $\eta_i$ is injective (see Claim \[claim-image-of-f-i\]).
Monodromy {#sec-monodromy}
=========
Recall that the Mukai lattice $\Lambda(S)$ is a rank $24$ integral lattice isometric to the orthogonal direct sum $E_8(-1)^{\oplus 2}\oplus U^{\oplus 4}$, where $E_8(-1)$ is the negative definite $E_8$ lattice and $U$ is the unimodular rank $2$ hyperbolic lattice [@mukai-hodge]. Recall that ${{\mathcal M}}$ is the moduli space ${{\mathcal M}}_H(v)$. Denote by $O^+\Lambda(S)_v$ the subgroup of isometries of the Mukai lattice, which send $v$ to itself and preserve the spinor norm. The [*spinor norm*]{} is the character $O\Lambda(S)\rightarrow \{\pm 1\}$, which sends reflections by $-2$ vectors to $1$ and reflections by $+2$ vectors to $-1$. The group $O^+\Lambda(S)_v$ acts on $\Lambda(S)$ and on $K(S)\cong \Lambda(S)\otimes_{{\mathbb Z}}{{\mathbb Q}}$ via the natural action.
Let $D_S:K(S)\rightarrow K(S)$ be given by $D_S(\lambda)=\lambda^\vee$.
\[thm-symmetries\]
1. ([@markman-monodromy-I], Theorem 1.6) There exist natural homomorphisms $$\begin{aligned}
{{mon}}\ : \ O^+\Lambda(S)_v & \longrightarrow & GL\left[H^*({{\mathcal M}})\right],
\\
\widetilde{{{mon}}} \ : \ O^+\Lambda(S)_v & \longrightarrow &
GL\left[K({{\mathcal M}})\right],\end{aligned}$$ introducing an action of $O^+\Lambda(S)_v$ on both $H^*({{\mathcal M}})$ and $K({{\mathcal M}})$ via monodromy operators. Denote the image of $g\in O^+\Lambda(S)_v$ by ${{mon}}_g$ and $\widetilde{{{mon}}}_g$.
2. \[thm-item-automorphic-factor\] ([@markman-monodromy-I], Theorem 3.10) The equation $$\label{eq-equivariance}
\widetilde{{{mon}}}_g(\phi(\lambda^\vee))=\phi\left([g(\lambda)]^\vee\right)
$$ holds for every $g\in O^+\Lambda(S)_v$, for all $\lambda\in \Lambda(S)$. Consequently, the composite homomorphism $$K(S) \ {\stackrel{D_S}{\longrightarrow}}
K(S) \ {\stackrel{\eta}{\longrightarrow}} H^*(M)
$$ is $O^+\Lambda(S)_v$ equivariant.[^2]
Set $w:=D_S(v)$. We have the orthogonal direct sum decomposition $$K(S) \ \ = \ \
{{\mathbb Q}}w \oplus w^\perp_{{{\mathbb Q}}}$$ into two distinct irreducible representations of $O^+\Lambda(S)_v$, where we consider a new action of $O^+\Lambda(S)_v$ on $K(S)$, i.e., the conjugate by $D_S$ of the old one. So $g\in O^+\Lambda(S)_v$ acts on $K(S)$ via $D_S\circ g\circ D_S$. Let $\pi_w : K(S)\rightarrow {{\mathbb Q}}w$ and $\pi_{w^\perp} : K(S)\rightarrow w^\perp_{{{\mathbb Q}}}$ be the orthogonal projections. Let $\eta_w:{{\mathbb Q}}w\rightarrow H^*({{\mathcal M}})$ be the restriction of $\eta$ to ${{\mathbb Q}}w$ and $\eta_{w^\perp}:w^\perp_{{\mathbb Q}}\rightarrow H^*({{\mathcal M}})$ the restriction of $\eta$ to $w^\perp_{{{\mathbb Q}}}$. We have $$\pi_w\circ \eta^\dagger \ = \ (\eta_w)^\dagger \ \ \ \mbox{and} \ \ \
\pi_{w^\perp}\circ \eta^\dagger \ = \ (\eta_{w^\perp})^\dagger.$$ Set $\eta_{i,w}:=h_i\circ\eta_w$ and $\eta_{i,w^\perp}:=h_i\circ
\eta_{w^\perp}$. Then $$\begin{aligned}
(\eta_{i,w})^\dagger & = & \pi_w\circ \eta^\dagger\circ e_{2n-i},
\\
(\eta_{i,w^\perp})^\dagger & = & \pi_{w^\perp}\circ \eta^\dagger\circ e_{2n-i}.\end{aligned}$$
\[claim-image-of-f-i\] The homomorphisms $f_i$ and $\eta_i$ in diagram (\[eq-diagram\]) have the same image in $H^{2i}({{\mathcal M}})$.
Clearly, $\eta_{i,w}$ is injective, if it does not vanish. We observe next that the same is true for $\eta_{i,w^\perp}$. This follows from the fact that $\eta_{i,w^\perp}$ is equivariant with respect to the action of the group $O^+\Lambda(S)_v$ (Theorem \[thm-symmetries\], part \[thm-item-automorphic-factor\]). Now $w^\perp_{{{\mathbb Q}}}$ is an irreducible representation of $O^+\Lambda(S)_v$. Hence, $\eta_{i,w^\perp}$ is injective, if and only if it does not vanish. We have $$\begin{aligned}
\eta_i&=&\eta_{i,w}\circ\pi_w+\eta_{i,w^\perp}\circ\pi_{w^\perp},
\\
(\eta_i)^\dagger&=&(\eta_{i,w})^\dagger+(\eta_{i,w^\perp})^\dagger, \ \mbox{and}
\\
f_i &=& \eta_{i,w}\circ (\eta_{i,w})^\dagger+
\eta_{i,w^\perp}\circ (\eta_{i,w^\perp})^\dagger.\end{aligned}$$ Furthermore, the image of $(\eta_{i,w})^\dagger$ is equal to ${{\mathbb Q}}w$, if $\eta_{w,i}$ does not vanish, and the image of $(\eta_{i,w^\perp})^\dagger$ is equal to $w^\perp_{{\mathbb Q}}$, if $\eta_{i,w^\perp}$ does not vanish. Hence, the image of $\eta_{i,w}\circ (\eta_{i,w})^\dagger$ is equal to the image of $\eta_{i,w}$ and the image of $\eta_{i,w^\perp}\circ (\eta_{i,w^\perp})^\dagger$ is equal to the image of $\eta_{i,w^\perp}$. The image of $f_i$ is thus equal to the sum of the images of $\eta_{i,w}$ and $\eta_{i,w^\perp}$. The latter is precisely the image of $\eta_i$.
(Of Proposition \[prop-surjectivity\]) Follows immediately from Theorem \[thm-generators\] and Claim \[claim-image-of-f-i\].
Proof of the main theorem {#sec-proof-of-main-thm}
=========================
We can now prove the main result of this note. We use the notations of section 2.
Let $X$ be a smooth projective variety of $K3^{[n]}$-type. According to Theorem \[thm-deformability\], there exists a smooth and proper family $p:{{\mathcal X}}\rightarrow C$ of irreducible holomorphic symplectic varieties, over a connected reduced projective curve $C$, points $t_1, t_2\in C$, isomorphisms ${{\mathcal M}}\cong p^{-1}(t_1)$ and $X\cong p^{-1}(t_2)$. Additionally, if $q:{{\mathcal X}}\times_C{{\mathcal X}}\rightarrow C$ is the natural morphism, there exists a flat section $s$ of the local system $R_{q_*}^*{{\mathbb Q}}$ through the class $\kappa([E^1])$ in $H^*({{\mathcal M}}\times {{\mathcal M}})$ which is algebraic in $H^*(X_t\times
X_t)$, for every projective fiber $X_t$, $t\in C$, of $p$.
Let us denote by $Z_i$ an algebraic cycle in $H^{2i}(X\times X)$ with cohomology class the degree $2i$ component of $\left(\pi_1^*\sqrt{td_X}\right)
s(t_2)\left(\pi_2^*\sqrt{td_X}\right)$, where $\pi_1$ and $\pi_2$ are the two projections $X\times X {{\rightarrow}}X$.
Using the cycles $Z_i$, we prove by induction on $i\leq n$ that $X$ satisfies the Lefschetz standard conjecture in degree $2i$ for every integer $i$ – recall that the cohomology groups of $X$ vanish in odd degrees. This is obvious for $i=0$.
Let $i\leq n$ be a positive integer. Assume that the Lefschetz conjecture holds for $X$ in degrees up to $2i-1$. Let us show that the morphism $$H^{4n-2i}(X) \ {\stackrel{[Z_i]_*}{\longrightarrow}} \ H^{2i}(X) \ \rightarrow
H^{2i}(X)/\left[A^{2i}(X)+{{\mathbb Q}}\cdot c_i(TX)\right]$$ is surjective. Since $p$ and $q$ are smooth, the morphism above is the fiber at $t_2$ of the morphism $$R^{4n-2i}p_*{{\mathbb Q}}{{\rightarrow}}R^{2i}p_*{{\mathbb Q}}$$ of local systems over $C$ induced by $s$, which implies that it is surjective at $t_2$ if and only if it is surjective at $t_1$. The fiber at $t_1$ of this morphism is induced by the class $\kappa([E^1])$, which shows that it is surjective by Corollary \[cor-comparison-between-kappa-and-f\].
Corollary \[recurrence\] now shows that $X$ satisfies the Lefschetz standard conjecture in degree $2i$, which concludes the proof.
Note that the proof of the main result of this note makes essential use of deformations of hyperkähler varieties along twistor lines, and that a general deformation of a hyperkähler variety along a twistor line is never algebraic, see [@huybrects-basic-results], 1.17. Though the standard conjectures deal with projective varieties, we do not know a purely algebraic proof of the result of this note.
[**Acknowledgements:**]{} The work on this paper began during the authors visit at the University of Bonn in May, 2010. We would like to thank Daniel Huybrechts for the invitations, for his hospitality, and for his contribution to the stimulating conversations the three of us held regarding the paper [@charles]. The first author also wants to thank Claire Voisin for numerous discussions.
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[^1]: Note that this subspace is zero unless $i$ is even.
[^2]: The appearance of $\lambda^\vee$ instead of $\lambda$ as an argument of $\phi$ in (\[eq-equivariance\]), as well as in equation (\[eq-Mukai-isomorphism\]) for Mukai’s isometry, is due to the fact that we use the Mukai pairing to identify $\Lambda(S)$ with its dual. So the class of the kernel $[{{\mathcal F}}]\otimes p_2^*\exp\left(\frac{-c_1(\phi'(v^\vee))}{2n-2}\right)$ of $\phi$ in $K(S\times {{\mathcal M}})\cong K(S)\otimes K({{\mathcal M}})$ is $O^+\Lambda(S)_v$ invariant with respect to the usual action of $O^+\Lambda(S)_v$ on the first factor, and the monodromy action on the second.
|
---
abstract: 'We show that the preliminary data of $p_{T}$ spectra of $\Omega^{-}$ and those of $\phi$ at midrapidity in inelastic events in $pp$ collisions at $\sqrt{s}=$ 13 TeV exhibit a property of constituent quark number scaling, which is a clear signal of quark combination mechanism at hadronization. We use a quark combination model under equal velocity combination approximation to systematically study the production of identified hadrons in $pp$ collisions at $\sqrt{s}$= 13 TeV. The midrapidity data of $p_{T}$ spectra of proton, $\Lambda$, $\Xi^{-}$, $\Omega^{-}$, $\phi$ and $K^{*}$ in inelastic events are well fitted by the model and the $p_{T}$ spectrum of strange quark and that of up/down quarks at hadronization are extracted. The strong $p_{T}$ dependence for data of $p/\phi$ ratio is well explained by the model, which suggests that the production of two hadrons with similar masses is determined by their quark contents instead of their masses as in statistical model. Using preliminary data of proton and $\phi$, $p_{T}$ spectra of other identified hadrons in different multiplicity classes in $pp$ collisions at $\sqrt{s}=$ 13 TeV are predicted. The preliminary data of yield ratios $\Lambda/\pi$, $\Xi^{-}/\pi$ and $\Omega^{-}/\pi$ as the function of midrapidity density of charged particle multiplicity show a strangeness-related hierarchy and are naturally understood by quark combination mechanism.'
address:
- 'School of Physics and Engineering, Qufu Normal University, Shandong 273165, China'
- 'Department of Physics, Jining University, Shandong 273155, China'
- 'School of Physics and Engineering, Qufu Normal University, Shandong 273165, China'
- 'Department of Physics, Jining University, Shandong 273155, China'
author:
- 'Jian-wei Zhang'
- 'Hai-hong Li'
- 'Feng-lan Shao'
- Jun Song
bibliography:
- 'refpp.bib'
title: 'New feature of soft strange hadron production in $pp$ collisions at $\sqrt{s}=$ 13 TeV '
---
Introduction\[sec1\]
====================
Most of hadrons produced in high energy collisions are of relatively low (transverse) momentum perpendicular to beam axis. Production of soft hadrons is mainly driven by soft QCD process and, in particular, non-perturbative hadronization. Experimental and theoretical studies of soft hadron production are important to understand the property of the soft parton system created in collisions and test/develop existing phenomenological models. Heavy ion physics at SPS, RHIC and LHC energies show the creation of quark gluon plasma (QGP) in early stage of collisions. In elementary $pp$ and/or $p\bar{p}$ collisions, it is usually presumed that QGP is not created, at least up to RHIC energies. However, recent measurements at LHC energies show a series of highlights of hadron production in $pp$ collisions such as ridge and collectivity behaviors [@Khachatryan:2010gv; @Khachatryan:2015lva; @Khachatryan:2016txc], the increased baryon/meson ratio and the increased strangeness [@Bianchi:2016szl; @ALICE:2017jyt; @Sarma:2017hvg]. Theoretical studies of these new phenomena mainly focus on how to build the new feature of small parton system relating to these observations by considering different mechanisms such as the color re-connection, string overlap and/or color rope [@Bautista:2015kwa; @Bierlich:2014xba; @Ortiz:2013yxa; @Christiansen:2015yqa], or by considering the creation of mini-QGP or phase transition [@Liu:2011dk; @Werner:2010ss; @Bzdak:2013zma; @Bozek:2013uha; @Prasad:2009bx; @Avsar:2010rf], etc.
In our latest works [@Song:2017gcz; @Shao:2017eok; @Song:2018tpv; @Li:2017zuj; @Gou:2017foe], we propose a new understanding of new features of hadron production (mainly of $p_{T}$ spectrum and yield) for the small quark/parton systems created in $pp$ and/or $p$-Pb collisions at LHC energy, that is, the change of hadronization mechanism from the traditional fragmentation to the quark (re-)combination. In quark (re-)combination mechanism (QCM), there exists some typical behaviors for the production of identified hadrons such as the enhanced baryon/meson ratio and quark number scaling of hadron elliptical flow at intermediate $p_{T}$. These behaviors are already observed in relativistic heavy ion collisions [@Adler:2003kg; @Abelev:2006jr; @Adare:2006ti] and, recently, are also observed in $pp$ and $p$-Pb collisions at LHC energies in high multiplicity classes [@Bianchi:2016szl; @Sarma:2017hvg; @Khachatryan:2014jra; @Khachatryan:2016txc]. In particular, a quark number scaling property for hadron transverse momentum spectra is firstly observed in $p-$Pb collisions at $\sqrt{s_{NN}}=$ 5.02 TeV [@Song:2017gcz].
Recently, ALICE collaboration reports the data of $p_{T}$ spectra of identified hadrons in different multiplicity classes in $pp$ collisions at $\sqrt{s}=$ 7 TeV [@Acharya:2018orn] and the preliminary data in inelastic events in $pp$ collisions at $\sqrt{s}=$ 13 TeV [@Bencedi:2018eye]. Here, we find, with a great exciting, a clear signal of quark number scaling property. Considering the production of baryon $\Omega^{-}\left(sss\right)$ and meson $\phi\left(s\bar{s}\right)$, their momentum distribution functions $f\left(p_{T}\right)\equiv dN/dp_{T}$ in QCM under equal velocity combination approximation read as $$\begin{aligned}
f_{\Omega}\left(p_{T}\right) & =\kappa_{\Omega}\left[f_{s}\left(\frac{p_{T}}{3}\right)\right]^{3},\label{eq:fpt_Omega}\\
f_{\phi}\left(p_{T}\right) & =\kappa_{\phi}f_{s}\left(\frac{p_{T}}{2}\right)f_{\bar{s}}\left(\frac{p_{T}}{2}\right)=\kappa_{\phi}\left[f_{s}\left(\frac{p_{T}}{2}\right)\right]^{2}.\label{eq:fpt_phi}\end{aligned}$$ Here, $\kappa_{\phi}$, $\kappa_{\Omega}$ are coefficients independent of momentum. $f_{s,\bar{s}}\left(p_{T}\right)$ is $s$ ($\bar{s}$) quark distribution at hadronization and we assume $f_{s}\left(p_{T}\right)=f_{\bar{s}}\left(p_{T}\right)$ in the center rapidity region at LHC energies. With above two formulas, we get a production correlation between $\Omega^{-}$ and $\phi$ in QCM $$f_{\phi}^{1/2}\left(2p_{T}\right)=\kappa_{\phi,\Omega}f_{\Omega}^{1/3}\left(3p_{T}\right)=\kappa_{\phi}^{1/2}f_{s}\left(p_{T}\right),\label{eq:qns}$$ where $\kappa_{\phi,\Omega}=\kappa_{\phi}^{1/2}/\kappa_{\Omega}^{1/3}$ is independent of momentum. In order to show this scaling property, we take following operation on the the $dN/dp_{T}dy$ data of $\Omega$ and $\phi$ at midrapidity [@Acharya:2018orn]: (i) divide the $p_{T}$ bin of $\Omega^{-}$($\phi$) by 3 (2) and (ii) take the $1/3$ ($1/2)$ power of the measured $dN/dp_{T}dy$ for $\Omega^{-}$($\phi$), and (iii) multiply $\left(dN_{\Omega}/dp_{T}dy\right)^{1/3}$ by a constant factor $\kappa_{\phi,\Omega}$ so that it is consistent with the scaled data of $\phi$ in magnitude. We show in Fig.\[fig1\] the results in different multiplicity classes in $pp$ collisions at $\sqrt{s}=$ 7 TeV. We see that in the high multiplicity classes, e.g., Fig.\[fig1\](a), the scaled data of $\Omega^{-}$ is consistent very well with those of $\phi$, and therefore the quark number scaling property holds well. This is a clear signal (direct evidence) of quark combination hadronization in $pp$ collisions at LHC. In the low multiplicity classes, Fig. \[fig1\](c) and (d), the scaled data of $\Omega^{-}$ are somewhat flatter than that of $\phi$ as $p_{T}\gtrsim1$ GeV/c and the quark number scaling property is broken to a certain extent. We note that this is probably due to the threshold of strange quark production [@Gou:2017foe].
![The scaling property for the $dN/dp_{T}dy$ data of $\Omega^{-}$ and $\phi$ at midrapidity in different multiplicity classes in $pp$ collisions at $\sqrt{s}=$ 7 TeV. The coefficient $\kappa_{\phi,\Omega}$ in four multiplicity classes is taken to be (1.76, 1.82, 1.83, 1.93), respectively. Data of $\Omega^{-}$ and $\phi$ are taken from Ref.[@Acharya:2018orn]. \[fig1\] ](fig1)
In Fig. \[fig2\], we show the scaled data of $\Omega^{-}$ and $\phi$ in $pp$ collisions at $\sqrt{s}=$ 7 and 13 TeV [@Abelev:2012hy; @Abelev:2012jp; @Acharya:2018orn; @Bencedi:2018eye] as a guide of energy dependence. We see that the quark number scaling property in inelastic events (summation of different multiplicity classes) in $pp$ collisions at $\sqrt{s}=$ 7 TeV is broken to a certain extent but it is well fulfilled in inelastic events in $pp$ collisions at $\sqrt{s}=$ 13 TeV. This suggests that the quark combination hadronization is more applicable at higher collision energies. Therefore, in this paper, we systematically study the $p_{T}$ distributions and yields of identified hadrons in $pp$ collisions at $\sqrt{s}=$ 13 TeV to further test the quark combination hadronization in $pp$ collisions at LHC energies.
![The scaling property for the $dN/dp_{T}dy$ data of $\Omega^{-}$ and $\phi$ at midrapidity in inelastic events in $pp$ collisions at $\sqrt{s}=$ 7 and 13 TeV. The coefficient $\kappa_{\phi,\Omega}$ is taken to be (2.0, 1.5), respectively. Data of $\Omega^{-}$ and $\phi$ are taken from Refs. [@Acharya:2018orn; @Bencedi:2018eye]. \[fig2\] ](fig2)
The paper is organized as follows: Sec. II briefly introduces a working model of quark (re)combination mechanism under equal combination approximation. Sect. III and Sec. IV present our results and relevant discussions in minimum-bias events and different multiplicity classes, respectively. A summary is given at last in Sec. V.
quark combination model under equal velocity combination approximation
======================================================================
Quark (re-)combination/coalescence mechanism was proposed in 1970s [@Bjorken:1973mh] and has many applications both in elementary $e^{-}e^{+}$, $pp$ and heavy-ion collisions [@Das:1977cp; @Xie:1988wi; @Hwa:1979pn; @Chiu:1978nc; @Hwa:1994uha; @Wang:1996jy; @Cuautle:1997ti]. In particular, ultra-relativistic heavy ion collisions create the deconfined hot quark matter of large volume whose microscopic hadronization process can be described by QCM naturally [@Greco:2003xt; @Fries:2003vb; @Molnar:2003ff; @Hwa:2002tu; @Shao:2004cn; @Song:2013isa]. In this section, we briefly introduce a working model proposed in previous works [@Song:2017gcz; @Gou:2017foe] within QCM framework under the equal velocity combination approximation. We take the constituent quarks and antiquarks as the effective degrees of freedoms of the soft parton system created in collisions just at hadronization. The combination of these constituent quarks and antiquarks with equal velocity forms the identified baryons and/or mesons.
momentum distributions and yields of identified hadrons
-------------------------------------------------------
The momentum distributions of identified baryon and meson read as $$\begin{aligned}
f_{B_{i}}\left(p_{B}\right) & =N_{B_{i}}\,f_{B_{i}}^{\left(n\right)}\left(p_{B}\right),\label{eq:fbi}\\
f_{M_{i}}\left(p_{M}\right) & =N_{M_{i}}\,f_{M_{i}}^{\left(n\right)}\left(p_{M}\right),\label{eq:fmi}\end{aligned}$$ where superscript $\left(n\right)$ denotes the distribution is normalized to one and $N_{B_{i}}$ ($N_{M_{i}}$) is the momentum-integrated multiplicity. Under the equal velocity combination approximation, or called comoving approximation, we have , for $B_{i}$$\left(q_{1}q_{2}q_{3}\right)$ $$f_{B_{i}}^{\left(n\right)}\left(p_{B}\right)=A_{B_{i}}\,f_{q_{1}}^{\left(n\right)}\left(x_{1}p_{B}\right)f_{q_{2}}^{\left(n\right)}\left(x_{2}p_{B}\right)f_{q_{3}}^{\left(n\right)}\left(x_{3}p_{B}\right),\label{eq:fnbi}$$ and for $M_{i}\left(q_{1}\bar{q}_{2}\right)$ $$f_{M_{i}}^{\left(n\right)}\left(p_{M}\right)=A_{M_{i}}f_{q_{1}}^{\left(n\right)}\left(x_{1}p_{M}\right)f_{\bar{q}_{2}}^{\left(n\right)}\left(x_{2}p_{M}\right),\label{eq:fnmi}$$ where $f_{q}^{\left(n\right)}\left(p\right)\equiv dn_{q}/dp$ is the momentum distribution of quarks normalized to one. $A_{B_{i}}^{-1}=\int{\rm d}p\prod_{i=1}^{3}f_{q_{i}}^{\left(n\right)}\left(x_{i}p\right)$ and $A_{M_{i}}^{-1}=\int{\rm d}pf_{q_{1}}^{\left(n\right)}\left(x_{1}p\right)f_{\bar{q}_{2}}^{\left(n\right)}\left(x_{2}p\right)$ are normalization coefficients for baryon $B_{i}$ and $M_{i}$, respectively. Momentum fractions $x$ are given by recalling momentum $p=m\gamma v\propto m$, $$x_{i}=m_{i}/\sum_{j}m_{j},$$ where indexes take $i,j=1,2,3$ for baryon and $i,j=1,2$ for meson. Quark masses are taken to be constituent masses $m_{s}=500$ MeV and $m_{u}=m_{d}=330$ MeV.
Multiplicities of baryon and meson $$\begin{aligned}
N_{B_{i}} & =N_{q_{1}q_{2}q_{3}}P_{q_{1}q_{2}q_{3}\rightarrow B_{i}},\label{eq:Nbi}\\
N_{M_{i}} & =N_{q_{1}\bar{q}_{2}}P_{q_{1}\bar{q}_{2}\rightarrow M_{i}}.\label{eq:Nmi}\end{aligned}$$ Here $N_{q_{1}q_{2}q_{3}}$ is the number of all possible three quark combinations relating to $B_{i}$ formation and is taken to be $6N_{q_{1}}N_{q_{2}}N_{q_{3}}$, $3N_{q_{1}}\left(N_{q_{1}}-1\right)N_{q_{2}}$ and $N_{q_{1}}\left(N_{q_{1}}-1\right)\left(N_{q_{1}}-2\right)$ for cases of three different flavors, two identical flavor and three identical flavor, respectively. Factors 6 and 3 are numbers of permutation relating to different quark flavors. $N_{q_{1}\bar{q}_{2}}=N_{q_{1}}N_{\bar{q}_{2}}$ is the number of all possible $q_{1}\bar{q}_{2}$ pairs relating to $M_{i}$ formation.
Considering the flavor independence of strong interaction, we assume the probability of $q_{1}q_{2}q_{3}$ forming a baryon and the probability of $q_{1}\bar{q}_{2}$ forming a meson are flavor independent, the combination probability can be determined with a few parameters
$$\begin{aligned}
P_{q_{1}q_{2}q_{3}\rightarrow B_{i}} & = & C_{B_{i}}\frac{\overline{N}_{B}}{N_{qqq}},\label{prob_B}\\
P_{q_{1}\bar{q}_{2}\rightarrow M_{i}} & = & C_{M_{i}}\frac{\overline{N}_{M}}{N_{q\bar{q}}}.\label{prob_M}\end{aligned}$$
Here $\overline{N}_{B}/N_{qqq}$ denotes the average (or flavor blinding) probability of three quarks combining into a baryon. $\overline{N}_{B}$ is the average number of total baryons and $N_{qqq}=N_{q}(N_{q}-1)(N_{q}-2)$ is the number of all possible three quark combinations with $N_{q}=\sum_{f}N_{f}$ the total quark number. $C_{B_{i}}$ is the probability of selecting the correct discrete quantum number such as spin relating to $B_{i}$ as $q_{1}q_{2}q_{3}$ is destined to form a baryon. Similarly, $\overline{N}_{M}/N_{q\bar{q}}$ approximately denotes the average probability of a quark and antiquark combining into a meson and $C_{M_{i}}$ is the branch ratio to $M_{i}$ as $q_{1}\bar{q}_{2}$ is destined to from a meson. $\overline{N}_{M}$ is total meson and $N_{q\bar{q}}=N_{q}N_{\bar{q}}$ is the number of all possible quark antiquark pairs for meson formation.
In this paper, we only consider the ground state $J^{P}=0^{-},\,1^{-}$ mesons and $J^{P}=(1/2)^{+},\,(3/2)^{+}$ baryons in flavor SU(3) group. For mesons $$C_{M_{j}}=\left\{ \begin{array}{ll}
\frac{1}{1+R_{V/P}} & \text{for }J^{P}=0^{-}\textrm{ mesons}\\
\frac{R_{V/P}}{1+R_{V/P}} & \textrm{for }J^{P}=1^{-}\textrm{ mesons},
\end{array}\right.$$ where the parameter $R_{V/P}$ represents the relative production weight of the $J^{P}=1^{-}$ vector mesons to that of the $J^{P}=0^{-}$ pseudo-scalar mesons of the same flavor composition; for baryons $$C_{B_{j}}=\left\{ \begin{array}{ll}
\frac{1}{1+R_{D/O}} & \textrm{for }J^{P}=({1}/{2})^{+}\textrm{ baryons}\\
\frac{R_{D/O}}{1+R_{D/O}} & \textrm{for }J^{P}=({3}/{2})^{+}\textrm{ baryons},
\end{array}\right.$$ except that $C_{\Lambda}=C_{\Sigma^{0}}={1}/{(2+R_{D/O})},~C_{\Sigma^{*0}}={R_{D/O}}/{(2+R_{D/O})},~C_{\Delta^{++}}=C_{\Delta^{-}}=C_{\Omega^{-}}=1$. Here, $R_{D/O}$ stands for the relative production weight of the $J^{P}=(3/2)^{+}$ octet to the $J^{P}=(1/2)^{+}$ decuplet baryons of the same flavor content. Here, $R_{V/P}$ is taken to be 0.45 by fitting the data of $K^{*}/K$ ratios in $pp$ collisions at $\sqrt{s}=$ 7 TeV and $p$-Pb collisions at $\sqrt{s_{NN}}=$ 5.02 TeV [@Adam:2016bpr] and $R_{D/O}$ is taken to be 0.5 by fitting the data of $\Xi^{*}/\Xi$ and $\Sigma^{*}/\Lambda$ [@Adamova:2017elh]. The fraction of baryons relative to mesons is $N_{B}/N_{M}\approx0.085$ [@Song:2013isa; @Shao:2017eok; @Gou:2017foe]. Using the unitarity constraint of hadronization $N_{M}+3N_{B}=N_{q}$, $N_{B_{i}}$ and $N_{M_{i}}$ can be calculated using above formulas for given quark numbers at hadronization.
We summarize the main underlying dynamics of this working model. Constituent quarks and antiquarks are assumed to be effective degrees of freedom of soft parton system at hadronization. The combination of these constituent quarks with equal velocity forms baryons and mesons. This is similar to that in (constituent) quark model, i.e., the summation of mass (and momentum) of constituent quarks properly constructs the mass (and momentum) of hadron. Model parameters $R_{V/P}$ and $R_{D/O}$ contain unclear non-perturbative dynamics and are obtained by fitting the relevant experimental data and are assumed to be relatively stable in different collision systems or collision energies. Also, the normalization of hadronization process is a prerequisite for quark combination, e.g. $N_{M}+3N_{B}=N_{q}$. Therefore, this working model is different from those popular parton recombination/coalescence models [@Greco:2003xt; @Fries:2003vb] which adopt the Wigner wave function method under instantaneous hadronization approximation.
Threshold effects in case of small quark numbers
------------------------------------------------
As quark numbers at hadronization are small, identified hadron production will suffer some threshold effects. For example, baryon production is forbidden for events with $N_{q}<3$. For events with $N_{s}<3$, $\Omega^{-}$ baryon production is forbidden. In $pp$ collisions at LHC energies, the event-averaged number of strange quarks $\langle N_{s}\rangle\lesssim1$ in mid-rapidity region ($|y|<0.5$) for inelastic events and not-too high multiplicity event classes. Therefore, the yield of $\Omega^{-}$ is no longer completely determined by the average number of strange quarks but is strongly influenced by the distribution of strange quark number. Similar case for $\Xi$ which needs two strange quarks, etc. Here we use $\mathcal{P}\left(\left\{ N_{q_{i}}\right\} ,\left\{ \langle N_{q_{i}}\rangle\right\} \right)$ to denote the distribution of quark numbers around the event averages, and obtain the averaged multiplicity of identified hadrons, $$\langle N_{h_{i}}\rangle=\sum_{\left\{ N_{q_{i}}\right\} }\mathcal{P}\left(\left\{ N_{q_{i}}\right\} ,\left\{ \langle N_{q_{i}}\rangle\right\} \right)N_{h_{i}}$$ where $N_{h_{i}}$ is given by Eqs. (\[eq:Nbi\]) and (\[eq:Nmi\]) and is the function of $\left\{ N_{q_{i}}\right\} $.
For simplicity, we assume flavor-independent quark number distributions $$\mathcal{P}\left(\left\{ N_{q_{i}}\right\} ,\left\{ \langle N_{q_{i}}\rangle\right\} \right)=\prod_{f}\mathcal{P}\left(N_{f},\langle N_{f}\rangle\right),$$ where $f$ runs over $u$, $d$, $s$ flavors. Here we neglect the fluctuation of net-charges and take $N_{f}=N_{\bar{f}}$ in each events. The distribution of $u$ and $d$ quarks is based on the Poisson distribution $Poi\left(N_{u(d)},\langle N_{u(d)}\rangle\right)$. As aforementioned discussion, we particularly tune strange quark distribution. Because in minimum bias events and small multiplicity classes in $pp$ collisions $\langle N_{s}\rangle\lesssim1$ and Poisson distribution $Poi\left(N_{s},\langle N_{s}\rangle\right)$ in this case has a long tail as $N_{s}\geq3$ which may over-weight the events with $N_{s}\geq3$, we distort the Poisson distribution by a suppression factor $\gamma_{s}$, that is $\mathcal{P}\left(N_{s},\langle N_{s}\rangle\right)=\mathcal{N}Poi\left(N_{s},\langle N_{s}\rangle\right)\times\left[\gamma_{s}\,\Theta\left(N_{s}-3\right)+\Theta\left(3-N_{s}\right)\right]$ where $\Theta\left(x\right)$ is the Heaviside step function and $\mathcal{N}$ is normalization constant. $\gamma_{s}$ is taken to be 0.8 in inelastic (INEL>0) events and various multiplicity classes.
There are other possible effects of small quark numbers. For example, in events with $N_{s}=N_{\bar{s}}=1$, because $s$ and $\bar{s}$ are most likely created from the same one vacuum excitation and therefore they are not likely to directly constitute color single-let and therefore $\phi$ production in these events is suppressed. In addition, quark momentum distributions are more or less dependent on the quark numbers (in other words, system size) and we neglect such dependence in events for the given multiplicity classes and its potential effects are studied in the future works.
Results in INEL events
=======================
We apply the above working model to describe the hadron production in the midrapidity range in $pp$ collisions. The quark momentum distribution function is reduced to $f_{q}\left(p\right)=f_{q}\left(p_{T},y\right)\equiv\frac{dN_{q}}{dy}f_{q}^{\left(n\right)}\left(p_{T}\right)$ with quark number density $dN_{q}/dy$ and (azimuthal integrated, one dimensional) quark $p_{T}$ spectrum $f_{q}^{\left(n\right)}\left(p_{T}\right)\equiv dn_{q}/dp_{T}$. In the following text, $N_{q}$ is also quoted as the number of quark in the unit rapidity range $|y|<0.5$. The notation of hadrons is similar.
quark $p_{T}$ distribution at hadronization
-------------------------------------------
Using the scaling properties in Eq. (\[eq:qns\]) and experimental data shown in Fig. \[fig2\], we can directly obtain the normalized $p_{T}$ distribution of strange quarks at hadronization, which can be parameterized in the form $$f_{q}^{\left(n\right)}\left(p_{T}\right)=\mathcal{N}_{q}\left(p_{T}+a_{q}\right)^{b_{q}}\left(1+\frac{\sqrt{p_{T}^{2}+M_{q}^{2}}-M_{q}}{n_{q}c_{q}}\right)^{-n_{q}},\label{eq:fq_para}$$ where $\mathcal{N}_{q}$ is the normalization constant and parameters $a_{s}=0.15$ GeV/c, $b_{s}=0.649$, $n_{s}=4.14$, $M_{s}=0.5$GeV and $c_{s}=0.346$. By fitting the data of other hadrons involving the up/down quarks, we also obtain the $p_{T}$ distribution of up/down quarks at hadronization. The corresponding parameter are $a_{u}=0.15$ GeV/c, $b_{u}=0.355$, $n_{s}=3.46$, $M_{s}=0.33$ GeV and $c_{s}=0.358$. In Fig. \[fig3\], we plot $f_{u}^{\left(n\right)}\left(p_{T}\right)$ and $f_{s}^{\left(n\right)}\left(p_{T}\right)$ as the function of $p_{T}$ and their ratio.
![The $p_{T}$ spectra of $u$ and $s$ quarks at midrapidity and the ratio between them in inelastic (INEL>0) events in $pp$ collisions at $\sqrt{s}=$ 13 TeV. \[fig3\]](fig3)
We emphasize that, by taking advantage of the quark number scaling property, this is the first time we can conveniently extract the momentum distributions of soft quarks at hadronization from the experimental data of hadronic $p_{T}$ spectra. The extracted quark $p_{T}$ spectra carry important information of soft parton system created in $pp$ collisions at LHC energy. First, the extracted $f_{u}^{\left(n\right)}\left(p_{T}\right)$ and $f_{s}^{\left(n\right)}\left(p_{T}\right)$ deviate from Boltzmann (or Fermi-Dirac) distribution in the low $p_{T}$ range. This indicates that thermalization may be not reached for the small parton system created in $pp$ collisions at LHC. Second, we see that the ratio $f_{s}^{\left(n\right)}\left(p_{T}\right)/f_{u}^{\left(n\right)}\left(p_{T}\right)$, Fig. \[fig3\] (b), increases at small $p_{T}$ and then saturates (only slightly decreases) with $p_{T}$. This property is similar to the observation in $pp$ collisions at $\sqrt{s}=$ 7 TeV [@Gou:2017foe] and in $p$-Pb collisions at $\sqrt{s_{NN}}=$ 5.02 TeV [@Song:2017gcz] and is also similar to the observation in heavy ion collisions at RHIC and LHC [@Shao:2009uk; @Wang:2013pnn; @Chen:2008vr]. These information of constituent quarks provides important constraint of developing more sophisticated theoretical models of soft-parton system in high energy collisions.
$p_{T}$ spectra of identified hadrons
--------------------------------------
In Fig. \[fig4\], we show the calculation results of identified hadrons proton, $\Lambda$, $\Xi^{-}$, $\Omega^{-}$, $\phi$ and $K^{*0}$ in inelastic (INEL>0) events in $pp$ collisions at $\sqrt{s}=$ 13 TeV using the quark spectra in Fig. \[fig3\] and quark numbers $\langle N_{u}\rangle=2.8$ and $\langle N_{s}\rangle=0.86$ in midrapidity region $|y|<0.5$. Here isospin symmetry $\langle N_{u}\rangle=\langle N_{d}\rangle$ is applied in $pp$ collisions at LHC energies. Solid lines are QCM results and symbols are (preliminary) data of ALICE collaboration [@Bencedi:2018eye]. We see that the data can bell fitted in general by QCM [^1] . Our result of magnitude (i.e, multiplicity) of $K^{*}$ is slightly smaller than the data. If we multiply the result of $K^{*0}$ spectrum by an constant factor and we will see the shape is in good agreement with the data.
Besides the scaling property between the $p_{T}$ spectra of $\Omega^{-}$ and $\phi$ shown in the introduction (Sec. \[sec1\]), $\left(p+\bar{p}\right)/\phi$ ratio as the function of $p_{T}$ can also give an intuitive picture of microscopic mechanism of hadron production. Proton and $\phi$ have similar masses but totally different quark contents. In the central (0-10% centrality) Pb-Pb collisions at $\sqrt{s_{NN}}=2.76$ TeV, the data of $\left(p+\bar{p}\right)/\phi$ ratio, black squares in Fig. \[fig5\], are almost flat with respect to $p_{T}$. This flat ratio is usually related to similar masses of proton and $\phi$ and is usually attributed to the strong radial flow and statistical hadronization under chemical/thermal equilibrium in relativistic heavy ion collisions. However, data of $\left(p+\bar{p}\right)/\phi$ in inelastic events in $pp$ collisions at $\sqrt{s}=$ 13 TeV, solid circles in Fig. \[fig5\], show a rapid decrease with the increasing $p_{T}$. This is an indication of out of thermal equilibrium in $pp$ collisions. Quark content is an important factor in describing the shape of hadron $p_{T}$ distribution. We see that the QCM result of $\left(p+\bar{p}\right)/\phi$ is in good agreement with the data of $pp$ collisions. Therefore, $pp$ data are more directly probe to the microscopic mechanism of hadron production at hadronization.
![Ratio $\left(p+\bar{p}\right)/\phi$ as the function of $p_{T}$ in inelastic events in $pp$ collisions at $\sqrt{s}=$ 13 TeV. Sold line is the result of QCM and symbols are (preliminary) data of ALICE collaboration.\[fig5\]](fig5)
Hyperons $\Lambda$, $\Xi^{0,-}$ and $\Omega^{-}$ contain one, two and three $s$ constituent quarks, respectively. Therefore, ratios $\Xi/\Lambda$ and $\Omega^{-}/\Xi$ can reflect the difference in momentum distribution between $u$($d$) quark and $s$ quark at hadronization. Fig. \[fig5-1\] shows our prediction of ratios $\Xi^{-}/\Lambda$ and $\Omega^{-}/\Xi^{-}$ as the function of $p_{T}$ in inelastic events in $pp$ collisions at $\sqrt{s}=$ 13 TeV. We see that two ratios increase with $p_{T}$ and then tend to saturate at intermediate $p_{T}\sim6$ GeV, which is directly due to the difference in quark $p_{T}$ spectra shown in Fig. \[fig3\](b).
![Prediction of ratios $\Xi^{-}/\Lambda$ and $\Omega^{-}/\Xi^{-}$ as the function of $p_{T}$ at midrapidity in inelastic events in $pp$ collisions at $\sqrt{s}=$ 13 TeV.\[fig5-1\]](ratio_Xi_Lam_nsd)
Predictions in different multiplicity classes
=============================================
Using the preliminary data of $p_{T}$ spectra of $\phi$, proton and $K^{*}$ in different multiplicity classes [@Sarma:2018pkp; @Dash:2018cjh], it is sufficient to determine the $p_{T}$ spectra of constituent quarks at hadronization. Fig. \[fq\_class\] shows the extracted quark $p_{T}$ spectra (using the parameterized form Eq. (\[eq:fq\_para\])) at midrapidity in different multiplicity classes. We see that the quark $p_{T}$ spectra in high multiplicity classes tend to be more similar to thermal behavior, which is related to the increasing multiple parton interactions in these event classes.
![The $p_{T}$ spectra of $u$ and $s$ quarks at midrapidity in different multiplicity classes in $pp$ collisions at $\sqrt{s}=$ 13 TeV. \[fq\_class\]](fq_class)
In Fig. \[fig8\], we show the fit of data of $p_{T}$ spectra of proton, $\phi$ and $K^{*0}$ [@Sarma:2018pkp; @Dash:2018cjh] using QCM and the prediction of of other identified hadrons in different multiplicity classes in $pp$ collisions at $\sqrt{s}=$ 13 TeV. Note that classes IV and V are combined for the $K^{*0}$ data and the same for our $K^{*}$ results. Beside directly comparing the prediction of single hadron spectra with the future data, we emphasize that QCM can be more effectively tested by some spectrum ratios and/or scaling. The first is to test whether the scaling property between the $p_{T}$ spectrum of $\Omega^{-}$ and that of $\phi$ holds for the data in different multiplicity classes. The second is to study the ratio $\Omega^{-}/\phi$ as the function of $p_{T}$. $\Omega^{-}/\phi$ ratio in QCM is solely determined by the strange quark $p_{T}$ spectrum at hadronization and the ratio usually exhibits a nontrivial $p_{T}$ dependence, as shown in Fig. \[fig9\](a), which is a typical behavior of Baryon/Meson ratio in QCM and is absent or unapparent in the traditional fragmentation picture. We also see that the ratio $\Omega^{-}/\phi$ in higher multiplicity classes can reach higher peak values than that in relatively low multiplicity classes and the peak position of $\Omega^{-}/\phi$ in higher multiplicity classes is also enlarged in compared with that in low multiplicity classes. The third is to study $p/\phi$ ratio as the function of $p_{T}$ to clarify the $p_{T}$ dependence is flavor originated or mass originated? Results of QCM is shown in Fig. \[fig9\](b) which decrease with $p_{T}$ and show relatively weak multiplicity dependence.
In Fig. \[fig10\], we show the $p_{T}$-integrated yields of identified hadrons in different multiplicity classes and compare them with the preliminary data in $pp$ collisions at $\sqrt{s}=$ 13 TeV [@Dash:2018cjh; @Vislavicius:2017lfr]. In general, results of QCM, solid lines, are in good agreement with the data (with maximum deviation about 10%).
![Prediction of ratios $\Omega^{-}/\phi$ and $p/\phi$ as the function of $p_{T}$ at midrapidity in $pp$ collisions at $\sqrt{s}=$ 13 TeV.\[fig9\]](fig7)
Yield ratios of different hadrons can significantly cancel the dependence of model parameters and/or model inputs. Therefore, they are more direct test of the basic physics of the model in confronting with the experimental data. In Fig. \[fig11\], we show the yield ratios of hyperons $\Omega^{-}$, $\Xi^{-}$ and $\Lambda$ to pions divided by the values in the inclusive INEL>0 events. Data of $pp$ collisions at $\sqrt{s}=$ 7 [@ALICE:2017jyt] and 13 TeV [@Dash:2018cjh] and those of $p$-Pb collisions at $\sqrt{s_{NN}}=$ 5.02 TeV [@ABELEV:2013zaa; @Abelev:2013haa] are all presented in order to get a visible tendency with respect to multiplicity of charged particles at mid-rapidity. Solid lines are numerical results of QCM, which are found to be in agreement with the data. We emphasize that such strangeness-related hierarchy behavior is closely related to the strange quark content of these hyperons during their production at hadronization, which can be understood easily via an analytical relation in QCM. Taking yield formulas Eqs. (\[eq:Nbi\]), (\[prob\_B\]) and considering the strong and electromagnetic decays, we have $$\begin{aligned}
N_{\Omega} & \approx\frac{\lambda_{s}^{3}}{\left(2+\lambda_{s}\right)^{3}}\overline{N}_{B},\\
N_{\Xi} & \approx\frac{3\lambda_{s}^{2}}{\left(2+\lambda_{s}\right)^{3}}\overline{N}_{B},\label{eq:NXi}\\
N_{\Lambda} & \approx\left(\frac{2+0.88R_{D/O}}{2+R_{D/O}}+0.94\frac{R_{D/O}}{1+R_{D/O}}\right)\frac{6\lambda_{s}}{\left(2+\lambda_{s}\right)^{3}}\overline{N}_{B},\\
& \approx\frac{7.73\lambda_{s}}{\left(2+\lambda_{s}\right)^{3}}\overline{N}_{B},\nonumber \end{aligned}$$ where we neglect the effects of small quark numbers and adopt the strangeness suppression factor $\lambda_{s}=\langle N_{s}\rangle/\langle N_{u}\rangle$. Because of complex decay contributions, pion yield has a complex expression [@Wang:2012cw] and here we can write $N_{\pi}=a_{\pi}\langle N_{q}\rangle$ with coefficient $a_{\pi}$ being a almost constant. Then the double ratios in Fig. \[fig11\] have simple approximate expressions $$\begin{aligned}
\frac{N_{\Omega}}{N_{\pi}}/\left(\frac{N_{\Omega}}{N_{\pi}}\right)_{INEL>0}^{pp} & \approx\frac{\lambda_{s}^{3}}{\left(2+\lambda_{s}\right)^{3}}/\frac{\lambda_{s}^{'3}}{\left(2+\lambda_{s}^{'}\right)^{3}},\label{eq:dr_Omega}\\
\frac{N_{\Xi}}{N_{\pi}}/\left(\frac{N_{\Xi}}{N_{\pi}}\right)_{INEL>0}^{pp} & \approx\frac{\lambda_{s}^{2}}{\left(2+\lambda_{s}\right)^{3}}/\frac{\lambda_{s}^{'2}}{\left(2+\lambda_{s}^{'}\right)^{3}},\label{eq:dr_Xi}\\
\frac{N_{\Lambda}}{N_{\pi}}/\left(\frac{N_{\Lambda}}{N_{\pi}}\right)_{INEL>0}^{pp} & \approx\frac{\lambda_{s}}{\left(2+\lambda_{s}\right)^{3}}/\frac{\lambda_{s}^{'}}{\left(2+\lambda_{s}^{'}\right)^{3}},\label{eq:dr_Lambda}\end{aligned}$$ where $\lambda_{s}^{'}$ is the strangeness suppression factor in INEL>0 events in $pp$ collisions. The dotted lines in Fig. \[fig11\] are results of above analytic formulas with a parameterized strangeness $\lambda_{s}=\lambda_{s}^{'}\left[1+0.165\log\left(\frac{\langle dN_{ch}/d\eta\rangle_{|\eta|<0.5}}{6.0}\right)\right]$ with $\lambda_{s}^{'}=0.31$. We see that the experimental data of these double ratios as $\langle dN_{ch}/d\eta\rangle_{|\eta|<0.5}\gtrsim10$ can be correctly described by analytical formulas Eqs. (\[eq:dr\_Omega\]-\[eq:dr\_Lambda\]). In small multiplicity classes $\langle dN_{ch}/d\eta\rangle_{|\eta|<0.5}\lesssim6$ finite quark number effects are not negligible and the analytic approximation is larger than the experimental data to a certain extent. Our numerical results have included finite quark number effects and are found to be more close to the data.
![Yield ratios to pions divided by the values in the inclusive INEL>0 events. Data of $pp$ collisions at $\sqrt{s}=$ 7 [@ALICE:2017jyt] and 13 TeV [@Dash:2018cjh] and in $p$-Pb collisions at $\sqrt{s_{NN}}=$ 5.02 TeV [@ABELEV:2013zaa; @Abelev:2013haa] are presented. Sold lines are numerical results of QCM and dotted lines are analytic approximation in QCM.\[fig11\]](fig11)
![Yield ratio $\Xi/\phi$ as a function of $\langle dN_{ch}/d\eta\rangle_{|\eta|<0.5}$. The preliminary data in $pp$ collisions at $\sqrt{s}=$ 13 TeV, solid squares, and in $p$-Pb collisions at $\sqrt{s_{NN}}=$ 5.02 TeV, solid circles, are taken from Refs. [@Dash:2018cjh; @Tripathy:2018ehz]. The solid line is the result of QCM. \[fig12\]](ratio_Xi_phi)
Yield ratio $\Xi/\phi$ is also influenced by the small quark number effects. If we neglect small quark number effects, we have $$N_{\phi}\approx\frac{R_{V/P}}{1+R_{V/P}}\frac{\lambda_{s}^{2}}{\left(2+\lambda_{s}\right)^{2}}\overline{N}_{M}$$ and using Eq. (\[eq:NXi\]) we get the ratio $$\frac{N_{\Xi}+N_{\bar{\Xi}}}{N_{\phi}}=2\frac{1+R_{V/P}}{R_{V/P}}\frac{3}{2+\lambda_{s}}R_{B/M}\approx\frac{1.7}{2+\lambda_{s}},$$ which slightly decreases with the increase of $\lambda_{s}$ and therefore will slightly decrease with the increase of multiplicity $\langle dN_{ch}/d\eta\rangle$ because $\lambda_{s}$ increases with $\langle dN_{ch}/d\eta\rangle$. This is in contradiction with the experimental data. However, considering small quark number effects in QCM will predict the correct behavior of the ratio $\Xi/\phi$, see the solid line in Fig. \[fig12\]. The formation of $\Xi^{-}$ needs not only two $s$ quarks but also a $d$ quark, which is different from the formation of $\phi$ needing only a $s$ and a $\bar{s}$. Therefore, in events of small multiplicity or small quark numbers, the formation of $\Xi^{-}$ will be suppressed to a certain extent (or forbidden occasionally) due to the need of one more light quark, in comparing with the formation of $\phi$. We see that the calculated ratio $\Xi/\phi$ using QCM increases with system multiplicity $\langle dN_{ch}/d\eta\rangle$ and the increased magnitude of the ratio is consistent with the experimental data of $pp$ collisions at $\sqrt{s}=$ 13 TeV and those of $p$-Pb collisions at $\sqrt{s_{NN}}=$ 5.02 TeV [@Dash:2018cjh; @Tripathy:2018ehz].
Summary
=======
We observed that the midrapidity data of $p_{T}$ spectra of $\Omega^{-}$ and $\phi$ in inelastic events (INEL>0) in $pp$ collisions at $\sqrt{s}=$ 13 TeV exhibit a constituent quark number scaling property. This is an important signal of the quark combination mechanism at hadronization in $pp$ collisions at $\sqrt{s}=$ 13 TeV. We applied the quark combination model with equal velocity approximation to understand the existing preliminary data of yields and $p_{T}$ spectra of soft strange hadrons in minimum bias events (INEL>0) and in different multiplicity classes. We find the model can well describe the existing preliminary data observed by ALICE collaboration. We also find other signals of quark combination hadronization. One signal is the $p/\phi$ ratio as the function of $p_{T}$ and the other is yield ratios $\Lambda/\pi$, $\Xi^{-}/\pi$ and $\Omega^{-}/\pi$ divided by the values in minimum bias events as the function of system multiplicity $\langle dN_{ch}/d\eta\rangle$ at midrapidity. These signals are mainly related to the hierarchy behavior caused by strange quark content during the production of those hadrons at hadronization. Therefore, our results suggest that the constituent quark degrees of freedom play an important role at the hadronization of small quark/parton system created in $pp$ collisions at $\sqrt{s}=$ 13 TeV.
This work is supported by the National Natural Science Foundation of China under Grant Nos. 11575100.
[^1]: Because the masses of pion and kaon are significantly smaller than the summed masses of their constituent (anti-)quarks, the current working model is not suitable to directly calculate the momentum spectra of pion and kaon. Therefore, results of pion and kaon are not shown in this paper.
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abstract: 'We report a magnetocaloric study of the peak effect and Bragg glass transition in a Nb single crystal. The thermomagnetic effects due to vortex flow into and out of the sample are measured. The magnetocaloric signature of the peak effect anomaly is identified. It is found that the peak effect disappears in magnetocaloric measurements at fields significantly higher than those reported in previous ac-susceptometry measurements. Investigation of the superconducting to normal transition reveals that the disappearance of the bulk peak effect is related to inhomogeneity broadening of the superconducting transition. The emerging picture also explains the concurrent disappearance of the peak effect and surface superconductivity, which was reported previously in the sample under investigation. Based on our findings we discuss the possibilities of multicriticality associated with the disappearance of the peak effect.'
author:
- 'N. D. Daniilidis'
- 'I. K. Dimitrov'
- 'V. F. Mitrovi[ć]{}'
- 'C. Elbaum'
- 'X. S. Ling'
title: Magnetocaloric Studies of the Peak Effect in Nb
---
\[Sec1\]Introduction
====================
One very fascinating result in condensed matter physics in recent decades is the understanding that, in spite of early predictions,[@IM] the long-range topological order associated with broken continuous symmetries can survive in systems with random pinning.[@natt; @GL] In bulk type-II superconductors with weak point-like disorder the existence of a novel Bragg glass phase has been predicted.[@GL] This reaffirmed experimental facts known since the 1970s, that vortex lattices in weak-pinning, bulk, type-II superconductors can produce pronounced Bragg peaks in neutron diffraction.[@Christen] Recent experiments[@Ling; @Troy] have shown that a genuine order-disorder transition occurs in vortex matter. This transition appears to be the underlying mechanism of the well-known anomaly of the peak effect[@PE] in the critical current near . However there are still many outstanding issues concerning the Bragg glass phase boundary and the nature of the disordered vortex state above the peak effect.
Previous studies in a Niobium single crystal have revealed an intriguing picture of the peak effect in weakly-pinned conventional superconductors. Neutron scattering has shown that a vortex lattice order-disorder transition occurs in the peak effect region. This transition shows hysteresis and is believed to be first order, separating a low temperature ordered phase from a high temperature disordered one.[@Ling] The hysteresis was not observed across the lower field part of the superconducting-to-normal phase boundary. Magnetic ac-susceptometry showed that at lower fields the peak effect disappears as well, indicating further connection between the peak effect and the order-disorder transition.[@Park] In addition, the line of surface superconductivity, $H_{\text{c3}}$, terminates in the vicinity of the region where the peak effect disappears. This picture is summarized in Fig.\[Fig1\].
It was thus proposed that the peak effect is the manifestation of a first-order transition which terminates at a multicritical point (MCP) where the peak effect line meets a continuous, Abrikosov transition, . The MCP would be bicritical if a third line of continuous transitions ends there. The transition lines considered as a possible third candidate were a continuous vortex glass transition, $T_{\text{c2}}$, and the line of surface superconductivity. Alternatively, the MCP would be tricritical if the disordered phase is a pinned liquid with no high field transition into the normal state.[@Park] Subsequently, the disappearance of the peak effect at low fields has also been reported in other systems.[@Adesso; @Jaiswal]
Thermodynamic considerations[@Park] suggest that the MCP is likely a bicritical point. Since bicriticality implies the existence of competing types of order in the vortex system the question of which of the two lines, $T_{\text{c2}}$ or $H_{\text{c3}}$, is relevant to the bicritical point has major importance. Its answer will provide insight into the disordered vortex state above the peak effect and the disordering transition itself. Evidently, the possible relevance of surface superconductivity to the destruction of bulk Bragg glass ordering, and hence the existence of the multicritical point, cannot be dismissed *a priori*. In fact it is well known that surface premelting plays an important role in solid-liquid transitions.[@SurfMelt]
\[ht\] ![[]{data-label="Fig1"}](Fig1 "fig:")
This issue could be resolved by repeating the ac magnetic susceptibility measurements after having suppressed surface superconductivity with appropriate surface treatment. Efforts to nondestructively achieve this, e.g. by elecroplating the sample surface with a ferromagnetic layer, proved unfruitful, possibly due to high oxygen content of the surface. To address the problem, we performed measurements of magnetocaloric effects on the Nb single crystal studied by Ling *et al.*[@Ling] and Park *et al.*[@Park]
Here we report a study of the peak effect using a magnetocaloric technique. Clear features associated with the peak effect, have been identified in the magnetocaloric data. We find that the superconducting to normal transition shows inhomogeneity broadening at all fields. The peak effect in the bulk critical current is found to disappear when it enters into the inhomogeneity broadened transition region. It is concluded that the concurrent disappearance of surface superconductivity and the peak effect is likely due to interference of surface superconductivity with the ac-susceptibility measurements. Nevertheless, the location of the MCP, as determined from magnetocaloric measurements, is in close vicinity to the location previously reported.[@Park] The transition across $T_{\text{c2}}$ seems to be the same as that at $H_{\text{c2}}$, and surface superconductivity plays only a coincidental role in the critical point. This result suggests that the disordered vortex state, which is represented by the shaded part in Fig.\[Fig1\], is a distinct thermodynamic phase. Bicriticality implies that this phase has an order parameter competing with that of the Bragg glass phase. We also discuss an alternative scenario, in which no critical point exists at any finite field or temperature.
The paper is organized as follows: In Sec.\[Sec2\] we review the basic principle of magnetocaloric measurements, and give the experimental details of the sample and the technique used. In Sec.\[Sec3\] we present our main results, discuss the consequences of irreversible and nonequilibrium effects on magnetocaloric measurements, and proceed to interpretation of the data and estimates of sample properties. Finally we summarize our findings and conclusions, and propose experimental work necessary to address the issues raised.
\[Sec2\]Experimental
====================
\[Sec2a\]Basic principle of magnetocaloric measurements
-------------------------------------------------------
### \[Sec2a1\] Heat flow in magnetocaloric measurements
In studies of vortex phases in bulk superconductors, various experimental techniques provide complementary pieces of information. Combining these in a consistent picture is a non-trivial task. Magnetic ac-susceptibility measurements are sensitive to screening currents, and thus to the location of peak effect features, but are not suitable for the identification and study of the mean field transition itself.[@Park] Commercial magnetometers are not suited for study of large samples. Calorimetric[@Lortz; @Daniilidis] and ultrasonic attenuation[@Shapira; @Dimitrov] measurements determine the upper critical field where bulk condensation of Cooper pairs occurs, but it remains unclear under what circumstances they also provide a peak effect signature. Moreover the combination of information obtained with different techniques has to rely on thermometer calibration issues. Furthermore, the dynamical measurements suffer from thermal gradients in the studied samples. Magnetocaloric measurements overcome these difficulties because they are sensitive to both the presence of bulk superconductivity and to dynamical, flux-flow related effects. Moreover they can be performed in quasi-adiabatic conditions, where virtually no thermal gradients are present in the sample. Finally, they can be easily performed using a common calorimeter.
The magnetocaloric effect is a special case of thermomagnetic effects in the mixed-state of superconductors which have long been known and investigated.[@Thermomagnetic] These arise from the coexistence of the superconducting condensate which is not involved in entropy-exchange processes for the superconductor, and the quasiparticles, localized in the vortex cores, which are entropy carriers. Due to the presence of localized quasiparticles in the vortex cores, vortex motion results in entropy transport, which causes measurable thermal effects. Specfically, during field increase, new vortices are created at the edge of the sample, quasiparticles inside the vortices absorb entropy from the atomic lattice and cause quasi-adiabatic cooling of the sample. Conversely, during field decrease, the exiting vortices release their entropy to the atomic lattice, causing quasi-adiabatic heating.
Typically, in an experiment, the amount of entropy carried by the vortices entering or leaving the sample is not the exact thermodynamic equilibrium vortex entropy. The reason is that the vortex assembly is not equilibrated due to pinning as well as metastability associated with the underlying first-order phase transition at the peak effect. In practice, the magnetocaloric cooling and heating due to vortex entry and exit can be understood formally in terms of the actual entropy exchange between the vortex and atomic lattices. A superconductor subject to a changing magnetic field and allowed to exchange heat with the environment, undergoes a temperature change.[@adiabatic] This process is described by the relation: $$\label{Eq1}
dQ_{abs}/dt=n\,T_{\text{s}}\,(\partial{s}/\partial{H})_{T}\,dH/dt+
n\,C_{\text{s}}\,dT_{\text{s}}/dt$$ where $dQ_{abs}/dt$ is the net rate of heat absorption, positive for absorption of heat by the sample, $n$ is the molar number of the superconductor, $T_{\text{s}}$ its temperature, $T_{\text{s}}\,(\partial{s}/\partial{H})_{T}$ the molar magnetocaloric coefficient, and $C_{\text{s}}$ the specific heat of the superconductor. The magnetocaloric term in the equation induces temperature changes. These are smeared out by the last term, describing the effect of specific heat. Nevertheless, in practice this last term is constrained to be negligible when magnetocaloric effects are measured. In our measurements, we use very low field ramp rates ($dH/dt$), which result in very low temperature change rates ($dT_{\text{s}}/dt$) causing the last term to be negligible.
A schematic of our setup is represented in Fig.\[Fig2\]a. In this setup, absorption or release of heat from the sample results in minute, but measurable, variation of its temperature. In a typical measurement, the sample temperature is first fixed at a selected value, $T_{\text{s0}}$. Subsequently, the field is ramped at a steady rate, resulting in quasi-adiabatic absorption or release of heat from the sample. The resulting sample temperature change is recorded. If we neglect irreversible and non-equilibrium effects, the temperature change of the sample () around its static value, $T_{\text{s0}}$, allows us to determine the molar magnetocaloric coefficient by use of: $$\label{Eq2}
-G_{\text{link}}\,\Delta T=n\,T_{\text{s}}\,(\partial{s}/\partial{H})_{T}\,dH/dt$$ where the sample temperature $T_{\text{{s}}}=T_{\text{s0}}+\Delta T$, differential thermal conductance of the heat link $G_{\text{link}}$, molar number for the sample $n$, and field ramp rate $dH/dt$ are all independently measured. The latter is positive for increasing fields, negative for decreasing fields.
### \[Sec2a2\]Irreversible and non-equilibrium effects
In deriving the above equations we assumed that heat is only exchanged between the superconductor and the environment and that the superconductor reaches its quasi-static state as the measurement is performed. In an actual experiment, non-equilibrium and irreversible effects need to be considered. The heat generated by the dissipative processes between the vortex lattice and atomic lattice leads to a modification of the left hand side of equations \[Eq1\]&\[Eq2\]. Non-equilibrium effects lead to a modification of the right hand side. While the effects of the Bean-Livingston surface barrier,[@BeanLiv] flux flow heating,[@FF] and the Bean critical state[@Bean] can be complex and subtle, a detailed analysis can be performed and it will be discussed in Sec.\[Sec3b1\]. We show that the magnetocaloric measurements allow us to shed light into the problem of the peak effect in Nb that would not have been possible with any other single technique. In concluding this section, we note that since Eq.\[Eq2\] is approximate, in what follows we refrain from using to symbolize the quantity related to the magnetocaloric temperature change, $\Delta T$. Instead, we choose $(ds/dH)$ to symbolize the measured “entropy” derivative.
\[Sec2b\]Sample and setup
-------------------------
We used a Nb single crystal sample with mass of $24.78\, \text{g}$ which was previously studied using SANS and ac-susceptometry.[@Ling; @Park] It has an imperfect cylindrical shape (radius $0.5\, \text{cm}$, length $2.47\, \text{cm}$) with the cylinder axis oriented parallel to the \[111\] crystallographic direction. It has a of $9.16\, \text{K}$ and upper critical field $(0)\approx5600\, \text{Oe}$, as previously reported.[@Ling; @Park] We performed a zero field specific heat measurement, shown in Fig.\[Fig2\]b. This was done with the heat-pulse (relaxation) technique. From this measurement we obtain the Ginzburg-Landau parameter $\kappa(0)=3.8$ which is higher than the previous estimate[@Park]. In addition we find a superconducting transition width of $83\,\text{mK}$. The residual resistivity ratio from $300\, \text{K}$ to $10\, \text{K}$ of this sample is measured to be 12, suggesting a significant amount of defects in this Nb crystal. This is consistent with the large $\kappa$ and upper critical field values.
\[ht\] ![[]{data-label="Fig2"}](Fig2 "fig:")
Our experimental setup is a homemade calorimeter and its idealized heat-flow diagram is shown in Fig.\[Fig2\]a. It consists of an oxygen-free, high-purity copper can which serves the role of a heat bath surrounding the sample. A piece of high-purity copper wire is used as a heat link between the sample and heat bath and the mechanical support for the sample is provided by nylon rods. During measurements, the heat bath is maintained at a temperature of $4.20\, \text{K}$ and a carbon glass thermometer (Lakeshore) monitors its temperature. A second thermometer (Cernox, Lakeshore) is directly attached with silver epoxy to the lower end of the sample for reading the sample temperature. To minimize electronic noise in reading the thermometer resistance, high-frequency sinusoidal excitation together with pulse-sensitive detection is used. A Manganin wire, which is non-inductively wound on the sample and secured with Stycast epoxy, serves as a heater. A 50-turn high-purity copper coil is directly wound on the sample for simultaneous ac magnetic susceptibility measurement. During measurements the vacuum in the calorimeter is maintained to lower than $1\, \text{micron}\, \text{Hg}$ by use of activated charcoal in thermal contact with the helium bath.
Calorimetric measurements are performed with the standard heat-pulse technique. The heat input to the sample is increased incrementally, with step duration of 60-100 sec. Every incremental increase of the heat input results in exponential temporal relaxation of the sample temperature to its new equilibrium value. The sample heat capacity is determined through the decay time of the exponential. This technique offers moderate resolution and is not suited for studying the thermodynamics of the peak effect. Nevertheless, it allows us to characterize our sample. In Fig.\[Fig2\]b, we show the data acquired in zero applied field, from which we measured the sample properties quoted above.
During magnetocaloric measurements, a constant heat input, $P_{in}$, is supplied to the sample through the manganin heater, fixing the temperature at a selected static value, $T_{\text{s0}}$. After the sample temperature has stabilized to $T_{\text{s0}}$, the magnetic field is ramped up, then down, at a constant rate. During a field ramp, the sample temperature as a function of field, $T_{\text{s}}(H)$, is recorded. The magnetocaloric signal, $\Delta T(H)$, has to be measured with respect to a *field dependent*, static sample temperature *reading*: $T_{\text{s0}}=T_{\text{s0}}(H)$. This is the thermometer reading obtained at a given field, $H$, in the absence of field ramping. In other words one has to determine $\Delta T(H)=T_{\text{s}}(H)-T_{\text{s0}}(H)$. An example of raw data, $\Delta T$ $vs.\;H$, is shown in Fig.\[Fig3\].
The field dependence of the $T_{\text{s0}}(H)$ thermometer reading is a result of the following two effects: First, the magnetoresistance of the thermometer used. Second, a changing temperature gradient across the sample, as its field dependent thermal conductivity changes. This gradient can become considerable in the Meissner state for the highest measured temperatures ($\approx 8 \,\text{mK}\,/\,\text{cm}$ at $8.33\, \text{K}$). Nevertheless, it is negligible (less than $0.5 \,\text{mK}\,/\,\text{cm}$) in the peak effect region which is the main focus of this paper. We stress here that the thermal gradient is a result of the external heat input $P_{in}$, not the magnetocaloric effect.
![[]{data-label="Fig3"}](Fig3)
A possible variation of the heat input with applied field, which could be due to magnetoresistance in the heater, was investigated. It was verified that the heat input from the manganin heater does not vary by more than 1 part in $10^{4}$ for the entire field range covered in any of our measurements.
The differential thermal conductance of the heat link between the sample and heat bath, $G_{\text{link}}$ to be used in Eq.\[Eq2\] above, is also measured at all temperatures of interest, for the entire range of applied fields. This is done through application of a small, pulsed heat input ($P_{in}\,\pm\,\delta P_{in}$) to the sample and recording of the resulting temperature change, at different applied fields. The link conductance is found to vary smoothly by no more than 1 part in 100 for the entire field range studied. This variation is insignificant compared to the random noise in the raw $\Delta T$ data, and shall not be considered further.
Finally, the field ramp rate is measured through a resistor connected in series with the magnet coil, which allows us to monitor the current through the magnet. For all of the measurements presented here the ramp rate is $0.92\, \text{Oe/sec}$, which is the lowest possible with our system. At higher ramp rates, for example $1.87\, \text{Oe/sec}$, giant flux jumps occur in the sample.
\[Sec3\] Results and discussion
===============================
\[Sec3a\]Magnetocaloric results
-------------------------------
### \[Sec3a1\]Main features in field scans
In Fig.\[Fig4\]a&b, we summarize the molar entropy derivative, $(ds/dH)$, measurements on increasing (a) and decreasing (b) fields, at different temperatures. $(ds/dH)$ is calculated from the $T_{\text{S}}(H)$ data following the procedure outlined above.
As indicated for the lowest temperature curve, $4.83\, \text{K}$ (the upper most curve in Fig.\[Fig4\]a) several important features deserve attention. On increasing fields a peak occurs at low field. This is marked by . It corresponds to the lowest field for vortex entry through a surface barrier. Its locus on the $H-T$ plane closely follows the thermodynamic field, but occurs slightly lower. This behavior is expected for a sample with finite demagnetizing factor and mesoscopic surface irregularities.[@deGennes] No corresponding peak is present on decreasing fields. Rather, a smoother increase of $(ds/dH)$ occurs as the field is lowered, before entry of part of the sample into the Meissner state where the magnetocaloric signal vanishes, as seen in figures \[Fig3\]&\[Fig4\]b.
In intermediate fields, we identify a novel feature which was not observed in previous magnetic susceptibility studies.[@Park] This appears as a knee in $(ds/dH)$, which shows larger negative slope as a function of field for fields lower than $H_{\text{knee}}$, as illustrated in Fig.\[Fig4\]c. The feature is the same for both field-ramp directions. With our setup we can identify the $H_{\text{knee}}$ feature up to $7.41\,\text{K}$. As the temperature is increased, the region between $H_{1}$ and $H_{\text{c2}}$ narrows and it becomes increasingly difficult to discern $H_{\text{knee}}$. Thus it is unclear how this new feature terminates, i.e. whether it ends on the line at around $H=1000\,\text{Oe}$, or if it continues to lower fields.
At high field, across $H_{\text{c2}}$, the equilibrium mean-field theory of Abrikosov predicts a step function for $(\partial m/\partial T)_{H}$.[@deGennes2] Thus, in the simplest picture one would expect a simple step function for the molar entropy derivative at $H_{\text{c2}}$. Instead, we observe that across the peak-effect regime, complex features of valley and peak appear in $(ds/dH)$ below the field marked . The valley in $(ds/dH)$ corresponds to the peak effect, as seen in Fig.\[Fig4\]. Similar features appear in decreasing field. At fields above the peak effect, the disappearance of the magnetocaloric effect marks the upper critical field, . As we will soon discuss, our magnetocaloric measurements indicate that the upper critical field shows inhomogeneity broadening, in agreement with the zero field calorimetric data mentioned in section \[Sec2\]C. In Fig.\[Fig4\] we mark with the upper end of the upper critical field. This is the value of field at which the magnetocaloric signal in the mixed state exceeds noise levels. Our technique is not sensitive to $H_{\text{c3}}$ effects and the data are featureless above .
\[!tb\] ![[]{data-label="Fig4"}](Fig4a "fig:") ![[]{data-label="Fig4"}](Fig4b "fig:") ![[]{data-label="Fig4"}](Fig4c "fig:")
### \[Sec3a2\]Identification of the peak effect
To verify the identification of the peak effect in our measurements, we performed simultaneous magnetocaloric and ac-susceptibility measurements, as shown in Fig.\[Fig5\]a. In the quasi-adiabatic magnetocaloric setup, the ac-amplitude used in this procedure has to be small. For large amplitudes inductive heating occurs in the mixed state on increasing *dc* field. Consequently the sample temperature increases rapidly by several degrees as the peak effect is crossed. This behavior is not surprising, but reaffirms that caution has to be taken when dynamic perturbations are combined with quasi-adiabatic measurements. We used an amplitude of $0.5\,\text{Oe}$ at $107\,\text{Hz}$ as a compromise between feasibility of the magnetocaloric measurement and resolution in the $\chi '$ results. The results are shown in Fig.\[Fig5\]a.
In this combined measurement we find that both the onset and the peak of the peak effect have corresponding features. Moreover, we find no clear change in $\chi '$ when the upper critical field, determined from the magnetocaloric measurement, is crossed. A slight change occurs in the slope of the $\chi '(H)$ curve across , but significant amount of screening, caused by surface superconductivity, remains when the bulk of the sample is in the normal state. This is a typical example of the inadequacy of ac-susceptometry in determining the upper critical field. Even with the use of larger ac fields the change in slope of $\chi '(H)$ turns into a shoulder which does not reveal the exact location or characteristics of the bulk superconducting transition.[@Park].
### \[Sec3a3\]Features of the peak effect
In light of the first-order transition underlying the peak effect, it is tempting to interpret the peak appearing in $(ds/dH)$ at the peak effect as a manifestation of the entropy discontinuity of the transition. Nevertheless, a simpler interpretation exists in the context of critical state screening.
In specific, the magnetocaloric signature of the peak effect is consistent with critical-state[@Bean] induced, non-equilibrium magnetization during the field ramps. The magnetocaloric valley-and-peak features in the peak-effect regime allow us to determine the positions of the onset, , the peak, , and the end, $H_{\text{end}}$, of the peak effect. We start with increasing field data. At low temperatures below $6.79\, \text{K}$ where the peak effect is observed, the magnetocaloric signal starts dropping at the onset, , of the peak effect. A minimum occurs in the vicinity of the peak of the peak effect, , and it is followed by a peak, see Fig.\[Fig5\]b. This indicates slowing down, then acceleration of vortex entry into the sample, as the critical state profile becomes steep, then levels, in the peak effect region. Finally the magnetocaloric signal gradually drops to zero in the region of the upper critical field.
On decreasing field a magnetically reversible region exists for fields between $H_{\text{end}}$ (the “end” of the peak effect) and . This is shown in Fig.\[Fig5\]b. Such behavior can be understood keeping in mind that the upper critical field in our sample is characterized by inhomogeneity broadening. We believe that the sliver of magnetic reversibility corresponds to the appearance of superconducting islands in our sample. These give rise to magnetocaloric effects, but they are isolated within the bulk and cannot support a screening supercurrent around the circumference of the sample, hence the reversible behavior. The “end” of the peak effect marks the onset of irreversibility and corresponds to a shoulder in the decreasing field curve. In the critical state screening picture, this occurs when continuous superconducting paths form around the sample and a macroscopic critical current is supported. As the field is lowered below $H_{\text{end}}$, flux exit is delayed due to the increase in critical current, until past the peak of the peak effect, when accelerated flux exit results in a peak in $(ds/dH)$ below .
\[ht\] ![[]{data-label="Fig5"}](Fig5a "fig:") ![[]{data-label="Fig5"}](Fig5b "fig:") ![[]{data-label="Fig5"}](Fig5c "fig:")
### \[Sec3a4\]Fluctuation peak in $(ds/dH)$
In both field-ramp directions, the features due to the peak effect become less pronounced for higher temperatures, and finally disappear even before the previously identified[@Park] critical point is reached. On increasing fields, for $T>7.18\,\text{K}$, a new peak appears in $(ds/dH)$ just below the critical field, shown in Fig.\[Fig5\]c. This peak feature has already been reported in calorimetric measurements in pure Nb and Nb$_{3}$Sn.[@Gough; @Lortz] It has been attributed to critical fluctuations in the superconducting order parameter which set in as the critical field is approached.[@Thouless] In our measurements we see the effect of critical-fluctuation entropy enhancement in . We expect the same peak to exist in the curves displaying the peak effect feature, but its presence will be obscured by the dramatic results of non-equilibrium magnetization discussed above. Interestingly a similar peak is *not* observed on the decreasing field data, Fig.\[Fig5\]c. Moreover, it is evident in Fig.\[Fig5\]c, that apart from hysteretic behavior over an approximately $100\,\text{Oe}$ wide region between and the behavior of the sample is reversible to within noise levels. This behavior occurs consistently at all temperatures where the peak effect is not observed.
### \[Sec3a5\]The superconducting to normal transition region
A very striking feature of our data is the invariance of the shape of the transition into (or out of) the bulk normal state with changing temperature, or critical field. For increasing fields, the transition into the normal state occurs between the fields and , where $(ds/dH)$ drops to zero monotonically, as shown in Fig.\[Fig6\]a. To illustrate this, in Fig.\[Fig6\] we show the *normalized* entropy derivatives as a function of field for temperatures ranging from $4.83\, \text{K}$ to $8.33\, \text{K}$, with an expanded view of the upper critical field region. The normalization has been performed such that the average of $(ds/dH)_{norm}$ over a $50\,\text{Oe}$ wide region below the onset of the peak effect equals unity. The curves have also been horizontally offset, on a $\Delta H=H-$ axis. The horizontal alignment can alternatively be performed by aligning the ordinate of either , or the part of the curve where the signal equals a given value, for example 0.1 in the normalized Y axis. All different criteria result in alignments differing by only a few Oe. The case is similar for decreasing fields.
In Fig.\[Fig6\]a we present increasing field data that display the peak effect on the normalized/offset axes. For comparison, one curve which does not display the peak effect has been included. This corresponds to T=7.41K. In Fig.\[Fig6\]b we present the corresponding decreasing field data. In Fig.\[Fig6\]c we show only curves without a peak effect, for both increasing and decreasing field. These figures illustrate the uniform characteristics of the transition between the mixed state and the normal state. This is most evident in Fig.\[Fig6\]c: All different curves collapse onto one uniform curve for each field ramp direction. In figures \[Fig6\]a&b, the occurrence of the peak effect results in variations of the magnetocaloric signal around this uniform transition. These variations are, as already discussed, consistent with critical state induced flux screening on the field ramps. The uniformity of the transition for all field values implies that critical fluctuation broadening of the transition plays a minor role in our sample. Rather, inhomogeneity broadening seems to be the cause for the observed behavior.
\[!t\] ![[]{data-label="Fig6"}](Fig6a "fig:") ![[]{data-label="Fig6"}](Fig6b "fig:") ![[]{data-label="Fig6"}](Fig6c "fig:")
As already stated in Sec.\[Sec2\], calorimetric measurements indicate inhomogeneity broadening to a width of approximately 83mK for the zero field transition in our sample. With this in mind, we conclude that for increasing fields the gradual disappearance of the magnetocaloric signal in the region between and corresponds to the gradual loss of bulk superconductivity in our sample. In all of our measurements, the width $H_{\text{c2}}^{\text{up}}-H_{0}$ is essentially constant around a mean of $74.1\pm1.9\, \text{Oe}$ (Fig.\[Fig6\]a&c), which translates to a width of $78.8\pm2.0\,\text{mK}$ on the temperature axis. This value is in good agreement with that obtained in the calorimetric measurement, given the finite temperature step of 10 to 15mK used in the latter. Based on the identification of the lower and upper limits of the upper critical field, we identify the location of the mean field transition, , to be in the midpoint of the to range, see Fig.\[Fig6\]a. Local variations in electronic properties in the sample cause broadening around this value. With this in mind we proceed to the discussion of the evolution of the peak effect.
### \[Sec3a6\]Disappearance of the peak effect in $(ds/dH)$
We already mentioned that in the magnetocaloric measurements the peak effect is not observed for temperatures above $7.18\,\text{K}$, or critical fields below $1718\,\text{Oe}$. The disappearance of the bulk peak effect at such high field seems to contradict the previous observation, from ac-susceptometry, that the peak effect occurs for fields as low as approximately $900\,\text{Oe}$.[@Park] Our current findings offer a resolution of this controversy.
\[!tb\] ![[]{data-label="Fig7"}](Fig7a "fig:") ![[]{data-label="Fig7"}](Fig7b "fig:")
To do this we trace the evolution of the positions of the onset and the peak of the peak effect versus in our increasing field data. We focus on increasing fields, because in these the range of the upper critical field transition between and is clearly discernible. The location of each of these features is identified by the intersection of two linear fits on different sections of the $(ds/dH)$ curve. Each fit is performed in a limited $\Delta H$ range on either side of the turning point where the feature occurs. For example, is defined by the intersection of two linear fits to the data, one roughly in the range of to and one in the range of to . In Fig. \[Fig7\]a we show the positions of , $H_{\text{knee}}$, , and in $H$ vs. $T$ axes. In Fig. \[Fig7\]b we mark the positions of the onset, the peak, and the end of the peak effect, as well as the lower end () and the midpoint () of the transition in $\Delta H$ vs. axes. It is evident in the figure that the peak effect disappears when it crosses over into the to range, where bulk superconductivity is partially lost due to sample inhomogeneity.
It is not clear whether the peak effect continues to exist with a reduced magnitude inside this region. It seems likely that its magnitude is reduced below detectable levels. This can be due to loss of superconductivity in regions of the sample and the resulting absence of bulk macroscopic critical current. However the peak effect continues to manifest itself in ac-susceptibility. This can be explained under the assumption that the screening supercurrent near the sample surface is assisted by the sheath of surface superconductivity. This picture also provides an explanation for the approximately simultaneous disappearance of the peak effect and surface superconductivity in previous studies.[@Park]
The observation with ac-susceptometry of a continuation of the peak effect line on the surface of the sample is an indication that the peak effect exists, though unobservable, at these lower fields, in isolated superconducting islands in the bulk of the sample as well. Moreover, as shown in Fig. \[Fig7\], the linear extrapolation of the onset and the peak of the peak effect, as well as , merge at a field of approximately $850\,\text{Oe}$, suggesting that this may indeed be the vicinity of the critical point where the first-order phase transition underlying the peak effect ends.
### \[Sec3a7\]The and $T_{\text{c2}}$ transition lines
The nature of the MCP which was previously identified by Park *et al.*[@Park] remained unresolved. Our current findings suggest that surface superconductivity plays only a coincidental role in the disappearance of the peak effect. Thus, the nature of the MCP is determined from the nature of the and $T_{\text{c2}}$ lines discussed in section \[Sec1\].
We have a means of comparing the and $T_{\text{c2}}$ transitions. The approximate location of the MCP in ac-susceptometry is at $T=$8.1K, $H=$900Oe. The disappearance of the peak effect below $1718\,\text{Oe}$ in the magnetocaloric measurements but only below approximately $900\,\text{Oe}$ in ac-susceptometry, allows us to compare the transitions into the bulk normal state on the two sides of the MCP. As shown in Fig.\[Fig6\]c, all the curves with $<1718\,\text{Oe}$ (or equivalently $T>7.18\,\text{K}$) for both increasing and decreasing fields collapse strikingly on two different curves. These data include transitions on both sides of the MCP. This suggests that the phase transitions out of the Bragg Glass () and disordered ($T_{\text{c2}}$) phases are of the same nature. In other words, well defined vortices exist in the disordered vortex state above the peak effect. Here “well defined” is taken to mean that their magnetocaloric signature is indistinguishable from the one obtained in the transition between the normal and the Bragg Glass phases. However, it has to be borne in mind that changes in critical behavior can be subtle and hard to identify in our sample which shows significant inhomogeneity broadening.
![Integrals of the rescaled $(ds/dH)$ for $\Delta H$ from $-150\,\text{Oe}$ to $30\,\text{Oe}$, for increasing ($\mathcal{F}_{up},\;\blacktriangle$) and decreasing ($\mathcal{F}_{down},\;\blacktriangledown$) fields. The peak effect feature was not observed at these temperatures. The error bars reflect the uncertainty in alignment of the magnetocaloric curves. Temperatures above 8.15K correspond to the line of transitions proposed by Park *et al.*[@Park] []{data-label="Fig8"}](Fig8)
We can overcome this difficulty and look for changes in critical behavior by integrating the normalized experimental curves. This way, a change or a trend in critical behavior obscured by inhomogeneity broadening will be more easily discerned as a trend in the computed integrals. The computed integrals of the normalized increasing ($\mathcal{F}_{up}$) and decreasing ($\mathcal{F}_{down}$) field curves around the region of the upper critical field are shown in Fig.\[Fig8\]. The integration has been performed between $\Delta H=-150\, \text{Oe}$ and $30\, \text{Oe}$ in the aligned axis. More explicitly: $$\mathcal{F}=\int_{\Delta H=-150}^{\Delta H=30}(ds/dH)_{normalized}\,d(\Delta H)\,.$$ The error bars arise from the alignment uncertainty. All temperatures refer to the $T_{\text{c2}}$ transition, except for the two marked with an asterisk which correspond to . We see no systematic trend in the result in any of the available temperatures. In conclusion, as far as the inhomogeneity of our sample allows us to discern, there is no detectable change between the low-field $H_{\text{c2}}$ transition and the high-field $T_{\text{c2}}$ transition to the normal state.
### \[Sec3a8\]End of the peak effect
Finally, we discuss the “end” of the peak effect. Its position in the phase diagram is shown in Fig.\[Fig7\]b. In our data $H_{\text{end}}$ occurs in a range between 32 and 39Oe below and slightly above the position of the average superconducting transition, . $H_{\text{end}}$ has been identified as the field at which a superconducting network which can support a macroscopic screening supercurrent forms inside the sample. The “end” feature occurs slightly above the midpoint of the to range, as shown in Fig.\[Fig7\]b. This indicates that the critical current appears when roughly half of the sample is in the mixed state while the rest is still in the normal state. This observation is very interesting but requires further investigation. The role of surface superconductivity in establishing macroscopic supercurrents in the superconducting network can be examined experimentally.
\[Sec3b\]Discussion
-------------------
### \[Sec3b1\]Surface barrier, flux-flow heating, critical state screening
We can extend the conclusions from our measurements by evaluating the results of non-equilibrium and irreversible processes in the magnetocaloric effect.
We start with the surface barrier. Its presence results in delay of flux entry into the sample on increasing fields, up to a field approximately equal to the thermodynamic critical field, $H_{C}$.[@deGennes] In addition, the surface barrier has the more subtle consequence of introducing an asymmetry between the measured on increasing and decreasing fields. On increasing fields, vortices have to enter the sample through an energy barrier in a vortex free region.[@Clem; @Burlachkov] In this process, energy is dissipated approximately at a rate $$\Phi_{0}\cdot\big(({H_{C}}^2+H^2)^{1/2}-H\big)\cdot (V/4\pi)\cdot dH/dt\,,$$ where $V$ is the sample volume and $H_{C}$ the thermodynamic critical field.
In our measurements, for example at $H\approx3000\, \text{Oe}$, this amounts to approximately $2\, \mu \text{W}$, and will reduce the (negative) observed on increasing fields by roughly $1\, \text{mK}$. On decreasing fields, the surface barrier has essentially no effect, and no irreversible heating is expected.[@Clem] An asymmetry of this kind is shown schematically in Fig.\[Fig9\]a, and it is present in our data, for example in Fig.\[Fig5\]b. Moreover, this type of asymmetry disappears for lower values of the critical field, where surface superconductivity has disappeared,[@Park] for example in Fig.\[Fig5\]c.
Flux flow heating of the sample also leads to a similar asymmetry between the ascending and descending field branches. On increasing fields the negative is reduced and on decreasing fields the positive increased. An order of magnitude estimate of flux-flow heating can be obtained on the basis of the Bardeen-Stephen model.[@BardSteph] For a cylindrical sample of radius $R$, length $L$, and for smooth field ramping at a rate $dH/dt$, one obtains $P_{ff}\approx10^{7}(dH/dt)^2\pi R^4 L/(8\rho_{ff})$. This turns out to be negligible for the field ramp rates, approximately 1Oe/sec, used in our measurements. We show the effect of this mechanism, grossly exaggerated, in Fig.\[Fig9\]b. Between the above two sources of irreversible heating, it is clear that low field ramp rates will render the latter ($\propto (dH/dt)^{2}$) negligible, but will not reduce the effect of the former which scales as $dH/dt$, as does the magnetocaloric .
\[ht\] ![[]{data-label="Fig9"}](Fig9 "fig:")
Next we examine the case of a non-equilibrium critical state profile outside the peak effect region. A critical current that monotonically decreases with field, i.e. away from the peak effect region, will result in the opposite asymmetry than the one just mentioned. This is so, because on increasing fields the critical state profile becomes less steep, resulting in faster loss of flux than the field ramp rate, and thus increased . The opposite occurs on decreasing fields, resulting in a lower than indicated by the equilibrium . This mechanism is shown schematically in Fig.\[Fig9\]d. A simplified calculation for cylindrical sample geometry, like the ones found in the literature, [@Bean; @Xu] yields an asymmetry factor in equal to: $$\label{Eq3}
1\pm(4\pi R/3c)(\partial J_{c}/\partial H)_{T}$$ for increasing ($-$) and decreasing ($+$) fields. Here, $R$ is the sample radius and $J_{c}(H)$ the $H$ dependent critical current. Moreover it is assumed that the critical state field profile does not result in significant modification of the critical current in the sample interior. Nevertheless, even in the case where the value of the critical current varies spatially inside the sample, $J_{c}=J_{c}(B(r))$, the same type of asymmetry results, with slightly modified value though. The derivation of the above factor is outlined in the appendix. This asymmetry is illustrated in Fig.\[Fig9\]c. This type of asymmetry is only observed in our data in the region of the upper critical field and the peak effect.
For example, in data without peak effect, as shown in Fig.\[Fig5\]c, the magnetocaloric curves are reversible outside a region between and but consistently show hysteresis in that region, see also Fig\[Fig6\]b. This behavior can be attributed to critical state screening. Then, the hysteresis which persists almost all the way up to indicates that the critical current is nonzero arbitrarily close to the upper critical field, and vanishes abruptly only at a distance of $10$ -$30\,\text{Oe}$ from . In addition, the lack of asymmetry in the increasing and decreasing field curves below the hysteretic region, implies that the critical current density varies very weakly with applied field. More specifically, the asymmetry in any of the data without peak effect is less than 5% of the magnitude of $(ds/dH)$. By Eq.\[Eq3\] we obtain $$(\partial J_{c}/\partial H)_{T}\,<\,(1/2)\cdot0.05\cdot(3c/4\pi R)\approx\,0.36\,\text{A}\,/\,\text{cm}^{2}\,\text{Oe}\,,$$ where the numerical result is expressed in conventional units for convenience.
### \[Sec3b2\]Critical current estimates
From the measurements we can estimate the value of the critical current before it vanishes in the upper critical field region, in the context of critical state screening. We assume that the entropy per vortex is essentially constant in the neighborhood of the upper critical field and the peak effect.[@note] Therefore changes in magnetic flux ($\Delta\Phi$) are proportional to changes in entropy, $\Delta\Phi\propto\Delta S$. Then an integral of $ds/dH$, such as $\mathcal{F}_{up}$ and $\mathcal{F}_{down}$, can be approximately taken to represent a change in magnetic flux in the sample. In the critical state model, we can relate changes in flux to the critical current.
At the temperatures where no peak effect is observed, we estimate the critical current in the neighborhood of . For simplicity we consider a linear field profile in the sample, with slope $$dH/dr=(4\pi/c)\,J_{c}\,,$$ where $r$ is the radial distance from the axis of the (cylindrical) sample. In the vicinity of simple integration gives a macroscopic magnetic flux difference ($\Delta\Phi$) between the increasing and decreasing field branch approximately equal to: $$\Delta \Phi=\left(1+\frac{1}{\zeta}\right)(16\pi^{2}/3c)\,J_{c}(H_{\text{hyst}})\,R^{3}\,.$$ see Eq.\[EqA2\] in the appendix. Due to the normalization performed for the integrals in Fig.\[Fig8\] the flux difference is also approximately given by: $$\Delta \Phi=(\mathcal{F}_{up}-\mathcal{F}_{down})\,\pi R^{2}\,.$$ These two relations, allow us to obtain estimates for the critical current in the sample before this collapses to zero in the upper critical field region. We show these in Fig.\[Fig10\], for temperatures above 7K, corresponding to the symbol for $J_{c}$. One should note that critical currents with the values given for $J_{c}$ in the figure, will lead to radial variation of the induction inside the sample by roughly 10Oe. Due to the constraint imposed on $(\partial J_{c}/\partial H)_{T}$, (see discussion above), the critical current density will vary inside the sample due to screened induction by less than 3.6A/cm$^{2}$. This represents only 10% of its value, and thus the assumption that the slope of the critical state profile (i.e. $J_{c}(r)$) is essentially constant is self consistent.
A similar procedure can be followed for the curves showing the peak effect, in order to estimate the critical current at the peak of the peak effect. This is most easily done for the decreasing field, curves, under the additional assumption that the critical current increases linearly from zero at $H_{\text{end}}$ to $J_{c}^{p}$ at . The integration relating the screened flux ($\Delta\Phi$) to these three quantities leads to the rather complicated equation: $$\label{Eq4}
\left\{ \Delta H-\frac{\Delta\Phi}{\pi R^2(1+\frac{1} {\zeta})} \right\}x^{2}-2\Delta Hx+2\Delta H=2\Delta H e^{-x}\;,$$ where $\Delta H =H_{\text{end}}-$, $x=(4\pi\,R\,J_{c}^{p})/(c\,\Delta H)$. The screened flux for a curve with peak effect is obtained via integration between and with respect to the universal curve of Fig.\[Fig6\]b (bottom). The resulting equations are solved numerically, and the results for $J_{c}^{p}$ are shown in Fig.\[Fig10\]. The value at $6.79\,\text{K}$ is not included, because the peak effect is barely observable at that temperature, which makes the procedure we outlined inapplicable.
![[]{data-label="Fig10"}](Fig10)
### \[Sec3b3\]Low field hysteresis: alternative interpretaion
We return to the low-field hysteresis near , illustrated in Fig.\[Fig5\]b. The combination of hysteretic and reversible behavior in the vicinity of is striking. Furthermore this behavior persists in all of the measurements which do not show the peak effect, down to an upper critical field of $676\,\text{Oe}$. In addition, we suggested earlier that the disappearance of the peak effect in our bulk measurements is due to loss of superconductivity in parts of the sample, while in ac-susceptometry it is due to the disappearance of surface superconductivity. Thus the possibility arises that the peak effect continues to exist at low fields, but it is not observable with the techniques used so far. In other words, our data raise the question whether the hysteretic order-disorder transition reported previously[@Ling] continues down to low fields, where no peak effect is observed, but hysteresis occurs over a range too narrow to be detectable in neutron scattering. In this scenario, the hysteresis seen in Fig.\[Fig5\]c is related to a first order phase transition. If this is the case the hysteresis presented here will be observable in high resolution calorimetric measurements.
### \[Sec3b4\]The “knee” feature
Finally we return to the newly identified “knee” feature, shown in Fig.\[Fig4\]c. From the occurrence of the knee in both ramp directions we conclude that it corresponds to an equilibrium feature of the thermodynamic behavior inside the Bragg Glass phase. To appreciate this argument, note that all three sources of irreversible and non-equilibrium effects discussed at the beginning of this subsection, will induce asymmetry on the $(ds/dH)$ curves for opposite field-ramp directions. For example, the two curves of Fig.\[Fig4\]c show asymmetry, as already mentioned. This is due to surface barrier-related heating on increasing field. On the other hand, symmetric trends in the measured $(ds/dH)$ curves have to be related to equilibrium behavior, and we thus conclude that $H_{\text{knee}}$ corresponds to an equilibrium feature. Neutron scattering did not reveal any structural changes in the vortex lattice[@Ling; @Park] around $H_{\text{knee}}$, which suggests that the nature of this feature is rather subtle. It would be interesting to investigate the corresponding part of the phase diagram for changes in dynamical response, as well as for a possible relation of this novel feature to the thermomagnetic instability in Nb.
\[Sec3c\]Summary
----------------
From the reported magnetocaloric-effect results, we gain a significant amount of novel information about the Nb sample previously studied using SANS and ac-susceptometry.
The upper critical field shows significant inhomogeneity broadening. The broad transition is related to the disappearance of the peak effect in bulk measurements. The peak effect disappears in the phase diagram when it enters the range where regions of the sample are normal. Yet, magnetic irreversibility, indicating nonzero critical currents, occurs when regions of the sample are normal. The peak effect is observable with ac-susceptometry at fields much lower than with magnetocaloric measurements. In the former case screening occurs locally, near the surface,[@Park] presumably assisted by a superconducting surface layer. This could indicate that the peak effect still occurs in the bulk of the sample in regions that are superconducting.
The low field transition from the Bragg Glass phase into the normal state seems to be the same as the high field transition between the structurally disordered[@Ling] vortex state and the normal state. Moreover, hysteresis occurs in magnetocaloric curves that do not display the peak effect, but in the neighborhood of this hysteresis the non-equilibrium and irreversible effect signatures that we discussed are absent. Finally a new feature corresponding to a knee in the magnetocaloric coefficient has been identified and its position mapped out in the phase diagram, Fig. \[Fig7\]a. These new findings allow us to refine the previously proposed peak effect phase diagram (Fig.\[Fig1\]), but also point to an alternative picture.
In the multicritical point picture, the magnetocaloric-effect results reported here have strong implications for understanding the nature of the multicritical point where the peak effect disappears.[@Park; @Adesso; @Jaiswal] A tricritical point can be ruled out since the change in slope between the and low field lines would lead to violation of the 180$^\circ$ rule[@Wheeler; @Park] imposed by thermodynamics. The fact that the magnetocaloric transition appears to be continuous and of the same character across both $T_{c2}$ and $H_{c2}$, suggests that the critical point is bicritical. Bicriticality implies the competition of two bulk vortex phases, an ordered Bragg glass[@natt; @GL] and a disordered vortex glass. The vortex glass phase is not necessarily superconducting in the sense of the original proposal,[@FisherI] but it has to be a genuine phase possessing an order parameter absent in the normal state,[@Menon] and *in competition* with that of the Bragg glass. In addition, it possesses superconducting phase rigidity even under partial loss of bulk (mixed state) superconductivity.
Finally, we wish to discuss an alternative picture in which the order-disorder transition does not end at the previously identified critical point. The experimental disappearance of the peak effect in magnetocaloric measurements may be due to sample inhomogeneity. In addition, the peak effect disappears in ac-susceptometry due to the disappearance of surface superconductivity.[@Park] Nevertheless, the vortex lattice disordering transition can continue to lower fields, where the peak effect is not detectable. Hysteresis related to a first order phase transition occurs over a narrow range in this part of the phase diagram, and the width of the hysteresis region grows appreciably only at higher fields where the peak effect is sufficiently far from the region. In this picture, no critical point associated with the disordering transition exists at any finite field or temperature.[@Menon]
\[Sec4\]Conclusions
===================
We have identified the magnetocaloric signatures of the peak effect in a Nb single crystal. In addition we classified and outlined the various sources of irreversible and non-equilibrium effects occurring in such measurements and offered ways for their evaluation. With these in mind, magnetocaloric measurements prove very useful in studies of the mixed state. They are suitable for studying the upper critical field region. They allow the identification and study of dynamical effects, such as the onset, the peak and the end of the peak effect, and provide estimates of the critical currents involved. Moreover they are useful in identification of changes in thermodynamic behavior deeper inside the mixed state, e.g. the “knee” feature identified here.
With the wealth of novel information obtained in our measurements we were able to shed light into the diasppearance of the peak effect in Nb. We refined the multicritical point scenario drawn previously.[@Park] In addition, based on experimental facts, we proposed an alternative scenario which can be experimentally tested.\
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We thank J. J. Rush and J. W. Lynn from NIST for providing the Nb crystal and acknowledge numerous helpful discussions with D. A. Huse, J. M. Kosterlitz and M. C. Marchetti. This work was supported by NSF under grant No. NSF-DMR 0406626.\
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Magnetic flux changes due to the critical state
===============================================
\[A1\]Asymmetry factor of $(ds/dH)$
-----------------------------------
First we derive the critical state asymmetry factor of $(ds/dH)$ discussed in Section \[Sec3b\]. We limit our discussion to cylindrical sample geometry, with the field applied along the cylindrical axis. The critical current is taken to depend on $B$, or equivalently on $H$. We focus on the region close to , where these are linearly related by : $B=(1+1/\zeta)H-H_{c2}/\zeta$, with $\zeta=\beta_{A}\,(2\kappa^{2}-1)$.[@deGennes2] Due to critical current screening, the local induction (or field) inside the sample is modified with respect to the applied field and depends on radial distance from the axis ($r$) of the sample: $H=H(r)$. From the Amp[è]{}re law the field variation resulting from the critical current is: $$\frac{dH}{dr}=\pm\frac{4\pi} {c}\,J_{c}\big(H(r)\big)\;,$$ for increasing ($+$) and decreasing ($-$) field. The critical current depends on $B$, but we make the simplifying assumption that for a given value of applied field $H_{a}$, the local induction inside the sample leads to a negligible critical current variation: $J_{c}(r)=J_{c}=const$. In addition we neglect demagnetizing effects. We then obtain: $$H(r)=H_{a}\pm(4\pi/c)\,J_{c}\cdot(r-R)\;,$$ where $R$ is the sample radius and again the solutions correspond to increasing ($+$) and decreasing ($-$) field. The corresponding magnetic induction is simply: $$B(r)=\left\{H_{a}\pm\frac{4\pi}{c}\,J_{c}\cdot(r-R)\right\}\cdot(1+1/\zeta)-H_{c2}/\zeta\;.$$ This is easily integrated over the cross sectional area of the sample, in order to obtain the magnetic flux through the sample: $$\label{EqA1}
\Phi=\int_{0}^{R}B(r)\,2\pi r\cdot dr\;,$$ to yield: $$\label{EqA2}
\Phi=\left\{H_{a}\,\left(1+\frac{1}{\zeta}\right)-H_{c2}/\zeta\mp\left(1+\frac{1}{\zeta}\right)\,\frac{4\pi R\,J_{c}}{3c}\right\}\cdot\pi
R^{2}\;.$$ Note that here the significance of the signs is reversed for increasing ($-$) and decreasing ($+$) field. The rate of change in magnetic flux through the sample for changes in applied field is: $$\frac{d\Phi}{dH_{a}}=\left\{1\mp\frac{4\pi R}{3c}\,\left(\frac{\partial J_{c}}{\partial H}\right)_{T}\right\}\cdot\left(
1+\frac{1} {\zeta} \right)\cdot\pi R^{2}\;.$$ This is proportional to the magnetocaloric signal, and includes the asymmetry factor given in Eq.\[Eq3\], with negative ($-$) sign for increasing field, positive ($+$) for decreasing.
\[A2\] Flux screening in the peak effect region
-----------------------------------------------
The critical current between and $H_{\text{end}}$ is taken to be: $$J_{c} =
\begin{cases}
0\,, & \text{if $H>H_{\text{end}}$} \\
J_{c}^{p}\left(\frac{H_{\text{end}}-H}{H_{\text{end}}-H{\text{p}}}\right)\, , & \text{if $H_{\text{p}}<H<H_{\text{end}}$}
\end{cases}$$ Then the Amp[è]{}re law for the field ($H_{\text{p}}<H<H_{\text{end}}$) inside the sample is: $$\frac{dH}{dr}=-\frac{4\pi\,J_{c}^{p}}{c}\cdot\left(\frac {H_{\text{end}}-H}{H_{\text{end}}-H_{\text{p}}}\right)$$ This is solved for applied field $H_{a}=H_{\text{p}}$, with the boundary condition $H(r=R)=H_{\text{p}}$ (again by neglecting demagnetizing effects), and yields: $$H(r)=H_{\text{end}}-(H_{\text{end}}-H_{\text{p}})\,e^{(r-R)/l}\;,$$ with $l^{-1}=(4\pi\,J_{c}^{p})/(c\,(H_{\text{end}}-H_{\text{p}}))$. From this, the local induction is obtained, and via integration over the sample cross-section, see Eq.\[EqA1\] the magnetic flux through the sample is: $$\begin{aligned}
\Phi=&&\left( 1+\frac {1}{\zeta} \right)\nonumber\\
&&\times\left\{ H_{\text{end}}\,\pi R^{2}-2\Delta H\pi(Rl-l^2)-
2\Delta H\pi l^2e^{-R/l}\right\}\nonumber\\
&&-H_{c2}\,\pi R^2/\zeta\,,\end{aligned}$$ where we used the notation $\Delta H=H_{\text{end}}-H_{\text{p}}$. The flux in the absence of screening ($J_{c}=0$) is: $$\Phi'=\left( 1+\frac{1}{\zeta} \right)\,H_{\text{p}}\,\pi R^2-H_{c2}\,\pi R^2/\zeta\,.$$ Therefore the amount of screened flux $\Delta\Phi=\Phi-\Phi'$, turns out to be: $$\begin{aligned}
\Delta\Phi=&&\pi R^2\,\left( 1+\frac {1}{\zeta} \right)\nonumber\\
&&\times\left\{\Delta H-2\,\Delta H\left(\frac{l}{R}-\frac{l^2}{R^2}\right)
-2\,\Delta H\frac{l^2}{R^2}\,e^{-R/l}\right\}\,.\end{aligned}$$ The additional definition $x=R/l$, straightforwardly leads to Eq.\[Eq4\]: $$\left\{ \Delta H-\frac{\Delta\Phi}{\pi R^2\left( 1+\frac {1}{\zeta} \right)} \right\}x^{2}-2\Delta H x+2\Delta H=2\Delta H e^{-x}\,.$$\
\
\
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|
---
abstract: |
A CeF$_3$ crystal produced during early R&D studies for calorimetry at the CERN Large Hadron Collider was exposed to a 24GeV/c proton fluence $\Phi_p=(2.78 \pm
0.20) \times 10^{13}\;\mathrm{cm^{-2}}$ and, after one year of measurements tracking its recovery, to a fluence $\Phi_p=(2.12 \pm 0.15) \times
10^{14}\;\mathrm{cm^{-2}}$. Results on proton-induced damage to the crystal and its spontaneous recovery after both irradiations are presented here, along with some new, complementary data on proton-damage in Lead Tungstate. A comparison with FLUKA Monte Carlo simulation results is performed and a qualitative understanding of high-energy damage mechanism is attempted.
title: |
A study of high-energy proton induced damage\
in Cerium Fluoride in comparison with measurements\
in Lead Tungstate calorimeter crystals
---
G. Dissertori, P. Lecomte, D. Luckey, F. Nessi-Tedaldi, F. Pauss Th. Otto, S. Roesler, Ch. Urscheler
Introduction {#s-INT}
============
The Large Hadron Collider (LHC) at CERN is expected to undergo a substantial upgrade in luminosity after the exploitation of its physics potential, to allow further exploration of the high energy frontier in particle physics. The higher luminosity will strengthen the requirements on performance of most detector components. With data taking extended over several years under an increased luminosity, detectors will be exposed to even larger ionising radiation and hadron fluences than at the LHC, and they will need to be upgraded as well. While the harvest of LHC proton-proton collision data is just starting, studies are already being performed on a LHC upgrade (superLHC) where calorimetry will have to perform adequately in a radiation environment and hadron fluences an order of magnitude more severe than at the LHC. It will thus be important to have results at hand on calorimetric materials able to withstand the anticipated radiation levels and particle fluences before making decisions on detector upgrades.
Concerning hadron effects on crystals used for calorimetry, the present study extends to Cerium Fluoride (CeF$_3$) and complements our earlier work performed on Lead Tungstate (PbWO$_4$) [@r-LTNIM; @r-pionNIM; @r-LYNIM]. In our earlier investigations we have shown that high-energy protons [@r-LTNIM] and pions [@r-pionNIM] cause a permanent, cumulative loss of Light Transmission in PbWO$_4$, while we observed no hadron-specific change in scintillation emission [@r-LYNIM]. The features of the observed damage hint at disorder that might be caused by fragments of heavy elements, Pb and W. Above a certain threshold, these can have a range up to 10 $\mu$m and energies up to $\sim$100 MeV, corresponding to a stopping power nearly 10000 higher than the one of minimum-ionising particles. The associated local energy deposition is capable of inducing the displacement of lattice atoms.
The qualitative understanding we gained of hadron damage in Lead Tungstate lead us to predict [@r-CAL08] that such hadron-specific damage contributions are absent in crystals, like Cerium Fluoride, consisting only of elements with $Z < 71$, which is the experimentally observed threshold for fission [@r-THR]. Studies on Cerium Fluoride are expected to help at the same time in understanding hadron damage to scintillating crystals and in possibly providing a viable material for calorimetry in an extreme environment as the superLHC will be.
Cerium Fluoride
===============
Cerium Fluoride is a scintillating crystal whose luminescence characteristics are known since the early studies by F.A. Kröger and J. Bakker [@r-KRO], who measured its emission spectrum and light decay time constants and understood the responsible transitions. Its properties as a scintillator were revealed by D. F. Anderson [@r-AND] and by W. W. Moses and S. E. Derenzo [@r-MOS], who attracted attention to its characteristics as a fast, bright and dense calorimetric medium for high-energy physics and positron-emission tomography applications.
Its density ($\rho=6.16\; {\mathrm{g/cm}}^3$), radiation length ($X_0 =
1.68$ cm), Molière radius ($R_M = 2.6$ cm), nuclear interaction length ($\lambda_I = 25.9$ cm) and refractive index ($n = 1.68$) make it a competitive medium for compact calorimeters. Its emission is centred at 340 nm, with decay time constants of 10 - 30 ns; it is insensitive to temperature changes (${\mathrm{dLY/dT}}\; (20^o$C$)=0.08\%/^o$C) as well as bright (4 - 10% of NaI($T\ell$)) and thus suitable for high-rate, high-precision calorimetry [@r-AND2].
In the nineties, this material was subject to an intense research program, that established its scintillation characteristics, its behavior in $\gamma$ [@r-KOB; @r-CCC1] and MeV-neutron irradiations [@r-CHI] and the capability for crystal growers to produce crystals of dimensions suitable for high-energy physics applications. It should be noted that Cerium is a readily available rare earth, which would allow containing raw material costs, were a mass production envisaged. Its melting point of $1430^o$C allows applying well-known crystal growth technologies. The ability to grow crystals beyond 30 cm length was demonstrated, as visible in Fig. \[f-PHOTO\], but for its use in a calorimeter, R&D would have to be resumed, since no commercial production of macroscopic crystals presently exists, although the material is still used, e.g. in the form of 10 $\mu$m nanoparticles, for neutron capture cross-section measurements [@r-STA].
The tysonite structure is complicated, with each Cerium atom surrounded by 11 Fluorines in the Edshammar’s polyhedron [@r-HYD] and the lower symmetry thus causes difficulties in the calculation of energy levels and defect structures [@r-MER]. Cerium Fluoride is superionic [@r-TRN], the fluorine ions having high mobility, which could account for part of its observed radiation hardness. It is also paramagnetic, a characteristics which influences the Faraday rotation in a magnetic field [@r-XU]. Fluorine ions labelled $F_1$ in literature [@r-F1] account for most of the Cerium Fluoride conductivity and hence their mobility might help repair defects. There is a vast literature on this and in particular on doping [@r-ROS] with Barium and Strontium, which have been observed to increase the conductivity even further [@r-PRI]. The composition which maximizes conductivity might maximize radiation hardness as well, as hypothesised in [@r-SOR], but establishing this would require a dedicated R&D. It should also be noticed that the Cerium Fluoride conductivity increases by a factor 4 between $20^o$C and $50^o$C temperature [@r-SOR], while its Light Output remains unaffected. Thus, for calorimetry applications in a hostile radiation environment, maintaining the crystals at a higher temperature could help minimising radiation damage. It is also pointed out in [@r-SOR] that such crystals are the best fluoride superionics for electro-chemical solid state devices such as fuel cells.
The performance of Cerium Fluoride was also studied with high-energy particle beams in prototype crystal matrices [@r-CEFTB], in particular since it was adopted as baseline calorimetric medium in the CMS [@r-LOI] and L3P [@r-L3P] Letters of Intent. Energy resolutions of the order of $0.5\%$ for electron energies of 50 GeV and higher were achieved, and it was observed how Cerium Fluoride appeared to be the best material at that time for homogeneous electromagnetic calorimetry at LHC and only the need for a very compact calorimeter justified relegating it behind Lead Tungstate as the preferred material.
Cerium Fluoride was also considered for the ANKE spectrometer upgrade at COSY and for medical imaging applications [@r-MO2]. In [@r-NOV], its energy response to electromagnetic probes was extended down to a few MeV in photon energy and a good time resolution, below 170 ps, was obtained using a time-of-flight technique.
We have performed a test of hadron effects in Cerium Fluoride with the expectation to yield a better understanding of the whole hadron damage issue in scintillating crystals, and to provide the community with a viable solution for the hadron fluences expected during operation at superLHC.
The crystal {#s-xtal}
===========
For this study, we have used a Barium-doped $\mathrm{CeF}_3$ crystal from Optovac [@r-optovac], which has parallelepipedic dimensions of $21
\times 16 \times 141\;\mathrm{mm}^3$ $(8.4 \;\mathrm{X_0})$. Its longitudinal Light Transmission (LT) before irradiation as a function of wavelength is shown in Fig. \[f-LT\]. One observes that the smoothness of the transmission curve is interrupted by several dents which are known to be due to the presence of $\mbox{Nd}^{3+}$ impurities [@r-CCC2] and are of no further concern to the present study. One also observes, in the light of Fig. 2 in Ref. [@r-CCC2], how the Barium doping translates into a characteristic Light Transmission band edge which sits right above 300 nm, i.e. $\sim$ 15 nm higher compared to crystals grown with undoped raw material.
The irradiations {#s-irrad}
================
The crystal was irradiated with 24GeV/c protons at the IRRAD1 facility[@r-IR1] in the T7 beam line of the CERN PS accelerator. The first irradiation was performed beginning of November 2007, with a flux $\phi_p= 1.16 \times 10^{12}$cm$^{-2}$h$^{-1}$. The proton fluence reached was $\Phi_p=(2.78 \pm 0.20) \times
10^{13}\;\mathrm{cm^{-2}}$. After one year of periodic measurements, where its spontaneous recovery at room temperature, in the dark, was tracked, a second irradiation was performed with a flux $\phi_p= 0.94 \times
10^{13}$cm$^{-2}$h$^{-1}$ and a similar measurement series was performed. The proton fluence reached with the second irradiation was $\Phi_p=(2.12 \pm 0.15) \times 10^{14}\;\mathrm{cm^{-2}}$. In both cases, the irradiation procedure described in [@r-LTNIM], where all details can be found, was followed: the proton beam was broadened to cover the whole crystal front face, and the fluence for each irradiation was determined through the activation of an aluminium foil covering the crystal front face.
Light Transmission measurements and results
===========================================
Longitudinal transmission curves at various intervals after proton irradiation are represented in Fig. \[f-LT\]. The earliest ones were taken as soon as it was possible to handle the crystal while keeping people’s exposure to radiation within regulatory safety limits, 18 days after the first irradiation and 62 days after the second one. From the LT curves, it is evident that the damage reduces Light Transmission at all wavelengths, while no transmission band-edge shift is observed after irradiation. This observation is consistent with our qualitative understanding [@r-IEEE], that the band-edge shift observed in Lead Tungstate [@r-LTNIM] and BGO [@r-BGO] after hadron irradiation must be due to disorder causing an Urbach-tail behavior [@r-URB; @r-ITO]. The absence of a band-edge shift in Cerium Fluoride is consistent with the anticipated lack of heavy fragments that can cause lattice disorder. The extreme steepness of the band-edge, which is preserved throughout the proton irradiations, is due to an allowed transition, as indicated in [@r-SCH]. \[s-LT\]
-- --
-- --
The longitudinal Light Transmission was repeatedly measured over time, to collect recovery data. The damage is quantified through the induced absorption coefficient as a function of light wavelength $\lambda$, defined as: $$\mu_{IND}(\lambda) = \frac{1}{\ell}\times \ln \frac{LT_0 (\lambda)}{LT (\lambda)}
\label{muDEF}$$ where $LT_0\; (LT)$ is the Longitudinal Transmission value measured before (after) irradiation through the length $\ell$ of the crystal.
Figure \[f-mu\] shows the profile of induced absorption as a function of wavelength, 150 days after each irradiation. We notice here the absence of the $\lambda^{-4}$ behavior we previously observed [@r-LTNIM] in Lead Tungstate. That behavior, peculiar to Rayleigh scattering, is a qualitative indication of the presence of very small regions of severe damage, as one expects to be caused by highly ionising fragments from nuclei break-up. The absence of a Rayleigh-scattering behavior in Cerium Fluoride is a further confirmation of our understanding. We also observe the presence of a yet unidentified absorption band, peaked around 400 nm, which does not recover with time, but is of no further concern, because it affects only a small fraction of the emitted light. The absorption band amplitude scales in a way which is consistent with a linear dependence on $\Phi_p$. The density of centres $N$ multiplied by the oscillator strength $f$ calculated according to [@r-DEX] for the second irradiation is $$N \times f \simeq 1.7 \times 10^{13} {\mbox{cm}}^{-3}.$$ A FLUKA [@r-fluka1; @r-fluka2] simulation of the irradiation, which is described in detail in section \[s-ACT\], yields an abundance $\rho
= 2.4\times 10^{-2} {\mbox{cm}}^{-3}$ of stable light nuclei (H and He) per impinging proton. Even assuming an oscillator strength $f = 1$, such an abundance does not account for the observed absorption band, and thus allows to exclude defects caused by hydrogen and helium. Typical $f$-values range from 1 to $10^{-3}$ and therefore we hypothesise that the observed absorption band might be linked to defects in the Cerium sub-lattice.
Furthermore, the dips due to Nd$^{3+}$ contamination mentioned in section \[s-xtal\] disappear when the induced absorption coefficients are evaluated, as it is evident in the plots of Fig. \[f-mu\]. This proves that such dips are not influenced by radiation, nor are hidden absorption bands present underneath them.
To examine damage recovery further, data analysis focused on the changes in Longitudinal Transmission at a wavelength of 340 nm, which corresponds to the peak emission of $\mbox{CeF}_3$ scintillation light, and is thus the relevant quantity for calorimetry.
-- --
-- --
The evolution of damage over time is shown in Fig. \[f-recLT\] for the two irradiations, where $\mu_{IND}(340\; \mbox{nm})$ is plotted over time. The data, taken over one year, are well fitted by a sum of a constant and two exponentials with time constants $\tau_i\;
(i=1,2)$: $$\mu_{IND}^{ j}(340\; \mathrm{nm},t_{\rm{rec}}) = \sum_{i=1}^{2} A_i^je^{-t_{\rm{rec}}/\tau_i} + A_3^j
\label{e-Afit}$$
where $t_{\rm{rec}}$ is the time elapsed since the irradiation, while $A_i^j \; (i=1,2)$ and $A_3^j$ are the amplitude fit parameters for the irradiation $j\; (j=1,2)$. Figure \[f-recLT\] shows a fit where the recovery time constants have been independently fitted for the two irradiations. The time constants obtained are compatible, yielding values $\tau_1 = 11\pm 2$ days and $\tau_2 = 70\pm 9$ days, with all the damage recovering. Such a result is very important if one considers Cerium Fluoride for superLHC calorimetry, in that one can expect to have a hadron damage which remains - due to its recovery time characteristics - all the time at a very small level compared to the cumulative damage amplitudes we measured for Lead Tungstate [@r-LTNIM]. In all this it should be pointed out, however, that our measurements are not sensitive to damage with a recovery time constant shorter than a few days, because the proton-irradiated crystal was initially too radioactive for safe handling during the first two weeks after irradiation, and thus no measurements were performed on it.
The dependence of damage on proton fluence is plotted in Fig. \[f-LIN\], in comparison with the values obtained for Lead Tungstate. The line therein is the fit from Fig. 15 in [@r-LTNIM] to $\mu_{IND}(420\;\mbox{nm})$ for Lead Tungstate, 150 days after irradiation. The white circles are measurements of $\mu_{IND}(420\;\mbox{nm})$ for the same Lead Tungstate crystals studied in [@r-LTNIM], taken 300 days after irradiation, showing how stable the long-term damage is in that crystal. Since in [@r-LTNIM] all crystals studied were produced by the Bogoroditzk Techno-Chemical Plant (BTCP) in Russia, we have performed an irradiation of a further crystal produced by a different supplier, the Shanghai Institute of Ceramics (black circle in Fig. \[f-LIN\]). The measurement for the SIC Lead Tungstate crystal is in perfect agreement with those for BTCP crystals, proving how hadron damage is not linked to fine details of doping, stoichiometry and related defects, nor to growth technology, but it is rather due to the effects the hadron cascade has on the bulk of the crystal.
It should be pointed out that all crystals tested in Ref. [@r-LTNIM] are 23 cm ($25.8\; X_0$) long, while the Cerium Fluoride crystal studied here is only $14.1$ cm ($8.4\; X_0$) in length. In the same plot, we have thus also superposed the proton damage measured in two Lead Tungstate crystals $7.5$ cm ($8.4\; X_0$) long, studied in [@r-pionNIM] after a proton irradiation where they were placed one behind the other. Also for these shorter crystals $\mu_{IND}(420\;\mbox{nm})$ is well consistent with the values measured for the longer crystals. This can be understood from the star density profiles for 20 - 24 GeV/c protons in Fig. 3 of ref. [@r-LTNIM], which are nearly flat over the length of the crystal, besides a small build-up over the initial 5 cm, which is reflected in a slightly smaller damage in one of the two crystals. It thus appears justified to compare damage measured in an $8.4\; X_0$ long Cerium Fluoride crystal with the existing measurements for 23 cm long Lead Tungstate crystals. The induced absorption $\mu_{IND}(340\;\mbox{nm})$ at the peak of scintillation emission measured in Cerium Fluoride 300 days after irradiation is thus also plotted in Fig. \[f-LIN\]. As easily understandable, a cumulative damage in Cerium Fluoride would be fitted by a line parallel to the one for Lead Tungstate in this doubly-logarithmic plot, which is not what we observe.
With the correlation of Fig. \[f-LIN\] extending over almost three orders of magnitude in fluence, the damage observed 150 days after irradiation is a factor 15 smaller in Cerium Fluoride than in Lead Tungstate for a fluence $\Phi_p=2.78 \times
10^{13}\;\mathrm{cm^{-2}}$ and a factor 30 smaller for $\Phi_p=2.12
\times 10^{14}\;\mathrm{cm^{-2}}$. The gap increases to a factor 25 and 124 respectively 300 days after irradiation, as also visible in Fig. \[f-LIN\]. However, one has to be aware of the fact that, were the induced absorption expressed in units of inverse radiation length, the gap would be reduced by a factor 1.5.
Light Output measurements and results {#s-LY}
=====================================
The relevant quantity for calorimeter operation is the Light Output (LO), and thus, in the present study we have verified that with the recovery of Light Transmission, also the Light Output is restored. For this purpose, we have taken Light Output spectra using a bialkaline 12-stage Photomultiplier (PM), and its anode charge was digitised using a charge-integrating ADC as described in [@r-LYNIM]. To identify a scintillation signal well above the background due to the intrinsic induced radioactivity after proton irradiation, we have triggered on cosmic muons traversing the crystal sideways, by means of two plastic scintillators.
-- --
-- --
Such muons are minimum-ionising, and leave, according to [@r-dedx], an energy deposit of 7.9 MeV/cm in the crystal. Due to the geometrical acceptance of the trigger scintillator setup, the mean path in the crystal is $2.3 \pm 0.2$ cm. and thus the muon energy deposit in average $18.2\pm 1.6$ MeV. The Light Output spectrum is shown in Fig. \[f-muon\] (top), which was acquired attenuating the photomultiplier signal by 26 db. The peak position is determined to be at channel $1276\pm 8$. Keeping the same PM gain, but with 0 db attenuation, we have acquired the spectrum in Fig. \[f-muon\] (bottom), by triggering on the Light Output from the crystal itself with a threshold set below the level of single photoelectrons thermally emitted by the photocathode: those yield the leftmost peak in the histogram. The peak at channel $225 \pm 1$ corresponds, if we determine its equivalent energy deposit scaling from the muon peak position, to $E_{\gamma} = 160 \pm 10$ keV. As will be evident from activation measurements and related FLUKA simulations described in the following sections, the dominant isotope created in the proton irradiation tests is $^{139}$Ce, whose electron capture decay to $^{139}$La is accompanied by an emission of a 165 keV photon. The peak we observe in the spectrum is in good agreement with the activity from this isotope.
A Light Output measurement using cosmic muons one year after the second irradiation allows us to determine a remaining loss of $ \Delta{\mathrm{LO / LO}} =
(11\pm 2) \%$. The measured fraction of induced absorption coefficient which has not recovered 1 year after the second irradiation (see Fig. \[f-mu\]), is $\mu_{IND}=0.33 \pm 0.04 \;\mbox{m}^{-1}$. A correlation between LO loss and induced absorption has been published in [@r-LYNIM] for 23 cm long Lead Tungstate crystals, and the correlation therein between LO loss and induced absorption coefficients is similar to the one we observe here. A precise comparison would require taking into account the different crystal dimensions and their influence on light collection. Our measurement however shows how the observed spontaneous recovery at room temperature of transmission loss after hadron irradiation in Cerium Fluoride up to $\Phi_p=(2.12 \pm 0.15) \times
10^{14}\;\mathrm{cm^{-2}}$ is accompanied, as expected, by an almost complete recovery of scintillation Light Output.
Activation measurements and results {#s-ACT}
===================================
Present irradiations
--------------------
Hadron irradiation causes the production of radioactive isotopes in the crystals. While most of them are short-lived, those with a long half-life are responsible for the remnant radioactivity and are relevant in case a calorimeter needs human intervention after exposure. It might thus be of interest to compare measurements of radio-activation in Cerium Fluoride to those in Lead Tungstate. The latter has been extensively studied through simulations and measurements in our early work [@r-LTNIM] and references therein. The measurements there agree with simulation results on average within 30% and never beyond a factor of 2, and confirm that radiation exposure is an important concern for a Lead Tungstate calorimeter used in intense hadron fluences. Activation measurements in Cerium Fluoride provide important practical information on access and handling possibilities for such a calorimeter if used at superLHC. The induced ambient dose equivalent rate (“dose”) ${\dot{H}^*(10)_{\rm ind}}$ was regularly measured according to the procedure described in [@r-LTNIM] with an Automess 6150AD6 [@r-AUTOMESS] at a distance of 4.5 cm from the long face of the crystal at its longitudinal centre. The reference point of the sensitive element in the 6150AD6 is reported to be 12mm behind the entrance window. Thus our actual distance was 5.7 cm from the crystal face. The measured dose as a function of cooling time is plotted in Fig. \[f-RA\]. The activation values are compatible with the scaling of fluences, if one takes into account that when the second irradiation was started, the crystal was still showing a remaining level of activation from the first one.
-- --
-- --
------------- ----------------------- ---------------- ----------------
Isotope $\tau_{\frac{1}{2}} $ Activity total $\gamma$
\[Bq/cm $^3$\] energy
$^3H$ 12.33 y $1017 \pm 3$ -
$^{88}Y$ 106.65 d $109 \pm 1$ 2.734 MeV
$ ^{109}Cd$ 462.6 d $114 \pm 1$ 0.088 MeV
$^{139}Ce$ 137.64 d $930 \pm 6$ 0.166 MeV
------------- ----------------------- ---------------- ----------------
: Dominating isotope activities in Cerium Fluoride from FLUKA simulations, 1 year after irradiation by 24 GeV/c protons up to a fluence $\Phi_p=2.78\times10^{13}$cm$^{-2}$ []{data-label="t-ACT"}
We have fitted the data, taken over one year, with a sum of a constant and two exponentials with time constants $\tau_i\; (i=1,2)$: $$\dot{H}^*(10)_{\rm ind}(t_{\rm{rec}}) = \sum_{i=1}^{2} D_i^je^{-t_{\rm{rec}}/\tau_i} + D_3^j,
\label{e-Dfit}$$ where $t_{\rm{rec}}$ is the time elapsed since the irradiation, while $D_i^j,\; (i=1,2)$ and $D_3^j$ are the amplitude fit parameters for irradiation $j\; (j=1,2)$. The time constants obtained for the two independent fits are compatible, with values $\tau_1 = 11$ days and $\tau_2 = 85$ days. Interestingly, the radio-activation recovery time constants are compatible within $2\sigma$ with those for the damage recovery determined in Sec. \[s-LT\]. While there is no evidence for a link between the two, we wonder whether the long-lived induced absorption component might be due to a self-irradiation of the crystal.
FLUKA Monte Carlo simulations were performed using the code Version 2008.3c.0 [@r-fluka1; @r-fluka2] and rely on the input parameters used in our $\mathrm{PbWO}_4$ study [@r-HUH; @r-LTNIM]. The beam profile was assumed to be squared ($3\times3\,\mathrm{cm^{2}}$) and uniformly distributed. The crystals was simulated according to the experimental setup. The FLUKA geometry includes also the back wall of the irradiation zone, i.e. the T7 beam line dump. Because the hadron shower induced from beam protons impinging on the crystal shows a significant forward direction, such that the integrated hadron fluence at the backside is roughly ten times more intense than the lateral fluence, the side walls were neglected.
For the dose measurements following the irradiation, the crystal has been removed from the irradiation zone and was kept for measurements in an area with low background. To simulate the two processes with different geometries, the FLUKA two-step method [@r-2step] was applied. In a first step (the irradiation) the produced radionuclides in the crystal, namely the $\gamma$ - and $\beta^+$ - emitters, were recorded. These provided the input for the second step, where the average ambient dose equivalent in a volume of $1\,\mathrm{cm^{3}}$ was calculated, using fluence to ambient dose equivalent conversion coefficients [@r-AMB74]. The center of the dose recording region was set laterally centered and at a distance of $5.7\,\mathrm{cm}$ from the crystal, according to the experimental settings. The FLUKA simulation results of the Cerium Fluoride activation at a few intervals after each of the two irradiations are shown in Fig. \[f-RA\]. One approximation was made in the simulation: a single irradiation was assumed in each case, while the same crystal was actually irradiated twice at one year’s interval. This might explain the slight dose underestimate from FLUKA at long cooling times after the second irradiation. Table \[t-ACT\] lists the isotopes expected to be still present in the crystal, according to FLUKA, one year after the irradiation up to a fluence $\Phi_p=2.78\times10^{13}$cm$^{-2}$ and with an activity larger than 10 Bq/cm$^3$. From the tabulated values, it is evident that the isotope that contributes with the largest activity is $^{139}$Ce. One may also notice that the longer recovery time constant fitted in Fig. \[f-RA\] for the radiation dose is of the same order of magnitude as the life time of $^{139}$Ce. Taking into account contributions from the other long-lived isotopes present, the agreement is quite reasonable.
Full-size crystals
------------------
A comparison between Cerium Fluoride and Lead Tungstate activation levels after hadron irradiation for full-size crystals is a relevant input to the selection of the calorimetric medium for a calorimeter upgrade. In Fig. \[f-decay\] we show the results from our FLUKA simulation for the irradiation and cooling conditions of [@r-LTNIM] for full-size Lead Tungstate crystals exposed to 20 GeV/c protons, compared to the measured activation and the FLUKA simulation performed therein. The agreement validates the present FLUKA simulations. Also shown in Fig. \[f-decay\] are the FLUKA results for the activation expected for full-size Cerium Fluoride crystals. Because of the smaller density and longer radiation length of Cerium Fluoride, a full-size calorimeter crystal would have to be 42 cm in length. Transverse dimensions of 24 mm x 24 mm correspond to the typical cross section yielding a similar $\eta - \phi$ granularity as for existing Lead Tungstate calorimeters. The FLUKA simulation, which was validated already for short Cerium Fluoride crystals (Fig. \[f-RA\]) and for Lead Tungstate (Fig. \[f-decay\]), was performed for such full-size dimensions and for the same fluence of $\Phi_p=1
\times 10^{13}\;\mathrm{cm^{-2}}$, yielding dose values shown as lozenges in Fig. \[f-decay\]. One observes that workers’ exposure to proton-irradiated, 26 X$_0$ long Cerium Fluoride crystals is expected to be always a factor of 2 to 3 lower than the one due to Lead Tungstate.
Conclusions {#s-CON}
===========
We have studied Cerium Fluoride as a possible scintillating crystal for calorimetry at the superLHC. This investigation was inspired by our earlier studies of Lead Tungstate, where we observed a hadron-specific, cumulative damage from charged hadrons. All characteristics of the damage in Lead Tungstate are consistent with an intense local energy deposition from heavy fragments. Measurements of absorption induced in $\mathrm{CeF}_3$ by 24 GeV/c protons up to fluences $\Phi_p=(2.78 \pm 0.20) \times
10^{13}\;\mathrm{cm^{-2}}$ and $\Phi_p=(2.12 \pm 0.15) \times
10^{14}\;\mathrm{cm^{-2}}$ show a Light Transmission damage which is not cumulative, is more than one order of magnitude smaller than in $\mathrm{PbWO}_4$ 6 months after irradiation, and — unlike $\mathrm{PbWO}_4$ — recovers further. The absence of a dominant Rayleigh-scattering component in $\mathrm{CeF}_3$ confirms our understanding, that in $\mathrm{PbWO}_4$ it is due to highly-ionising fission fragments as produced in crystals with elements above Z=71. The scintillation Light Output in $\mathrm{CeF}_3$ is observed to recover by 90% over 1 year, and the remaining loss is consistent with the induced absorption still present.
With its extreme resistance to hadron-induced damage, manifested through a modest induced absorption which recovers with time, low raw material costs, high light yield and negligible temperature dependence, Cerium Fluoride is an excellent candidate for medical imaging applications and for calorimetry at superLHC or in any high hadron fluence environment.
Acknowledgements {#acknowledgements .unnumbered}
================
We are indebted to R. Steerenberg, who provided us with the required CERN PS beam conditions for the proton irradiations. We are deeply grateful to M. Glaser, who operated the proton irradiation facility and provided the Aluminium foil dosimetry.
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D. Kruk, J. Phys. Condens. Matter 18 (2006) 1725-1741. N. I. Sorokin and B. P. Sobolev, Crystallography Reports 52 (2007) 842-863. E. Auffray, F.N.-T., P. L. et al., Nucl. Instr. and Meth. A 378 (1996) 171-178 The CMS Collaboration, “CMS Letter of Intent”, CERN/LHCC 92-003, LHCC/I 1, (1992, CERN, Geneva, Switzerland). The L3P Collaboration, “L3P Letter of Intent”, CERN/LHCC 92-005, LHCC/I 3, (1992, CERN, Geneva, Switzerland). W. W. Moses, S. Derenzo et al., J. Lumin. 59 (1994) 89-100. R. Novotny et al., Nucl. Instr. and Meth. A 486 (2002) 131-135 Optovac, North Brookfield, USA, later acquired by Corning Advanced Material, Corning, USA. Crystal Clear Collab., E.Auffray et al., Nucl. Instr. and Meth. A 383 (1996) 367-390 M. Glaser [*et al*]{}, Nucl. Instr. and Meth. A 426 (1999) 72. G. Dissertori, D. Luckey, P. Lecomte, F. Nessi-Tedaldi, F. Pauss, [ *“Studies of hadron damage in Lead Tungstate and Cerium Fluoride Crystals for HEP Calorimetry”*]{}, paper N55-1, IEEE/NSS 2008 Conference Record. M. Kobayashi et al., Nucl. Instr. and Meth. A 206 (1983) 107-117 F. Urbach, Phys. Rev. 91 (1953) 1324. M. Itoh et al., Phys. Stat. Sol. 231 (2002) 595-600. M.Schneegans, Nucl. Instr. and Meth. A344 (1994) 47-56 D. L. Dexter, Phys. Rev. 101 (1956) 48. A. Ferrari, P. R. Sala, A. Fasso, J. Ranft, [*”FLUKA: a multi-particle transport code”*]{}, CERN 2005-10 (2005), INFN TC 05/11, SLAC-R-773. G. Battistoni, S. Muraro, P. . Sala, F. Cerutti, A. Ferrari, S. Roesler, A. Fasso, J. Ranft, [*”The FLUKA code: Description and benchmarking”*]{}, Proc. of the Hadronic Shower Simulation Workshop 2006, Fermilab 6-8 September 2006;\
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---
abstract: 'The quadrupole interaction between the Rydberg electronic states of a Rydberg ion and the radio frequency electric field of the ion trap is analyzed. Such a coupling is negligible for the lowest energy levels of a trapped ion but it is important for a trapped Rydberg ion due to its large electric dipole moment. This coupling cannot be neglected by the standard rotating-wave approximation because it is comparable to the frequency of the trapping electric field. We investigate the effect of the quadrupole coupling by performing a suitable effective representation of the Hamiltonian. For a single ion we show that in this effective picture the quadrupole interaction is replaced by rescaled laser intensities and additional Stark shifts of the Rydberg levels. Hence this detrimental quadrupole coupling can be efficiently compensated by an appropriate increase of the Rabi frequencies. Moreover, we consider the strong dipole-dipole interaction between a pair of Rydberg ions in the presence of the quadrupole coupling. In the effective representation we observe reducing of the dipole-dipole coupling as well as additional spin-spin interaction.'
author:
- 'Lachezar S. Simeonov'
- 'Nikolay V. Vitanov'
- 'Peter A. Ivanov'
title: 'Compensation of the trap-induced quadrupole interaction in trapped Rydberg ions'
---
Introduction
============
Ion trap system is one of the leading platforms in quantum information technologies [@Blatt2008; @Haffner2012; @Schneider2012]. The ability to control and read out the external and internal degrees of freedom of the trapped ions with high accuracy leads to experimental implementation of various entangled states [@Sackett2000; @Haffner2005; @Leibfried2005; @Monz2011] and quantum gates with high fidelity [@Leibfried2003; @Ballance2016; @Gaebler2016]. However, the control of trapped ions becomes more difficult for multiple ions. Indeed, the phonon mode structure becomes too complicated for entanglement operations and the storage capacity is rendered limited. One way to overcome this limit is to use array of ion traps which store a small number of ions [@Kielpinski2002; @Monroe2013; @Weidt2016]. Another approach is based on using trapped *Rydberg* ions. In this approach instead of using the common phonon mode for entanglement, the strong *dipole-dipole interaction* may be used for implementation of entangled states and quantum gates [@Li2014], as well as for quantum simulation [@Li2012].
The strongly interacting Rydberg atoms offer a promising platform for quantum computation and simulation [@Saffman2010; @Killian2007; @Nguyen2018]. One hope that one may use the advantages of *both* trapped ions (individual addressability, entanglement operations with small errors, etc.) *and* the strong *long* range interaction of Rydberg ions. However this novel system suffers from some disadvantages. For example, stray electric and magnetic fields due to the trap could alter the dipole moment of the Rydberg ion. Despite that, trapped Rydberg ions have been recently experimentally accomplished [@Feldker2015]. The Floquet sidebands due to quadrupole interaction as well as modification of the trapping potential due to the strong polarization of the $^{88}\text{Sr}^{+}$ Rydberg ion have been observed [@Higgins2017] .
In Ref. [@Feldker2015; @Bachor2016] the Rydberg levels are excited using a single-photon excitation with vacuum ultraviolet laser light at 122 nm. However, this is quite difficult to handle experimentally. Another experimental approach is to use $^{88}\text{Sr}^{+}$ Rydberg ions [@Higgins2017]. In that case the Rydberg ions are excited by two-photon transitions at 243 and 309 nm respectively. Though the $^{88}\text{Sr}^{+}$ ions are excited more easily to Rydberg levels, the $nD_{3/2}$ Rydberg states are coupled by the quadrupole field of the trap, see Fig. \[FIG1\]. These undesirable transitions may transfer population out of the Rydberg state.
![(a) Level scheme of the Rydberg $^{88}\text{Sr}^{+}$ ion. The levels $nD_{3/2}$ and $n^{\prime}P_{1/2}$ are Rydberg levels. Two laser fields are applied which drive the transitions $|1\rangle\leftrightarrow|2\rangle$ and $|2\rangle\leftrightarrow|3\rangle$ with Rabi frequencies $\Omega_{1,2}$ and detunings $\Delta_{2,3}$. The quadrupole field couples the levels $|2\rangle$ and $|4\rangle$ with peak Rabi frequency $\Omega$. This transition oscillates with radio trap frequency $\omega$. The level $|4\rangle$ has an energy shift $\Delta_{4}$. (b) Using the effective picture, the system is reduced to three state ladder system with rescaled Rabi frequencies $\Omega_{1,2}^{\prime}$ and detunings $\Delta_{2}^{\prime}$.[]{data-label="FIG1"}](Fig1.eps){width="45.00000%"}
In this paper we consider the effect of the quadrupole coupling on the coherent dynamics of Rydberg trapped ions. Such an effect arises due to the composite nature of the Rydberg ion and causes undesired excitation of electronic transitions driven by the radio-frequency electric field of the Paul trap. We consider single Rydberg ion with one and two Rydberg state manifolds subject to the quadrupole coupling. We show that as long as the radio trap frequency $\omega$ of these quadrupole transitions is *sufficiently large*, the negative effect is averaged and traced out. To see that, we perform a suitable unitary transformation and investigate the system into a different picture. We show that the effect of the quadrupole coupling is merely to *rescale* the Rabi frequencies $\Omega_{i}$ which drive the transition between the Rydberg levels. The quandrupole interaction also induces an energy shift of the respective Rydberg levels. Moreover, we consider the strong dipole-dipole interaction between the Rydberg ions in the presence of quadrupole coupling. We show that the effect of the coupling is to reduce the strength of the dipole-dipole interaction. We also find that the quadrupole coupling induces residual dipole-dipole interaction which can be neglected only in the rotating-wave approximation.
The paper is organized as follows. In Section \[sec1\] we provide discussion of the effect of the radio-frequency electric field on the electronic Rydberg transition. In Sec. \[sec2\] we introduce the suitable unitary transformation which allows to treat the effect of the oscillating quadrupole coupling. In Sec. \[sec3\] we discuss the single trap ion with one and two Rydberg state manifolds sensitive to the quadrupole coupling. In Sec. \[sec4\] we consider two trapped Rydberg ions interacting via strong dipole-dipole coupling. Finally, in Sec. \[sec5\] we summarize our findings.
The level system of $^{88}\text{Sr}^{+}$ trapped ion {#sec1}
====================================================
Our quantum system consists of a single trapped Rydberg ion. Although the method is applicable for any Rydberg ion we consider for concreteness Rydberg $^{88}\text{Sr}^{+}$ ion with the level structure shown on Fig. \[FIG1\]. The Rabi frequency $\Omega_{1}$ drives the two-photon transition between the states $|1\rangle$ and $|2\rangle$. State $|2\rangle$ belongs to a Rydberg $nD_{3/2}$ manifold with detuning $\Delta_{2}$. We apply an additional laser field with Rabi frequency $\Omega_{2}$ which couples level $|2\rangle$ and level $|3\rangle$ with detuning $\Delta_{3}$. The latter is part of $n^{\prime}P_{1/2}$ manifold. The interaction Hamiltonian becomes ($\hbar=1$) $$\begin{aligned}
\hat{H}_{0}&=\Delta_{2}|2\rangle\langle 2|+\Delta_{3}|3\rangle\langle 3|+\Delta_{4}|4\rangle\langle 4|\notag\\
&+\left(\Omega_{1}|1\rangle\langle 2|+\Omega_{2}|2\rangle\langle 3|+ \text{H.c.}\right).\label{HamInt1}\end{aligned}$$
Let us consider the typical length scales of the trapped Rydberg ion. The external trapping frequency $\omega$ is of the order of MHz. To this frequency there corresponds a so called oscillator length $a_{\text{o}}$, which is roughly the *localization length* of the ion around its *equilibrium* position. For $\omega\sim \text{MHz}$ we have $a_{\text{o}}\sim 10 \;\text{nm}$. On the other hand, the size of the Rydberg *orbit* $a_{\text{Ry}}$ is proportional to $n^{2}$, where $n$ is the principal quantum number. For Rydberg states $a_{\text{Ry}}\sim 100\;\text{nm}$. Thus, it follows that $a_{\text{Ry}}\gg a_{\text{o}}$. Therefore the Rydberg ion can no longer be considered as a point-like particle but rather as a *composite* object [@Muller2008] and its internal structure *must* be taken into account. Indeed, as shown in Refs. [@Muller2008; @Kaler2011], the electric field of the Paul trap may excite *internal electronic* transitions which are no longer negligible contrary to the ordinary trapped ions.
The Paul trap electric field can be written as
$$\Phi(\textbf{r},t)=\alpha\cos{(\omega t)}(x^{2}-y^{2})-\beta\left(x^{2}+y^{2}-2z^{2}\right),$$
where $\alpha$ and $\beta$ are electric field gradients and $\omega$ is the radio-frequency of the Paul trap [@Singer2010]. In the customary ion traps, this electric field does *not* couple internal electronic states. The ion in that case can be considered as a point particle. However, in the case of Rydberg ions, it will couple electronic transitions. The coupling $\hat{H}_{\text{e}}$ of the above electric field is given by $\hat{H}_{\text{e}}=e\Phi(\textbf{r},t)$, where $e$ is the electronic charge. Generally, this quadrupole coupling *cannot* couple (to first order) states in the manifold $nX_{J}$ for $J=1/2$ for any $X=S,P,D,...$. Such transitions are only allowed for $J>1/2$ due to selection rules. However, states in the manifold $nD_{J}$ ($J=3/2$ or $J=5/2$) are coupled even to first order by the quadrupole field. It turns out, that the time dependent interaction with the quadrupole is [@Higgins2017] $$\hat{V}(t)=\hbar\Omega\cos{(\omega t)}\sum_{m_{J}=1/2}^{3/2}\{|nLJ(m_{J}-2)\rangle\langle nLJm_{J}|+\text{H.c.}\}\label{Quadrupole1},$$ where $\Omega$ is the effective Rabi frequency for the quadruple coupling which oscillates with the trap frequency $\omega$. This transition may lead to a leak of population to an undesirable state $|4\rangle$ as is shown in Fig. \[FIG1\]. Unfortunately rotating wave approximation is not applicable because the effective Rabi frequency $\Omega$ is comparable with the trap frequency $\omega$ [@Higgins2017]. In the next section we shall propose solution to this problem.
General theory of the effective picture {#sec2}
=======================================
First, let us rewrite Eq. ([\[Quadrupole1\]]{}) for the $^{88}\text{Sr}^{+}$ ion, $$\hat{V}(t)=\hat{v}e^{\i\omega t}+\hat{v}^{\dag}e^{-\i\omega t},$$ where $$\hat{v}=\frac{\Omega}{2}\left(|2\rangle\langle 4|+|4\rangle\langle 2|\right).\label{v-single}$$ Including the quadrupole interaction the total Hamiltonian becomes $$\hat{H}=\hat{H}_{0}+\hat{V}(t).\label{TotalHam}$$ As we mentioned above the interaction $\hat{V}(t)$ may lead to leak of population out of Rydberg state $|2\rangle$ which spoils the single as well as the two qubit operators. In the following we perform a suitable unitary transformation. We shall designate this new quantum picture as an effective picture.
In order to derive the effective picture we perform a time dependent unitary transformation $\hat{U}(t)=e^{{\i}\hat{K}(t)}$ to the state vector $|\psi\rangle$ such that $|\tilde{\psi}\rangle=\hat{U}(t)|\psi\rangle$, where $\hat{K}(t)$ is an hermitian operator. Our goal is to choose $\hat{K}(t)$ such that the effective Hamiltonian $\hat{H}_{\text{eff}}=\hat{U}\hat{H}\hat{U}^{\dag}+i (\partial_{t}\hat{U})\hat{U}^{\dag}$ becomes a time-independent to *any* desired order of $\omega^{-1}$. Method for averaging of the rapidly oscillating terms was proposed in [@James2007], which however is not suitable for our case since it requires knowledge of the spectrum of $\hat{H}_{0}$. Wee derive $\hat{K}(t)$ following the method presented in [@Goldman2014; @Rahav2003] (see the Supplement for an overview of the derivation). Here we simply state the result $$\begin{aligned}
\hat{K}(t)=\omega^{-1}\hat{K}_{1}(t)+\omega^{-2}\hat{K}_{2}(t)+O(\omega^{-3}),\label{K(t)}\end{aligned}$$ where $$\begin{aligned}
\hat{K}_{1}(t)=2 \hat{v}\sin(\omega t),\quad \hat{K}_{2}(t)=-2i[\hat{v},\hat{H}_{0}]\cos(\omega t).\label{K(t)Corr}\end{aligned}$$ We find that the effective Hamiltonian becomes $$\hat{H}_{\rm eff}=\hat{H}_{0}+\omega^{-2}[[\hat{v},\hat{H}_{0}],\hat{v}]+O(\omega^{-4})\label{HEFFF},$$ which is indeed time-independent to $O(\omega^{-4})$.
Single Trapped Rydberg Ion {#sec3}
==========================
Single manifold coupled by the quadrupole interaction
-----------------------------------------------------
![(a) Time evolution of the probabilities $P_{1}(t)$ and $P_{3}(t)$ for the four-level system. We compare the probabilities derived from the original Hamiltonian (solid lines) and the effective Hamiltonian for $P_{1}(t)$ (blue dots) and $P_{3}(t)$ (red triangles). The red dashed line is the solution for $P_{1}(t)$ assuming rotating wave approximation. The parameters are set to $\Omega/2\pi=12$ MHz, $\omega/2\pi=20$ MHz, $\Omega_{i}/2\pi=2$ MHz, $\Delta_{2}/2\pi=\Delta_{3}/2\pi=2.0$ MHz, $\Delta_{4}/2\pi=1.0$ MHz. (b) Probability $P_{2}(t)$ (black solid line) compared with the effective solution (red line).[]{data-label="FIG3"}](Fig33_new.eps){width="45.00000%"}
In this subsection we consider the single trapped Rydberg ion with *one* Rydberg manifold coupled by the quadrupole coupling, see Fig. \[FIG1\].
Substituting Eqs. and into Eq. , we obtain the following effective Hamiltonian, $$\begin{aligned}
\hat{H}_{\text{eff}}&=\Delta_{2}^{\prime}|2\rangle\langle 2|+\Delta_{3}|3\rangle\langle 3|+\Delta_{4}^{\prime}|4\rangle\langle 4|\notag\\
&+\left(1-\frac{\Omega^{2}}{4\omega^{2}}\right)\left(\Omega_{1}|1\rangle\langle 2|+\Omega_{2}|2\rangle\langle 3|+\text{H.c.}\right).\label{Heff1}\end{aligned}$$ Interestingly, we observe that the quadrupole interaction between states $|2\rangle$ and $|4\rangle$ is removed. However the new Rabi frequencies in the effective picture are rescaled (renormalized) with the same factor $\left(1-\Omega^{2}/(4\omega^{2})\right)$. Therefore in order to compensate the quadrupole interaction one needs to merely increase the laser intensities with the factor $\left(1-\Omega^{2}/(4\omega^{2})\right)^{-1}$. Additionally, we find that the quadrupole interaction caused an energy shift of the states $|2\rangle$ and $|4\rangle$ such that the laser detuning becomes $\Delta_{2}^{\prime}=\Delta_{2}\{1-\frac{\Omega^{2}}{2\omega^{2}}\left(1-\frac{\Delta_{4}}{\Delta_{2}}\right)\}$ and respectively $\Delta_{4}^{\prime}=\Delta_{4}\{1-\frac{\Omega^{2}}{2\omega^{2}}\left(1-\frac{\Delta_{2}}{\Delta_{4}}\right)\}$.
In Fig. \[FIG3\] we compare the exact dynamics governed by the full Hamiltonian (\[TotalHam\]) and the effective Hamiltonian (\[Heff1\]). As can be seen very good agreement is observed. We also show the effective dynamics which is obtained by standard rotating-wave approximation (RWA). As expected, RWA significantly deviates from the exact solution. This is due to the fact that $\Omega$ and $\omega$ are of the same order of magnitude, namely $\Omega=0.6\omega$. In Fig. \[FIG3\](b) we plot the population of the level $|2\rangle$ which is subject of the strong quadrupole interaction. Because of that the time evolution of the population contains fast and slow components where the latter can be described within the effective picture. Figure \[d\] shows the frequency scan of the populations $P_{1,3}$ at fixed interaction time. The exact and the effective solutions are almost indiscernible.
After a lengthy calculation it can be shown that the next correction to the effective Hamiltonian Eq. (\[HEFFF\]) is *not* $O(\omega^{-3})$ but is $O(\omega^{-4})$. This explains why the agreement in Fig. \[FIG3\] is quite accurate.
Two Rydberg manifolds coupled by the quadrupole coupling
--------------------------------------------------------
![(a) Probability $P_{1}$ at time $t=0.5$ $\mu$s versus the laser detuning $\Delta_{2}$. The exact solution with the Hamiltonian (solid lines) is compared with the solution with the effective Hamiltonian (blue circles). The red dashed line is the solution for $P_{1}$ assuming rotating wave approximation. (b) Same but for population $P_{3}$. Solid line is the exact result and the red triangle is the effective solution.[]{data-label="d"}](d_new.eps){width="45.00000%"}
We extend the discussion by including higher angular momentum Rydberg states such as $n^{\prime}P_{3/2}$ states, see Fig. \[FIG2\](a). In that case the quadrupole Hamiltonian couples not only states $|2\rangle$ and $|4\rangle$ but also states $|3\rangle$ and $|5\rangle$. We shall show that in the effective picture the quadrupole coupling is again removed.
In this case the Hamiltonian is again of the type $\hat{H}=\hat{H}_{0}+\hat{V}(t)$. However, here $$\begin{aligned}
\hat{H}_{0}=&\Delta_{2}|2\rangle\langle2|+\Delta_{3}|3\rangle\langle3|+\Delta_{4}|4\rangle\langle4|+\Delta_{5}|5\rangle\langle5|\notag\\
&+(\Omega_{1}|1\rangle\langle2|+\Omega_{2}|2\rangle\langle3|+{\rm H.c.})\end{aligned}$$ and $\hat{V}(t)=\hat{v}e^{\i\omega t}+\text{H.c.}$, where $\hat{v}$ is given by $$\hat{v}=\frac{\Omega}{2}|2\rangle\langle4|+\frac{\bar{\Omega}}{2}|3\rangle\langle5|+{\rm H.c.},$$ with $\Omega$ and $\bar{\Omega}$ being the effective Rabi frequencies for the quadrupole interaction.
The expression Eq. (\[HEFFF\]) for the effective Hamiltonian as well as Eqs. (\[K(t)\]) and (\[K(t)Corr\]) for $\hat{K}$ remain valid. Thus, we obtain $$\hat{H}_{\text{eff}}=\hat{H}_{1}+\hat{H_{2}}.\label{Heff2}$$ Here $$\hat{H}_{1}=\Delta_{4}^{\prime}|4\rangle\langle 4|+\Delta_{5}^{\prime}|5\rangle\langle 5|+\Omega_{3}^{\prime}\left(|4\rangle\langle 5|+|5\rangle\langle 4|\right),$$ and $$\begin{aligned}
\hat{H}_{2}&=&\Delta_{2}^{\prime}|2\rangle\langle2|+\Delta_{3}^{\prime}|3\rangle\langle3|+\Omega_{1}\left(1-\frac{\Omega^{2}}{4\omega^{2}}\right)(|1\rangle\langle2|\notag\\
&&+|2\rangle\langle1|)+\Omega_{2}\left(1-\frac{\Omega^{2}}{4\omega^{2}}-\frac{\bar{\Omega}^{2}}{4\omega^{2}}\right)\left(|2\rangle\langle3|+|3\rangle\langle2|\right).\end{aligned}$$ This result means that the initial five-level coupled system is reduced to two uncoupled ladders, see Fig. \[FIG2\](b). The first ladder is a two level system consisting of states $|4\rangle$ and $|5\rangle$ driven by effective Rabi frequency $\Omega_{3}^{\prime}=\frac{\Omega\bar{\Omega}\Omega_{2}}{2\omega^{2}}$. This transition is caused by the virtual chain of transitions between the states $|4\rangle\leftrightarrow |2\rangle\leftrightarrow|3\rangle\leftrightarrow|5\rangle$. This explains why $\Omega_{3}^{\prime} \varpropto \Omega\Omega_{2}\bar{\Omega}$. Additionally, the quadrupole interaction causes energy shift of the level $|5\rangle$ such that we have $\Delta_{5}^{\prime}=\Delta_{5}\{1-\frac{\bar{\Omega}^{2}}{2\omega^{2}}\left(1-\frac{\Delta_{3}}{\Delta_{5}}\right)\}$. The second ladder consists of three states $|1\rangle$, $|2\rangle$ and $|3\rangle$. The effect of the quadruple interaction is to rescale the respective Rabi frequencies and detunings $\Delta_{2}^{\prime}$, $\Delta_{3}^{\prime}=\Delta_{3}\{1-\frac{\bar{\Omega}^{2}}{2\omega^{2}}\left(1-\frac{\Delta_{5}}{\Delta_{3}}\right)\}$ as is shown in Fig. \[FIG2\](b). As long as the initial population is in state $|1\rangle$, the population will remain in the second ladder, described by $\hat{H}_{2}$.
![(a) Rydberg levels for a quadrupole coupling when *both* Rydberg manifolds, $nD_{3/2}$ and $n^{\prime}P_{3/2}$ are coupled by the quadrupole trap field with Rabi frequencies $\Omega$ and $\bar{\Omega}$. Both quadrupole couplings oscillate with radio trap frequency $\omega$. (b) Effective quantum system is reduced into two *uncoupled* systems. The first system consists of the levels $|4\rangle$ and $|5\rangle$ which are driven by Rabi frequency $\Omega_{3}^{\prime}=\frac{\Omega\bar{\Omega}\Omega_{2}}{2\omega^{2}}$. The other system is formed by the states $|i\rangle$ $i=1,2,3$ in a ladder configuration driven by the rescaled Rabi frequencies $\Omega_{1}^{\prime}=\Omega_{1}\left(1-\frac{\Omega^{2}}{4\omega^{2}}\right)$ and $\Omega_{2}^{\prime}=\Omega_{2}\left(1-\frac{\Omega^{2}}{4\omega^{2}}-\frac{\bar{\Omega}^{2}}{4\omega^{2}}\right)$ and detunings $\Delta_{2}^{\prime}$, $\Delta_{3}^{\prime}$.[]{data-label="FIG2"}](Fig3.eps){width="1.0\columnwidth"}
In Fig. \[FIG4\] we show the resonance oscillations of the probability $P_{3}(t)$. We observe that the initial prepared population in state $|2\rangle$ exhibits Rabi oscillations where the exact solution is very closed to the effective picture. Although the quadrupole coupling between the states $|2\rangle$ and $|4\rangle$ is very strong and comparable with the radio trap frequency the corresponding probability is slightly affected.
Two Rydberg ions interacting with dipole-dipole interaction {#sec4}
===========================================================
![Time evolution of the probability $P_{3}(t)$. The Rydberg states $|2\rangle$ and $|3\rangle$ are coupled by quadrupole interaction with the states $|4\rangle$ and $|5\rangle$ with coupling strengths $\Omega/2\pi=12$ MHz and $\bar{\Omega}/2\pi=4$ MHz. The trap radio frequency is set to $\omega/2\pi=20$ MHz. The other parameters are $\Omega_{2}/2\pi=2$ MHz, $\Omega_{1}=0$, $\Delta_{i}=0$ ($i=2,3,4$). The solid line is the exact result and the dashed blue circles is the solution using the effective Hamiltonian (\[Heff2\]). The red dashed line is the solution using rotating-wave approximation.[]{data-label="FIG4"}](fig4.eps){width="1.0\columnwidth"}
In this section, we extend the discussion including the dipole-dipole interaction. We consider an ion chain consisting of two Rydberg ions. The generalization for chain with $N$ ions is straightforward. The full Hamiltonian is quite complicated, see for example Ref. [@Muller2008]. However, under certain rather plausible approximations the Hamiltonian can be reduced to [@Muller2008] $$\hat{H}=\hat{H}_{0}+\hat{V}(t)+\hat{H}_{\text{dd}}.\label{twoions}$$ Here $\hat{H}_{0}$ is given by $$\begin{aligned}
\hat{H}_{0}&=\sum_{j=1}^{2}\{\Delta_{2}|2_{j}\rangle\langle 2_{j}|+\Delta_{3}|3_{j}\rangle\langle 3_{j}|+\Delta_{4}|4_{j}\rangle\langle 4_{j}|\}\notag\\
&+\left(\Omega_{1}|1_{j}\rangle\langle 2_{j}|+\Omega_{2}|2_{j}\rangle\langle 3_{j}|+ \text{H.c.}\right)\}.\end{aligned}$$ This is the single-ion Hamiltonian without the quadrupole interaction, see Fig. \[FIG1\] and Eq. .
The quadrupole interaction is again of the type $\hat{V}(t)=\hat{v}e^{\i\omega t}+\text{H.c.}$, where $$\hat{v}=\frac{\Omega}{2}\left(|2_{1}\rangle\langle 4_{1}|+|2_{2}\rangle\langle 4_{2}|+\text{H.c.}\right).$$ Lastly, the term $\hat{H}_{\text{dd}}$ is the dipole-dipole interaction. It is given by [@Muller2008] $$\hat{H}_{\text{dd}}=\frac{\hat{d}_{1}^{(x)}\hat{d}_{2}^{(x)}+\hat{d}_{1}^{(y)}\hat{d}_{2}^{(y)}-2\hat{d}_{1}^{(z)}\hat{d}_{2}^{(z)}}{8\pi\epsilon_{0}|z_{0}^{(1)}-z_{0}^{(2)}|^{3}}.$$ Here $\hat{d}_{j}^{(\alpha)},\;\alpha=x,y,z$ is the $\alpha$ component of the operator of the dipole moment for the $j$th ion, $\epsilon_{0}$ is the permittivity of the vacuum and $z_{0}^{(j)}$ is the equilibrium position of the $j$th ion along the $z$ axis. We can project this dipole-dipole interaction upon the basis states. Next we perform an optical RWA which is fulfilled as long as the Bohr transition frequencies of the Rydberg levels are much higher than the all Rabi frequencies, such that we obtain $$\hat{H}_{\text{dd}}=\lambda\left(|2_{1}3_{2}\rangle\langle 3_{1}2_{2}| +|3_{1}2_{2}\rangle\langle 2_{1}3_{2}|\right) ,\label{Hdd}$$ where $$\lambda = \frac{|\langle 2|\hat{d}_{x}|3\rangle|^{2}+|\langle 2|\hat{d}_{y}|3\rangle|^{2}-2|\langle 2|\hat{d}_{z}|3\rangle|^{2}}{8\pi\epsilon_{0}|z_{0}^{(1)}-z_{0}^{(2)}|^{3}}.$$ Here $\lambda$ is the strength of the Rydberg dipole-dipole interaction. Only the matrix elements of $\hat{d}_{\alpha},\;\alpha=x,y,z$ between *Rydberg* states (state $|2\rangle$ and state $|3\rangle$) have been used, since the other matrix elements are negligible. The reason is that the overlap between the wave-function of the ground state $|1\rangle$ and a Rydberg wave-function is negligible. The dipole-dipole coupling resembles the XX Heisenberg spin-spin interaction. Indeed, setting $\Omega_{1}=0$ one can introduce the spin rising $\sigma^{+}_{j}=|3_{j}\rangle\langle2_{j}|$ and lowering $\sigma^{-}_{j}=|2_{j}\rangle\langle3_{j}|$ operators such that the dipole-dipole interaction can be rewritten as $\hat{H}_{\rm dd}=\lambda(\sigma^{x}_{1}\sigma^{x}_{2}+\sigma^{y}_{1}\sigma^{y}_{2})$, where $\sigma_{j}^{\alpha}$ are the Pauli matrices.
![(a) Coherent exchange of spin excitation versus the interaction time. We compare the exact solution for the probabilities to observe states $|2_{1}3_{2}\rangle$ and $|3_{1}2_{2}\rangle$ (solid lines) with the effective Hamiltonian for $P_{23}(t)$ (blue triangles) and $P_{32}(t)$ (red circles). The parameters are set to $\Omega/2\pi=8.0$ MHz, $\omega/2\pi=30$ MHz, $\Delta_{2}/2\pi=1.0$ MHz, $\lambda/2\pi=7.0$ MHz, and $\Omega_{2}/2\pi=2.0$ MHz. (b) The same but initially the system is prepared in the state $|3_{1}4_{2}\rangle$. The solid line is the exact solution and the dashed blue squares is the solution with the effective Hamiltonian. The dashed line shows the probability $P_{34}(t)$ assuming rotating wave approximation.[]{data-label="FIG5"}](ss.eps){width="1.0\columnwidth"}
Combining $\hat{\tilde{H}}_{0}=\hat{H}_{0}+\hat{H}_{\text{dd}}$, the total Hamiltonian becomes again of the type $\hat{H}=\hat{\tilde{H}}_{0}+\hat{V}(t)$. Therefore the expression (\[HEFFF\]) for the effective Hamiltonian as well as Eqs. (\[K(t)\]) and (\[K(t)Corr\]) for $\hat{K}$ remain valid. Using Eq. the effective Hamiltonian becomes $$\begin{aligned}
\hat{H}_{\text{eff}}&=\sum_{j=1}^{2}\{\Delta_{2}^{\prime}|2_{j}\rangle\langle 2_{j}|+\Delta_{3}^{\prime}|3_{j}\rangle\langle 3_{j}|+\Delta_{4}^{\prime}|4_{j}\rangle\langle 4_{j}|\}\notag\\
&+\left(1-\frac{\Omega^{2}}{4\omega^{2}}\right)(\Omega_{1}|1_{j}\rangle\langle 2_{j}|+\Omega_{2}|2_{j}\rangle\langle 3_{j}|+\text{H.c.})\}\notag\\
&+\lambda\left(1-\frac{\Omega^{2}}{2\omega^{2}}\right)\left(|2_{1}3_{2}\rangle\langle 3_{1}2_{2}|+|3_{1}2_{2}\rangle\langle 2_{1}3_{2}|\right)\notag\\
&+\frac{\lambda\Omega^{2}}{2\omega^{2}}\left(|3_{1}4_{2}\rangle\langle 4_{1}3_{2}|+|4_{1}3_{2}\rangle\langle 3_{1}4_{2}|\right).\label{Heff3}\end{aligned}$$ In Fig. \[FIG5\], we compare the exact solution with Hamiltonian (\[twoions\]) with the solution using the effective Hamiltonian (\[Heff3\]). Due to the strong dipole-dipole interaction the system exhibits coherent exchange of spin excitations described by the XX Heisenberg spin model. The quadropule interaction leads to rescaling of the dipole-dipole coupling by the factor $\left(1-\Omega^{2}/(2\omega^{2})\right)$, i.e., $\lambda \rightarrow\lambda\left(1-\frac{\Omega^{2}}{2\omega^{2}}\right)$. The single ion Rabi frequencies are again renormalized with the same factor $\left(1-\Omega^{2}/(4\omega^{2})\right)$, i.e., $\Omega_{i}\rightarrow\Omega_{i}\left(1-\frac{\Omega^{2}}{4\omega^{2}}\right),\; i=1,2$. Additionally, the quadropule interaction induces residual dipole-dipole coupling between the states $|3_{i}4_{j}\rangle$ and $|4_{i}3_{j}\rangle$ described by the last term in (\[Heff3\]). This coupling spoils the XX-type Heisenberg interaction between the Rydberg levels $|2_{i}3_{j}\rangle$ and $|3_{i}2_{j}\rangle$. In general, the residual dipole-dipole interaction can not be ignored except in the limit $\omega\gg\Omega$ where the RWA can be applied. Finally, we consider the dipole-dipole interaction between microwave dressed Rydberg ions. Such a dressing creates additional term in the dipole-dipole interaction (\[Hdd\]) which couples the states $|2_{i}2_{j}\rangle\langle 2_{i}2_{j}|$ with coupling strengths $\mu$ (see the Supplement for more details). Because of that we find that the residual terms due to the quadrupole interaction are of order of $\mu(\Omega^{2}/2\omega^{2})$.
Note that the whole technique is valid so long as $\omega\gtrsim\Omega_{i},\; i=1,2$ *as well as* $\omega\gtrsim\lambda$. The last condition $\lambda\lesssim\omega$ puts a lower limit on the frequency $\omega$. However the experimenter can increase $\omega$ above this limit. In addition, numerical simulations show that even for $\lambda=\omega/2$, the effective Hamiltonian remains quite correct. Therefore a long ranged dipole-dipole interaction of strength of $\sim 10\div 20\;\text{MHz}$ is still viable. In addition, by increasing the radio frequency $\omega$ more powerful interaction $\lambda$ can be used and the effective Hamiltonian is still applicable. For instance for $\omega = 2\pi\times 40\;\text{MHz}$ dipole-dipole interaction of the order of $20\;\text{MHz}$ can be achieved.
Conclusion {#sec5}
==========
In this paper we have shown that the quadrupole interaction which causes a reduction in the dipole moment in a trapped Rydberg ion can be dealt with by increasing the Rabi frequencies. To show that we have applied an unitary transformation. In this new picture, dubbed ’effective picture’, see Fig. \[FIG1\], the Rabi frequencies are renormalized with the same factor $\left[1-\Omega^{2}/(4\omega^{2})\right]$. Therefore by increasing the laser intensities with the factor $\left[1-\Omega^{2}/(4\omega^{2})\right]^{-1}$, the negative effect of the quadrupole interaction $\Omega\cos{\omega t}$ can be removed. In addition, we have extended the discussion, see Fig. \[FIG2\], when *both* Rydberg manifolds are coupled by the quadrupole part of the trapping electric field. On the right side of Fig. \[FIG2\] one observes that the effective five level system is *decoupled* and the Rabi frequencies are altered by *different* factors. Therefore even in that case, the negative effect of the quadrupole interaction can be removed. One merely has to rescale the laser intensities by different magnitudes. We have extended the discussion to an ion chain of two ions and we have shown that the Rabi frequencies are renormalized as well as the dipole-dipole coupling is modified. The latter is out of experimental control. However the reduction of the dipole-dipole coupling is only a few per cent for reasonable experimental parameters [@Higgins2017], while the renormalization can be dealt with by increasing the Rabi frequency as was shown in the single ion case.
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Acknowledgments {#acknowledgments .unnumbered}
===============
This work has been supported by the ERyQSenS, Bulgarian Science Fund Grant No. DO02/3.
Author Contributions {#author-contributions .unnumbered}
====================
L. S. S. developed the concept. The main calculations and numerical simulations are performed by L. S. S. and P. A. I. N. V. V. contributed to the analysis of the results. All authors wrote the manuscript.
Additional Information {#additional-information .unnumbered}
======================
**Competing financial interests**: The authors declare no competing financial interests.
|
---
abstract: 'Using the new equipment of the Shanghai Tian Ma Radio Telescope, we have searched for carbon-chain molecules (CCMs) [**towards**]{} five outflow sources and six Lupus I starless dust cores, including one region known to be characterized by warm carbon-chain chemistry (WCCC), Lupus I-1 (IRAS 15398-3359), and one TMC-1 like cloud, Lupus I-6 (Lupus-1A). Lines of [HC$_3$N]{} $J=2-1$, [HC$_5$N]{} $J=6-5$, [HC$_7$N]{} $J=14-13$, $15-14$, $16-15$ and [C$_3$S]{} $J=3-2$ were detected in all the targets except in the outflow source L1660 and the starless dust core Lupus I-3/4. The column densities of nitrogen-bearing species range from 10$^{12}$ to 10$^{14}$ cm$^{-2}$ and those of C$_3$S are about 10$^{12}$ cm$^{-2}$. Two outflow sources, I20582+7724 and L1221, could be identified as new carbon-chain–producing regions. Four of the Lupus I dust cores are newly identified as early quiescent and dark carbon-chain–producing regions similar to Lup I-6, which together with the WCCC source, Lup I-1, indicate that [ carbon-chain-producing regions are popular in Lupus I which [ can be regard as]{} a Taurus like molecular cloud complex]{} in our Galaxy. The column densities of [C$_3$S]{} are larger than those of [HC$_7$N]{} in the three outflow sources I20582, L1221 and L1251A. Shocked carbon-chain chemistry (SCCC) is proposed to explain the abnormal high abundances of [C$_3$S]{} compared with those of nitrogen-bearing CCMs. Gas-grain chemical models support the idea that shocks can fuel the [ environment]{} of those sources with enough $S^+$ thus driving the generation of S-bearing CCMs.'
author:
- |
Yuefang Wu[$^{1,2}$]{}[^1], Xunchuan Liu[$^{1,2}$]{}, Xi Chen[$^{3,4}$]{}, Lianghao Lin[$^{1,5,6}$]{}, Jinghua Yuan[$^{7}$]{}, Chao Zhang[$^{1,2,8}$]{}, Tie Liu,[$^{9,10}$]{} Zhiqiang Shen[$^{3}$]{}, Juan Li[$^{3}$]{}, Junzhi Wang[$^{3}$]{}, Sheng-Li Qin[$^{8}$]{}, Kee-Tae Kim[$^{9}$]{}, Hongli Liu[$^{11}$]{}, Lei Zhu[$^{7}$]{}, Diego Madones[$^{12,13}$]{}, Natalia Inostroza[$^{14}$]{}, C. Henkel[$^{15,16,17}$]{}, Tianwei Zhang[$^{1,2}$]{}, Di Li[$^{7,18,19}$]{}, Jarken Esimbek[$^{17,20}$]{} and Qinghui Liu[$^{3}$]{}\
$^1$Department of Astronomy, School of Physics, Peking University, 100871 Beijing, China\
$^2$Kavili Institute for Astronomy and Astrophysics, Peking University, 100871 Beijing, China\
$^3$Shanghai Astronomical Observatory, Chinese Academy of Sciences, Shanghai 200030, China\
$^4$Center for Astrophysics, GuangZhou University, Guangzhou 510006, China\
$^5$School of Astronomy and Space Sciences, University of Science and Technology of China, 96 Jinzhai Road, Hefei, 230026, China\
$^6$Purple Mountain Observatory and Key Laboratory of Radio Astronomy, Chinese Academy of Sciences, 8 Yuanhua Road, Nanjing, 210034, China\
$^7$National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100101, China\
$^8$Department of Astronomy, Yunnan University, Kunming, 650091, China\
$^9$Korea Astronomy and Space Science Institute, 776 Daedeokdae-ro, Yuseong-gu, Daejeon 34055, Korea\
$^{10}$East Asian Observatory, 660 North A$\arcmin$ ohoku Place, Hilo, HI 96720, USA\
$^{11}$Department of Physics, The Chinese University of Hong Kong, Shatin, NT, Hong Kong SAR\
$^{12}$Department of Astronomy, University of Chile, Casilla 36-D, Santiago, Chile\
$^{13}$ Centre for Astrochemical Studies, Max-Planck-Institute for Extraterrestrial Physics, Giessenbachstrasse 1, 85748, Garching, Germany\
$^{14}$ Núcleo de Astroquímicay Astrofísica, Instituto de Ciencias Químicas Aplicadas, Facultad de Ingeniería, Universidad Autónoma de Chile Av.\
Pedro de Valdivia 425, Providencia, Santiago de Chile.\
$^{15}$Max-Planck Institut für Radioastronomie, Auf Dem Hügel 69, 53121 Bonn, Germany\
$^{16}$Astronomy Department, Faculty of Science, King Abdulaziz University, PO Box 80203, Jeddah, 21589, Saudi Arabia\
$^{17}$Xinjiang Astronomical Observatory, Chinese Academy of Sciences, 830011, Urumqi, China\
$^{18}$Key Laboratory of Radio Astronomy, Chinese Academy of Science, Nanjing, China\
$^{19}$University of Chinese Academy of Sciences, Beijing 100049, China\
$^{20}$Key Laboratory of Radio Astronomy, Chinese Academy of Sciences, Urumqi 830011, China
bibliography:
- 'ms.bib'
title: 'Carbon-Chain Molecules in Molecular Outflows and Lupus I Region –New Producing Region and New Forming Mechanism'
---
\[firstpage\]
ISM: molecules – ISM: abundance – stars: formation – ISM: Jets and outflows – ISM: kinematics and dynamics
Introduction {#sec_intro}
============
Carbon-chain molecules are important components of the interstellar gas. They are major players in hydrocarbon chemistry and sensitive indicators of evolutionary states of molecular regions due to their wide range [ of]{} masses and large permanent dipole moments [@2001ApJ...558..693D; @2011JChPh.135x4310I; @2012MNRAS.419..238B]. They can also trace dynamic processes in molecular regions including material infall [@2013MNRAS.436.1513F]. After being first detected in [ the]{} massive star-forming region Sgr B2 [@1971ApJ...163L..35T; @1976ApJ...205L.173A], CCMs were found in interstellar [ clouds]{} with different evolutionary states. A number of small carbon-chain molecules such as C$_2$H, C$_3$H$_2$ and C$_3$H were detected in diffuse clouds . These molecules were also found in photodissociation regions . Their abundances were explained by a carbon-chain producing mechanism in which the photo-erosion of UV-irradiated large carbonaceous compounds could feed the ISM with small carbon clusters or molecules . CCMs have also been also detected in dense molecular cores.
In the early phase of stellar evolution, a number of cold and dark cores were found as abundant producing regions of CCMs. Among them, TMC-1 is one of the best studied regions where sulfur-, oxygen- and deuterium-bearing CCM species were found . The most recent one is HC$_5$O detected by @2017ApJ...843L..28M. It is a good test source of chemical models of cold dark cores [@1984MNRAS.207..405M; @2014MNRAS.437..930L]. In [[email protected]], @1992ApJ...392..551S studied a sample consisting of 27 starless dark cores and 22 star-forming cores. @2010ApJ...718L..49S identified Lupus 1A (Lup I-6) as the “TMC-1 like” cloud in the Lupus region. Such cores are in an early stage of chemical evolution [@1992ApJ...392..551S; @2006ApJ...646..258H].
In star-forming cores, CCMs become deficient [@2008ApJ...672..371S]. Abundances of S-bearing species such as C$_2$S are much lower than those in cold and dark cores [@1992ApJ...392..551S]. In fact, depletions of S-bearing species start much earlier in the densest regions. In the starless core L1544, observations with BIMA [ have]{} revealed that C$_2$S emission [@1999ApJ...518L..41O] shows a shell like structure, while at the center N$_2$H$^+$ emission is concentrated [@1999ApJ...513L..61W]. [ Recently @2018MNRAS.478.5514V detected 21 S-bearing species towards L1544 and found that a strong depletion was needed to explain the results]{}. @1992ApJ...392..551S found that C$_\mathrm{n}$S and C$_\mathrm{n}$H (n=1-3) reached their [ maximum]{} at A$_V$ $\leq$1.2 in the gas phase. Among the sources they selected, protostellar cores such as L1489, L1551 and L1641N were not detected or only marginally detected in C$_2$S and C$_3$S. [ However among 16 deeply embedded low mass protostars, C$_2$S and [C$_3$S]{} were detected in 88$\%$ and 38$\%$ of the targets respectively [@2018ApJ...863...88L]]{}.
High excitation lines of carbon-chain molecules were detected toward low-mass star-forming cores. Lines such as [C$_4$H]{} (N=9-8), [CH$_3$CCH]{} (J=5-4, K=2) and [C$_4$H$_2$]{} $J=10_{0,10}-9_{0,9}$ were detected in L1527 and IRAS 15398-3359 [@2008ApJ...672..371S; @2009ApJ...697..769S]. The CCM chemistry in these regions is different from that in early cold and dark cores, thus a new mechanism called warm carbon-chain chemistry (WCCC) was suggested by @2008ApJ...672..371S. High excitation linear hydrocarbons in these regions are formed by reactions of CH$_4$, [ i.e. from]{} molecules that have been sublimated from warm dust grains [@2008ApJ...672..371S; @2008ApJ...681.1385H]. Another possible reason for the high abundance of CCMs in the WCCC sources is the shorter time scale of prestellar collapse in these cores compared with those in other protostellar cores, which results in the survival of CCMs [@2008ApJ...672..371S; @2009ApJ...697..769S]. Besides high-excitation species, N-bearing CCMs are also abundant in WCCC sources. Lines of [HC$_3$N]{}, [HC$_5$N]{}, [HC$_7$N]{} and even HC$_9$N were detected in L1527. [HC$_5$N]{}- J= 32-31 with very high excitation energy (67.5 K) was detected in [ a]{} second WCCC source IRAS 15398-3359 [@2009ApJ...697..769S]. However, the S-bearing species in WCCC sources are less abundant than in common star-forming cores [@2008ApJ...672..371S; @1992ApJ...392..551S].
Despite this notable progress, CCMs remain not yet fully understood. How do the emissions of CCMs evolve as they undergo longer periods of heating by protostellar sources when compared to WCCC sources? Are the emissions of CCMs affected by dynamical feedback from the protostars such as molecular outflows and jets? Are the N-bearing species still abundant there? Will the S-bearing species be quenched? For the cores at early stellar phase, so far the carbon-chain producing regions detected are mainly located in the Taurus molecular complex [@2009ApJ...699..585H; @1992ApJ...392..551S; @1981ApJ...244...45S]. Are there other CCM rich molecular complexes similar to the Taurus complex in our Galaxy?
In this paper, we present observations of [HC$_3$N]{}, [HC$_5$N]{} and [HC$_7$N]{} as well as [C$_3$S]{} in the 15.2 to 18.2 GHz frequency range toward five molecular outflow sources and six Lupus I starless dust cores to investigate their emissions and production mechanisms. The Lup I cores were numbered with the numbers of the dust cores of . A known WCCC core, IRAS 15398-3359 (Lup I-1), and the TMC-1 like core, Lupus 1A (Lup1-6), are included to explore their CCM emissions and to compare the derived properties with those obtained from other samples. The observed sources are listed in . Parameters related to the observed lines are given in , which are taken from the “Splatalogue” molecular database [^2]. Our observation is introduced in . In [ Sects.]{} 3 and 4 we present results and discussions. provides a summary.
Observation {#sec_obs}
===========
The observations were carried out with the Tian Ma Radio Telescope (TMRT) of the Shanghai Observatory. The TMRT is a newly built 65-m diameter fully steerable radio telescope located in the western outskirts of Shanghai [@2016ApJ...824..136L]. The pointing accuracy is better than 10, and the main beam efficiency is 0.60 [ in]{} the 12-18 GHz band [@2015AcASn..56...63W; @2016ApJ...824..136L]. The front end is a cryogenically cooled receiver covering the frequency range of 11.5$-$18.5 GHz. An FPGA-based spectrometer based upon the design of the Versatile GBT Astronomical Spectrometer (VEGAS) was employed as the Digital backend system (DIBAS) [@2012AAS...21944610B]. For molecular line observations, DIBAS supports a variety of observing modes, including 19 single-subband modes and 10 eight-subband modes. The center frequency of each subband is tunable to an accuracy of 10 kHz. For our observations mode 22 was adopted. Each of the eight subbands in two banks (Bank A and Bank B) has a bandwidth 23.4 MHz and 16384 channels. The velocity resolution is [ by]{} a little larger than the channel spacing which is 0.028 km s$^{-1}$ in the 15 GHz band and 0.023 km s$^{-1}$ in the 18 GHz band respectively. [ The calibration uncertainty is 3 percent .]{}
As mentioned above, among the observed sources there are five outflow sources and six Lupus I starless dust cores. The numbers of the Lupus I cores labeled by have been adopted. Lupus is abbreviated as Lup. Source names, alternative names, positions, distances, references and observational notes are listed in columns 1 to 7 of .
The measured carbon-chain molecular transitions are listed in . Columns 2-6 show the transitions covered by the 16 subbands, and their frequencies, the upper energies as well as the Einstein transition coefficients for spontaneous emission.
The half power beam widths (HPBWs) of the TMRT beam at our observed frequencies range from 52 to 60 arcsec, and are listed in the Column 6 of . shows regions covered within a single beam towards each source. For outflow sources, the beam can cover the blue and red lobes of outflows either entirely or at least their main parts, including the driving objects, H$_2$ outflows and optical jets (also see Sect. \[sec\_JS\_Phy\]). The Herschel 250 $\mu$m images of the outflow sources have sizes less than or similar to our beam size, and the beam can cover them too (). Our beam can also cover the Herschel 250 $\mu$m images of the Lupus cores entirely ( and ).
The rms noise (Part II of ) of IRAS 20582+7724 and L1221 (8 mK $-$ 21 mK) is much lower than the corresponding values of the remaining sources (30 mK $-$ 100 mK) because supplementary observations were made towards these two sources on Nov. 9, 2017 (). For observations in the 16-18 GHz range towards cores Lup I-1 and Lup I-7/8/9, the rms noise is higher (100 mK $-$ 200 mK) because they were measured on 2016 March 21 and March 24 under relatively poor weather conditions.
The package GILDAS including CLASS and GREG [@2000ASPC..217..299G] was used to reduce the data and draw the spectra.
Results {#sec_results}
=======
Detected lines
---------------
Lines in the observed band were detected in all our targets except towards the outflow source L1660 and the dust core Lup I-3/4. The spectral lines of [HC$_3$N]{} $J=2-1$ for the detected sources are shown in . Five hyperfine components of [HC$_3$N]{} $J=2-1$ were well resolved for all the detected sources. Panels (a)-(i) of present the remaining spectra of the observed transitions for each detected source.
In IRAS 20582+7702 (hereafter I20582) and L1221, lines of [HC$_3$N]{} and [HC$_5$N]{} were detected. It is the first time to detect CCMs in these two sources. Lines of [HC$_7$N]{} were weak or not detected. Emission of [C$_3$S]{} $J=3-2$ was detected and turned out to be stronger than the emissions of the three rotation transitions of [HC$_7$N]{} for these two sources.
In L1251A, CCMs were [ also reported]{} at higher frequency transitions by @2011ApJ...730L..18C but were detected [ in]{} the observed band for the first time. From (c) one can see that the $J=6-5$ line of [HC$_5$N]{} was detected and the hyperfine component $F=5-4$ was resolved. The $J=14-13,~15-14$ and $~16-15$ transitions of [HC$_7$N]{} were detected. Emission of [C$_3$S]{} $J=3-2$ is stronger than emissions of the three transitions of [HC$_7$N]{} too.
The observed lines of Lup I-1 (IRAS 15398-3359) are the strongest among all the observed sources including outflows and dust cores of Lupus I. The only exception is the emission of [C$_3$S]{}, which is slightly weaker than that of L1251A. Emissions of [HC$_7$N]{} $J=14-13,~15-14,~16-15$ from this core are all stronger than that of [C$_3$S]{} $J=3-2$.
In the Lupus I region, besides outflow source Lup I-1, six starless cores were mapped in the dust continuum at 850 [$\mu$m]{} . Lup I-6 was found as a “TMC-1 like cloud” and named as Lupus-1A by @2010ApJ...718L..49S. All the observed transitions are detected in the six cores except Lup I-3/4, which is therefore not displayed in [ Figures 1 and 2]{}. Lup I-7/8/9 and Lup I-11 show strongest emission of [HC$_3$N]{} J=2-1 while Lup I-2 shows weakest. The [HC$_5$N]{} hyperfine components $F=7-6,~6-5$, and $5-4$ are well resolved in Lup I-6, Lup I-7/8/9 and Lup I-11. The emission of [C$_3$S]{} is strongest in Lup I-11 and weakest in Lup I-6. However, emission of [C$_3$S]{} $J=3-2$ is weaker than that of the three rotation transitions of [HC$_7$N]{} for all the five detected Lupus I cores. Emissions of our searched lines were not detected in two sources, L1660 and Lup I-3/4. L1660 is an outflow source and possesses an H$_2$ jet . There are no Herschel data within 70 arcmin of L1660. The reason for the non-detection of CCMs needs to be further examined. Cores Lup I-3/4 are associated with a bright emission nebula (B77) and a reflection nebula (GN 15.42.0) within about 35 . Its rather advanced evolutionary state and hot environment seem to be adverse to the production of CCMs.
Line parameters
---------------
All the resolved hyperfine structure (HFS) of detected spectral lines was fitted with independent Gaussian [ functions]{}. The observed parameters including [ Local Standard of Rest center velocity V$_\mathrm{lsr}$, peak temperature $T_\mathrm{MB}$, and full width to half maximum (FWHM) line width]{} as well as the integrated area are given in Part I-IV of respectively.
From as well as [ Figures 1 and 2]{}, one can see that the V$_\mathrm{LSR}$ of transitions of different molecules agrees with each other quite well. The line widths of [ the ]{}outflow sources I20582 and L1221 are the widest, while the line widths of [ the]{} cores Lup I-7/8/9 and Lup I-11 are narrower, which are still with high ratios to the velocity resolution. The main beam brightness temperatures T$_\mathrm{MB}$ of the [HC$_3$N]{} $J=2-1,~F=3-2$ lines are the highest among all the detected transitions for each source. [ The emissions of this transition are quite strong in all the Lupus I starless cores. However, the outflow Lup I-1 has the strongest emission, while the emissions of the other three outflows are all much weaker than those of Lup I-1 and the starless cores.]{}
The T$_\mathrm{MB}$ of this transition of the five Lupus I starless dust cores ranges from 1.71 K (Lup I-2) to 4.10 K (Lup I-7/8/9). The outflow Lup I-1, a WCCC source, has the highest T$_\mathrm{MB}$ of this transition (5.11 K) among the detected sources. However, for the three outflow sources I20582, L1221 and L1251A, the T$_\mathrm{MB}$ values of [HC$_3$N]{} $J=2-1,~F=3-2$ are much lower than those of Lup I-1 and the five Lupus I starless dust cores.
For the outflow source Lup I-1 and the 5 Lupus I starless dust cores, three hyperfine components of [HC$_5$N]{} $J=6-5$ were well or partially resolved. While for the three outflow sources I20582, L1221 and L1251A, the emissions of [HC$_5$N]{} are weaker than those of the Lup I-1 and the 5 Lupus I starless dust cores. The T$_\mathrm{MB}$ of the three rotational transitions of [HC$_7$N]{} ranges from 0.18 to 0.49 K for the Lupus cores. For the three outflow sources the emissions of these lines are not or only marginally detected.
[C$_3$S]{} emission was detected in all our sources except L1660 and Lup I-3/4. The emissions of [C$_3$S]{} are weaker than those of all the N-bearing molecules in Lup I-1 and the five Lupus I starless dust cores. However, for the three outflow sources, the emission peaks of [C$_3$S]{} are higher than those of [HC$_7$N]{} $14-13$, $15-14$ and $16-15$.
Column densities
----------------
Column densities of the observed molecular species were calculated assuming local thermodynamic equilibrium (LTE) with the solution of the radiation transfer equation [@1991ApJ...374..540G; @2015PASP..127..266M].
$$\begin{aligned}
N= \frac{3k}{8\pi^3v}\frac{Q}{S_{ij}\mu^2}
\frac{J(T_{ex})exp(\frac{E_{up}}{kT_{ex}})}{J(T_{ex})-J(T_{bg})} \frac{\tau}{1-e^{-\tau}}
\int T_r d\upsilon
\end{aligned}$$
$$\begin{aligned}
J(T)=(exp^{\frac{hv}{kT}}-1)^{-1}
\end{aligned}$$
where $B$, $\mu$, and $Q$ are the rotational constant, the permanent dipole moment and the partition function adopted from “Splatalogue” .
[ Excitation temperatures can be derived from the HFS fittings of [HC$_3$N]{} $J=2-1$ (denoted as T$_{ex}$(HC$_3$N)) with the fitting program in GILDAS/CLASS[^3]. Five hyperfine lines of HC3N J=2-1 are detected towards all target sources. Optical depths ($\tau$(HC$_3$N)) and excitation temperature (T$_{ex}$(HC$_3$N)) are listed in column 2 and 3 of respectively. However, this method can not be applied to sources I20582 and L1251A, where the HFS of [HC$_3$N]{} indicates optically thin emission and the uncertainty introduced by the beam filling factor can not be ignored.]{} The observed HFS lines of [HC$_5$N]{} J=6-5, F=5-4, 6-5 and 7-6 are optically thin. [HC$_5$N]{} J=6-5, F=5-4, and 7-6 were not detected toward I20582 and L1221 while J=6-5, F=6-5 was not detected in L1251A, Lup I-2, and Lup I-5. The three rotation lines of [HC$_7$N]{} J=14-13, 15-14, and 16-15 have similar E$_{up}$ [ (Table 2)]{}. Furthermore the [HC$_7$N]{} lines of J=14-13 in L1221, and J=15-14 in I20582 as well as J=16-15 in L1221 do not have enough S/N. Therefore one can not derived excitation temperature from the observed molecular lines of [HC$_5$N]{} and [HC$_7$N]{} .
Dust temperature ($T_\mathrm{d}$) as well as column densities of hydrogen molecules are derived from SED fitting of Herschel data at 70, 160, 250, 350, and 500 $\mu$m (See Appendix \[sec\_appendix\]) and are listed in . For comparisons, dust parameters of TMC-1 are also included. T$_\mathrm{d}$ of Lup I-1 is 13.9 K (see ), which is close to the molecular gas temperature of this source and of L1527 derived by @2009ApJ...697..769S [@2008ApJ...672..371S]. The dust temperatures T$_\mathrm{d}$ are also listed in column 4 of for the purpose of comparison. In Lup I-7/8/9, T$_\mathrm{d}$ is lower than T$_{ex}$(HC$_3$N) by 4.3 K, i.e. 10.2 K versus 14.5 K () which is the largest difference between these two temperature values of all the sources. This may be due to the fact that our Lup I -7/8/9 spectra are a combination of Lup I-7, Lup I-8 and Lup I-9 which are all inside our TMRT beam (see Table 3 of , and Figure A1). According to the dust temperatures of these three cores derived from Herschel data are all 13 K individually, which is close to our T$_{ex}$(HC$_3$N). The TMC-1 like cloud Lup I-6 is characterized by T$_{ex}$(HC$_3$N) $\sim$7.0 K and T$_\mathrm{d}$ $\sim$10 K, which are rather close to each other compared with an excitation temperature of 7.3$\pm$1.0 K given by @2010ApJ...718L..49S on the basis of C$_6$H and the dust temperature of 13 K given by . The difference of these two values results in an error of derived column density related to the estimation of T$_{ex}$, but the error is not larger than 15 percent.
Thus, we assume in the following that T$_\mathrm{ex}$ equals the dust temperature (T$_\mathrm{d}$) under local thermodynamic equilibrium (LTE) conditions [@2012ApJ...756...60S]. [ The column densities derived from each detected line are given in Part I of ]{} for all the sources. One can see that for every species in each source, the column densities derived from different line components tend to be close to each. [ Part II of lists]{} the unweighed average of the column densities derived from different hyperfine components or rotation lines, which are adopted as the column density of a detected species. From , one can see that among different species, the column density of [HC$_3$N]{} is the highest. The WCCC source Lup I-1 shows the highest value. Lup I-5 and Lup I-11 also have column densities close to Lup I-1. The three outflow sources I20582, L1221 and L1521A have the lowest [HC$_3$N]{} column densities. The [C$_3$S]{} column densities of the WCCC source Lup I-1 and the five Lupus I cores are all lower than column densities of their N-bearing species. However, [C$_3$S]{} column densities of the three outflow sources are higher than their [HC$_7$N]{} column densities.
Abundance
---------
With the column densities of molecules presented in , [ abundances relative to $N$(H$_2$) (see )]{} of detected species were obtained, which are given in . presents the changes of abundances among N-bearing LCCMs HC$_\mathrm{2n+1}$N (n=1-3) and between N-bearing species and [C$_3$S]{}. For N-bearing species HC$_\mathrm{2n+1}$N (n=1-3), the abundance decreases while n increases. WCCC source Lup I-1 and the five Lupus I starless cores show the highest abundances. Abundances of N-bearing species are lower in the other three outflow sources. For the molecule [C$_3$S]{}, the abundance in L1251A is the highest among all the detected sources. The abundance of [C$_3$S]{} in I20582 or L1221 is also quite high compared with that of the TMC-1 like cloud Lup I-6. The three outflow sources all have abundances of [C$_3$S]{} larger than that of [HC$_7$N]{}, especially in the case of L1251A in which the abundance of [C$_3$S]{} is even comparable to that of [HC$_5$N]{}.
presents the abundance ratios of all the species for all the sources. The ratios of x([HC$_3$N]{})/x([HC$_5$N]{}) for the Lup I starless cores and WCCC source Lup I-1 are all rather close to each other, with an average value of $\sim$3.8. However the ratios of the other three outflow sources are 6.8 on the average and higher than those of the starless cores and the Lup I-1 outflow source. The ratios of x([HC$_5$N]{})/x([HC$_7$N]{}) are 3.6 on the average for all the starless cores and the Lup I-1 outflow source. While the other three outflow sources [ show an average ratio of]{} 4.6. [ For the ratio of x([C$_3$S]{})/x([HC$_7$N]{}), the starless cores and the outflow source Lup I-1 [ have a ratio of]{} 0.4 on the average and the other three outflow sources 3.4, showing the largest difference among the compared ratios. These results indicate that the changes of the CCM emissions with the carbon length in Lup I starless cores and the outflow source Lup I-1 are different from those of the other three outflow sources. The three panels of Figure 4 clearly show that the correlations between each ratio of x([HC$_3$N]{})/x([HC$_5$N]{}), x([HC$_5$N]{})/x([HC$_7$N]{}) and x([C$_3$S]{})/x([HC$_7$N]{}) are different for the starless cores/Lup I-1 outflow source and the other three outflow sources. The difference between the x([C$_3$S]{})/x([HC$_7$N]{}) of the Lup I starless cores/Lup I-1 outflow source and the other three outflow sources is the largest.]{}
Discussion {#sec_discussions}
==========
According to their different relative values of abundances of N-bearing CCMs and [C$_3$S]{}, the detected sources can be divided into three groups.
- **Group CC** includes all the five cold and quiescent dark cores in [ the]{} Lup I region. Their abundances of N-bearing species decrease [ with the length of the carbon chains]{}. The ratio x(HC$_{2n+1}$N)/x(HC$_{2n+3}$N) (n=1-2) is 3.7 on the average. Their ratio of [ x([C$_3$S]{} )/x([HC$_7$N]{} ) is 0.4]{} on the average.
- **Group WC** contains Lup I-1 only, which is a known WCCC source. Its ratios of x(HC$_{2n+1}$N)/x(HC$_{2n+3}$N) (n=1-2) are about the same as those of Group CC. However the ratios of [ x([C$_3$S]{})/x([HC$_7$N]{}) is 0.3, larger]{} than those of Group CC, which means that [C$_3$S]{} is more deficient in Lup I-1 than in Group CC sources.
- **Group JS** consists of I20582, L1221 and L1251A, all with jets and shocks. Their abundances of N-bearing species x(HC$_{2n+1}$N) decline faster with the increase of ‘n’ than the other two group sources. Their [ x([C$_3$S]{} /x([HC$_7$N]{} )) ratio]{} is 3.4 on the average, which is in contrast to the same ratios of the sources in the other two groups and it has not been seen previously.
Below we discuss physical conditions and chemical contents for each group.
Lupus I starless dust cores
---------------------------
Lupus I is a nearby molecular cloud complex. The distance measured by is 155$\pm$8 pc. For our Lup I cores, except Lup I-1, all are dark and without associated stellar sources within radii of one arcmin except Lup I-5 which [ contains]{} an object surveyed with Spitzer offset by $\sim$ 0.5 $\arcmin$ from the center . The Herschel 250 images of Lup I cores are shown in the bottom panel of Figure A1. The dust temperatures of these cores range from 10.0 to 11.9 K, with an average value of 11.0 K which is similar to that of the cyanopolyyne peak in TMC-1, 10.6 K ().
The dust cores Lup I-2 and Lup I-5 are located in the gas core C6 where [HC$_3$N]{} J=3-2 and 10-9 were detected [@2012MNRAS.419..238B]. Lup I-6 was identified as a TMC-1 like cloud and is located in the C3 core where [HC$_3$N]{} J=10-9 was detected [@2010ApJ...718L..49S; @2012MNRAS.419..238B]. The gas core C3 also includes the dust core Lup I-7/8/9, which was detected by our TMRT observations too. Lup I-11 is in the gas core C8 where [HC$_3$N]{} J=3-2 and 10-9 were detected [@2012MNRAS.419..238B]. It is also starless with IRAS 15422-3414 [ located 62]{} arcsec away.
From Figs. \[fig\_hctn\] and \[fig\_spectra\] and , one can see that CCM emissions of the four dust cores, Lup I-2, Lup I-5, Lup I-7/8/9 and Lup I-11 are all quite strong. Emissions of three (except Lup I-2) of them are even stronger than those of Lup I-6 (Lupus-IA) [@2010ApJ...718L..49S]. These results demonstrate that these four starless dust cores detected with LABOCA, Herschel and Planck are all CCM emission cores. The detections of fruitful CCM emissions in Lupus I starless cores as well as in Lup I-1, the second WCCC source [@2009ApJ...697..769S], indicate that Lupus I is a CCM rich region similar to Taurus.
There is no apparent ongoing star formation in these Lup I cores yet, except Lup I-1. The chemistry of these cores in Lupus I belong to that of a cold and quiescent phase. In this early phase carbon atoms and ions needed to produce carbon-chain molecules arise from photodissociation and photoionization during the diffuse phase of the cloud. [ In such cloud]{} the C and C$^+$ from [ a more diffuse earlier stage of evolution have not been incorporated into CO yet]{} [@1992ApJ...392..551S].
[ All these starless cores and the outflow source Lup I-1 indicate [ that]{} the Lup I is [ a]{} rich CCM and popular carbon-chain–producing region, which [ may match that of the Taurus molecular complex located at a similar distance from the Sun]{}. This region is located at the edge of the youngest subgroup (Upper-Scorpius) of the Scorpius-Centaurus OB association. It borders the expanding HI shell around the Upper-Scorpius at the north-east side which may represent the source of carbon atoms and ions to form CCMs in the Lup I region .]{}
The abundance ratios of N-bearing species x(HC$_{2n+1}$N)/x(HC$_{2n+3}$N) of Lup I starless cores are 3.8 and 3.5 (n=1,2) (Sect 3.4), which is not inconsistent with the values of 2-3 for n=1,2 in dark clouds .
About the ratio of x([C$_3$S]{})/x([HC$_7$N]{}), the values of the Lup I starless cores range from 0.3 to 0.5 with an average value of 0.4. It is noteworthy that the x([C$_3$S]{})/x([HC$_7$N]{}) ratio of Lup I-5 is the smallest, which is indicated by the blue point below the blue line in the right panel of Figure 4.
The [C$_3$S]{} emission of this core seems to be influenced by a [ Spitzer detected]{} c2d source at the 0.5$\arcmin$ from the core center [ ]{}. [ Regarding the abundance ratio of [ a source previously detected with the TMRT,]{} Serpens South 1a [@2016ApJ...824..136L], the x([HC$_3$N]{})/x([HC$_5$N]{}) and x([HC$_5$N]{})/x([HC$_7$N]{}) ratios are 6.5 and 2.0 respectively, and 4.3 on the average, which is close to those of the Lup I starless cores. The ratio of x([C$_3$S]{})/x([HC$_7$N]{}) is 1.0 which is larger than those of all the Lup I cores. Serpens South 1a is located in the Serpens South Cluster where young stellar objects are embedded. Infall motion was detected with [HC$_7$N]{} and NH$_3$. This may be related to the change of the x([C$_3$S]{})/x([HC$_7$N]{}) ratio.]{}
WCCC source Lup I-1
-------------------
Lup I-1 is a WCCC source [@2009ApJ...697..769S] second only to L1527 [@2008ApJ...672..371S]. WCCC sources contain stellar envelope regions with a slightly elevated temperature of $\sim$30 K. In this warmer environment, CH$_4$ can be evaporated from the dust grain mantles into the surrounding gas, and reacts with C$^+$ to produce carbon-chain molecules [@2008ApJ...681.1385H; @2009ApJ...697..769S; @1991MNRAS.249...69B; @1984MNRAS.207..405M]. Lup I-1 contains a young class 0 stellar object . The bolometric temperature is 52 K, higher than that of L1527 (44 K) . Besides the normal radiation of the protostar, a recent accretion burst likely happened during the last 10$^2$ to 10$^3$ yr. This burst might lead to an increase of the stellar luminosity by a factor of 100, which would [ heat the dust and enhance WCCC emission in this core [@2008ApJ...672..371S]. This was listed as one of the interesting perspectives raised by the recent episodic accretion burst by @2013ApJ...779L..22J]{}.
Lup I-1 shows a young molecular outflow driven by [ a]{} Class 0 object. The outflows traced with [ CO]{} $J=3-2$ and $6-5$ [ have an average]{} dynamical time $1.1\times10^3$ yr. The dynamic time given by the lower-J CO line ($J=2-1$) is 2$\times$10$^3$ yr [@1996PASJ...48..489T]. These time scales are all about one order of magnitude smaller than those of the WCCC source L1527 ($1.1\times10^4$ yr) .
The x([HC$_3$N]{})/x([HC$_5$N]{}) and x([HC$_5$N]{})/x([HC$_7$N]{}) ratios of Lup I-1 are similar to those of the Lup I starless cores. However the x([HC$_3$N]{})/x([HC$_5$N]{}) ratio of the median values [ of 16]{} embedded low mass protostars is 8.7, and that of Orion is 13$\pm$6 , showing that the changes of emissions from N-bearing species [ with given length of the carbon chain differ]{} in different star formation regions. In low-mass star formation regions, the cyanopolyynes are usually quite deficient but still closely related [@2018ApJ...863...88L]. The different x([HC$_3$N]{})/x([HC$_5$N]{}) [ ratios]{} in starless cores and outflow sources may be caused by the different dominant pathways between [HC$_3$N]{} and [HC$_5$N]{} .
The [C$_3$S]{} column density of Lup I-1 is lower than those of [HC$_3$N]{}, [HC$_5$N]{} and [HC$_7$N]{}, and the ratio of x([C$_3$S]{})/x([HC$_7$N]{}) is as small as that of Lup I-5, i. e. the smallest among our targets.
It shows that [ the]{} [C$_3$S]{} abundance in Lup I-1 does not increase as much as in the other three outflow sources (see Sect. \[sec\_JS\_Chem\]) though HH185 was found within the confines of the outflow region [@1996PASJ...48..489T]. The reason may be that the CCMs retained from the early core phase keep the abundances of these species at high levels, which might possibly happen during a fast collapse of the cloud’s core like in Lup I-1 [@2008ApJ...672..371S]. Another reason may be that the outflow is very young and there is not enough time for the formation of S-bearing LCCMs from shock induced S$^+$.
Excitation conditions and chemistry in Group JS {#sec_JS}
-----------------------------------------------
### [ Group JS]{} {#sec_JS_Phy}
Three molecular outflow sources, I20582, L1221 and L1251A are included in the Group JS.
In I20582, molecular outflows were detected in the $J=1-0$ lines of CO and $^{13}$CO . The outflow age is $\sim10^5$ yr [@1997ApJ...491..653T]. HH 199B1 to HH 199B3 are at or near the CO blue lobe detected with the OVRO interferometer [@2004ApJ...612..342A; @1995ApJ...454..345B]. HH 199R1-R5 are located in the north-east and are distributed up to 8 away from I20582. There is a chain of 2.122 [$\mu$m]{} H$_2$ jet knots through the center and aligned in the south-east to north-west direction [@2004ApJ...612..342A; @1995ApJ...454..345B]. The enriched infrared and optical characteristics indicate strong shocks in this source. The observation by @1995ApJ...454..345B revealed strong S$^+$ emission in HH 199B1-B3, B6 and HH 199R2. The peaks of the dense gas detected with CS J=2-1 coincide with the HH 199B1-B3 objects and the CO blue lobe peak, where there is a lack of quiescent gas traced by c-C$_3$H$_2$ [@1997ApJ...491..653T]. L1221 has the highest dust temperature $15.1\pm0.2$ K among all the LCCM detected sources. CO $J=1-0$ outflows with an age $\sim 5 \times 10^4$ yr were detected with the 45-m of Nobeyama Radio Observatory (NRO) [@1991ApJ...377..510U]. It was showing a U-shape distribution resulting from the interaction between the outflow and the surrounding gas [@1991ApJ...377..510U]. This core therefore appears to be cometary and contains the infrared source IRAS 22266+6845. There is a close binary consisting of two infrared sources located in the east and west of the IRAS source and the eastern one seems to drive the outflow [@2005ApJ...632..964L]. The source was detected in K$^\prime$ band and has a south-eastern extension in the same direction as the eastern lobe of the outflow, showing shocked H$_2$ emission. HH 363 seen in H$\alpha$ and S$^+$ is also associated with the outflow [@1997IAUS..182P..51A]. Recently, three IRS sources were revealed with Spitzer IRAC/MIPS data and ground based submillimeter emissions. IRS 1 and IRS 2 are Class I objects and IRS 3 is a Class 0 stellar object. IRS 1 contains an arc and IRS 3 a jet. IRS 3 also exhibits 3-6 cm emission which indicates shock ionization due to the interaction between the jet from IRS 3 and the surrounding gas [@2009ApJ...702..340Y].
Finally, in the outflow source L1251A, a collimated molecular outflow was detected with CO $J=2-1$ line observation by @2010ApJ...709L..74L. The dynamical timescale of this outflow is $5.2\times 10^4$ yr assuming an inclination angle of 70$^\circ$ [@2010ApJ...709L..74L]. Spitzer IRAC observations revealed four infrared sources IRS 1-IRS 4. The driving source is IRS 3, a Class 0 object. From this object a collimated long infrared jet originated. Our observed position is 12 arcsec east and 9 arcsec south to IRS 3 (see ). The detected strong Spitzer IRAC observations revealed a jet with a bipolar structure extending from north to south though IRS 3. The jet may originate from a paraboloidal shock [@2010ApJ...709L..74L]. [ About the N-bearing species in these outflows, the average values of x([HC$_3$N]{})/x([HC$_5$N]{}) and x([HC$_5$N]{})/x([HC$_7$N]{}) are 6.8 and 4.6 respectively, which are both larger than those of the Lup I starless cores and the WCCC source Lup I-1, showing that the abundance of N-bearing species decreases faster with carbon length in these outflows than in starless cores and WCCC sources. The ratio of x([HC$_3$N]{})/x([HC$_5$N]{}) is somewhat lower than the median value (8.7) of 16 low mass protostars detected by @2018ApJ...863...88L. shows that in the starless cores, x([C$_3$S]{}) is two or more times lower than x([HC$_7$N]{}). The x([C$_3$S]{}) in the WCCC source Lup I-1 is also three times less than x([HC$_7$N]{}). While in the other three outflow sources the x([C$_3$S]{}) are 2-5 times of the x([HC$_7$N]{}). These results demonstrate that the x([C$_3$S]{})/x([HC$_7$N]{}) values for the three outflows are larger than those of starless cores and the WCCC source almost one order of magnitude higher than their average value.]{}
### Chemical mechanism of Group JS sources {#sec_JS_Phy}
The [ abnormal high abundance of [C$_3$S]{} compared with those of nitrogen-bearing LCCMs in JS sources]{} has not been seen before. Previous studies revealed that the column densities of C$_2$S have good positive correlation with those of [HC$_3$N]{} and [HC$_5$N]{} in quiescent dark cores [@1992ApJ...392..551S]. The column densities of [C$_3$S]{} are 3-5 times lower than those of C$_2$S . High C$_2$S and [C$_3$S]{} abundances were found towards Taurus dark clouds, and there is still a good correlation between column densities of C$_2$S and [HC$_5$N]{}. In star formation regions, S-bearing CCMs were at best only marginally detected [@1992ApJ...392..551S], indicating that the results of our three outflow sources are unusual. WCCC theory was a possible explanation for the large hydrocarbon abundances such as [C$_4$H]{} and C$_6$H present in L1251A [@2011ApJ...730L..18C]. However, WCCC can not explain the [ abundances of]{} [C$_3$S]{} in the three outflow sources, since the relative intensities of N-bearing and S-bearing molecules in WCCC sources are close to those in early cold quiescent cores as listed in .
In the observations of @2011ApJ...730L..18C towards L1251A, the [C$_3$S]{} column density is lower than that of [HC$_7$N]{}, which is contrary to our results. Their target point locates at R.A.(2000)=22:30:40.4, Dec.(2000)=75:13:46, while ours ( and ) is more than one arcmin closer to IRS3 and the jet [@2011ApJ...730L..18C; @2010ApJ...709L..74L]. [ The associated outflow may heat the surroundings and contribute in inhibiting the depletion of S-molecules .]{} We speculate that the high abundances of [C$_3$S]{} are relevant to the jets and shocks in the three sources.
These aspects adequately demonstrate that sulfur ions are produced during shock processes developed in these three sources. Shock processes may lead to the reduction of CCMs. However, shocks can fuel the environments with plenty of S$^+$ and thus driving the generation of S-bearing CCMs including [C$_3$S]{}. The regions with most abundant S-bearing CCMs do not necessarily completely coincide with the shocked regions since sulfur elements will only enter into S-bearing CCMs through mild chemical reactions. Unlike in cold dark clouds and WCCC sources, shock induced chemistry plays a major role in performances of N- and S-bearing CCMs in these three [ outflow]{} sources. This is a new chemistry and we name [ it shocked]{} carbon-chain chemistry (SCCC). The characteristics of our SCCC sources are as following:
1\. Their emissions of the N-bearing species are generally weaker than those in early cold and dark cores and WCCC sources. Taking our samples including three SCCC sources and six of the early cold/WCCC sources into account only, the highest column density of [HC$_3$N]{} characterizing the SCCC sources is close to the lowest value of the early cold/WCCC sources.
2\. They have relatively strong [C$_3$S]{} emission. Different from the early cold and dark cores and WCCC sources, the column densities of [C$_3$S]{} of SCCC sources exceed those of [HC$_7$N]{}, and this could be used as a criterion to identify SCCC source. 3. The sources are associated with molecular outflows and infrared/optical jets. The dynamic time scales range from 5$\times$10$^4$ to 10$^5$ yr.
4\. Emissions of ionization species, especially S$^+$, are enhanced in shocked regions.
Model test {#sec_JS_Chem}
----------
We modeled the chemistry of WCCC/SCCC sources by treating them as homogeneous, isotropic clouds. In this simulation, a single point chemical network was run under an ordinary differential equation solver DVODE [@doi:10.1137/0910062] with most physical parameters fixed, n(H$_2$) = 10$^5$ cm$^{-3}$, A$_V$ = 5 mag, $\sigma$$_g$ = 0.03 $\mu$, and cosmic-ray ionization rate $\gamma$ = $1.2\times10^{-17}s^{-1}$ [@2004ApJ...617..360L]. The metal abundances were adopted as the low-metal abundance case of @1982ApJS...48..321G, except for that of sulfur. In star-forming regions, the total abundance of sulfur-bearing species is $\sim$ $10^{-8}$ [@2016ApJ...824..136L], and this value was adopted instead of the widely used $8\times 10^{-8}$ . All the elements were initially ionized except for the hydrogen atoms. The network of chemical reactions as well as the binding energies of the species were adopted from the UMIST Database for Astrochemistry (http://www.udfa.net) as described in . The network consists of 6173 gas phase reactions and 195 reactions on grain surfaces involving 467 atomic and molecular species. The accretion and desorption rates were calculated according to @1992ApJS...82..167H, assuming a sticking coefficient of [ unity]{}. The temperatures of the gas and dust were assumed to be well coupled. The changes of the fractional abundances of HC$_{2n+1}$N (n=1-3) and [C$_3$S]{} with time are shown in the left panel of .
The temperature, represented by the dotted green line in the left panel of , stays as 15 K till $10^5$ yr after the beginning of modeling. Apparently abundances of N- and S-bearing CCMs would increase first and then get depleted significantly before $10^5$ yr. During this period, the abundance of [C$_3$S]{} is positively correlated with those of N-bearing CCMs. These characteristics agree with the observations in dark clouds and star-forming sources [@1992ApJ...392..551S].
As the temperature increases to 50 K at $1\times 10^5$ yr, abundances of N-bearing species increase quickly because of the reactions between the durable nitrogen atoms and the evaporated molecules such as CH$_\mathrm{4}$ and $\mathrm{C_2H_2}$. Soon after, abundances of N-bearing CCMs drop slowly for the lack of sustained supplements of precursors. The enhancements of N-bearing CCMs with shorter carbon [ chains]{}, especially [HC$_3$N]{}, are more significant when compared with heavier N-bearing LCCMs. Such [ a]{} behavior can explain why WCCC sources such as L1527 and our Group JS sources have relatively large x(HC$_{2n+1}$N)/x(HC$_{2n+3}$N). However, the abundance of [C$_3$S]{} is not enhanced much during this period because sulfur atoms have [ not been consumed yet, are not as effectively incorporated]{} as nitrogen atoms, and the downward trend of [C$_3$S]{} can not be reversed. These results indicate that, besides hydrocarbons such as [C$_4$H]{} detected in WCCC sources, the N-bearing CCMs will also be largely enhanced in the early phase of WCCC. It can explain the variances between the younger WCCC source Lup I-1 [@2009ApJ...697..769S] and the older one L1527 [@2008ApJ...672..371S] whose abundances of N-bearing CCMs are nearly an order of magnitude lower than those of the former one.
Shocks can be the driving sources for a persistent supply of sulfur atoms and ions. The right panel of shows the [ evolution]{} of the abundances of some important species after a J-shock simulated by the MHD shock code of . The simulation ran with a magnetic field parameter b = 0.1, a pre-shock density n$_H$ = 10$^5$ cm$^{-3}$, and shock speed, $v_s$ = 5 km s$^{-1}$ as described in @1992ApJS...82..167H. The abundances of $N^+$, $C^+$ and S$^+$ would peak at 5$\times$$10^2$ yr after the shock, and then decrease because of the piling up of the after-shock fluid and contribute in inhibiting the depletion rates. In realistic cases in molecular clouds, after-shock material can be spread diffusely and bring the shock-induced ions into the environment. A simple order of magnitude estimation can be made to support the idea that the shock can provide enough S$^+$ to drive the evolution of sulfur-bearing carbon-chain molecules. Supersonic shocks with a velocity $v_{e}$ = 5 km s$^{-1}$ are powerful enough to spur sulfur elements from grain surfaces and to release shocked gas into the environment with S$^+$ at abundance $f_s=10^{-9}$ (the right panel of ). For a jet with a mass flux ($S_s$) $10^{-7}$ $M_{\sun}$/yr and a velocity ($v_s$) 300 km s$^{-1}$, the rate of the supplement of sulfur ions to the environment can be estimated as $$f_s(\frac{v_s}{v_e})^2\frac{S_s}{M_{env}}\ \sim\ 10^{-20}\ s^{-1}\ per\ H\ atom$$ in which the envelope mass, $M_{env}$, is taken as 1 solar mass.
After $2\times 10^5$ yr, a supplement of S$^+$ is applied at a rate of $10^{-20}$ $s^{-1}$ per H atom. The abundance of [C$_3$S]{} rises by one order of [ magnitude and quickly exceeds]{} that of [HC$_7$N]{}. The artificial supplement of S$^+$ is critical for explaining the emission [ characteristics]{} of S- and N-bearing CCMs in SCCC sources, since WCCC alone can not reproduce the high abundance of [C$_3$S]{}. The abundance of [C$_3$S]{} even exceeds the emission of [HC$_5$N]{} at 2.5$\times$10$^5$ yr if the duration of supplementing of S$^+$ is not limited. The source L1251A is close to that point with a [C$_3$S]{} column density comparable to that of [HC$_5$N]{}. The time duration between this point and the start of the SCCC process (5$\times$10$^4$ yr) is comparable to the dynamical timescale of L1251A (5.2$\times$10$^4$ yr; Sect \[sec\_JS\_Phy\]). The column density of [C$_3$S]{} may exceed that of [HC$_5$N]{} in more evolved SCCC sources. Shocks are more effective in supplying S$^+$ than C$^+$ and N$^+$ (see the right panel of ). Our sources in [ the]{} JS group are all associated with HH objects and jets harboring [ H$\alpha$ lines and features in emission]{}, and shocked regions are reservoirs of $S^+$ [@1995ApJ...454..345B; @1997IAUS..182P..51A; @2010ApJ...709L..74L]. The precursors of N-bearing CCMs such as N and C$^+$ are more abundant compared with S$^+$ (see the left panel of ) which makes the shock induced enhancements of N-bearing LCCMs relatively ineffective. On the other hand, the pre-consuming of precursors such as CH$_4$ in [ the]{} WCCC process would further restrain the enhancements of N-bearing LCCMs. The abundances of N-nearing CCMs are nearly unaffected if supplement rates of C$^+$ ($10^{-21}$ s$^{-1}$ per H atom) and N$^+$ ($10^{-22}$ s$^{-1}$ per H atom) are added in the gas-grain model after $2\times 10^5$ yr. These characteristics imply that sulfur-containing species are unique tracers of shocks.
Summary {#sec_summary}
=======
Using the new TMRT telescope of the Shanghai Observatory, 11 sources including five outflow sources and six Lupus I starless dust cores were observed measuring [HC$_3$N]{} $J=2-1$, [HC$_5$N]{} $J=6-5$, [HC$_7$N]{} $J=14-13,~15-14,~16-15$ and [C$_3$S]{} $J=3-2$. Four of the five outflow sources, I20582, L1221, L1251A and Lup I-1 were detected. Among them, I20582 (IRAS20582+7724) and L1221 were newly detected as carbon-chain–producing regions. In the six starless cores of the Lupus I region all the observed transitions were detected except Lup I-3/4. For our detected sources, the excitation temperatures derived from hyperfine component fitting of [HC$_3$N]{} $J=2-1$ are consistent with the dust temperatures derived from Herschel data. Column densities are calculated. Abundances were derived and analyzed for different kinds of species and sources. The main results are as [ follows]{}:
1\. Emission of [C$_3$S]{} $J=3-2$ in the three outflow sources I20582, L1221 and L1521A was found stronger than their [HC$_7$N]{} emissions, which can not be explained by chemistry of early cold and dark cores and warm carbon-chain chemistry (WCCC) sources. Shock carbon-chain chemistry (SCCC) is suggested. In SCCC sources, shocks fuel the environments with abundant S$^+$ and thus drive the generation of S-bearing CCMs including [C$_3$S]{}.
2\. In the other outflow source Lup I-1 all the observed lines are strong. However, emission of the [C$_3$S]{} is still weaker than those of the detected N-bearing species in this source, similar to the cases in the starless cores in the Lupus region. In young SCCC sources such as L1251A, WCCC can still play a role in local regions.
3\. Emissions of N- and S-bearing CCMs of starless cores in Lupus I are pretty strong, and those of Lup I-11, Lup I-5 and Lup I-7/8/9 are even stronger than those of the TMC-1 like cloud Lup I-6 (Lupus-1A). The results of these cores and Lup I-1 show that Lupus I is a carbon-chain molecule([ CCM]{}) rich complex similar to the Taurus complex.
4\. Our gas-grain chemical model shows that SCCC is necessary to explain the observations toward the three above mentioned outflow sources since the WCCC can enhance the abundances of N-bearing CCMs while it does not strongly influence the S-bearing CCMs. Shock feedback by protostars can provide enough S$^+$ to drive SCCC. Our results demonstrate that the chemistry of S- and N-bearing species can be different in molecular outflow/jet sources. More observations are needed to further explore SCCC.
Acknowledgements {#acknowledgements .unnumbered}
================
We are grateful to the staff of SHAO and PMO Qinghai Station. We also thank Shonghua Li, Kai Yang and Bingru Wang for their assistance during observation period. This project was supported by the grants of the National Key R&D Program of China No. 2017YFA0402600, NSFC Nos. 11433008, 11373009, 11503035, 11573036 and U1631237, and the China Ministry of Science and Technology under State Key Development Program for Basic Research (No.2012CB821800), and the Top Talents Program of Yunnan Province (2015HA030). D.Madones acknowledges support from CONICYT project Basal AFB-170002. Natalia Inostroza acknowledges CONICYT/PCI/REDI170243.
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Name Ra(J2000) Dec(J2000) D(pc) Notes Reference Observation date
----------------- ------------- ------------- ------- -------------------------- ------------ ------------------------- -- --
L1660 07:20:06.75 -24:02:20.9 1000 Molecular outflow [$^{a}$]{} 2016.03.24
IRAS 20582+7724 20:57:10.6 +77:35:46 200 Molecular outflow, L1228 [$^{b}$]{} 2016.01.25/26,2017,11,9
L1221 22:28:02.7 +69:01:13 200 Molecular outflow [$^{c}$]{} 2016.01.26,2017,11,9
L1251A 22:30:35.0 +75:14:00 330 Molecular outflow [$^{d}$]{} 2015.12.16, 2016.01.25
Lup I-1 15:43:01.68 -34:09:08.9 155 Molecular outflow,WCCC [$^{e}$]{} 2016.03.21
Lup I-2 15:44:59.8 -34:17:09 155 In Lup1 C6 [$^{f}$]{} 2016.01.25
Lup I-3/4 15:45:14.8 -34:17:02.7 155 In Lup1 C7 [$^{f}$]{} 2016.03.21
Lup I-5 15:45:03.80 -34:17:57.3 155 In Lup1 C6 [$^{f}$]{} 2016.03.21
Lup I-6 15:42:52.4 -34:07:54 155 Lupus-1A, in Lup1 C3 [$^{g}$]{} 2016.01.25
Lup I-7/8/9 15:42:44.06 -34:08:30.4 155 In Lup1 C3 [$^{f}$]{} 2016.03.24
Lup I-11 15:45:25.10 -34:24:01.8 155 In Lup1 C8 [$^{f}$]{} 2016.03.24
\
[$^{a}$ and the references therein]{}; [$^{b}[email protected]]{}; [$^{c}[email protected]]{}; [$^{d}[email protected]]{}; [$^{e}$]{}; [$^{f}$]{}; [$^{g}$]{}
----------- ------------ ------------ ---------------------------- ------------- ------------------------------------------------------------------ --
Molecular Transition freq.(MHz) $Log_{10}(A_{ij}\ s^{-1})$ $E_{up}(K)$ HPBWs (")\
[HC$_3$N]{}& J=2-1 F=1-1 & 18198.3745 & -7.26550 & 1.30990 & 52\
& J=2-1 F=3-2 & 18196.3104 & -6.88533 & 1.30995 & 52\
& J=2-1 F=2-1 & 18196.2169 & -7.01030 & 1.30980 & 52\
& J=2-1 F=1-0 & 18195.1364 & -7.14068 & 1.31003 & 52\
& J=2-1 F=2-2 & 18194.9195 & -7.48753 & 1.30988 & 52\
[HC$_5$N]{}& J=6-5 F=7-6 & 15975.9831 & -6.86356 & 2.68359 & 60\
& J=6-5 F=6-5 & 15975.9663 & -6.87581 & 2.68345 & 60\
& J=6-5 F=5-4 & 15975.9336 & -6.87816 & 2.68359 & 60\
[HC$_7$N]{}& J=16-15 & 18047.9697 & -6.11305 & 7.36235 & 53\
& J=15-14 & 16919.9791 & -6.19805 & 6.49617 & 56\
& J=14-13 & 15791.9870 & -6.28888 & 5.68424 & 60\
[C$_3$S]{}& J=3-2 & 17342.2564 & -6.44743 & 1.66464 & 55\
----------- ------------ ------------ ---------------------------- ------------- ------------------------------------------------------------------ --
Lines Transition I20582 L1221 L1251A LupusI-1 LupusI-2 LupusI-5 LupusI-6 LupusI-7/8/9 LupusI-11
------------- ------------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- -------------- -----------
[HC$_3$N]{} J=2-1 F=1-1 -8.1(1) -4.53(4) -3.92(3) 5.11(2) 4.88(3) 4.97(3) 5.13(2) 5.13(2) 4.38(2)
J=2-1 F=3-2 -8.04(3) -4.49(3) -3.96(2) 5.10(2) 4.87(2) 4.95(2) 5.12(2) 5.14(2) 4.37(2)
J=2-1 F=2-1 -8.06(4) -4.51(4) -3.95(2) 5.10(2) 4.86(2) 4.94(2) 5.12(2) 5.14(2) 4.37(2)
J=2-1 F=1-0 -8.01(3) -4.47(4) -3.97(3) 5.09(2) 4.86(3) 4.94(3) 5.11(2) 5.14(2) 4.37(2)
J=2-1 F=2-2 -8.12(6) -4.46(5) -3.96(4) 5.09(2) 4.84(3) 4.99(3) 5.14(2) 5.13(2) 4.37(2)
[HC$_5$N]{} J=6-5 F=7-6 -3.79(3) 5.08(2) 4.94(2) 5.09(2) 5.11(2) 5.14(2) 4.39(2)
J=6-5 F=6-5 -8.00(4) -4.59(5) 5.08(2) 5.11(2) 5.14(2) 4.39(2)
J=6-5 F=5-4 -3.97(3) 5.09(4) 4.89(2) 4.96(2) 5.11(3) 5.14(2) 4.38(2)
[HC$_7$N]{} J=16-15 -7.95(7) -4.00(3) 5.10(2) 4.87(3) 4.96(3) 5.11(2) 5.14(2) 4.39(2)
J=15-14 -4.49(9) -3.96(4) 5.09(2) 4.83(3) 4.95(2) 5.10(3) 5.14(2) 4.39(2)
J=14-13 -7.73(8) -4.09(6) 5.08(2) 4.82(3) 4.94(2) 5.12(2) 5.12(2) 4.37(2)
[C$_3$S]{} J=3-2 -8.07(5) -4.55(5) -3.98(2) 5.12(3) 4.71(4) 4.94(3) 5.10(4) 5.14(3) 4.37(2)
[HC$_3$N]{} J=2-1 F=1-1 0.05(2) 0.09(2) 0.37(8) 1.5(2) 0.38(8) 0.7(2) 0.59(9) 1.0(1) 0.88(9)
J=2-1 F=3-2 0.28(2) 0.30(2) 1.57(8) 5.1(2) 1.71(9) 2.9(2) 2.04(9) 4.1(1) 3.50(9)
J=2-1 F=2-1 0.15(2) 0.21(2) 0.83(8) 3.2(2) 1.08(9) 1.7(2) 1.39(9) 2.6(1) 2.30(9)
J=2-1 F=1-0 0.08(2) 0.13(2) 0.36(8) 1.6(2) 0.48(9) 0.9(2) 0.82(9) 1.3(1) 1.17(9)
J=2-1 F=2-2 0.06(2) 0.08(2) 0.22(8) 1.1(2) 0.39(9) 0.55(2) 0.63(9) 1.2(1) 0.97(9)
[HC$_5$N]{} J=6-5 F=7-6 0.20(3) 1.28(5) 0.53(4) 1.01(5) 0.62(5) 0.93(5) 1.00(5)
J=6-5 F=6-5 0.061(7) 0.057(8) 1.08(5) 0.47(5) 0.79(5) 0.90(5)
J=6-5 F=5-4 0.14(3) 0.92(5) 0.34(4) 0.61(6) 0.42(5) 0.70(5) 0.81(5)
[HC$_7$N]{} J=16-15 0.037(8) 0.08(4) 0.76(7) 0.28(5) 0.54(7) 0.40(5) 0.55(5) 0.65(5)
J=15-14 0.026(8) 0.10(4) 8.3(7) 0.26(5) 0.54(7) 0.30(5) 0.53(5) 0.63(6)
J=14-13 0.030(8) 0.06(4) 0.75(7) 0.31(5) 0.65(7) 0.38(5) 0.44(4) 0.72(6)
[C$_3$S]{} J=3-2 0.049(8) 0.036(8) 0.26(4) 0.22(6) 0.11(4) 0.25(6) 0.10(5) 0.13(4) 0.32(4)
[HC$_3$N]{} J=2-1 F=1-1 1.0(2) 0.4(1) 0.36(5) 0.19(2) 0.45(4) 0.33(4) 0.22(3) 0.18(3) 0.22(3)
J=2-1 F=3-2 0.85(4) 0.74(4) 0.34(2) 0.25(2) 0.42(2) 0.42(2) 0.26(2) 0.21(2) 0.23(4)
J=2-1 F=2-1 0.90(7) 0.80(6) 0.36(3) 0.25(2) 0.39(3) 0.41(3) 0.24(3) 0.20(2) 0.22(2)
J=2-1 F=1-0 1.0(1) 0.54(8) 0.35(3) 0.23(3) 0.44(4) 0.41(3) 0.20(3) 0.18(3) 0.21(2)
J=2-1 F=2-2 0.6(1) 0.5(1) 0.40(5) 0.24(3) 0.40(4) 0.50(5) 0.18(3) 0.17(3) 0.22(3)
[HC$_5$N]{} J=6-5 F=7-6 0.21(2) 0.67(3) 0.59(2) 0.19(3) 0.18(2) 0.23(2)
J=6-5 F=6-5 1.31(8) 0.8(1) 0.56(3) 0.27(3) 0.19(3) 0.16(2) 0.18(2)
J=6-5 F=5-4 0.39(4) 0.23(2) 0.37(3) 0.39(3) 0.18(2) 0.16(2) 0.18(2)
[HC$_7$N]{} J=16-15 0.9(1) 0.35(7) 0.29(2) 0.43(4) 0.42(4) 0.18(2) 0.19(2) 0.25(2)
J=15-14 0.64(3) 0.31(5) 0.26(2) 0.46(4) 0.42(3) 0.25(3) 0.20(2) 0.27(2)
J=14-13 0.6(2) 0.6(1) 0.31(2) 0.49(3) 0.41(3) 0.22(3) 0.25(2) 0.26(2)
[C$_3$S]{} J=3-2 1.4(2) 0.8(1) 0.32(3) 0.24(3) 0.54(2) 0.30(4) 0.3(1) 0.35(5) 0.24(3)
[HC$_3$N]{} J=2-1 F=1-1 0.041(8) 0.033(6) 0.14(2) 0.30(2) 0.18(2) 0.25(2) 0.14(1) 0.19(1) 0.20(1)
J=2-1 F=3-2 0.26(1) 0.23(1) 0.57(2) 1.38(5) 0.77(3) 1.27(4) 0.55(2) 0.90(3) 0.85(3)
J=2-1 F=2-1 0.13(1) 0.14(1) 0.32(1) 0.84(3) 0.45(2) 0.78(3) 0.36(2) 0.55(2) 0.54(2)
J=2-1 F=1-0 0.066(7) 0.053(7) 0.13(1) 0.40(2) 0.23(2) 0.41(3) 0.18(1) 0.25(1) 0.27(1)
J=2-1 F=2-2 0.031(6) 0.035(7) 0.10(1) 0.30(2) 0.17(1) 0.30(3) 0.12(1) 0.21(1) 0.23(1)
[HC$_5$N]{} J=6-5 F=7-6 0.12(1) 0.28(1) 0.38(1) 0.64(2) 0.12(1) 0.18(1) 0.24(1)
J=6-5 F=6-5 0.077(4) 0.058(5) 0.31(1) 0.096(6) 0.013(6) 0.174(7)
J=6-5 F=5-4 0.041(4) 0.23(1) 0.13(1) 0.25(1) 0.080(5) 0.118(6) 0.158(6)
[HC$_7$N]{} J=16-15 0.026(4) 0.031(6) 0.24(1) 0.13(1) 0.24(1) 0.077(6) 0.109(6) 0.168(6)
J=15-14 0.013(3) 0.031(5) 0.23(1) 0.126(8) 0.24(1) 0.078(7) 0.110(6) 0.18(1)
J=14-13 0.014(3) 0.038(7) 0.25(1) 0.16(1) 0.28(1) 0.089(6) 0.116(6) 0.199(9)
[C$_3$S]{} J=3-2 0.069(7) 0.024(3) 0.088(6) 0.054(7) 0.064(7) 0.079(8) 0.033(8) 0.046(6) 0.083(6)
-------------- --------------------- ------------------ -----------
Sources $\tau$([HC$_3$N]{}) Tex([HC$_3$N]{}) T$_d$
K K
I20582 $<$0.1 – 13.5(0.3)
L1221 0.28(0.19) 14.02(5.11) 15.1(0.2)
L1251A $<$0.1 – 12.4(0.2)
LupusI-1 0.84(0.08) 15.79(1.69) 13.9(0.1)
LupusI-2 0.55(0.11) 9.52(1.32) 11.5(0.2)
LupusI-5 0.72(0.13) 11.93(1.71) 11.2(0.1)
LupusI-6 1.43(0.13) 7.03(0.52) 10.0(0.2)
LupusI-7/8/9 0.85(0.08) 14.50(1.33) 10.2(0.1)
LupusI-11 1.07(0.08) 11.39(0.77) 11.9(0.8)
-------------- --------------------- ------------------ -----------
Lines Transition I20582 L1221 L1251A LupusI-1 LupusI-2 LupusI-5 LupusI-6 LupusI-7/8/9 LupusI-11
------------- ------------- ----------- ---------- ----------- ----------- ---------- ----------- ----------- -------------- ----------- --
[HC$_3$N]{} J=2-1 F=1-1 21(4) 20(3) 69(6) 157(8) 83(6) 112(11) 57(4) 80(4) 98(5)
J=2-1 F=3-2 23.8(0.7) 28(1) 49.4(1.0) 131(1) 63(1) 104(2) 42.1(0.8) 69.7(0.8) 72.1(0.9)
J=2-1 F=2-1 22(1) 30(1) 52(1) 148(3) 69(2) 118(3) 50(1) 79(1) 85(1)
J=2-1 F=1-0 25(2) 25(3) 49(4) 157(7) 79(5) 139(8) 56(3) 80(3) 95(3)
J=2-1 F=2-2 16(3) 22(4) 46(5) 156(10) 78(6) 135(12) 52(3) 89(4) 107(4)
[HC$_5$N]{} J=6-5 F=7-6 7.8(0.6) 35(1) 24(1) 40(1) 12.7(0.5) 19.1(0.5) 27.8(0.8)
J=6-5 F=6-5 3.6(0.6) 3.2(0.8) 46(1) 11.9(0.6) 16.4(0.6) 23.6(0.8)
J=6-5 F=5-4 6.7(0.7) 40(1) 20(1) 39(1) 11.8(0.7) 17.5(0.7) 25.4(0.8)
[HC$_7$N]{} J=16-15 1.1(0.2) 1.3(0.2) 10.0(0.3) 5.2(0.3) 9.8(0.4) 3.1(0.2) 4.4(0.2) 6.9(0.2)
J=15-14 0.7(0.2) 1.4(0.2) 10.2(0.4) 5.4(0.3) 10.2(0.4) 3.3(0.3) 4.6(0.2) 7.8(0.3)
J=14-13 0.7(0.1) 1.8(0.3) 12.2(0.4) 7.6(0.4) 13.0(0.5) 4.0(0.3) 5.2(0.2) 9.3(0.3)
[C$_3$S]{} J=3-2 3.5(0.4) 1.4(0.2) 4.2(0.2) 2.8(0.4) 2.9(0.3) 3.5(0.4) 1.4(0.3) 2.0(0.3) 3.9(0.2)
[HC$_3$N]{} 21(2) 25(2) 53(3) 150(6) 74(4) 121(7) 52(2) 79(3) 91(3)
[HC$_5$N]{} 3.6(0.6) 3.2(0.8) 7.3(0.6) 40(1) 22(1) 39(1) 12.1(0.6) 17.7(0.6) 25.6(0.8)
[HC$_7$N]{} 0.9(0.2) 0.7(0.1) 1.5(0.3) 10.8(0.4) 6.1(0.3) 11.0(0.4) 3.5(0.2) 4.7(0.2) 8.0(0.3)
[C$_3$S]{} 3.5(0.4) 1.4(0.2) 4.2(0.2) 2.8(0.4) 2.9(0.3) 3.5(0.4) 1.4(0.3) 2.0(0.3) 3.9(0.2)
Sources [HC$_3$N]{} [HC$_5$N]{} [HC$_7$N]{} [C$_3$S]{}
-------------- ------------- ------------- ------------- ------------
I20582 10(1) 1.7(0.3) 0.42(0.07) 1.7(0.2)
L1221 16(1) 2.2(0.4) 0.5(0.1) 1.01(0.08)
L1251A 26(1) 3.6(0.3) 0.7(0.1) 2.1(0.1)
LupusI-1 40(1) 11.0(0.3) 2.92(0.10) 0.76(0.10)
LupusI-2 25(1) 7.7(0.4) 2.1(0.1) 1.0(0.1)
LupusI-5 43(2) 14.2(0.5) 3.9(0.2) 1.3(0.1)
LupusI-6 13(1) 3.1(0.2) 0.89(0.06) 0.36(0.09)
LupusI-7/8/9 23(1) 5.0(0.2) 1.36(0.06) 0.56(0.07)
LupusI-11 61(2) 17.1(0.5) 5.3(0.2) 2.6(0.2)
-------------- ----------------- --------- ----------------- --------- ----------------- ---------
Sources x([HC$_3$N]{}) x([HC$_5$N]{}) x([C$_3$S]{})
/x([HC$_5$N]{}) Average /x([HC$_7$N]{}) Average /x([HC$_7$N]{}) Average
I20582 5.8 4.0 5.0
L1221 7.3 5.0 2.0
L1251A 7.3 6.8 4.9 4.6 3.3 3.4
LupusI-1 3,8 3.8 3.7 3.7 0.3 0.3
LupusI-2 3.3 3.6 0.5
LupusI-5 3.1 3.5 0.3
LupusI-6 4.3 3.5 0.4
LupusI-7/8/9 4.5 3.8 0.4
LupusI-11 3.6 3.8 3.2 3.5 0.5 0.4
-------------- ----------------- --------- ----------------- --------- ----------------- ---------
Dust Temperatures and Column Densities {#sec_appendix}
======================================
All the sources observed except L1660 have *Herschel* data at 70, 160, 250, 350, and 500 [$\mu$m]{} in the Herschel Science Archive, where we extracted Level 2 or Level 2.5 images. These data, which are from the Gould Belt Survey and a few other Herschel key programs and open time programs, have resolutions of 8.$\!\!\arcsec$4, 13.$\!\!\arcsec$5, 18.$\!\!\arcsec$1, 24.$\!\!\arcsec$9 and 36.$\!\!\arcsec$4 at 70, 160, 250, 350, and 500 [$\mu$m]{} respectively$^,$, and the corresponding pixel sizes are 3.$\!\!\arcsec$2, 3.$\!\!\arcsec$2, 6.$\!\!\arcsec$0, 10.$\!\!\arcsec$0 and 14.$\!\!\arcsec$0. On the basis of these multi-bands far-IR data, we obtained dust temperatures and column densities.
Background Removal
------------------
Background/foreground emission was removed using the *CUPID-findback* algorithm of the *Starlink* suite[@2014ASPC..485..391C]. The algorithm constructs the background iteratively from the original image. At first, a filtered form of the input data is produced by replacing every input pixel by the minimum of the input values within a rectangular box centered on the pixel. This filtered data is then filtered again, using a filter that replaces every pixel value by the maximum value in a box centered on the pixel. Then each pixel in this filtered data is replaced by the mean value in a box centered on the pixel. The same box size is used for the first three steps. The final background estimate is obtained via some corrections and iterations by comparisons with the initial input data. More details about the algorithm can be found on the online document of *findback*. As a key parameter, the box has been assigned to be the source size at 250 which was measured based on the area with emission higher than 50% of the peak intensity for each target.
Flux Measurements
-----------------
For each source, we firstly determined the source size based on the emission at 250 () via fitting an ellipse to the region with emission higher than 50% of the peak intensity around the target. Then fluxes at other bands were obtained by integrating the emission encompassed by the ellipse if the ellipse [ was]{} larger than the beam. Otherwise fluxes referring to the beam centered at the source have been used. The source size at 250 and fluxes at Herschel bands are given in 2ed-7th column of .
Spectral Energy Distribution Fitting
------------------------------------
The fluxes at *Herschel* bands have been modeled using single temperature gray-bodies, $$\label{eq-gb}
S_\nu=B_\nu(T) (1-e^{-\tau_\nu})\Omega$$ where the Planck function $B_\nu(T)$ is modified by [ the]{} optical depth $$\tau_\nu = \mu_\mathrm{H_2}m_\mathrm{H}\kappa_\nu N_\mathrm{H_2}/R_\mathrm{gd}.$$ Here, $\mu_\mathrm{H_2}=2.8$ is the mean molecular weight adopted from , $m_\mathrm{H}$ is the mass of a hydrogen atom, $N_\mathrm{H_2}$ is the column density, $R_\mathrm{gd}=100$ is the gas to dust ratio. The dust opacity $\kappa_\nu$ can be expressed as a power law of frequency, $$\kappa_\nu=5.9\left(\frac{\nu}{850~\mathrm{GHz}}\right)^\beta~\mathrm{cm^2g^{-1}}.$$ with $\kappa_\nu(\mathrm{850~GHz})=5.9~\mathrm{cm^2g^{-1}}$ adopted from . The $\Omega$ in is the solid angle of the target. The free parameters are the dust temperature, dust emissivity index $\beta$ and column density.
The fluxes at 70 [$\mu$m]{} were excluded from the SED fitting because emission at this wavelength may arise from a warmer dust component and a large fraction of the 70 [$\mu$m]{} emission originates from very small grains, where the assumption of a single equilibrium temperature is not valid.
The fitting was performed with the Levenberg-Marquardt algorithm provided in the python package *lmfit* [@2016ascl.soft06014N]. Fitted SED curves were shown in . The resulting dust temperatures and column densities are listed in columns 8 and 9 of .
{width="110mm"}
{width="140mm"}
---------------- -------------- --------------- --------------- --------------- --------------- ------------------ ----------------------- -------------------------
[Source]{} [S$_{70}$]{} [S$_{160}$]{} [S$_{250}$]{} [S$_{350}$]{} [S$_{500}$]{} [Size$_{250}$]{} [T$_\mathrm{dust}$]{} [N$_\mathrm{H_2}$]{}
[Jy]{} [Jy]{} [Jy]{} [Jy]{} [Jy]{} [arcsec]{} [K]{} [$10^{22}$ cm$^{-2}$]{}
IRAS20582+7724 9.04 18.14 17.28 8.94 3.89 43.16 13.5(0.3) 2.1(0.2)
L1221 6.22 5.34 5.28 3.23 1.40 21.91 15.1(0.2) 2.0(0.1)
L1251A 1.75 8.35 9.77 5.50 2.53 40.67 12.4(0.2) 2.0(0.2)
Lupus1-1 21.49 59.44 54.10 29.07 10.67 55.00 13.9(0.1) 3.7(0.1)
Lupus1-2 0.19 2.24 5.61 4.86 3.44 36.00 11.5(0.2) 2.9(0.2)
Lupus1-3/4 8.86 31.73 27.13 15.37 5.83 57.96 15.0(0.7) 1.3(0.2)
Lupus1-5 0.25 4.00 7.67 6.79 3.32 42.69 11.2(0.1) 2.8(0.2)
Lupus1-6 0.04 3.34 7.26 5.22 2.87 43.00 10.0(0.2) 3.9(0.4)
Lupus1-7/8/9 0.00 4.86 10.00 7.79 3.80 50.42 10.2(0.1) 3.5(0.2)
Lupus1-11 0.32 3.31 5.50 3.21 2.08 39.20 11.9(0.8) 1.5(0.5)
TMC-1 0.14 16.95 36.78 31.92 16.96 100.00 10.6(0.1) 3.2(0.2)
---------------- -------------- --------------- --------------- --------------- --------------- ------------------ ----------------------- -------------------------
\
\[lastpage\]
[^1]: E-mail: [email protected]
[^2]: [www.splatalogue.net](www.splatalogue.net)
[^3]: <http://www.iram.fr/IRAMFR/GILDAS/doc/html/class-html>
|
---
abstract: 'Let $0<r<1/4$, and $f$ be a non-vanishing continuous function in $|z|\leq r$, that is analytic in the interior. Voronin’s universality theorem asserts that translates of the Riemann zeta function $\zeta(3/4 + z + it)$ can approximate $f$ uniformly in $|z| < r$ to any given precision $\varepsilon$, and moreover that the set of such $t \in [0, T]$ has measure at least $c(\varepsilon) T$ for some $c(\varepsilon) > 0$, once $T$ is large enough. This was refined by Bagchi who showed that the measure of such $t \in [0,T]$ is $(c(\varepsilon) + o(1)) T$, for all but at most countably many $\varepsilon > 0$. Using a completely different approach, we obtain the first effective version of Voronin’s Theorem, by showing that in the rate of convergence one can save a small power of the logarithm of $T$. Our method is flexible, and can be generalized to other $L$-functions in the $t$-aspect, as well as to families of $L$-functions in the conductor aspect.'
address:
- 'Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, ON, M3J1P3 Canada'
- 'Department of Mathematics and Statistics, University of Montreal Pavillon André-Aisenstadt, PO Box 6128, Centre-ville Station Montreal, Quebec H3C 3J7'
- 'Department of Mathematics McGill University 805 Sherbrooke Street West Montreal, Quebec H3A 0G4'
author:
- Youness Lamzouri
- Stephen Lester
- Maksym Radziwill
title: An effective universality theorem for the Riemann zeta function
---
[^1]
Introduction
============
In 1914 Fekete constructed a formal power series $\sum_{n = 1}^{\infty} a_n x^n$ with the following *universal* property: For any continuous function $f$ on $[-1,1]$ (with $f(0) = 0$) and given any $\varepsilon > 0$ there exists an integer $N > 0$ such that $$\sup_{-1 \leq x \leq 1} \Big | \sum_{n \leq N} a_n x^n - f(x) \Big | < \varepsilon.$$ In the 1970’s Voronin [@Voronin] discovered the remarkable fact that the Riemann zeta-function satisfies a similar universal property. He showed that for any $r < \tfrac 14$, any non-vanishing continuous function $f$ in $|z| \leq r$, which is analytic in the interior, and for arbitrary $\varepsilon>0$, there exists a $T > 0$ such that $$\label{universal}
\max_{|z| \leq r} \Big | \zeta(\tfrac 34 + i T + z) - f(z) \Big | < \varepsilon.$$ Voronin obtained a more quantitative description of this phenomena, stated below.
Let $0 < r < \tfrac 14$ be a real number. Let $f$ be a non-vanishing continuous function in $|z|\leq r$, that is analytic in the interior. Then, for any $\varepsilon > 0$, $$\label{LIMINF}
\liminf_{T \rightarrow \infty}
\frac{1}{T} \cdot \textup{meas} \Big \{ T \leq t \leq 2T: \max_{|z| \leq r}
\Big | \zeta(\tfrac 34 + it + z) - f(z) \Big | < \varepsilon \Big \}
> 0,$$ where $\textup{meas}$ is Lebesgue’s measure on $\mathbb{R}$.
There are several extensions of this theorem, for example to domains more general than compact discs (such as any compact set $K$ contained in the strip $1/2 < {\textup{Re}}(s) < 1$ and with connected complement), or to more general $L$-functions. For a complete history of this subject, we refer the reader to [@Matsumoto].
The assumption that $f(z) \neq 0$ is necessary: if $f$ were allowed to vanish then an application of Rouche’s theorem would produce at least $\asymp T$ zeros $\rho = \beta + i \gamma$ of $\zeta(s)$ with $\beta > \tfrac 12 + \varepsilon$ and $T \leq \gamma \leq 2T$, contradicting the simplest zero-density theorems.
Subsequent work of Bagchi [@Bagchi] clarified Voronin’s universality theorem by setting it in the context of probability theory (see [@Kowalski] for a streamlined proof). Viewing $\zeta(\tfrac 34 + it + z)$ with $t \in [T, 2T]$ as a random variable $X_T$ in the space of random analytic functions (i.e $X_T(z) = \zeta(\tfrac 34 + i U_T + z)$ with $U_T$ uniformly distributed in $[T, 2T]$), Bagchi showed that as $T \rightarrow \infty$ this sequence of random variables converges in law (in the space of random analytic functions) to a random Euler product, $$\zeta(s, X) := \prod_{p} \Big (1 - \frac{X(p)}{p^s} \Big )^{-1}$$ with $\{X(p)\}_p$ a sequence of independent random variables uniformly distributed on the unit circle (and with $p$ running over prime numbers). This product converges almost surely for ${\textup{Re}}(s) > \tfrac 12$ and defines almost surely a holomorphic function in the half-plane ${\textup{Re}}(s) > \sigma_0$ for any $\sigma_0 > \tfrac 12$ (see Section 2 below). The proof of Voronin’s universality is then reduced to showing that the support of $\zeta(s+3/4, X)$ in the space of random analytic functions contains all non-vanishing analytic $f : \{ z : |z| < r\} \rightarrow \mathbb{C} \backslash \{0\}$. Moreover it follows from Bagchi’s work that the limit in Voronin’s universality theorem exists for all but at most countably many $\varepsilon > 0$.
In this paper, we present an alternative approach to Bagchi’s result using methods from hard analysis. As a result we obtain, for the first time, a rate of convergence in Voronin’s universality theorem. We also give an explicit description for the limit in terms of the random model $\zeta(s, X)$.
\[thm:main\] Let $0 < r < \tfrac 14$. Let $f$ be a non-vanishing continuous function on $|z|\leq (r+1/4)/2$ that is holomorphic in $|z| < (r+1/4)/2$. Let $\omega$ be a real-valued continuously differentiable function with compact support. Then, we have $$\begin{aligned}
\frac{1}{T}\int_{T}^{2T}\omega\left(\max_{|z| \leq r} |\zeta(\tfrac 34 + it + z) - f(z)| \right)dt
& = {\mathbb{E}}\left(\omega\left(\max_{|z|\leq r}|\zeta(\tfrac 34 + z, X)-
f(z)|\right)\right)\\
&+ O\left((\log T)^{-\frac{(3/4-r)}{11}+o(1)} \right),\end{aligned}$$ where the constant in the $O$ depends on $f, \omega$ and $r$.
If the random variable $Y_{r, f}=\max_{|z|\leq r}|\zeta(\tfrac 34 + z, X)-f(z)|$ is absolutely continuous, then it follows from the proof of Theorem \[thm:main\] that for any fixed $\varepsilon>0$ we have $$\begin{aligned}
&\frac{1}{T} \cdot \textup{meas} \Big \{ T \leq t \leq 2T: \max_{|z| \leq r}
\Big | \zeta(\tfrac 34 + it + z) - f(z) \Big | < \varepsilon \Big \}\\
&= \mathbb{P}\left(\max_{|z|\leq r}\big|\zeta(\tfrac 34 + z, X)-
f(z)\big|<\varepsilon\right)
+ O\left((\log T)^{-\frac{(3/4-r)}{11}+o(1)} \right).\end{aligned}$$ Unfortunately, we have not been able to even show that $Y_{r, f}$ has no jump discontinuities. We conjecture the latter to be true, and one might even hope that $Y_{r, f}$ is absolutely continuous.
A slight modification of the proof of Theorem \[thm:main\] allows for more general domains than the disc $|z| \leq r$. Furthermore, if $\omega \geq \mathbf{1}_{(0, \varepsilon)}$ (where $\mathbf{1}_{S}$ is the indicator function of the set $S$), then it follows from Voronin’s universality theorem that the main term in Theorem \[thm:main\] is positive. Explicit lower bounds for the limit in (in terms of $\varepsilon$) are contained in the papers of Good [@Good] and Garunkstis [@Garunkstis].
Our approach is flexible, and can be generalized to other $L$-functions in the $t$-aspect, as well as to “natural” families of $L$-functions in the conductor aspect. The only analytic ingredients that are needed are zero density estimates, and bounds on the coefficients of these $L$-functions (the so-called Ramanujan conjecture). In particular, the techniques of this paper can be used to obtain an effective version of a recent result of Kowalski [@Kowalski], who proved an analogue of Voronin’s universality theorem for families of $L$-functions attached to $GL_2$ automorphic forms. In fact, using the zero-density estimates near $1$ that are known for a very large class of $L$-functions (including those in the Selberg class by Kaczorowski and Perelli [@KP], and for families of $L$-functions attached to $GL_n$ automorphic forms by Kowalski and Michel [@KM]), one can prove an analogue of Theorem \[thm:main\] for these $L$-functions, where we replace $3/4$ by some $\sigma<1$ (and $r<1-\sigma$). The main idea in the proof of Theorem \[thm:main\] is to cover the boundary of the disc $|z|\leq r$ with a union of a growing (with $T$) number of discs, while maintaining a global control of the size of $|\zeta'(s + z)|$ on $|z| \leq r$. It is enough to focus on the boundary of the disc thanks to the maximum modulus principle. The behavior of $\zeta(s + z)$ with $z$ localized to a shrinking disc is essentially governed by the behavior at a single point $z = z_i$ in the disc. This allows us to reduce the problem to understanding the joint distribution of a growing number of shifts $\log\zeta(s + z_i)$ with the $z_i$ well-spaced, which can be understood by computing the moments of these shifts and using standard Fourier techniques.
It seems very difficult to obtain a rate of convergence which is better than logarithmic in Theorem \[thm:main\]. We have at present no understanding as to what the correct rate of convergence should be.
Key Ingredients and detailed results {#sec:propositions}
====================================
We first begin with stating certain important properties of the random model $\zeta(s, X)$. Let $\{X(p)\}_p$ be a sequence of independent random variables uniformly distributed on the unit circle. Then we have $$-\log\left(1-\frac{X(p)}{p^s}\right)=\frac{X(p)}{p^s}+ h_X(p,s),$$ where the random series $$\label{ErrorRandom}
\sum_{p} h_X(p,s),$$ converges absolutely and surely for ${\textup{Re}}(s)>1/2$. Hence, it (almost surely) defines a holomorphic function in $s$ in this half-plane. Moreover, since ${\mathbb{E}}(X(p))=0$ and ${\mathbb{E}}(|X(p)|^2)=1$, then it follows from Kolmogorov’s three-series theorem that the series $$\label{MainRandom}
\sum_{p}\frac{X(p)}{p^s}$$ is almost surely convergent for ${\textup{Re}}(s)>1/2$. By well-known results on Dirichlet series, this shows that this series defines (almost surely) a holomorphic function on the half-plane ${\textup{Re}}(s)>\sigma_0$, for any $\sigma_0>1/2$. Thus, by taking the exponential of the sum of the random series in and , it follows that $\zeta(s, X)$ converges almost surely to a holomorphic function on the half-plane ${\textup{Re}}(s)>\sigma_0$, for any $\sigma_0>1/2$.
We extend the $X(p)$ multiplicatively to all positive integers by setting $X(1)=1$ and $X(n):= X(p_1)^{a_1}\cdots X(p_k)^{a_k}, \text{ if } n= p_1^{a_1}\dots p_k^{a_k}.$ Then we have $$\label{orthogonality}
{\mathbb{E}}\left(X(n)\overline{X(m)}\right)=\begin{cases} 1 & \text{if } m=n,\\
0 & \text{otherwise}.\\
\end{cases}$$ Furthermore, for any complex number $s$ with ${\textup{Re}}(s)>1/2$ we have almost surely that $$\zeta(s, X)= \sum_{n=1}^{\infty} \frac{X(n)}{n^s}.$$
To compare the distribution of $\zeta(s+it)$ to that of $\zeta(s, X)$, we define a probability measure on $[T, 2T]$ in a standard way, by $$\mathbb{P}_T(S):= \frac{1}{T}\textup{meas}(S), \text{ for any } S\subseteq [T, 2T].$$
The idea behind our proof of effective universality is to first reduce the problem to the discrete problem of controlling the distribution of many shifts $\log \zeta(s_j + it)$ with all of the $s_j$ contained in a compact set inside the strip $\tfrac 12 < {\textup{Re}}(s) < 1$. One of the main ingredients in this reduction is the following result which allows us to control the maximum of the derivative of the Riemann zeta-function. This is proven in Section \[sec:derivative\].
\[ControlDerivative\] Let $0<r<1/4$ be fixed. Then there exist positive constants $b_1$, $b_2$ and $b_3$ (that depend only on $r$) such that $$\mathbb{P}_T \left( \max_{|z|\leq r} |\zeta'(\tfrac 34 + it + z)| > e^{V}
\right) \ll \exp\bigg( -b_1 V^{\frac{1}{1-\sigma(r)}}(\log V)^{\frac{\sigma(r)}{1-\sigma(r)}}\bigg)$$ where $\sigma(r)=\tfrac34-r$, uniformly for $V$ in the range $b_2<V \leq b_3 (\log T)^{1-\sigma}/(\log \log T)$.
We also prove an analogous result for the random model $\zeta(s, X)$, which holds for all sufficiently large $V$.
\[DerRandom\] Let $0<r<1/4$ be fixed and $\sigma(r)=\tfrac34-r$. Then there exist positive constants $b_1$ and $b_2$ (that depend only on $r$) such that for all $V>b_2$ we have $$\mathbb{P}\left(\max_{|z| \leq r} |\zeta'(\tfrac 34 + z, X)| > e^V\right) \ll \exp\bigg( -b_1 V^{\frac{1}{1-\sigma(r)}}(\log V)^{\frac{\sigma(r)}{1-\sigma(r)}}\bigg).$$
Once the reduction has been accomplished, it remains to understand the joint distribution of the shifts $\{\log \zeta(s_1 + it), \log \zeta(s_2 + it), \dots, \log \zeta(s_J + it)\}$ with $J \rightarrow \infty$ as $T \rightarrow \infty$ at a certain rate, and $s_1, \ldots, s_J$ are complex numbers with $\tfrac 12 < {\textup{Re}}(s_j) < 1$ for all $j \leq J$. Heuristically, this should be well approximated by the the joint distribution of the random variables $\{\log \zeta(s_1, X),\log \zeta(s_2, X), \dots, \log \zeta(s_J, X)\}$. In order to establish this fact (in a certain range of $J$), we first prove, in Section \[sec:moments\], that the moments of the joint shifts $\log\zeta(s_j+it)$ are very close to the corresponding ones of $\log\zeta(s_j, X)$, for $j\leq J$.
\[MomentsShifts\] Fix $1/2<\sigma_0<1$. Let $s_1, s_2, \dots, s_k, r_1, \dots, r_{\ell}$ be complex numbers in the rectangle $\sigma_0<{\textup{Re}}(z)<1$ and $|{\textup{Im}}(z)|\leq T^{(\sigma_0-1/2)/4}$. Then, there exist positive constants $c_3, c_4, c_5 $ and a set $\mathcal{E}(T)\subset [T,2T]$ of measure $\ll T^{1-c_3}$, such that if $k, \ell \leq c_4\log T/\log\log T$ then $$\begin{aligned}
&\frac{1}{T} \int_{[T,2T]\setminus \mathcal{E}(T)}\left(\prod_{j=1}^k\log\zeta(s_j+it)\right)\left(\prod_{j=1}^{\ell}\log\zeta(r_j-it)\right)dt\\
&= {\mathbb{E}}\left(\left(\prod_{j=1}^k\log\zeta(s_j,X)\right)\left(\prod_{j=1}^{\ell}\log\overline{\zeta(r_j, X)}\right)\right)+ O\left(T^{-c_5}\right).\end{aligned}$$
Having obtained the moments we are in position to understand the characteristic function, $$\Phi_T(\mathbf{u}, \mathbf{v}):= \frac1T \int_T^{2T} \exp\left(i\left(\sum_{j=1}^J (u_j {\textup{Re}}\log\zeta(s_j+it)+ v_j {\textup{Im}}\log\zeta(s_j+it))\right)\right)dt,$$ where $\mathbf{u}=(u_1,\dots, u_J)\in \mathbb{R}^J$ and $\mathbf{v}=(v_1,\dots, v_J)\in \mathbb{R}^J$. We relate the above characteristic function to the characteristic function of the probabilistic model, $$\Phi_{\textup{rand}}(\mathbf{u}, \mathbf{v}):= {\mathbb{E}}\left( \exp\left(i\left(\sum_{j=1}^J (u_j {\textup{Re}}\log\zeta(s_j, X)+ v_j {\textup{Im}}\log\zeta(s_j, X))\right)\right)\right).$$ This is obtained in the following theorem, which we prove in Section \[sec:characteristic\].
\[characteristic\] Fix $1/2<\sigma<1$. Let $T$ be large and $J\leq (\log T)^{\sigma}$ be a positive integer. Let $s_1, s_2, \dots, s_J$ be complex numbers such that $\min({\textup{Re}}(s_j))=\sigma$ and $\max(|{\textup{Im}}(s_j)|)<T^{(\sigma-1/2)/4}$. Then, there exist positive constants $c_1(\sigma), c_2(\sigma)$, such that for all $ \mathbf{u}, \mathbf{v} \in \mathbb{R}^J$ such that $\max(|u_j|), \max(|v_j|)\leq c_1(\sigma) (\log T)^{\sigma}/J$ we have $$\Phi_T(\mathbf{u}, \mathbf{v})= \Phi_{\textup{rand}}(\mathbf{u}, \mathbf{v})+ O\left(
\exp\left(-c_2(\sigma)\frac{\log T}{\log\log T}\right)\right).$$
Using this result, we can show that the joint distribution of the shifts $\log \zeta(s_j+it)$ is very close to the corresponding joint distribution of the random variables $\log \zeta(s_j, X)$. The proof depends on Beurling-Selberg functions. To measure how close are these distributions, we introduce the discrepancy $\mathcal{D}_T(s_1, \ldots, s_J)$ defined as $$\begin{aligned}
\sup_{(\mathcal R_1, \ldots, \mathcal R_J) \subset \mathbb C^J} \bigg| \mathbb P_T\bigg( \log \zeta(s_j+it) \in \mathcal R_j, \forall j \le J \bigg)-
\mathbb P\bigg( \log \zeta(s_j, X) \in \mathcal R_j, \forall j \le J \bigg)\bigg| \end{aligned}$$ where the supremum is taken over all $(\mathcal R_1, \ldots, \mathcal R_J) \subset \mathbb C^J$ and for each $j=1,\ldots, J$ the set $\mathcal R_j$ is a rectangle with sides parallel to the coordinate axes. Our next theorem, proven in Section \[sec:distribution\], states a bound for the above discrepancy. This generalizes Theorem 1.1 of [@LLR], which corresponds to the special case $J=1$.
\[discrep\] Let $T$ be large, $\tfrac12<\sigma<1$ and $J \le (\log T)^{\sigma/2}$ be a positive integer. Let $s_1,s_2, \ldots, s_J$ be complex numbers such that $$\tfrac12 < \sigma:=\min_j({\ensuremath{\operatorname{Re}}}(s_j)) \le \max_j({\ensuremath{\operatorname{Re}}}(s_j)) <1
\quad \mbox{and} \quad \max_j(|{\ensuremath{\operatorname{Im}}}(s_j)|) < T^{ (\sigma-\frac12)/4}.$$ Then, we have $$\begin{aligned}
\mathcal{D}_T(s_1, \ldots, s_J) \ll \frac{J^2 }{(\log T)^{\sigma}}.\end{aligned}$$
With all of the above tools in place we are ready to prove Theorem \[thm:main\]. This is accomplished in the next section.
Effective universality: Proof of Theorem \[thm:main\] {#sec:proof}
=====================================================
In this section, we will prove Theorem \[thm:main\] using the results described in Section \[sec:propositions\]. First, by the maximum modulus principle, the maximum of $|\zeta(\tfrac 34 + it + z) -
f(z)|$ in the disc $\{z: |z|\leq r\}$ must occur on its boundary $\{z: |z|=r\}$. Our idea consists of first covering the circle $|z| = r$ with $J$ discs of radius $\varepsilon$ and centres $z_j$, where $z_j\in \{z: |z|=r\} $ for all $1\leq j\leq J$, and $J \asymp 1/\varepsilon$. We call each of the discs $\mathcal{D}_j$. Then, we observe that $$\label{ComparisonSupMax}
\max_{j\leq J}|\zeta(\tfrac 34 + it + z_j) -
f(z_j)|\leq \max_{|z| \leq r} |\zeta(\tfrac 34 + it + z) - f(z)|\leq \max_{j\leq J}\max_{z\in \mathcal{D}_j} |\zeta(\tfrac 34 + it + z) - f(z)|.$$ Using Proposition \[ControlDerivative\], we shall prove that for all $j\leq J$ (where $J$ is a small power of $\log T$) we have $$\max_{z\in \mathcal{D}_j} |\zeta(\tfrac 34 + it + z) - f(z)|\approx |\zeta(\tfrac 34 + it + z_j)-
f(z_j)|$$ for all $t\in [T, 2T]$ except for a set of points $t$ of very small measure. We will then deduce that the (weighted) distribution of $\max_{|z| \leq r} |\zeta(\tfrac 34 + it + z) - f(z)|$ is very close to the corresponding distribution of $\max_{j\leq J}|\zeta(\tfrac 34 + it + z_j)-
f(z_j)|$, for $t\in [T, 2T]$. We will also establish an analogous result for the random model $\zeta(s, X)$ along the same lines, by using Proposition \[DerRandom\] instead of Proposition \[ControlDerivative\]. Therefore, to complete the proof of Theorem \[thm:main\] we need to compare the distributions of $\max_{j\leq J}|\zeta(\tfrac 34 + it + z_j)-f(z_j)|$ and $\max_{ j\leq J}|\zeta(\tfrac 34 + z_j, X)-
f(z_j)|$. Using Theorem \[discrep\] we prove
\[DistributionMax\] Let $T$ be large, $0<r<1/4$ and $J\leq (\log T)^{(3/4-r)/7}$ be a positive integer. Let $z_1, \dots, z_J$ be complex numbers such that $|z_j|\leq r$. Then we have $$\begin{aligned}
&\left|\mathbb{P}_T\left(\max_{j\leq J}|\zeta(\tfrac 34 + it + z_j)-
f(z_j)|\leq u\right)-\mathbb P\left(\max_{ j\leq J}|\zeta(\tfrac 34 + z_j, X)-
f(z_j)|\leq u\right)\right|\\
&\ll_u \frac{(J\log\log T)^{6/5}}{(\log T)^{(3/4-r)/5}}.\end{aligned}$$
Fix a positive real number $u$. Let $\mathcal{A}_J(T)$ be the set of those $t$ for which $|\arg\zeta(\tfrac 34 + it + z_j)|\leq \log\log T$ for every $j\leq J$. Since ${\textup{Re}}(\tfrac 34 + it + z_j)\geq \tfrac34-r$ and ${\textup{Im}}(\tfrac 34 + it + z_j)=t+O(1)$, then it follows from Theorem 1.1 and Remark 1 of [@La] that for each $j\leq J$ we have $$\label{LargeDeviationArg}
\mathbb{P}_T\left(|\arg \zeta(\tfrac 34 + it + z_j)|\geq \log\log T\right)
\ll \exp\left(-(\log\log T)^{(\tfrac14+r)^{-1}}\right)\ll \frac{1}{(\log T)^4}.$$ Therefore, we obtain $$\mathbb{P}_T\left([T, 2T]\setminus\mathcal{A}_J(T)\right)\leq \sum_{j=1}^J
\mathbb{P}_T\left(|\arg \zeta(\tfrac 34 + it + z_j)|\geq \log\log T\right)\ll \frac{J}{(\log T)^4}\ll \frac{1}{(\log T)^2},$$ and this implies that $$\label{mainterm}
\begin{aligned}\mathbb{P}_T\left(\max_{j\leq J}|\zeta(\tfrac 34 + it + z_j)-
f(z_j)|\leq u\right)&=\mathbb{P}_T \left(\max_{ j\leq J} |\zeta(\tfrac 34 + it + z_j) - f(z_j)
| \leq u \ , \ t \in \mathcal{A}_J(T)\right)\\
&+O\left(\frac{1}{(\log T)^2}\right).
\end{aligned}$$
For each $j\leq J$ consider the region $$\mathcal U_j=\left\{ z:
|e^z - f(z_j)| \leq u \ , \ |{\textup{Im}}(z) | \leq \log\log T \right\}.$$ We cover $\mathcal U_j$ with $K \asymp {\ensuremath{\operatorname{area}}}(\mathcal U_j) / \varepsilon^2 \asymp \log \log T/\varepsilon^2$ squares $\mathcal{R}_{j, k}$ with sides of length $\varepsilon=\varepsilon(T)$, where $\varepsilon$ is a small positive parameter to be chosen later. Let $\mathcal K_j$ denote the set of $k \in \{1, 2, \ldots, K\}$ such that the intersection of $\mathcal R_{j,k}$ with the boundary of $\mathcal U_j$ is empty and write $\mathcal K_j^c$ for the relative complement of $\mathcal K_j$ with respect to $\{1, 2, \ldots, K\}$. Note that $ |\mathcal K_j^c| \asymp \log \log T/\varepsilon$. By construction, $$\left( \bigcup_{k \in \mathcal K_j} \mathcal R_{j,k} \right)
\subset
\mathcal U_j \subset \left( \bigcup_{k \le K} \mathcal R_{j,k} \right).$$ Therefore (\[mainterm\]) can be expressed as $$\mathbb{P}_T \Big ( \forall j \leq J, \forall k \leq K:
\log \zeta(\tfrac 34 + it + z_j) \in \mathcal{R}_{j,k} \Big) +
\mathcal{E}_1$$ where by Theorem \[discrep\] $$\begin{split}
\mathcal{E}_1 \ll & \sum_{j \leq J} \sum_{k \in \mathcal K_j^c}
\mathbb{P}_{T} \Big ( \log \zeta(\tfrac 34 + it + z_j) \in \
\mathcal{R}_{j,k} \Big ) \\
\ll& \sum_{j \leq J} \sum_{k \in \mathcal K_j^c} \left(
\mathbb{P}_{T} \Big ( \log \zeta(\tfrac 34+z_j,X) \in \
\mathcal{R}_{j,k} \Big ) +\frac{1}{(\log T)^{3/4-r}} \right) \\
\ll& J \cdot \frac{ \log \log T}{\varepsilon} \left( \varepsilon^2+\frac{1}{ (\log T)^{3/4-r}} \right),
\end{split}$$ and in the last step we used the fact that $\log \zeta(s, X)$ is an absolutely continuous random variable (see for example Jessen and Wintner [@JeWi]). We conclude that $$\label{CoveringRectanglesZeta}
\begin{aligned}
\mathbb{P}_T \Big (\max_{j\leq J} |\zeta(\tfrac 34 + it + z_j) - f(z_j)
| \leq u \Big) & = \mathbb{P}_T \Big ( \forall j \leq J,
\forall k \leq K:
\log \zeta(\tfrac 34 + it + z_j) \in \mathcal{R}_{j,k} \Big ) \\
& + O\left( \varepsilon J \log\log T + \frac{J\log\log T}{\varepsilon (\log T)^{3/4-r}} \right).
\end{aligned}$$ Additionally, it follows from Theorem \[discrep\] that the main term of this last estimate equals $$\label{eq:mainterm}
\mathbb{P} \Big ( \forall j \leq J, \forall k \leq K:
\log \zeta(\tfrac 34 + z_j, X) \in \mathcal{R}_{j, k} \Big) +
O \left( \frac{J^2 (\log\log T)^2}{\varepsilon^4(\log T)^{3/4-r}} \right).$$
We now repeat the exact same argument but for the random model $\zeta(s, X)$ instead of the zeta function. In particular, instead of we shall use that $$\mathbb{P} \left(|\arg \zeta(\tfrac 34 + z_j, X)|\geq \log\log T\right)
\ll \exp\left(-(\log\log T)^{(\tfrac14+r)^{-1}}\right)\ll \frac{1}{(\log T)^4},$$ which follows from Theorem 1.9 of [@La]. Thus, similarly to we obtain $$\begin{aligned}
\mathbb{P} \Big ( \forall j \leq J, \forall k \leq K:
\log \zeta(\tfrac 34 + z_j, X) \in \mathcal{R}_{j, k} \Big) &= \mathbb{P}
\left(\max_{j\leq J} |\zeta(\tfrac 34 + z_j, X) - f(z_j)
| \leq u\right)\\
& + O\left( \varepsilon J \log\log T + \frac{J\log\log T}{\varepsilon (\log T)^{3/4-r}} \right).\end{aligned}$$ Combining the above estimate with and we conclude that $$\begin{aligned}
\mathbb{P}_T \Big (\max_{j\leq J} |\zeta(\tfrac 34 + it + z_j) - f(z_j)
| \leq u \Big) & = \mathbb{P}
\left(\max_{j\leq J} |\zeta(\tfrac 34 + z_j, X) - f(z_j)
| \leq u\right) \\
& + O\left( \frac{J^2 (\log\log T)^2}{\varepsilon^4(\log T)^{3/4-r}}+ \varepsilon J \log\log T\right).\end{aligned}$$ Finally, choosing $$\varepsilon= \left(\frac{J\log\log T}{(\log T)^{3/4-r}}\right)^{1/5}$$ completes the proof.
We wish to estimate $$\label{toestimate}
\frac{1}{T}\int_{T}^{2T}\omega\left(\max_{|z| \leq r} |\zeta(\tfrac 34 + it + z) - f(z)| \right)dt$$ with $f$ an analytic non-vanishing function, and where $\omega$ is a continuously differentiable function with compact support.
Recall that the maximum of $|\zeta(\tfrac 34 + it + z) -
f(z)|$ on the disc $\{z: |z|\leq r\}$ must occur on its boundary $\{z: |z|=r\}$, by the maximum modulus principle. Let $\varepsilon \leq (1/4-r)/4$ be a small positive parameter to be chosen later, and cover the circle $|z| = r$ with $J\asymp 1/\varepsilon$ discs $\mathcal{D}_j$ of radius $\varepsilon$ and centres $z_j$, where $z_j\in \{z: |z|=r\} $ for all $j\leq J$.
Let $\mathcal{S}_V(T)$ denote the set of those $t \in [T, 2T]$ such that $$\max_{|z| \leq (r+1/4)/2} |\zeta'(\tfrac 34 + it + z)| \leq e^V$$ where $V\leq \log\log T$ is a large parameter to be chosen later, and let $L := \max_{|z| \leq (r+1/4)/2} |f'(z)|$. Then for $t \in \mathcal{S}_V(T)$, and for all $z\in \mathcal{D}_j$ we have $$\label{SmallDistance}
\begin{aligned}
& \Big|\zeta(\tfrac 34 +it+z)-f(z) - \big(\zeta(\tfrac 34 + it+ z_j)-f(z_j)\big)\Big|
= \left|\int_{z_j}^{z} \zeta'(\tfrac 34 + it+ s)-f'(s) ds\right| \\
& \leq |z - z_j| \cdot \left(\max_{|z| \leq (r+1/4)/2} |\zeta'(\tfrac 34 + it+ z)|+L\right)
\leq \varepsilon (e^V+L) \leq C\varepsilon e^V,
\end{aligned}$$ for some large absolute constant $C$, depending at most on $L$. Define $$\theta(t):= \max_{|z| \leq r} |\zeta(\tfrac 34 + it + z) - f(z)|- \max_{j\leq J}|\zeta(\tfrac 34 + it + z_j) -
f(z_j)|.$$ Then, it follows from and that for all $t\in \mathcal{S}_V(T)$ we have $$\label{BoundErrorSupMax}
0\leq \theta(t)\leq C\varepsilon e^V.$$ Therefore, using this estimate together with Proposition \[ControlDerivative\] and the fact that $\omega$ is bounded, we deduce that equals $$\label{SplitSmall}
\begin{aligned}
&\frac{1}{T}\int_{t\in \mathcal{S}_V(T)}\omega\left(\max_{j\leq J}|\zeta(\tfrac 34 + it + z_j)-
f(z_j)|+\theta(t)\right)dt+ O\left(e^{-V^2}\right)\\
&=\frac{1}{T}\int_{t\in \mathcal{S}_V(T)}\omega\left(\max_{ j\leq J}|\zeta(\tfrac 34 + it + z_j)-
f(z_j)|\right)dt+ O\left(|\mathcal{E}_2|+ e^{-V^2}\right)\\
&=\frac{1}{T}\int_{T}^{2T}\omega\left(\max_{j\leq J}|\zeta(\tfrac 34 + it + z_j)-
f(z_j)|\right)dt+ O\left(|\mathcal{E}_2|+ e^{-V^2}\right),\\
\end{aligned}$$ where $$\mathcal{E}_2= \frac{1}{T}\int_{t\in \mathcal{S}_V(T)} \int_0^{\theta(t)} \omega'\left(\max_{j\leq J}|\zeta(\tfrac 34 + it + z_j)-
f(z_j)|+x\right) dx \cdot dt
\ll \varepsilon e^V,$$ using the fact that $\omega'$ is bounded on $\mathbb{R}$ together with .
Furthermore, observe that $$\label{SmoothProb}
\begin{aligned}
&\frac{1}{T}\int_{T}^{2T}\omega\left(\max_{j\leq J}|\zeta(\tfrac 34 + it + z_j)-
f(z_j)|\right)dt\\
=&
-\frac{1}{T}\int_{T}^{2T}\int_{\max_{j\leq J}|\zeta(\frac 34 + it + z_j)-
f(z_j)|}^{\infty}\omega'(u) du \cdot dt\\
=&
-\int_{0}^{\infty}\omega'(u) \cdot \mathbb P_T\left(\max_{j\leq J}|\zeta(\tfrac 34 + it + z_j)-
f(z_j)|\leq u\right) du.
\end{aligned}$$ Since $\omega$ has a compact support, then $\omega'(u)=0$ if $u>A$ for some positive constant $A$. Furthermore, it follows from Proposition \[DistributionMax\] that for all $0\leq u\leq A$ we have $$\begin{aligned}
\mathbb P_T\left(\max_{j\leq J}|\zeta(\tfrac 34 + it + z_j)-
f(z_j)|\leq u\right)&= \mathbb P\left(\max_{j\leq J}|\zeta(\tfrac 34 + z_j, X)-
f(z_j)|\leq u\right)\\
& +O\left(\frac{(J\log\log T)^{6/5}}{(\log T)^{(3/4-r)/5}}\right).\end{aligned}$$ Inserting this estimate in gives that $$\label{ApproximationMAX}
\begin{aligned}
&\frac{1}{T}\int_{T}^{2T}\omega\left(\max_{j\leq J}|\zeta(\tfrac 34 + it + z_j)-
f(z_j)|\right)dt\\
&= -\int_{0}^{\infty}\omega'(u) \cdot \mathbb P\left(\max_{j\leq J}|\zeta(\tfrac 34 + z_j, X)-
f(z_j)|\leq u\right) du+ O\left(\frac{(J\log\log T)^{6/5}}{(\log T)^{(3/4-r)/5}}\right)\\
&= {\mathbb{E}}\left(\omega\left(\max_{j\leq J}|\zeta(\tfrac 34 + z_j, X)-
f(z_j)|\right) \right)+ O\left(\frac{(J\log\log T)^{6/5}}{(\log T)^{(3/4-r)/5}}\right).\\
\end{aligned}$$
To finish the proof, we shall appeal to the same argument used to establish , in order to compare the (weighted) distributions of $\max_{j\leq J}|\zeta(\tfrac 34 + z_j, X)-f(z_j)|$ and $\max_{|z|\leq r}|\zeta(\tfrac 34 + z, X)-
f(z)|$. Let $\mathcal{S}_V(X)$ denote the event corresponding to $$\max_{|z| \leq (r+1/4)/2} |\zeta'(\tfrac 34 + z, X)| \leq e^V,$$ and let $\mathcal{S}_V^c(X)$ be its complement. Then, it follows from Proposition \[DerRandom\] that $\mathbb{P}\big(\mathcal{S}_V^c(X)\big)\ll \exp(-V^2).$ Moreover, similarly to one can see that for all outcomes in $\mathcal{S}_V(X)$ we have, for all $z\in \mathcal{D}_j$ $$\Big|\zeta(\tfrac 34+z, X)-f(z) - \big(\zeta(\tfrac 34 + z_j, X)-f(z_j)\big)\Big|
= \left|\int_{z_j}^{z} \zeta'(\tfrac 34 + s, X)-f'(s) ds\right| \ll \varepsilon e^V.$$ Thus, since the maximum of $|\zeta(\tfrac 34 + z, X) - f(z)|$ for $|z|\leq r$ occurs (almost surely) on the boundary $|z|=r$, then following the argument leading to , we conclude that $$\begin{aligned}
&{\mathbb{E}}\left(\omega\left(\max_{|z|\leq r}|\zeta(\tfrac 34 + z, X)-
f(z)|\right)\right)\\
&={\mathbb{E}}\left( \mathbf 1_{\mathcal S_V(X)} \, \omega\left(\max_{|z|\leq r}|\zeta(\tfrac 34 + z, X)-
f(z)| \right) \right)+ O\left(e^{-V^2}\right)\\
&={\mathbb{E}}\left( \mathbf 1_{\mathcal S_V(X)} \, \omega\left(\max_{j\leq J}|\zeta(\tfrac 34 + z_j, X)-
f(z)| \right) \right)+ O\left(\varepsilon e^V+ e^{-V^2}\right)\\
&= {\mathbb{E}}\left(\omega\left(\max_{j\leq J}|\zeta(\tfrac 34 + z_j, X)-
f(z)|\right)\right)+ O\left(\varepsilon e^V+ e^{-V^2}\right).\\\end{aligned}$$ Finally, combining this estimate with and , and noting that $J\asymp 1/\varepsilon $ we deduce that $$\begin{aligned}
\frac{1}{T}\int_{T}^{2T}\omega\left(\max_{|z| \leq r} |\zeta(\tfrac 34 + it + z) - f(z)| \right)dt&={\mathbb{E}}\left(\omega\left(\max_{|z|\leq r}|\zeta(\tfrac 34 + z, X)-
f(z)|\right)\right)\\
&+O\left(\varepsilon e^V+ e^{-V^2}+O\left(\frac{(\log\log T)^{6/5}}{\varepsilon^{6/5}(\log T)^{(3/4-r)/5}}\right) \right).\end{aligned}$$ Choosing $\varepsilon=(\log T)^{-(3/4-r)/11}$ and $V=2\sqrt{\log\log T}$ completes the proof.
Controlling the derivatives of the zeta function and the random model: Proof of Propositions \[ControlDerivative\] and \[DerRandom\] {#sec:derivative}
====================================================================================================================================
By Cauchy’s theorem we have $$|\zeta'(\tfrac 34 + it + z)| \leq \frac{1}{\delta}\max_{|s-z|=\delta} |\zeta(\tfrac 34 + it + s)|,$$ and hence we get $$\label{Cauchy}
\max_{|z|\leq r} |\zeta'(\tfrac 34 + it + z)| \leq
\frac{1}{\delta} \max_{|s| \leq r + \delta} |\zeta(\tfrac 34 + it + s)|.$$ Therefore, it follows that $$\label{derivative}
\begin{split}
\mathbb{P}_T \left( \max_{|z| \leq r} |\zeta'(\tfrac 34 + it + z)| >
e^V \right) &\leq
\mathbb{P}_T \left( \max_{|s| \leq r + \delta} |\zeta(\tfrac 34 + it + s)| >
\delta e^V \right) \\
&=\mathbb{P}_T \left( \max_{|s| \leq r + \delta} \log |\zeta(\tfrac 34 + it + s)| >
V+\log \delta \right).
\end{split}$$ To bound the RHS we estimate large moments of $\log \zeta(\tfrac34+it+s)$. This is accomplished by approximating $\log \zeta(\tfrac 34 + it + s)$ by a short Dirichlet polynomial, uniformly for all $s$ in the disc $\{|s|\leq r+\delta\}$. Using zero density estimates and large sieve inequalities, we can show that such an approximation holds for all $t\in [T, 2T]$, except for an exceptional set of $t$’s with very small measure. We prove
\[UnifShortDirichlet\] Let $0<r<1/4$ be fixed, and $\delta= (1/4-r)/4$. Let $y\leq \log T$ be a real number. There exists a set $\mathcal{I}(T)\subset [T, 2T]$ with $\text{meas}(\mathcal{I}(T))\ll T^{1-\delta} y(\log T)^5$, such that for all $t\in [T, 2T]\setminus\mathcal{I}(T)$ and all $|s|\leq r+\delta$ we have $$\log \zeta(\tfrac 34 +it+s)=\sum_{ n \le y} \frac{\Lambda(n)}{n^{\frac34+it+s} \log n}+O\left(
\frac{(\log y)^2 \log T}{y^{(1/4-r)/2}}\right).$$
To prove this result, we need the following lemma from Granville and Soundararajan [@GrSo].
\[ApproxShortEuler\] Let $y\geq 2$ and $|t|\geq y+3$ be real numbers. Let $1/2\leq \sigma_0< 1$ and suppose that the rectangle $\{z: \sigma_0<\textup{Re}(z)\leq 1, |\textup{Im}(z)-t|\leq y+2\}$ is free of zeros of $\zeta(z)$. Then for any $\sigma$ with $\sigma_0+2/\log y<\sigma\leq 1$ we have $$\log \zeta(\sigma+it)=\sum_{n\leq y}\frac{\Lambda(n)}{n^{\sigma+it}\log n} +O\left(\log |t| \frac{(\log y)^2}{y^{\sigma-\sigma_0}}\right).$$
Let $\sigma_0= 1/2+\delta$. For $j=1, 2$ let $\mathcal{T}_j$ be the set of those $t \in [T, 2T]$ for which the rectangle $$\{ z : \sigma_0 < \textup{Re}(z) \leq 1 , |\textup{Im}(z) - t| < y+ 1+ j \}$$ is free of zeros of $\zeta(z)$. Then, note that $\mathcal{T}_2\subseteq \mathcal{T}_1$, and for all $t\in\mathcal{T}_2 $, we have $t+{\textup{Im}}(s)\in \mathcal{T}_1$ for all $|s|\leq r+\delta $. Hence, by Lemma \[ApproxShortEuler\] we have $$\log \zeta(\tfrac 34 +it+s) = \sum_{n \leq y} \frac{\Lambda(n)}{n^{3/4+ it+s}
\log n} + O \left( \frac{(\log y)^2 \log T}{y^{(1/4-r)/2}}\right),$$ for all $t\in \mathcal{T}_2$ and all $|s|\leq r+\delta$. Let $N(\sigma, T)$ be the number of zeros of $\zeta(s)$ in the rectangle $\sigma<{\textup{Re}}(s)\leq 1$ and $|{\textup{Im}}(s)|\leq T$. By the classical zero density estimate $N(\sigma, T)\ll T^{3/2-\sigma}(\log T)^5$ (see for example Theorem 9.19 A ot Titchmarsh [@Ti]) we deduce that the measure of the complement of $\mathcal{T}_2$ in $[T, 2T]$ is $\ll T^{1- \delta}y(\log T)^5$.
We also require a minor variant of Lemma 3.3 of [@LLR], whose proof we will omit.
\[lem:momentbd\] Fix $1/2<\sigma<1$, and let $s$ be a complex number such that ${\textup{Re}}(s)=\sigma,$ and $|{\textup{Im}}(s)|\le1$. Then, for any positive integer $k \le \log T/(3 \log y)$ we have $$\frac{1}{T} \int_{T}^{2T} \left| \sum_{n \le y} \frac{\Lambda(n)}{n^{s+it} \log n} \right|^{2k} \, dt \ll \left(\frac{c_8 k^{1-{\sigma}}}{(\log k)^{\sigma}} \right)^{2k}$$ and $$\mathbb E \left(\left| \log \zeta(s,X) \right|^{2k}\right)
\ll \left(\frac{c_8 k^{1-{\sigma}}}{(\log k)^{\sigma}} \right)^{2k}$$ for some positive constant $c_8$ that depends at most on $\sigma$.
Let $\delta=e^{-V/2}$. Taking $y=(\log T)^{5(1/4-r)^{-1}}$ in Lemma \[UnifShortDirichlet\] gives for all $t\in [T, 2T]$ except for a set with measure $\ll T^{1- (1/4-r)/5}$ that $$\label{ZeroDensityApp}
\log \zeta(\tfrac 34 +it+s) = \sum_{n \leq y} \frac{\Lambda(n)}{n^{3/4+ it+s}
\log n}+ O \left(\frac{1}{\log T}\right),$$ for all $|s|\leq r+\delta$. Furthermore, it follows from Cauchy’s integral formula that $$\left( \sum_{n \le y} \frac{\Lambda(n)}{n^{3/4+it+s} \log n} \right)^{2k}=
\frac{1}{2\pi i} \int_{|z|=r+2\delta} \left( \sum_{n \le y} \frac{\Lambda(n)}{n^{3/4+it+z} \log n} \right)^{2k} \frac{dz}{z-s}.$$ Applying Lemma \[lem:momentbd\] we get that $$\label{eq:cauchy}
\begin{split}
\frac{1}{T}
\int_T^{2T}
\left(\max_{|s| \le r+\delta} \left| \sum_{n \le y} \frac{\Lambda(n)}{n^{3/4+it+s} \log n} \right| \right)^{2k} \, dt
\ll & \frac{1}{\delta} \int_{|z|=r+2\delta} \frac{1}{T} \int_{T}^{2T} \left| \sum_{n \le y} \frac{\Lambda(n)}{n^{3/4+it+z} \log n} \right|^{2k} \, dt |dz| \\
\ll & e^{V/2} \left(c_8(r) \frac{k^{1-\sigma'(r)}}{(\log k)^{\sigma'(r)}} \right)^{2k}
\end{split}$$ where $\sigma'(r)=\tfrac34-r-2\delta$, and $k \le c_9 \log T/\log \log T$, for some sufficiently small constant $c_9>0$. We now choose $k=\lfloor c_6(r) V^{\frac{1}{(1-\sigma(r))}} (\log V)^{\frac{\sigma(r)}{1-\sigma(r)}} \rfloor$ (so that $k^{\sigma'(r)} \asymp k^{\sigma(r)}$) where $c_6(r)$ is a sufficiently small absolute constant. Using and along with Chebyshev’s inequality and the above estimate we conclude that there exists $c_7(r)>0$ such that $$\begin{split}
\mathbb P_T\bigg( \max_{|z| \le r} |\zeta'(\tfrac34+it+z)| > e^V \bigg)
\ll & \mathbb P_T\bigg( \max_{|s| \le r+\delta} \left| \sum_{n \le y} \frac{\Lambda(n)}{n^{3/4+it+s} \log n} \right| > \frac{V}{4} \bigg)+T^{1- (1/4-r)/5} \\
\ll & e^{V/2} \bigg(\frac{4}{V} \cdot c_8(r)\frac{ k^{1-\sigma'(r)}}{ (\log k)^{\sigma'(r)}}\bigg)^{2k}+T^{1- (1/4-r)/5}\\
\ll & \exp\left(-c_6 V^{\frac{1}{1-\sigma(r)}} (\log V)^{\frac{\sigma(r)}{1-\sigma(r)}}\right)
\end{split}$$ for $V \le c_7 (\log T)^{1-\sigma(r)}/\log \log T$.
We now prove Proposition \[DerRandom\] along the same lines. The proof is in fact easier than in the zeta function case, since we can compute the moments of $\log \zeta(s, X)$, for any $s$ with ${\textup{Re}}(s)>1/2$.
Let $\delta= e^{-V/2}$. Since $\zeta(\tfrac34+s, X)$ is almost surely analytic in $|s|\leq r+2\delta$, then by Cauchy’s estimate we have almost surely that $$\max_{|z|\leq r} |\zeta'(\tfrac 34 + z, X)| \leq
\frac{1}{\delta} \max_{|s|\leq r + \delta} |\zeta(\tfrac 34 + s, X)|.$$ Therefore, we obtain $$\label{derivativeRand}
\begin{split}
\mathbb{P} \left( \max_{|z|\leq r} |\zeta'(\tfrac 34 + z, X)| >
e^V \right)\leq &
\mathbb{P} \left( \max_{|s|\leq r+\delta} |\zeta(\tfrac 34 + s, X)| >
\delta e^V \right) \\
\le & \mathbb{P} \left( \max_{|s|\leq r+\delta} |\log \zeta(\tfrac 34 + s, X)| >
\frac{V}{2} \right).
\end{split}$$
Let $k$ be a positive integer. By $\log \zeta(\tfrac34+s,X)$ converges almost surely to a holomorphic function in $|s|\leq r+2\delta$. Using Cauchy’s integral formula as in , we obtain almost surely that $$\left(\max_{|s|\leq r+\delta} |\log \zeta(\tfrac 34 + s, X)|\right)^{2k}
\ll \frac{1}{\delta} \int_{|z|=r+2\delta} \left| \log \zeta(\tfrac 34 + z, X)\right|^{2k}\cdot |dz|.$$ Hence, applying Lemma \[lem:momentbd\] we get $$\label{BoundProbSubHarm}
\begin{aligned}
\mathbb{P} \left(\max_{|s|\leq r+\delta} |\log \zeta(\tfrac 34 + s, X)| >
V/2 \right)
&\leq \left( \frac{2}{V}\right)^{2k} \cdot {\mathbb{E}}\left(\left(\max_{|s|\leq r+\delta} |\zeta(\tfrac 34 + s, X)| \right)^{2k} \right)\\
& \ll \left( \frac{2}{V}\right)^{2k} e^{V/2} \int_{|z|=r+2\delta} {\mathbb{E}}\left(|\log \zeta(\tfrac 34 + z, X)|^{2k}\right) \cdot |dz| \\
& \ll e^{V/2} \left(\frac{2 c_8(r) k^{1-\sigma'(r)}}{V (\log k)^{\sigma'(r)}}\right)^{2k},
\end{aligned}$$ where $\sigma'(r)=\tfrac34-r-2\delta$. Let $\sigma(r)=\tfrac34-r$ and take $k=\lfloor c_6 V^{\frac{1}{1-\sigma(r)}} (\log V)^{\frac{\sigma(r)}{1-\sigma(r)}} \rfloor$, where $c_6$ is sufficiently small (note that $k^{\sigma'(r)} \asymp k^{\sigma(r)}$), then apply to complete the proof.
Moments of joint shifts of $\log \zeta(s)$: Proof of Theorem \[MomentsShifts\] {#sec:moments}
===============================================================================
The proof of Theorem \[MomentsShifts\] splits into two parts. In the first part we derive an approximation to $$\prod_{j = 1}^{k} \log \zeta(s_j + it)$$ by a short Dirichlet polynomial. In the second part we compute the resulting mean-values and obtain Theorem \[MomentsShifts\].
Approximating $\prod_{j = 1}^{k} \log \zeta(s_j + it)$ by short Dirichlet polynomials
-------------------------------------------------------------------------------------
Fix $1/2<\sigma_0<1$, and let $\delta:=\sigma_0-1/2$. Let $k\leq \log T$ be a positive integer and $s_1, s_2, \dots, s_k$ be complex numbers (not necessarily distinct) in the rectangle $\{ z: \sigma_0\leq {\textup{Re}}(z)<1, \text{ and }|{\textup{Im}}(z)|\leq T^{\delta/4}\}$. We let ${\mathbf s}=(s_1,\dots, s_k)$, and define $$F_{{\mathbf s}}(n)= \sum_{\substack{n_1,n_2,\dots,n_k\geq 2\\ n_1n_2\cdots n_k=n}}\prod_{\ell=1}^k \frac{\Lambda(n_{\ell})}{n_{\ell}^{s_{\ell}}\log(n_{\ell})}.$$ Then for all complex numbers $z$ with ${\textup{Re}}(z)>1-\sigma_0$ we have $$\prod_{\ell=1}^k\log\zeta(s_{\ell}+z)= \sum_{n=1}^{\infty}\frac{F_{{\mathbf s}}(n)}{n^z}.$$ The main result of this subsection is the following proposition.
\[Approx\] Let $T$ be large, $s_1, \dots, s_k$ be as above, and $\mathcal{E}(T)$ be as in Lemma \[ExceptionalSet\] below. Then, there exist positive constants $a(\sigma_0), b(\sigma_0)$ such that if $k\leq a(\sigma_0)(\log T)/\log\log T$ and $t\in [T, 2T]\setminus \mathcal{E}(T)$ then $$\prod_{j=1}^k\log\zeta(s_j+it) = \sum_{n\leq T^{\delta/8}} \frac{F_{\mathbf s}(n)}{n^{it}}+ O\left(T^{-b(\sigma_0)}\right).$$
This depends on a sequence of fairly standard lemmas which we now describe.
\[DirichletC\] With the same notation as above, we have $$|F_{{\mathbf s}}(n)|\leq \frac{(2\log n)^k}{n^{\sigma_0}}.$$
We have $$|F_{{\mathbf s}}(n)|\leq \frac{1}{n^{\sigma_0}(\log 2)^k}\sum_{\substack{n_1,n_2,\dots,n_k\geq 2\\ n_1n_2\cdots n_k=n}}\prod_{\ell=1}^k \Lambda(n_{\ell})\leq \frac{2^k}{n^{\sigma_0}}\left(\sum_{m|n}\Lambda (m)\right)^k\leq \frac{(2\log n)^k}{n^{\sigma_0}}.$$
\[BoundLogZ\] Let $y\geq 2$ and $|t|\geq y+3$ be real numbers. Suppose that the rectangle $\{z: \sigma_0-\delta/2<{\textup{Re}}(z)\leq 1, |{\textup{Im}}(z)-t|\leq y+2\}$ is free of zeros of $\zeta(z)$. Then, for all complex numbers $s$ such that ${\textup{Re}}(s)\geq \sigma_0-\delta/4$ and $|{\textup{Im}}(s)|\leq y$ we have $$\log\zeta(s+it)\ll_{\sigma_0} \log|t|.$$
This follows from Theorem 9.6 B of Titchmarsh.
\[ExceptionalSet\] Let $s_1, \dots, s_k$ be as above. Then, there exists a set $\mathcal{E}(T)\subset [T, 2T]$ with measure $\text{meas}(\mathcal{E}(T))\ll T^{1-\delta/8}$, and such that for all $t\in [T, 2T]\setminus \mathcal{E}(T)$ we have $\zeta(s_j+it+z)\neq 0$ for every $1\leq j\leq k$ and every $z$ in the rectangle $\{z: -\delta/2<{\textup{Re}}(z)\leq 1, |{\textup{Im}}(z)|\leq 3T^{\delta/4}\}$.
For every $1\leq j\leq k$, let $\mathcal{E}_j(T)$ be the set of $t\in [T, 2T]$ such that the rectangle $\{z: -\delta/2<{\textup{Re}}(z)\leq 1, |{\textup{Im}}(z)|\leq 3T^{\delta/4}\}$ has a zero of $\zeta(s_j+it+z)$. Then, by the classical zero density estimate $N(\sigma, T)\ll T^{3/2-\sigma}(\log T)^5$, we deduce that $$\textup{meas}(\mathcal{E}_j(T))\ll T^{\delta/4} T^{3/2-\sigma_0+\delta/2}(\log T)^5 < T^{1-\delta/4}(\log T)^5.$$ We take $\mathcal{E}(T)=\cup_{j=1}^k \mathcal{E}_j(T)$. Then $\mathcal{E}(T)$ satisfies the assumptions of the lemma, since $\textup{meas}(\mathcal{E}(T))\ll T^{1-\delta/4}(\log T)^6\ll T^{1-\delta/8}$.
We are now ready to prove Proposition \[Approx\].
Let $x=\lfloor T^{\delta/8}\rfloor +1/2$. Let $c=1-\sigma_0+ 1/\log T$, and $Y=T^{\delta/4}$. Then by Perron’s formula, we have for $t\in [T, 2T]\setminus \mathcal{E}(T)$ $$\frac{1}{2\pi i}\int_{c-iY}^{c+iY} \left(\prod_{j=1}^k\log\zeta(s_j+it+z)\right)\frac{x^z}{z}dz=
\sum_{n\leq x} \frac{F_{{\mathbf s}}(n)}{n^{it}}+ O\left(\frac{x^c}{Y}\sum_{n=1}^{\infty} \frac{|F_{{\mathbf s}}(n)|}{n^{c}|\log(x/n)|}\right).$$ To bound the error term of this last estimate, we split the sum into three parts: $n\leq x/2$, $x/2<n<2x$ and $n\geq 2x$. The terms in the first and third parts satisfy $|\log(x/n)|\geq \log 2$, and hence their contribution is $$\ll \frac{x^{1-\sigma_0}}{Y} \sum_{n=1}^{\infty}\frac{|F_{{\mathbf s}}(n)|}{n^{c}}\leq \frac{x^{1-\sigma_0}}{Y} \left(\sum_{n=1}^{\infty}\frac{\Lambda(n)}{n^{\sigma_0+c}\log n}\right)^k\leq \frac{x^{1-\sigma_0}(2\log T)^k}{Y}\ll T^{-b(\sigma_0)},$$ or some positive constant $b(\sigma_0)$, if $a(\sigma_0)$ is sufficiently small. To handle the contribution of the terms $x/2<n<2x$, we put $r=x-n$, and use that $|\log(x/n)|\gg |r|/x$. Then by Lemma \[DirichletC\] we deduce that the contribution of these terms is $$\ll \frac{x^{1-\sigma_0}(3\log x)^k}{Y}\sum_{r\leq x}\frac{1}{r}\ll \frac{x^{1-\sigma_0}(3\log x)^{k+1}}{Y}\ll T^{-b(\sigma_0)}.$$ We now move the contour to the line ${\textup{Re}}(s)=-\delta/4$. By Lemma \[ExceptionalSet\], we do not encounter any zeros of $\zeta(s_j+it+z)$ since $t\in [T, 2T]\setminus \mathcal{E}(T)$. We pick up a simple pole at $z=0$ which leaves a residue $\prod_{j=1}^k\log\zeta(s_j+it)$. Also Lemma \[BoundLogZ\] implies that for any $z$ on our contour we have $$|\log\zeta(s_j+it+z)|\leq c(\sigma_0) \log T,$$ for all $j$ where $c(\sigma_0)$ is a positive constant. Therefore, we deduce that $$\frac{1}{2\pi i}\int_{c-iY}^{c+iY} \left(\prod_{j=1}^k\log\zeta(s_j+it+z)\right)\frac{x^z}{z}dz= \prod_{j=1}^k\log\zeta(s_j+it) + E_1,$$ where $$\begin{aligned}
E_1&=\frac{1}{2\pi i} \left(\int_{c-iY}^{-\delta/4-iY}+ \int_{-\delta/4-iY}^{-\delta/4+iY}+ \int_{-\delta/4+iY}^{c+iY}\right) \left(\prod_{j=1}^k\log\zeta(s_j+it+z)\right)\frac{x^z}{z}dz\\
&\ll \frac{x^{1-\sigma_0}(c(\sigma_0)\log T)^k}{Y}+ x^{-\delta/4}(c(\sigma_0)\log T)^k\log Y\ll T^{-b(\sigma_0)},\end{aligned}$$ as desired.
An Asymptotic formula for the moment of products of shifts of $\log\zeta(s)$
----------------------------------------------------------------------------
Let $\mathcal{E}_1(T)$ and $\mathcal{E}_2(T)$ be the corresponding exceptional sets for ${\mathbf s}$ and ${\mathbf r}$ respectively as in Lemma \[ExceptionalSet\], and let $\mathcal{E}(T)= \mathcal{E}_1(T)\cup \mathcal{E}_2(T)$. First, note that if $t\in [T,2T]\setminus \mathcal{E}(T)$ then by Proposition \[Approx\] and Lemma \[BoundLogZ\] we have $$\left|\sum_{n\leq x} \frac{F_{\mathbf s}(n)}{n^{it}}\right|\ll (c(\sigma_0)\log T)^{k}, \text{ and } \left|\sum_{m\leq x} \frac{F_{\mathbf r}(m)}{m^{-it}}\right|\ll (c(\sigma_0))\log T)^{\ell},$$ for some positive constant $c(\sigma_0)$. Let $x=T^{(\sigma_0-1/2)/8}$. Then, it follows from Proposition \[Approx\] that $$\label{MomentsProduct}
\begin{aligned}
&\frac{1}{T} \int_{[T,2T]\setminus \mathcal{E}(T)}\left(\prod_{j=1}^k\log\zeta(s_j+it)\right)\left(\prod_{j=1}^{\ell}\log\zeta(r_j-it)\right)dt\\
&= \frac{1}{T} \int_{[T,2T]\setminus \mathcal{E}(T)} \left(\sum_{n\leq x} \frac{F_{\mathbf s}(n)}{n^{it}}\right)\left(\sum_{m\leq x} F_{\mathbf r}(m)m^{it}dt\right) dt + O\left(T^{-b(\sigma_0)}(\log T)^{\max(k, \ell)}\right)\\
&= \frac{1}{T} \int_T^{2T} \left(\sum_{n\leq x} \frac{F_{\mathbf s}(n)}{n^{it}}\right)\left(\sum_{m\leq x} F_{\mathbf r}(m)m^{it}\right)dt + O\left(T^{-b(\sigma_0)/2}\right).
\end{aligned}$$
Furthermore, we have
$$\label{eq:mvt}
\frac{1}{T} \int_T^{2T} \left(\sum_{n\leq x} \frac{F_{\mathbf s}(n)}{n^{it}}\right)\left(\sum_{m\leq x} F_{\mathbf r}(n)m^{it}\right)dt= \sum_{m,n\leq x} F_{\mathbf s}(n)F_{\mathbf r}(m)\frac{1}{T} \int_T^{2T}\left(\frac{m}{n}\right)^{it}dt.$$
The contribution of the diagonal terms $n=m$ equals $\sum_{n\leq x} F_{\mathbf s}(n)F_{\mathbf r}(n)$. On the other hand, by Lemma \[DirichletC\] the contribution of the off-diagonal terms $n\neq m$ is $$\label{eq:offdiag}
\ll \frac{1}{T}\sum_{\substack{m,n\leq x \\ m \neq n}} \frac{(2\log n)^k (2\log m)^{\ell}}{(mn)^{\sigma_0}}\frac{1}{|\log(m/n)|}\ll \frac{x^{3-2\sigma_0}(2\log x)^{k+\ell}}{T}\ll T^{-1/2},$$ since $|\log(m/n)|\gg 1/x$.
Furthermore, it follows from that $$\label{eq:randmvt}
{\mathbb{E}}\left(\prod_{j=1}^k\log\zeta(s_j,X)\right)\left(\prod_{j=1}^{\ell}\log\overline{\zeta(r_j, X)}\right)= \sum_{n=1}^{\infty} F_{\mathbf s}(n)F_{\mathbf{r}}(n)= \sum_{n\leq x} F_{\mathbf s}(n)F_{\mathbf{r}}(n)+E_2,$$ where $$E_2\leq \sum_{n>x}\frac{(2\log n)^{k+\ell}}{n^{2\sigma_0}}.$$ Since the function $(\log t)^{\beta}/t^{\alpha}$ is decreasing for $t>\exp(\beta/\alpha)$, then with the choice $\alpha=(2\sigma_0-1)/2$ we obtain $$E_2\leq \frac{(2\log x)^{k+\ell}}{x^{\alpha}}\sum_{n>x}\frac{1}{n^{1+\alpha}}\ll \frac{(2\log x)^{k+\ell}}{x^{2\alpha}}\ll x^{-\alpha}.$$ Combining this with , , and completes the proof.
The characteristic function of joint shifts of $\log \zeta(s)$ {#sec:characteristic}
==============================================================
Let $\mathcal{E}(T)$ be as in Theorem \[MomentsShifts\]. Let $N=[\log T/(C(\log\log T))]$ where $C$ is a suitably large constant. Then, $\Phi_T(\mathbf{u}, \mathbf{v})$ equals $$\begin{aligned}
\label{Taylor}
& \nonumber \frac1T \int_{[T, 2T]\setminus \mathcal{E}(T)} \exp\left(i\left(\sum_{j=1}^J (u_j {\textup{Re}}\log\zeta(s_j+it)+ v_j {\textup{Im}}\log\zeta (s_j+it))\right)\right)dt +O\left(T^{-c_3}\right)\\
& =\sum_{n=0}^{2N-1} \frac{i^n}{n!} \cdot \frac1T \int_{[T, 2T]\setminus \mathcal{E}(T)}\left(\sum_{j=1}^J (u_j {\textup{Re}}\log\zeta(s_j+it)+ v_j {\textup{Im}}\log\zeta (s_j+it))\right)^ndt + E_3,\end{aligned}$$ where $$E_3 \ll T^{-c_3}+ \frac{1}{(2N)!}\left(\frac{2c_1(\log T)^{\sigma}}{J}\right)^{2N}\frac1T \int_{[T, 2T]\setminus \mathcal{E}(T)} \left(\sum_{j=1}^J|\log\zeta(s_j+it)|\right)^{2N}dt.$$ Now, by Theorem \[MomentsShifts\] along with Lemma \[lem:momentbd\], we obtain that for all $1\leq j\leq J$ $$\label{BoundM}
\frac1T \int_{[T, 2T]\setminus \mathcal{E}(T)} |\log\zeta(s_j+it)|^{2N}dt \ll
{\mathbb{E}}\left(|\log\zeta(s_j, X)|^{2N}\right)\leq \left(\frac{c_8(\sigma) N^{1-\sigma}}{(\log N)^{\sigma}}\right)^{2N},$$ for some positive constant $c_8=c_8(\sigma).$ Furthermore, by Minkowski’s inequality we have $$\frac1T \int_{[T, 2T]\setminus \mathcal{E}(T)} \left(\sum_{j=1}^J|\log\zeta(s_j+it)|\right)^{2N}dt \leq \left(c_8 J \frac{N^{1-\sigma}}{(\log N)^{\sigma}}\right)^{2N}.$$ Therefore, we deduce that for some positive constant $c_{9}=c_{9}(\sigma)$, we have $$E_3\ll T^{-c_3} + \left(c_{9}\frac{(\log T)^{\sigma}}{(N\log N)^{\sigma}}\right)^{2N}\ll e^{-N}.$$
Next, we handle the main term of . Let $\tilde{u_j}=(u_j+iv_j)/2$ and $\tilde{v_j}=(u_j-iv_j)/2$. Then by Theorem \[MomentsShifts\] we obtain $$\begin{aligned}
&\frac1T \int_{[T, 2T]\setminus \mathcal{E}(T)}\left(\sum_{j=1}^J (u_j {\textup{Re}}\log\zeta(s_j+it)+ v_j {\textup{Im}}\log\zeta(s_j+it))\right)^ndt\\
&\frac1T \int_{[T, 2T]\setminus \mathcal{E}(T)}\left(\sum_{j=1}^J (\tilde{u_j} \log\zeta(s_j+it)+ \tilde{v_j} \log\zeta(s_j-it))\right)^ndt\\
&=\sum_{\substack{k_1,\dots, k_{2J}\geq 0\\ k_1+\cdots+k_{2J}=n}}{n\choose k_1, k_2, \dots, k_{2J}}\prod_{j=1}^J\tilde{u_j}^{k_j} \prod_{\ell=1}^J\tilde{v_{\ell}}^{k_{J+\ell}}\\
& \quad \quad \times \frac1T \int_{[T, 2T]\setminus \mathcal{E}(T)} \prod_{j=1}^J(\log\zeta(s_j+it))^{k_j}\prod_{\ell=1}^J (\log\zeta(s_\ell-it))^{k_{J+\ell}}dt\\
&= \sum_{\substack{k_1,\dots, k_{2J}\geq 0\\ k_1+\cdots+k_{2J}=n}}{n\choose k_1, k_2, \dots, k_{2J}}\prod_{j=1}^J\tilde{u_j}^{k_j} \prod_{\ell=1}^J\tilde{v_{\ell}}^{k_{J+\ell}}\\
& \quad \quad \times {\mathbb{E}}\left(\prod_{j=1}^J(\log\zeta(s_j, X))^{k_j}\prod_{\ell=1}^J (\log\zeta(s_\ell, X))^{k_{J+\ell}}\right) +O\Big(T^{-c_5} \big(2c_1(\log T)^{\sigma}\big)^n\Big),\\
&= {\mathbb{E}}\left(\left(\sum_{j=1}^J (u_j {\textup{Re}}\log\zeta(s_j, X)+ v_j {\textup{Im}}\log\zeta(s_j, X))\right)^n\right) +O\Big(T^{-c_5} \big(2c_1(\log T)^{\sigma}\big)^n\Big).\end{aligned}$$
Inserting this estimate in , we derive $$\begin{aligned}
\Phi_T(\mathbf{u}, \mathbf{v})&=\sum_{n=0}^{2N-1} \frac{i^n}{n!}{\mathbb{E}}\left(\left(\sum_{j=1}^J (u_j {\textup{Re}}\log\zeta(s_j, X)+ v_j {\textup{Im}}\log\zeta(s_j, X))\right)^n\right) + O\Big(e^{-N}\Big)\\ &= \Phi_{\text{rand}}(\mathbf{u}, \mathbf{v}) +E_4.\end{aligned}$$ where $$E_4\ll e^{-N} + \frac{1}{(2N)!}\left(\frac{2c_1(\log T)^{\sigma}}{J}\right)^{2N}{\mathbb{E}}\left(\left(\sum_{j=1}^J | \log\zeta(s_j, X)|\right)^{2N}\right)\ll e^{-N}$$ by and Minkowski’s inequality. This completes the proof.
Discrepancy estimates for the distribution of shifts {#sec:distribution}
====================================================
The deduction of Theorem \[discrep\] from Theorem \[characteristic\] uses Beurling-Selberg functions. For $z\in \mathbb C$ let $$H(z) =\bigg( \frac{\sin \pi z}{\pi} \bigg)^2 \bigg( \sum_{n=-\infty}^{\infty} \frac{{\ensuremath{\operatorname{sgn}}}(n)}{(z-n)^2}+\frac{2}{z}\bigg)
\qquad\mbox{and} \qquad K(z)=\Big(\frac{\sin \pi z}{\pi z}\Big)^2.$$ Beurling proved that the function $B^+(x)=H(x)+K(x)$ majorizes ${\ensuremath{\operatorname{sgn}}}(x)$ and its Fourier transform has restricted support in $(-1,1)$. Similarly, the function $B^-(x)=H(x)-K(x)$ minorizes ${\ensuremath{\operatorname{sgn}}}(x)$ and its Fourier transform has the same property (see Vaaler [@Vaaler] Lemma 5).
Let $\Delta>0$ and $a,b$ be real numbers with $a<b$. Take $\mathcal I=[a,b]$ and define $$F_{\mathcal I} (z)=\frac12 \Big(B^-(\Delta(z-a))+B^-(\Delta(b-z))\Big).$$ The function $F_{\mathcal I}$ has the following remarkable properties. First, it follows from the inequality $B^-(x) \le {\ensuremath{\operatorname{sgn}}}(x) \le B^+(x)$ that $$\label{l1 bd}
0 \le \mathbf 1_{\mathcal I}(x)- F_{\mathcal I}(x)\le K(\Delta(x-a))+K(\Delta(b-x)).$$ Additionally, one has $$\label{Fourier}
\widehat F_{\mathcal I}(\xi)=
\begin{cases}\widehat{ \mathbf 1}_{\mathcal I}(\xi)+O\Big(\frac{1}{\Delta} \Big) \mbox{ if } |\xi| < \Delta, \\
0 \mbox{ if } |\xi|\ge \Delta.
\end{cases}$$ The first estimate above follows from and the second follows from the fact that the Fourier transform of $B^-$ is supported in $(-1,1)$. Before proving Theorem \[discrep\] we first require the following lemmas.
\[lem:functionbd\] For $x \in \mathbb R$ we have $
|F_{\mathcal I}(x)| \le 1.
$
It suffices to prove the lemma for $\Delta=1$. Also, note that we only need to show that $F_{\mathcal I}(x) \ge -1$. From the identity $$\sum_{ n=-\infty}^{\infty} \frac{1}{(n-z)^2}=
\left(\frac{\pi}{\sin \pi z}\right)^2$$ it follows that for $y \ge 0$ $$\label{eq:Hid}
H(y)=1-K(y)G(y)$$ where $$G(y)=2y^2 \sum_{m=0}^{\infty} \frac{1}{(y+m)^2}-2y-1.$$ In Lemma 5 of [@Vaaler], Vaaler shows for $y \ge 0$ that $$\label{eq:Gbd}
0 \le G(y) \le 1.$$ Also, note that for each $m \ge 1$, and $0<y \le 1$ one has $\frac{m}{(y+m)^3} \le \int_{m-1}^m \frac{t}{(y+t)^3} \, dt$ so that for $0<y \le 1$ $$\label{eq:Gdec}
G'(y)=4y \sum_{m \ge 1} \frac{m}{(y+m)^3}-2 \le 4y \int_0^{\infty} \frac{t}{(y+t)^3} \, dt-2 = 0.$$
First consider the case $a\le x \le b$. By we get that in this range $$F_{\mathcal I}(x)=\frac12 \left(2- K(x-a)(G(x-a)+1)-K(b-x)(G(b-x)+1) \right),$$ which along with implies $F_{\mathcal I}(x) \ge -1$ for $a \le x \le b$. Now consider the case $x<a$. Since $H$ is an odd function and imply $$\begin{split}
F_{\mathcal I}(x)=& \frac12 \left(K(x-a)(G(a-x)-1)-K(b-x)(G(b-x)+1) \right) \\
\ge & \frac12\left( -K(x-a)-2K(x-b)\right),
\end{split}$$ which is $\ge -1$ if $K(x-b) \le 1/2$. If $K(x-b) \ge 1/2$ we also have $K(x-a)>K(x-b)$ and $0<b-x < 1$. By this and we have in this range as well that $$F_{\mathcal I}(x) \ge \frac12 \left( K(x-b)(G(a-x)-G(b-x)-2)\right) \ge -1.$$ Hence, $F_{\mathcal I}(x)\ge -1$ for $x<a$. The remaining case when $x>b$ follows from a similar argument.
\[upper bd\] Fix $1/2<\sigma<1$, and let $s$ be a complex number such that ${\ensuremath{\operatorname{Re}}}(s)=\sigma$ and $|{\ensuremath{\operatorname{Im}}}(s)| \le T^{\frac14\cdot(\sigma-\frac12)}$. Then there exists a positive constant $c_1(\sigma)$ such that for $|u| \le c_1(\sigma)(\log T)^{\sigma}$ we have $$\Phi_T(u,0) \ll \exp\left( \frac{-u}{5 \log u} \right) \quad
\mbox{ and } \quad \Phi_T(0,u) \ll \exp\left( \frac{-u}{5 \log u} \right).$$
By a straightforward modification of Lemma 6.3 of [@LLR] one has that $$\mathbb E \bigg( \exp\Big(i u {\ensuremath{\operatorname{Re}}} \log \zeta(s, X)\Big) \bigg) \ll \exp\bigg(-\frac{u}{5 \log u} \bigg),$$ and $$\mathbb E \bigg( \exp\Big(i u {\ensuremath{\operatorname{Im}}} \log \zeta(s, X)\Big) \bigg) \ll \exp\bigg(-\frac{u}{5 \log u} \bigg).$$ Using the first bound and applying Theorem \[characteristic\] with $J=1$ establishes the first claim. The second claim follows similarly by using the second bound and Theorem \[characteristic\].
First, we claim that it suffices to estimate the discrepancy over $(\mathcal R_1, \ldots, \mathcal R_J)$ such that for each $j$ we have $\mathcal R_j \subset [-\sqrt{\log T}, \sqrt{\log T}] \times [-\sqrt{\log T}, \sqrt{\log T}]$. To see this consider $( \widetilde{\mathcal R_1}, \ldots, \widetilde{\mathcal R_J})$ where $\widetilde{\mathcal R_j}=\mathcal R_j \cap [-\sqrt{\log T}, \sqrt{\log T}] \times [-\sqrt{\log T}, \sqrt{\log T}] $. It follows that $$\notag
\begin{split}
&\bigg|\mathbb P_T \bigg( \log \zeta(s_j+it) \in \mathcal R_j, \forall j \le J \bigg)
-\mathbb P_T \bigg( \log \zeta(s_1+it) \in \widetilde{\mathcal R_1},\log \zeta(s_j+it) \in \mathcal R_j, 2 \le j \le J \bigg)
\bigg| \\
&\ll \mathbb P_T \bigg( |\log \zeta(s_1+it)| \ge \sqrt{\log T} \bigg) \ll \exp\Big(-\sqrt{\log T}\Big),
\end{split}$$ where the last bound follows from Theorem 1.1 and Remark 1 of of [@La]. Repeating this argument gives $$\bigg|\mathbb P_T \bigg( \log \zeta(s_j+it) \in \widetilde{\mathcal R_j}, \forall j \le J )
-\mathbb P_T \bigg( \log \zeta(s_j+it) \in \mathcal R_j, \forall j \le J \bigg)
\bigg| \ll J \exp\Big(-\sqrt{\log T}\Big).$$ Similarly, $$\bigg|\mathbb P \bigg( \log \zeta(s_j,X) \in \widetilde{\mathcal R_j}, \forall j \le J )
-\mathbb P \bigg( \log \zeta(s_j,X) \in \mathcal R_j, \forall j \le J \bigg)
\bigg| \ll J \exp\Big(-\sqrt{\log T}\Big).$$ Hence, the error from restricting to $( \widetilde{\mathcal R_1}, \ldots, \widetilde{\mathcal R_J})$ is negligible and establishes the claim.
Let $\Delta=c_1(\sigma) (\log T)^{\sigma}/J$ and $\mathcal R_j=[a_j,b_j]\times[c_j, d_j]$ for $j=1, \ldots, J$, with $|b_j-a_j|,|d_j-c_j| \le 2\sqrt{\log T}$. Also, write $\mathcal I_j=[a_j,b_j]$ and $ \mathcal J_j=[c_j,d_j]$. By Fourier inversion, , and Theorem \[characteristic\] we have that $$\label{long est}
\begin{split}
&\frac1T \int_T^{2T} \prod_{j=1}^J F_{\mathcal I_j} \Big( {\ensuremath{\operatorname{Re}}} \log \zeta(s_j+it)\Big)
F_{\mathcal J_j}\Big( {\ensuremath{\operatorname{Im}}} \log \zeta(s_j+it)\Big) \, dt\\
&
=\int_{\mathbb R^{2J}} \bigg(\prod_{j=1}^J \widehat{F}_{\mathcal I_j} (u_j)
\widehat{F}_{\mathcal J_j}( v_j)\bigg) \Phi_T(\mathbf u, \mathbf v) \, d\mathbf u \, d\mathbf v \\
&
= \int\limits_{\substack{|u_j|,|v_j| \le \Delta \\ j=1,2, \ldots, J}} \bigg(\prod_{j=1}^J \widehat{F}_{\mathcal I_j} (u_j)
\widehat{F}_{\mathcal J_j}( v_j)\bigg) \Phi_{{\ensuremath{\operatorname{rand}}}}(\mathbf u, \mathbf v) \, d\mathbf u \, d\mathbf v +O\left(\left(2\Delta\sqrt{\log T}\right)^{2J} \exp\Big(-
\frac{c_2 \log T}{\log \log T}\Big)\right)\\
&
=\mathbb E \bigg( \prod_{j=1}^J F_{\mathcal I_j} \Big( {\ensuremath{\operatorname{Re}}} \log \zeta(s_j,X)\Big)
F_{\mathcal J_j}\Big( {\ensuremath{\operatorname{Im}}} \log \zeta(s_j,X)\Big) \bigg)+O\left( \exp\left(-
\frac{c_2 \log T}{2\log \log T}\right)\right).
\end{split}$$
Next note that $\widehat K(\xi)=\max(0,1-|\xi|)$. Applying Fourier inversion, Theorem \[characteristic\] with $J=1$, and Lemma \[upper bd\] we have that $$\notag
\begin{split}
\frac1T \int_T^{2T} K\Big( \Delta \cdot \Big({\ensuremath{\operatorname{Re}}} \log \zeta(s+it)-\alpha\Big)\Big) \, dt
=&\frac{1}{\Delta}\int_{-\Delta}^{\Delta}\Big(1-\frac{|\xi|}{\Delta}\Big) e^{-2\pi i \alpha \xi} \Phi_T(\xi,0) \, d\xi
\ll \frac{1}{\Delta},
\end{split}$$ where $\alpha$ is an arbitrary real number and $s \in \mathbb C$ satisfies $\sigma \le {\ensuremath{\operatorname{Re(s)}}} <1$ and $|{\ensuremath{\operatorname{Im}}}(s)|< T^{\frac14(\sigma-\frac12)}$. By this and we get that $$\label{K bd}
\frac1T \int_T^{2T} F_{\mathcal I_1}\Big({\ensuremath{\operatorname{Re}}} \log \zeta(s_1+it)\Big) \, dt
=\frac1T \int_{T}^{2T} \mathbf 1_{\mathcal I_1}\Big({\ensuremath{\operatorname{Re}}} \log \zeta(s_1+it)\Big) dt+O(1/\Delta).$$ Lemma \[lem:functionbd\] implies that $|F_{\mathcal I_j}(x)|, |F_{\mathcal J_j}(x)| \le 1$ for $j=1,\ldots, J$. Hence, by this and $$\notag
\begin{split}
&\frac1T \int_T^{2T} \prod_{j=1}^J F_{\mathcal I_j} \Big( {\ensuremath{\operatorname{Re}}} \log \zeta(s_j+it)\Big)
F_{\mathcal J_j}\Big( {\ensuremath{\operatorname{Im}}} \log \zeta(s_j+it)\Big) \, dt \\
&=\frac1T \int_T^{2T} \mathbf 1_{\mathcal I_1} \Big( {\ensuremath{\operatorname{Re}}} \log \zeta(s_j+it)\Big)
F_{\mathcal J_1}\Big( {\ensuremath{\operatorname{Im}}} \log \zeta(s_j+it)\Big) \\
&\qquad \qquad \qquad \times \prod_{j=2}^J F_{\mathcal I_j} \Big( {\ensuremath{\operatorname{Re}}} \log \zeta(s_j+it)\Big)
F_{\mathcal J_j}\Big( {\ensuremath{\operatorname{Im}}} \log \zeta(s_j+it)\Big) \, dt+O(1/\Delta).
\end{split}$$ Iterating this argument and using an analog of for ${\ensuremath{\operatorname{Im }}} \log \zeta(s+it)$, which is proved in the same way, gives $$\label{one}
\begin{split}
&\frac1T \int_T^{2T} \prod_{j=1}^J F_{\mathcal I_j} \Big( {\ensuremath{\operatorname{Re}}} \log \zeta(s_j+it)\Big)
F_{\mathcal J_j}\Big( {\ensuremath{\operatorname{Im}}} \log \zeta(s_j+it)\Big) \, dt \\
&\qquad \qquad \qquad
=\mathbb P_T\Bigg(\log \zeta(s_j+it) \in \mathcal R_j, \forall j \le J\bigg) +O\left(\frac{J}{\Delta}\right).
\end{split}$$ Similarly, it can be shown that $$\label{two}
\begin{aligned}
\mathbb E \bigg( \prod_{j=1}^J F_{\mathcal I_j} \Big( {\ensuremath{\operatorname{Re}}} \log \zeta(s_j,X)\Big)
F_{\mathcal J_j}\Big( {\ensuremath{\operatorname{Im}}} \log \zeta(s_j,X)\Big) \bigg)
&=\mathbb P\Bigg(\log \zeta(s_j,X) \in \mathcal R_j, \forall j \le J \bigg) \\
&\ \ \ +O\left(\frac{J}{\Delta}\right).
\end{aligned}$$ Using and in completes the proof.
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[^1]: The first and third authors are partially supported by Discovery Grants from the Natural Sciences and Engineering Research Council of Canada.
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---
abstract: 'We present a comprehensive study of the magnetic properties of the long-range ordered quasi-one dimensional $J_{1}$-$J_{2}$ systems with a newly developed torque equilibrium spin-wave expansion approach, which can describe the spin Casimir and magnon decay effects in a unified framework. While the framework does not lose the generality, our discussion will be restricted to two representative systems, each of which has only one type of inter-chain coupling ($J_{3}$ or $J_{4}$) and is referred to the $J_{3}$- or $J_{4}$-system respectively. In spite of the long-range spiral order, the dynamical properties of these systems turn out to be highly nontrivial due to the incommensurate noncollinear spin configuration and the strong quantum fluctuation effects enhanced by the frustration and low-dimensionality. Both the systems show prominent spin Casimir effects induced by the vacuum fluctuation of the spin waves and related modification of the ordering vector, Lifshitz point’s position and sublattice magnetization. In addition to these static properties, the dynamical behaviors of these systems are also remarkable. Significant and spontaneous magnon decay effects are manifested in the quantum corrections to the excitation spectrum, including the broadening of the spectrum linewidth and downward renormalization of the excitation energy. Furthermore, the excitation spectrum appears to be very sensitive to the types of the inter-chain coupling and manifests three distinct features: (i) the magnon decay patterns between $J_{3}$- and $J_{4}$-system are very different, (ii) the renormalized spectrum and the overall decay rate of the $J_{3}$- and $J_{4}$-systems show very different sensitivity to the magnetic anisotropy, and (iii) there is a nearly flat mode in the renormalized magnon spectrum of the $J_{4}$-system along the X-M direction. By adjusting the strength of magnetic anisotropy and varying the approximation scheme, it is revealed that these striking distinct features are quite robust and have deep connection with both the spin Casimir and the magnon decay effects. Thus these special consequences of the inter-chain coupling on the spin wave dynamics may be served as a set of probes for different types of inter-chain couplings in experiments. At last, to guide experimental measurements such as inelastic neutron scattering in realistic materials and complement our theoretical framework, we develop the analytical theory of the dynamical structure factor within the torque equilibrium formulism and provide the explicit results of the quasi-one dimensional $J_{1}$-$J_{2}$ systems.'
author:
- 'Z. Z. Du, H. M. Liu, Y. L. Xie, Q. H. Wang and J. -M. Liu'
title: |
Magnetic excitations in quasi-one dimensional helimagnets:\
Magnon decays and influence of the inter-chain interactions
---
Introduction
============
Frustration and related phenomena in low-dimensional quantum antiferromagnets have been a subject of great interest because of the elusive quest for magnetically disordered phases with highly entangled ground states: quantum spin liquids. Relentless efforts have been made along the way and a large number of frustrated magnetic systems featuring exotic phases have been discovered and studied. One of the simplest and most investigated frustrated systems is the one-dimensional magnets with a ferromagnetic (FM) nearest-neighbor interaction and an antiferromagnetic (AFM) next-nearest-neighbor interactions, both of which are of intra-chain type. The majority of theoretical studies on this model are devoted to the ground state phase diagram in the pure one-dimensional case, which includes various exotic quantum phases such as dimer, vector chiral, and spin multi-polar states. As a matter of fact, real compounds exhibit besides the significant frustrated intra-chain couplings also relatively weak inter-chain couplings, which can efficiently suppress the quantum fluctuation effect and lead to magnetic long-range order below a Neel temperature $T_{N}$. In this case, with the frustrated intra-chain couplings, the ground state of the system becomes the noncollinear long-range ordered one, which is usually considered to be classical thus has received relatively less attention especially for the excitation spectral properties. However, there is a significant variety of experimental systems related to this areas of interests, in many of which the excitation spectra have been investigated by various experiments including the inelastic neutron and resonant inelastic x-ray scattering.
On the other hand, in spite of the magnetic long-range order, the spectral properties of the quasi-one dimensional $J_{1}$-$J_{2}$ systems are actually highly nontrivial due to the strong quantum fluctuation effect enhanced by the frustrated intra-chain couplings and low-dimensionality. Additionally, there are several important features that make the dynamic properties of these systems significantly rich. First, the noncollinear nature of the spiral state causes the mixed transverse and longitudinal fluctuations, which can further induce the spontaneous magnon decay effects including the strong renormalization of the spin-wave spectrum and finite magnon lifetime even at zero temperature. As a consequence, the magnetic dynamics of the system are qualitatively different from the results obtained within the linear spin-wave theory (LSWT), which is usually used to analyze the experimental results of the excitation spectra. Second, although the quasi-one dimensionality of these systems indicates an intra-chain coupling dominated magnetic dynamics, the inter-chain coupling can be considerably important as well. However, different from the large intra-chain coupling which is known with reasonable precision, the accurate information about the inter-chain coupling is usually lacking and sometimes which type of the couplings is the leading one remains controversial due to frustration. One of the interesting questions is the dependence of the dynamic properties on the inter-chain couplings and whether this dependence is sensitive enough to serve as a probe of different types of inter-chain couplings. Last but not least, the incommensurate nature of the noncollinear state leads to the spin Casimir effect generated by the zero point fluctuation of the spin wave. The spin Casimir effect is actually quite general and has different physical meanings in different circumstances, a detailed explanation can be found in our previous work. In general, it represents the macroscopic force or torque that is generated by the vacuum fluctuation of quantum spin systems. In this article, the spin Casimir effect specifically means the quantum fluctuation induced modification of the ordering vector. In the presence of spin Casimir effect, the conventional spin-wave expansion scheme is plagued with various non-physical singularities and divergences, and the associated spin-wave analysis has to be performed within an alternative formulism.
In this paper we present a comprehensive study of the spin-wave dynamics of the long-range ordered quasi-one dimensional $J_{1}$-$J_{2}$ system with a newly developed spin-wave expansion approach: the torque equilibrium spin wave theory (TESWT). This approach can be considered as an extension of the conventional nonlinear spin-wave theory, in which the spin Casimir effect is treated as a self-consistent manner and the associated expansion results are free from such singularities and divergences. The details of this expansion formulism have been discussed at length in our previous work, which mainly focus on the spin Casimir effect induced singular and divergent problems in the conventional spin-wave expansion scheme and the explicit formulism of the torque equilibrium expansion approach. In the present work, on the other hand, we extend this expansion formulism to a more realistic multi-parameter case and systematically investigate both the static and dynamic magnetic properties of the systems with the combined spin Casimir and magnon decay effects. It represents a substantial headway towards a comprehensive and advanced theoretical framework of spin wave dynamics in more realistic one-dimensional antiferromagnets with intra-chain and inter-chain interactions. To investigate the consequence of different inter-chain couplings, we consider two most common types ($J_{3}$ and $J_{4}$) in weakly coupled chain systems (see Fig. 1). Furthermore, we also study the cases with different strengths of magnetic anisotropy to test the robustness and investigate the physical origin of the distinct features caused by different types of inter-chain couplings.
{width="8.6cm"}\
In prior to the presentation of our formulation and computations, a brief highlight of the main conclusion is given here. First, the spin Casimir torque induced modification of the ordering vector is obtained by solving the torque equilibrium equation within the one-parameter renormalization approximation. Surprisingly, the magnetic anisotropy induced shift of the FM/spiral quantum phase transition point (Lifshitz point) can be qualitatively manifested in the ordering vector results. As a comparison, standard calculation of this Lifshitz point modification is also performed within both the conventional spin-wave theory (CSWT) and TESWT. Other than that, the sublattice magnetization in each case is obtained within both spin-wave theories as well. Besides these static properties, the dynamic magnetic properties are also investigated carefully. The quantum corrections to the excitation spectrum are obtained using both the on-shell and off-shell approximations with the one-loop magnon self-energy and significant magnon decay effects are manifested. Furthermore, in both the on-shell and off-shell cases the spin-wave spectrum appears to be very sensitive to the types of the inter-chain couplings and the magnetic anisotropy. Interestingly, several remarkable distinct features manifest in the renormalized spectrum of the $J_{3}$- and $J_{4}$-systems, such as the qualitatively different decay patterns, the very dissimilar sensitivities to the magnetic anisotropy and the appearance of a nearly flat mode in the $J_{4}$-systems. Moreover, these features turn out to be deeply related to both the spin Casimir effect and magnon decay effect, and are expected to be robust. These features may be served as a set of probes for different types of inter-chain couplings.
A surprising but method dependent feature is the “sudden non-decay region” in the off-shell approximated spectrum of the isotropic $J_{4}$-system, which is further analyzed by introducing the poles function and two-magnon density of states. Our analysis indicates that the appearance of this “sudden non-decay region” is in fact a consequence of the degeneration between the “bonding” and “antibonding” single-magnon states, and thus likely only an artifact of our one-loop approximation. To verify the influence of the inter-chain couplings on the excitation spectrum and further clarify the methodology associated problems, the spectral function of each system is also obtained, in which the degeneration of single-magnon states is clearly demonstrated. Furthermore, to guide experimental inelastic neutron scattering measurements and complement our theoretical framework, we develop the analytical theory of the dynamical structure factor $\mathcal{S}(\textbf{k},\varepsilon)$ within the torque equilibrium formulism and provide the explicit results for $\mathcal{S}(\textbf{k},\varepsilon)$ of the quasi-one dimensional $J_{1}$-$J_{2}$ systems.
The rest of this paper is organized as follows. In Section II we give a short introduction to the quasi-one dimensional $J_{1}$-$J_{2}$ system and the corresponding model Hamiltonian that we shall investigate. Section III provides a brief review of the nonlinear spin-wave theory and the newly developed torque equilibrium extension. The spin Casimir torque induced modification of the ordering vector, the magnetic anisotropy induced shift of the quantum Lifshitz point and sublattice magnetization of each system is considered in the Section IV. Section V is devoted to the calculation of quantum corrections to the spin-wave spectrum within both the on-shell approximation and the off-shell one. And in Section VI we further investigate the spectral function to verify the excitation spectrum results and clarify the methodology associated problems. Additionally, we develop the analytical theory of the dynamical structure factor within the torque equilibrium formulism for completeness and present the explicit numerical results in Section VII. Finally, we draw our discussions and conclusions in Section VIII.
Model Hamiltonian
=================
As a minimum model, the quasi-one dimensional FM-AFM frustrated $J_{1}$-$J_{2}$ Heisenberg model can be realized in a wide range of materials. One of the best studied family of compounds is the edge-shared chain cuprates, which have attracted much interest recently due to the extremely rich phase diagram with spin multipolar phases observed in high magnetic field. Additionally, this fascinating family can be extended to a more general one, in which Cu is replaced with a general magnetic ion with d orbital and the frustrated $J_{1}$-$J_{2}$ Heisenberg model is realized as follows. The coupling $J_{1}$ is mediated by the superexchange through the $p$ orbital of the O$^{2-}$ ions and thus strongly depends on the M-O-M bond angle $\theta$. For $\theta$=$90^{\circ}$, the superexchange process via O$^{2-}$ ions requires the exchange through quasi-orthogonal orbitals on O$^{2-}$, which dictates that the coupling is weakly ferromagnetic. However, for a structure with $\theta$ distorted away from this high symmetry, the AFM exchange coupling becomes stronger and consequently $J_{1}$ turns from the FM coupling to the AFM coupling at some critical angle $\theta_{c}$. On the other hand, the coupling $J_{2}$ is mediated by the super-superexchange through M-O-O-M path, and thus is usually AFM with small magnitude that comparable to $J_{1}$. Usually, this model can also be considered as a spin ladder with frustrated zigzag coupling, as shown in the right panel of Fig. 2.
{width="8cm"}\
In addition to the frustrated intra-chain coupling, an inter-chain coupling is unavoidably present in real materials. In spite of its weakness in quasi-one dimensional systems, the decisive role of the inter-chain coupling in suppressing the quantum fluctuation is well-known from the Mermin-Wagner-Coleman theorem. In the absence of frustration, the inter-chain coupling can be determined quite accurately through analyzing T$_{N}$ with Quantum Monte Carlo (QMC) studies. While in the cases with non-negligible frustration, the QMC is hindered by the so-called sign problem, and a determination of the exchange parameters becomes more difficult. As a consequence, in a frustrated quasi-one dimensional system, accurate information about the inter-chain couplings is usually lacking. Thus, it is instructive to investigate the dependence of dynamic properties on the inter-chain coupling and see whether this dependence is sensitive enough to serve as a probe of different types of inter-chain couplings.
In this paper, we employ a rather simple effective model for the edge-shared chain magnets with the Hamiltonian written as $$\hat{\mathcal{H}}=\hat{\mathcal{H}}_{\parallel}+\hat{\mathcal{H}}_{\perp}+\hat{\mathcal{H}}_{\Delta}$$ where $$\begin{aligned}
&&\hat{\mathcal{H}}_{\parallel}=\sum\limits_{i}J_{1}\textbf{S}_{i}\cdot\textbf{S}_{i+a}+J_{2}\textbf{S}_{i}\cdot\textbf{S}_{i+2a} \nonumber\\
&&\hat{\mathcal{H}}_{\perp}=\sum\limits_{i}J_{3}\textbf{S}_{i}\cdot\textbf{S}_{i+c}+J_{4}(\textbf{S}_{i}\cdot\textbf{S}_{i+a+c}+\textbf{S}_{i}\cdot\textbf{S}_{i+a-c}) \nonumber\\
&&\hat{\mathcal{H}}_{\Delta}=(\Delta-1)\sum\limits_{i}J_{1}S^{b}_{i}S^{b}_{i+a}+J_{2}S^{b}_{i}S^{b}_{i+2a}\end{aligned}$$ Here $\hat{\mathcal{H}}_{\parallel}$ represents the frustrated $J_{1}$-$J_{2}$ Heisenberg chain with $J_{1}<0$ and $J_{2}>0$. And $\hat{\mathcal{H}}_{\perp}$ represents the inter-chain coupling, which includes two representative types ($J_{3}$ and $J_{4}$) as demonstrated in Fig. 1. For the sake of simplicity, both $J_{3}$ and $J_{4}$ are assumed to be ferromagnetic, although an extension to the AFM case is very straightforward. Other than that, we consider an extension of the frustrated $J_{1}$-$J_{2}$ Heisenberg model to the $XXZ$ model with anisotropy of the easy-plane type $(0\leq\Delta\leq 1)$. As we shall see, this term can efficiently suppress the magnon decay region despite it gives no contribution to the classical energy and does not affect the cubic magnon vertexes at all. Under this circumstance, the classical ground state of the system is a spiral state lying in the $a$-$c$ plane and the classical ordering vector $\textbf{Q}_{cl}$=$(Q_{cl},0,0)$ is given by $$Q_{cl}=\arccos\Big(-\frac{J_{1}+2J_{4}}{4J_{2}}\Big)$$ for $|J_{1}+2J_{4}|<4J_{2}$. For the cases with $|J_{1}+2J_{4}|\geq4J_{2}$, the ground state becomes the FM one. This result is determined by minimizing the classical ground state energy and plotted in the left panel of Fig. 2, which will be modified once the quantum fluctuation is considered.
For the sake of general interest and highlighting the consequence of different types of inter-chain couplings, we would like to choose the cases with $S$=1/2 as the main focus of this work, which corresponds to the edge-shared chain cuprates. However, we address that our theoretical framework is of generality and can be applied to systems with arbitrary spin length.
Torque Equilibrium Spin Wave Theory
===================================
The spin wave theory or spin-wave expansion approach is based on the assumption that a long range ordered state exists as the ground state and the quantum fluctuation about this classical saddle point is small. Thus, this theory is expected to be less effective for quantum spin systems with low dimensionality or strong frustration, in which quantum fluctuation becomes more important. Surprisingly though, this spin-wave expansion approach is proven to be quite successful in describing the zero-temperature physics of a number of frustrated low-dimensional spin systems. The calculation results show that the anhamonic terms which are usually considered to be weak are actually very important. Consequently, it is necessary to consider the nonlinear effects of the spin waves in these systems. However, the conventional spin-wave formulism breaks down generally for those noncollinear ordered systems where the spin Caimir effect causes the shift of the classical saddle point. To fix this issue, we have developed a modified spin-wave expansion approach named as TESWT in Ref. 31. In this approach, the spin Casimir effect is treated in a self-consistent way, and the spin-wave expansion results are free from singularities and divergences and consistent with previous numerical results.
In the subsequent two subsections we discuss the standard noncollinear spin wave expansion approach and the torque equilibrium formulism. Here we use the basic notations as used in Ref. 31, in which an explicit explanation and derivation of these formulas can be found.
Nonlinear spin wave theory
--------------------------
The standard noncollinear spin wave theory begins by rewriting the spin-$S$ magnetic Hamiltonian for the system from the laboratory frame $(a,b,c)$ to the twisted frame $(x,y,z)$ associated with the classical ground state configuration of the spins as $$\begin{aligned}
\hat{\mathcal{H}}&=&\sum\limits_{ij}\Big[\Delta J_{ij} S^{y}_{i}S^{y}_{j}+J_{ij}\cos\theta_{ij}(S^{x}_{i}S^{x}_{j}+S^{z}_{i}S^{z}_{j})\nonumber\\
&&+J_{ij}\sin\theta_{ij}(S^{x}_{i}S^{z}_{j}-S^{z}_{i}S^{x}_{j})\Big]\end{aligned}$$ with $\theta_{ij}$=$\theta_{j}$-$\theta_{i}$ is the angle between two neighboring spins, which is determined by the ordering vector of the system. And, without loss of generality, all spins are assumed to lie in the $x$-$z$ plane. Then, in proceeding with the hermite Holstein-Primakoff transformation of the spin operators into bosons, followed by the Bogolyubov transformation diagonalizing the harmonic part of the bosonic Hamiltonian, we obtained the following effective Hamiltonian: $$\begin{aligned}
\hat{\mathcal{H}}_{eff}&=&\sum\limits_{\textbf{k}}\Big[(2S\varepsilon_{\textbf{k}}+\delta\varepsilon_{\textbf{k}})b^{\dagger}_{\textbf{k}}b_{\textbf{k}}-\frac{O_{\textbf{k}}}{2}(b_{\textbf{k}}b_{-\textbf{k}}+b^{\dagger}_{\textbf{k}}b^{\dagger}_{-\textbf{k}})\Big]\nonumber\\
&&+i\sqrt{2S}\sum\limits_{\textbf{k},\textbf{p}}\Big[\frac{1}{2!}\Gamma_{1}(\textbf{p},\textbf{k}-\textbf{p};\textbf{k})b_{\textbf{k}}b^{\dagger}_{\textbf{k}-\textbf{p}}b^{\dagger}_{\textbf{p}} \nonumber\\
&&+\frac{1}{3!}\Gamma_{2}(\textbf{p},-\textbf{k}-\textbf{p};\textbf{k})b^{\dagger}_{\textbf{p}}b^{\dagger}_{-\textbf{k}-\textbf{p}}b^{\dagger}_{\textbf{k}}-\textrm{H.c.}\Big]\end{aligned}$$ Here $\varepsilon_{\textbf{k}}$ represents the harmonic magnon energy spectrum and is given by $$\varepsilon_{\textbf{k}}=\sqrt{A^{2}_{\textbf{k}}-B^{2}_{\textbf{k}}}$$ with $$\begin{aligned}
A_{\textbf{k}}&=&\frac{1}{2}(\Delta J_{\textbf{k}}+\eta_{\textbf{k}}-2J_{\textbf{Q}})\nonumber\\
B_{\textbf{k}}&=&\frac{1}{2}(\Delta J_{\textbf{k}}-\eta_{\textbf{k}})\end{aligned}$$ and $$\begin{aligned}
J_{\textbf{k}}&=&J_{1}\cos k_{x}+J_{2}\cos2k_{x}+J_{3}\cos k_{y} \nonumber\\
&&+2J_{4}\cos k_{x}\cos k_{y} \nonumber\\
\eta_{\textbf{k}}&=&\frac{1}{2}(J_{\textbf{k}-\textbf{Q}}+J_{\textbf{k}+\textbf{Q}})\end{aligned}$$
The rest quadratic terms in the effective Hamiltonian come from the Hartree-Fock decoupling of the quartic interaction terms with $$\begin{aligned}
\delta\varepsilon_{\textbf{k}}&=&(u^{2}_{\textbf{k}}+v^{2}_{\textbf{k}})\delta A_{\textbf{k}}-2u_{\textbf{k}}v_{\textbf{k}}\delta B_{\textbf{k}} \nonumber\\
O_{\textbf{k}}&=&(u^{2}_{\textbf{k}}+v^{2}_{\textbf{k}})\delta B_{\textbf{k}}-2u_{\textbf{k}}v_{\textbf{k}}\delta A_{\textbf{k}}\end{aligned}$$ where $u_{\textbf{k}}$ and $v_{\textbf{k}}$ are the Bogolyubov transformation coefficients, which are under conditions $u^{2}_{\textbf{k}}-v^{2}_{\textbf{k}}=1$, $$u^{2}_{\textbf{k}}+v^{2}_{\textbf{k}}=\frac{A_{\textbf{k}}}{\varepsilon_{\textbf{k}}},~~~~2u_{\textbf{k}}v_{\textbf{k}}=\frac{B_{\textbf{k}}}{\varepsilon_{\textbf{k}}}$$ and $$\begin{aligned}
\delta A_{\textbf{k}}&=&A_{\textbf{k}}+\sum\limits_{\textbf{p}}\frac{1}{\varepsilon_{\textbf{p}}}\Big[A_{\textbf{p}}(A_{\textbf{k}-\textbf{p}}-A_{\textbf{k}}-A_{\textbf{p}}-B_{\textbf{k}-\textbf{p}}) \nonumber\\
&&+B_{\textbf{p}}(\frac{B_{\textbf{k}}}{2}+B_{\textbf{p}})\Big]\nonumber\\
\delta B_{\textbf{k}}&=&B_{\textbf{k}}-\sum\limits_{\textbf{p}}\frac{1}{\varepsilon_{\textbf{p}}}\Big[B_{\textbf{p}}(A_{\textbf{k}-\textbf{p}}-\frac{A_{\textbf{k}}}{2}-A_{\textbf{p}}-B_{\textbf{k}-\textbf{p}}) \nonumber\\
&&+A_{\textbf{p}}(B_{\textbf{k}}+B_{\textbf{p}})\Big]\end{aligned}$$ The cubic interaction terms which vanish in collinear magnetic systems are given by $$\begin{aligned}
\Gamma_{1}(\textbf{1},\textbf{2};\textbf{3})&=&\frac{-1}{2\xi}\Big[\zeta_{\textbf{1}}\kappa_{\textbf{1}}(\gamma_{\textbf{2}}\gamma_{\textbf{3}}+\kappa_{\textbf{2}}\kappa_{\textbf{3}})
+\zeta_{\textbf{2}}\kappa_{\textbf{2}}(\gamma_{\textbf{1}}\gamma_{\textbf{3}} \nonumber\\
&&+\kappa_{\textbf{1}}\kappa_{\textbf{3}})+\zeta_{\textbf{3}}\kappa_{\textbf{3}}(\gamma_{\textbf{1}}\gamma_{\textbf{2}}-\kappa_{\textbf{1}}\kappa_{\textbf{2}})\Big]
\nonumber\\
\nonumber\\
\Gamma_{2}(\textbf{1},\textbf{2};\textbf{3})&=&\frac{1}{2\xi}\Big[\zeta_{\textbf{1}}\kappa_{\textbf{1}}(\gamma_{\textbf{2}}\gamma_{\textbf{3}}-\kappa_{\textbf{2}}\kappa_{\textbf{3}})
+\zeta_{\textbf{2}}\kappa_{\textbf{2}}(\gamma_{\textbf{1}}\gamma_{\textbf{3}}
\nonumber\\
&&-\kappa_{\textbf{1}}\kappa_{\textbf{3}})+\zeta_{\textbf{3}}\kappa_{\textbf{3}}(\gamma_{\textbf{1}}\gamma_{\textbf{2}}-\kappa_{\textbf{1}}\kappa_{\textbf{2}})\Big]\end{aligned}$$ with $$\zeta_{\textbf{k}}=\frac{1}{2}(J_{\textbf{k}-\textbf{Q}}-J_{\textbf{k}+\textbf{Q}})$$ and $$\xi=\sqrt{\varepsilon_{\textbf{1}}\varepsilon_{\textbf{2}}\varepsilon_{\textbf{3}}},~~~
\kappa_{\textbf{i}}=\sqrt{A_{\textbf{i}}+B_{\textbf{i}}},~~~
\gamma_{\textbf{i}}=\sqrt{A_{\textbf{i}}-B_{\textbf{i}}}$$ where $\textbf{i}\in(\textbf{1},\textbf{2},\textbf{3})$ and $\textbf{1}, \textbf{2}...$ denote $\textbf{k}_{1}, \textbf{k}_{2}...$.
It is obvious that the spin-wave expansion contributes to the corrections of the ground state energy as well. With the vacuum energy modified from $E_{cl}$ to $E_{vac}$, the ordering vector of the system should be determined by minimizing $E_{vac}$ via $\delta E_{vac}/\delta \textbf{Q}=0$. The modification of the classical ordering vector is actually a shift of the classical saddle point due to the zero-point fluctuation. However, the harmonic spin-wave spectrum function $\varepsilon_{k}$ is only well-defined at $\textbf{Q}=\textbf{Q}_{cl}$, and thus the variation is normally treated approximately as an expansion around $\textbf{Q}_{cl}$. The conventional $1/S$ order expansion result is $$\textbf{Q}=\textbf{Q}_{cl}+\textbf{Q}_{1}$$ with $$\textbf{Q}_{1}=-\frac{1}{2S}\Bigg[\frac{\partial^{2}J_{\textbf{Q}}}{\partial \textbf{Q}^{2}}\Bigg]^{-1}
\sum\limits_{\textbf{k}}\frac{A_{\textbf{k}}+B_{\textbf{k}}}{\varepsilon_{\textbf{k}}}\cdot\frac{\partial J_{\textbf{k}+\textbf{Q}}}{\partial \textbf{Q}}\Bigg|_{\textbf{Q}_{cl}}$$ This result seems reasonable and is usually treated as the new ordering vector of the system. However, this direct expansion is actually divergent at the spiral/Neel Lifshitz point, thus can not be considered as a correction to the classical ordering vector. More than that, this direct expansion procedure further leads to various divergent results and thus invalidates the CSWT. As a consequence, some modifications have to be made to obtain a consistent spin-wave description of the system.
The torque equilibrium formulism
--------------------------------
The torque equilibrium formulism is based on the spin Casimir interpretation of the saddle point shifting problem. The spin Casimir torque that accounts for the shift of the saddle point is defined as $$\textbf{T}_{sc}(\textbf{Q})=\sum\limits_{\textbf{k}}\Bigg\langle\Psi_{vac}\Bigg|\frac{\partial\hat{\mathcal{H}}_{sw}}{\partial \textbf{Q}}\Bigg|\Psi_{vac}\Bigg\rangle$$ where $|\Psi_{vac}\rangle$ represents the quasi-particle vacuum state and $\hat{\mathcal{H}}_{sw}$ denotes the spin wave Hamiltonian before the Bogoliubov transformation. Notice that $\textbf{T}_{sc}$ is a function of $\textbf{Q}$ defined on bonds and represents the tendency of modification to the relative orientation of each spin. The saddle point condition is given by the torque equilibrium condition $$\textbf{T}_{cl}(\textbf{Q})+\textbf{T}_{sc}(\textbf{Q})=0$$ where $\textbf{T}_{cl}(\textbf{Q})$=$\partial E_{cl}/\partial\textbf{Q}$ represents the classical spin torque.
The core ingredient of the torque equilibrium formulism is to map the original spin system to a new spin system that has the same symmetry and set of exchange integrals as the original one. Consequently, this new spin system is nothing but the original one with a different parameter $J_{i}$ denoted as $\widetilde{J_{i}}$. The new system with $\widetilde{J_{i}}$ has classical ordering vector identical with the modified one in the old system $\widetilde{\textbf{Q}}_{cl}=\textbf{Q}$. Thus, the old system with shifted saddle point can be described by $$\hat{\mathcal{H}}_{sw}(J_{i},\textbf{Q})=\widetilde{\mathcal{H}}_{sw}(\widetilde{{J_{i}}},\textbf{Q})+\mathcal{H}_{sw}^{c}$$ with $$\mathcal{H}_{sw}^{c}=\hat{\mathcal{H}}_{sw}(J_{i},\textbf{Q})-\widetilde{\mathcal{H}}_{sw}(\widetilde{{J_{i}}},\textbf{Q})$$ Note that $\mathcal{H}_{sw}$ can be written as series of terms with different orders such as $\hat{\mathcal{H}}_{2}$,$\hat{\mathcal{H}}_{3}$,$\hat{\mathcal{H}}_{4}$ and so on, it is obvious that $\mathcal{H}_{sw}^{c}$ can be recast in the same form as well and its exact expression is fixed with physical renormalization conditions, which regularizes all the divergences as the counter-terms introduced in quantum field theory. The explicit proof of the divergent cancelation can be found in Ref. 31, and here we only list the final results for latter convenience. Given that we are only interested in the results at $1/S$ order, the higher order terms such as $\mathcal{H}^{c}_{3}$ and $\mathcal{H}^{c}_{4}$ can be neglected and we obtain $$\widetilde{\mathcal{H}}_{sw}=\widetilde{\mathcal{H}}_{2}+\mathcal{H}^{c}_{2}+\widetilde{\mathcal{H}}_{3}+\widetilde{\mathcal{H}}_{4}$$ where $\widetilde{R}$ represents $R(\widetilde{J_{i}},\textbf{Q})$ and $R^{c}$=$R-\widetilde{R}$. It is obvious that $\mathcal{H}^{c}_{2}$ is of the same order as $\widetilde{\mathcal{H}}_{4}$ and thus treated as perturbation. As a consequence, the spin Casimir torque in the torque equilibrium condition becomes $\widetilde{\textbf{T}}_{sc}(\textbf{Q})$ and the torque equilibrium equation turns into $$\textbf{T}_{cl}(\textbf{Q})+\widetilde{\textbf{T}}_{sc}(\textbf{Q})=0$$
Based on this renormalization condition and treating $\mathcal{H}_{2}^{c}$ as perturbation, the one-loop torque equilibrium effective Hamiltonian reads $$\begin{aligned}
\widetilde{\mathcal{H}}_{eff}&=&\sum\limits_{\textbf{k}}\Bigg\{(2S\widetilde{\varepsilon}_{\textbf{k}}+\delta\widetilde{\varepsilon}_{\textbf{k}})b^{\dagger}_{\textbf{k}}b_{\textbf{k}}
-\frac{\widetilde{O}_{\textbf{k}}}{2}(b_{\textbf{k}}b_{-\textbf{k}}+b^{\dagger}_{\textbf{k}}b^{\dagger}_{-\textbf{k}})\nonumber\\
&&+2S\Big[\varepsilon^{c}_{\textbf{k}}b^{\dagger}_{\textbf{k}}b_{\textbf{k}}
-\frac{O^{c}_{\textbf{k}}}{2}(b_{\textbf{k}}b_{-\textbf{k}}+b^{\dagger}_{\textbf{k}}b^{\dagger}_{-\textbf{k}})\Big]\Bigg\}\nonumber\\
\nonumber\\
&&+i\sqrt{2S}\sum\limits_{\textbf{k},\textbf{p}}\Big[\frac{1}{2!}\widetilde{\Gamma}_{1}(\textbf{p},\textbf{k}-\textbf{p};\textbf{k})b_{\textbf{k}}b^{\dagger}_{\textbf{k}-\textbf{p}}b^{\dagger}_{\textbf{p}} \nonumber\\
&&+\frac{1}{3!}\widetilde{\Gamma}_{2}(\textbf{p},-\textbf{k}-\textbf{p};\textbf{k})b^{\dagger}_{\textbf{p}}b^{\dagger}_{-\textbf{k}-\textbf{p}}b^{\dagger}_{\textbf{k}}-\textrm{H.c.}\Big]\end{aligned}$$ with $$\begin{aligned}
\varepsilon^{c}_{\textbf{k}}&=&(\widetilde{u}^{2}_{\textbf{k}}+\widetilde{v}^{2}_{\textbf{k}})A^{c}_{\textbf{k}}-2\widetilde{u}_{\textbf{k}}\widetilde{v}_{\textbf{k}}B^{c}_{\textbf{k}} \nonumber\\
O^{c}_{\textbf{k}}&=&(\widetilde{u}^{2}_{\textbf{k}}+\widetilde{v}^{2}_{\textbf{k}})B^{c}_{\textbf{k}}-2\widetilde{u}_{\textbf{k}}\widetilde{v}_{\textbf{k}}A^{c}_{\textbf{k}}\end{aligned}$$
According to this torque equilibrium effective Hamiltonian and the standard diagrammatic technique for bosons at zero temperature, the bare magnon propagator can be defined as $$G^{-1}_{0}(\textbf{k},\varepsilon)=\varepsilon-2S\widetilde{\varepsilon}_{\textbf{k}}+i0^{+}$$ Different from the standard nonlinear spin-wave expansion results, we obtain besides the two frequency independent Hartree-Fock contributions to the normal and anomalous self-energies $$\Sigma^{a}_{hf}(\textbf{k})=\delta\widetilde{\varepsilon}_{\textbf{k}},~~~
\Sigma^{b}_{hf}(\textbf{k})=-\widetilde{O}_{\textbf{k}}$$ another two frequency independent contributions $$\Sigma^{a}_{c}(\textbf{k})=2S\varepsilon^{c}_{\textbf{k}},~~~
\Sigma^{b}_{c}(\textbf{k})=-2SO^{c}_{\textbf{k}}$$ These two terms are actually of order $O(S^{0})$ in spite of the $2S$ coefficient. The last but not the least, the most important normal self-energies contributed from the cubic vertexes are $$\begin{aligned}
\Sigma^{a}_{3}(\textbf{k},\varepsilon)&=&\frac{1}{2}\sum_{\textbf{p}}\frac{|\widetilde{\Gamma}_{1}(\textbf{p};\textbf{k})|^{2}}
{\varepsilon-\widetilde{\varepsilon}_{\textbf{p}}-\widetilde{\varepsilon}_{\textbf{k}-\textbf{p}}+i0^{+}} \nonumber\\
\Sigma^{b}_{3}(\textbf{k},\varepsilon)&=&-\frac{1}{2}\sum_{\textbf{p}}\frac{|\widetilde{\Gamma}_{2}(\textbf{p};\textbf{k})|^{2}}
{\varepsilon+\widetilde{\varepsilon}_{\textbf{p}}+\widetilde{\varepsilon}_{\textbf{k}+\textbf{p}}-i0^{+}}\end{aligned}$$ and the anomalous self-energies contributed from the cubic vertexes are $$\begin{aligned}
\Sigma^{c}_{3}(\textbf{k},\varepsilon)&=&-\frac{1}{2}\sum_{\textbf{p}}
\frac{\widetilde{\Gamma}_{1}(-\textbf{k},\textbf{p})\widetilde{\Gamma}_{2}(\textbf{k},\textbf{p})}{\varepsilon+\widetilde{\varepsilon}_{\textbf{p}}+\widetilde{\varepsilon}_{\textbf{k}+\textbf{p}}-i0^{+}} \nonumber\\
\Sigma^{d}_{3}(\textbf{k},\varepsilon)&=&\frac{1}{2}\sum_{\textbf{p}}
\frac{\widetilde{\Gamma}_{1}(\textbf{k},\textbf{p})\widetilde{\Gamma}_{2}(-\textbf{k},\textbf{p})}{\varepsilon-\widetilde{\varepsilon}_{\textbf{p}}-\widetilde{\varepsilon}_{\textbf{k}-\textbf{p}}+i0^{+}}\end{aligned}$$ The diagrammatic representations of the these self-energies can be found in Fig.3 and Fig.5 of Ref. 31. All these lowest order results provide a basis for the systematic pertuabative calculations of various static and dynamic magnetic properties of the system.
Static properties
=================
The pure one dimensional $J_{1}$-$J_{2}$ model has been studied theoretically with much success over the last two decades relying on the availability of many exact results and the absence of the size constrain of the density matrix renormalization group (DMRG) calculation. As a consequence, most of the studies on the quasi-one dimensional models are carried out by perturbating the one dimensional results with weak inter-chain coupling using the bosonization method. On the contrary, the experimental results of realistic materials indicate that most of the quasi-one dimensional systems are in fact long range ordered at sufficiently low temperature. Thus, it seems that the spin wave description of the weakly coupled frustrated chain systems is legitimate at least at zero temperature with sufficiently strong inter-chain coupling. However, to the best of our knowledge, a reliable spin wave analysis of these systems is still lacking because the LSWT results are unreliable for neglecting the strong quantum fluctuation effect and the conventional $1/S$ expansion scheme is plagued with divergent problems due to the incommensurate noncollinear spin configuration. Under this circumstance, the fitting results of the experimental measurements obtained based on the conventional spin wave analysis turn out to be very inaccurate and can lead to controversial conclusions about the corresponding magnetic models.
In the subsequent three subsections, we shall investigate the static properties of the quasi-one dimensional $J_{1}$-$J_{2}$ systems within both the linear approximated CSWT and the TESWT. Our results indicate that the linear approximated TESWT provides a relatively more accurate description and thus may be served as an efficient parameter fitting tool for the incommensurate noncollinear ordered magnetic systems. The emergence of the spin Casimir torque and corresponding modification of the ordering vector are investigated in the first subsection. The standard calculation of the magnetic anisotropy induced shift of the FM/spiral Lifshitz point and the sublattice magnetization is performed in the second and last subsections respectively.
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Ordering vector
---------------
The quantum fluctuation induced renormalization of the classical ordering vector is a widespread phenomenon in noncollinear ordered antiferromagnets, especially for the systems with incommensurate spin correlation. This modification indicates that the quantum fluctuation can induce a shift of the saddle point due to the spin Casimir effect. To incorporate this effect within the spin wave theory, the spin torque equilibrium condition has to be fulfilled. From our previous definition, the spin Casimir torque in our system can be easily obtained as $$\textbf{T}_{sc}(\textbf{Q})
=\frac{S}{2}\sum\limits_{\textbf{k}}\frac{A_{\textbf{k}}+B_{\textbf{k}}}{\varepsilon_{\textbf{k}}}\cdot\frac{\partial J_{\textbf{k}+\textbf{Q}}}{\partial \textbf{Q}}$$ Note that $\textbf{T}_{sc}(\textbf{Q})$ represents the quantum fluctuation induced modification of the classical ordering vector and depends on all the parameters in the spin-wave Hamiltonian. As a consequence, the quantum fluctuation modified ordering vector $\textbf{Q}$ that is obtained by solving the torque equilibrium equation $$\frac{\partial J_{\textbf{Q}}}{\partial \textbf{Q}}=-\frac{1}{2S}\sum\limits_{\textbf{k}}\frac{\widetilde{A}_{\textbf{k}}+\widetilde{B}_{\textbf{k}}}{\widetilde{\varepsilon}_{\textbf{k}}}\cdot\frac{\partial \widetilde{J}_{\textbf{k}+\textbf{Q}}}{\partial \textbf{Q}}$$ also depends on all the parameters. Based on this observation, in principle, we should consider the renormalization of all these parameters when the torque equilibrium equation is solved. However, as argued in Ref. 31, the renormalization of more than one parameter is tedious and usually unnecessary. Thus, here we only consider the renormalization of the dominant intra-chain parameter $\alpha$=$J_{1}/J_{2}$ and neglect the renormalization of the other small parameters. This is expected to be a good approximation for the quasi-one dimensional cases where the intra-chain coupling is much greater than the inter-chain coupling. Additionally, the validity of this approximation can be easily verified by testing the sensitivity of the solution of the equation ($\widetilde{\alpha}$ and $\textbf{Q}$) to the other parameters.
Other than the verification of our approximation scheme, the dependence of the quantum ordering vector on various parameters is of interest on its own right. To explicitly show the role of each parameter, we consider two representative systems, each of which has only one type of inter-chain coupling. First, we consider the isotropic system with the direct FM inter-chain coupling $J_{3}$. The ordering vector in dependence of $J_{2}$ with different $J_{3}$ is demonstrated in the insert of Fig. 3(a). It is clearly shown that the ordering vector is drastically modified from it’s classical value by the quantum fluctuation effect, which indicates the necessity of including the spin Casimir contribution. Additionally, the quantum ordering vector $\textbf{Q}$ is mainly dependent of $J_{2}$ and does not show strong sensitivity to the inter-chain coupling and the results with different $J_{3}$ are nearly identical. As the magnetic anisotropy is introduced, the sensitivity of the quantum ordering vector to $\Delta$ is slightly stronger compared to $J_{3}$ especially near the FM/spiral Lifshitz point as shown in Fig. 3(b). Interestingly, this sensitivity is actually a consequence of the shift of the magnetic anisotropy induced modification of the Lifshitz point. This is different from the classical case, where the magnetic anisotropy does not affect the Lifshitz point at all. Furthermore, this is also different from the CSWT description, in which the magnetic anisotropy induced shift of the Lifshitz point can only be obtained by comparing the ground state energy as we shall discuss in the next subsection. Actually, it is a special property of the TESWT that the ordering vector results can indicate the shift of the phase boundary, such as the case in Ref. 31 the ordering vector can manifest a possible quantum order by disorder (QObD) effect. However, once $J_{2}$ is away from the Lifshitz point, this sensitivity is not obvious. Thus, our one parameter renormalization scheme turns out to be a good approximation for the $J_{3}$-systems if not too close to the Lifshitz point.
Next we discuss the systems with the crossed FM inter-chain coupling $J_{4}$. The situation here is more complicated than that in the $J_{3}$-systems because $J_{4}$ can induce modification of the ordering vector already at the classical level, as shown in Fig. 2. As a consequence, all the results we obtain on this system have mixed classical and quantum contributions. To eliminate the classical effect and show the quantum part more clearly, we also plot the difference between the quantum ordering vector $\textbf{Q}$ and its classical counterpart $\textbf{Q}_{cl}$. As shown in Fig. 3(c), the isotropic $J_{4}$-systems show relatively strong sensitivity to the inter-chain coupling than the $J_{3}$-systems. Nevertheless, the quantum corrections to the classical ordering vector can still be considered to be insensitive to the inter-chain coupling for cases of $J_{2}$ away from the Lifshitz point. On the other hand, the effect of the magnetic anisotropy on the $J_{4}$-systems shows very similar behavior to that on the $J_{3}$-systems away from the Lifshitz point. However, the behavior around the Lifshitz point is very different due to the mixed classical and quantum contribution to the modification of the Lifshitz point. To the convenience of comparison, we plot in Fig.4 the Lifshitz points’ position of each anisotropic $J_{3}$- and $J_{4}$-system that read from Fig. 3(b) and (d) to demonstrate the effect of the magnetic anisotropy on the modification of the Lifshitz point.
Combined with the results of these two representative systems, it is obvious that our one-parameter renormalization scheme can be considered as a good approximation for both situations, at least in regions away from the Lifshitz point. Moreover, our TESWT results show good consistency with previous numerical results obtained with the coupled cluster method, different from the classical prediction but less developed than one would expect from the pure one-dimensional results. This is exactly the situation that one usually encounters in the experimental parameter fitting processes for quasi-one dimensional incommensurate noncollinear ordered magnets: the LSWT results are too classical while the pure one dimensional DMRG results seem to be too quantum. Other than that, our TESWT predictions become identical with the CSWT results once the spin Casimir effect is absent. Thus it seems that our TESWT can provide a rather accurate prediction of the ordering vector in a general sense.
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FM/spiral Lifshitz point
------------------------
In both the quantum and classical models of the FM-AFM frustrated $J_{1}$-$J_{2}$ chain systems, the presence of a Lifshitz point at $J_{2}$=$|J_{1}|/4$ is well known. In the classical case, this Lifshitz point describes a zero temperature transition from the ferromagnetic state to spiral state, whose position is purely determined by the exchange couplings thus independent on the magnetic anisotropy. As the quantum fluctuation is considered, the magnetic long range order may break down and the Lifshitz point describes a general transition from a commensurate phase to incommensurate phase. Interestingly, in this case the position of this general Lifshitz point becomes sensitive to the magnetic anisotropy, as shown in both the analytical and numerical studies of the pure one-dimensional frustrated zigzag $XXZ$ model. In this subsection, we consider a relatively simple case where the long range magnetic order exists on both sides of the Lifshitz point within the framework of spin wave theory. As we have mentioned in the previous subsection, the phase boundary of different long range ordered states is deduced by comparing the energy of the states on each side. For the sake of the completeness of discussion and the consistency of description, the magnetic anisotropy induced shift of the Lifshitz point is obtained within both the conventional and torque equilibrium schemes.
In the ferromagnetic phase, the quantum fluctuation induced corrections to the ground state energy are absent and the energy reads $E_{FM}$=$S^{2}J_{\textbf{0}}$. On the other hand, the quantum fluctuation can remarkably modify the ground state energy in the spiral phase. In the conventional spin-wave approach, the first order corrected ground state energy in the spiral phase is $$E_{S}=S(S+1)J_{\textbf{Q}_{cl}}+S\sum_{\textbf{k}}\varepsilon_{\textbf{k}}$$ while in the torque equilibrium formulism it reads $$\widetilde{E}_{S}=S(S+1)J_{\textbf{Q}}+S\sum_{\textbf{k}}\widetilde{\varepsilon}_{\textbf{k}}$$ The main difference between the two expressions is that in the TESWT we use the quantum ordering vector $\textbf{Q}$ instead of the classical ones. Other than that, the vacuum fluctuation energy in the torque equilibrium formulism is considered with the renormalized parameters $\widetilde{J}_{i}$.
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The resultant phase diagram is shown in Fig. 4(a) and (b), corresponding to the anisotropic $J_{3}$- and $J_{4}$-system respectively. It is clearly shown that in the classical limit, the FM/spiral phase boundary does not depend on the anisotropic energy $\Delta$. At the same time, the magnetic anisotropy induced modification of the phase boundary is also demonstrated within the torque equilibrium spin-wave formulism: the Lifshitz point shifts towards the spiral phase as the anisotropic energy increases. Moreover, as we’ve mentioned in the last subsection, the ordering vector obtained through the torque equilibrium equation can also sense the anisotropic energy induced shift of the Lifshitz point, although the phase boundary is not quantitatively the same with the formal TESWT results. Surprisingly, however, the phase boundary is irrelevant with the $XXZ$ anisotropy within the conventional spin-wave description. And this situation persists even when higher order perturbation to the ground state energy is considered. Thus, the CSWT results indicate that the quantum phase boundary is exactly the same with the classical one. As a matter of fact, the spin wave theory can only offer a qualitative rather than quantitative description of a quantum critical point. Consequently, neither the result obtained within the conventional scheme nor that from the torque equilibrium scheme is the actual FM/spiral phase boundary. Nevertheless, the qualitative dependence of the Lifshitz point’s position on the magnetic anisotropy indicated by TESWT results should be correct. More than that, our TESWT results are close to the finite size calculation results of the one dimensional frustrated zigzag $XXZ$ model.
Sublattice magnetization
------------------------
Based on the quantum phase diagram that we have obtained, in this subsection we turn to the investigation of the sublattice magnetization. The sublattice magnetization defines the validity region of the spin wave representation, which usually serves as the order parameter for general long-range ordered magnetic states. In the classical limit, the linear spin wave approximation becomes exact and the sublattice magnetization is nothing but the spin length. The quantum fluctuation effect, on the other hand, tends to reduce the sublattice magnetization from its classical value in the long range ordered phases through the zero point fluctuation of spin waves until the breakdown of the magnetic ordered ground state. However, in the FM state, the zero point fluctuation does not exist, and thus correspondingly the sublattice magnetization always remains its classical value at zero temperature. As a consequence, here we only need to calculate the sublattice magnetization in the spiral phase.
In the spiral phase, as a matter of fact, an unbalanced spin Casimir torque can induce divergence in the second order correction to the sublattice magnetization in the spin wave analysis. As a result, the full one-loop calculation of the sublattice magnetization can only be performed in the torque equilibrium formulism. However, the full one-loop calculation of the sublattice magnetization is quite tedious while the resultant second order corrected result does not show much difference from the linear approximated results as shown in Ref. 31. Consequently, to investigate the multi-parameter dependence of the sublattice magnetization and to compare the results obtained within both CSWT and TESWT, here we only consider the linear spin wave results in both theories. In the conventional spin-wave approach, the first order corrected sublattice magnetization is $$\langle S\rangle=S\Big[1-\frac{1}{2S}\big(\sum_{\textbf{k}}\frac{A_{\textbf{k}}}{2\varepsilon_{\textbf{k}}}-1\big)\Big]$$ while in the torque equilibrium formulism it reads $$\langle \widetilde{S}\rangle=S\Big[1-\frac{1}{2S}\big(\sum_{\textbf{k}}\frac{\widetilde{A}_{\textbf{k}}}{2\widetilde{\varepsilon}_{\textbf{k}}}-1\big)\Big]$$ The main difference between these two expressions also lies in the choice of the ordering vector and the corresponding exchange parameters.
The numerical results are shown in Fig. 5, in which the sub-plots (a) and (b) correspond to the $J_{3}$-systems and the sub-plots (c) and (d) correspond to the $J_{4}$-systems. Different from the quantum ordering vector, the sublattice magnetization shows quite strong sensitivity to the inter-chain coupling and magnetic anisotropy. In the ferromagnetic phase, the sublattice magnetization is simply the spin length of the system and in our case $S$=1/2. As $J_{2}$ increases from the Lifshitz point, the sublattice magnetization of each system reduces due to strong quantum fluctuation effect enhanced by the onset of the intra-chain frustration. This reduction is more drastic in the isotropic $J_{4}$-systems with small inter-chain coupling, e.g. in the isotropic $J_{4}$-system with $J_{4}$=-0.1, the sublattice magnetization vanishes at $J_{2}$=2 (1.25) within the CSWT (TESWT) description. A zero sublattice magnetization represents the breakdown of the spin wave expansion around the present classical saddle point. And in our case, this indicates that the systems may run into a spin liquid state. Moreover, within both the conventional and torque equilibrium descriptions, there is a little kink near the Lifshitz point in the sublattice magnetization results of anisotropic $J_{3}$-and $J_{4}$-systems. Compared with the continuum results of the isotropic systems, the appearance of the kink may indicate that the spin wave description is inadequate for a shifted Lifshitz point. However, away from the Lifshitz point, both the CSWT and TESWT provide very reasonable results. The suppression of the quantum fluctuation is clearly shown as the anisotropic parameter $\Delta$ and inter-chain coupling increases. More than that, the $J_{4}$-systems always have relatively smaller sublattice magnetization compared with the $J_{3}$-systems with the same magnitude of inter-chain coupling and magnetic anisotropy. Thus, it seems that the quantum fluctuation is stronger in the $J_{4}$-systems compared with the $J_{3}$-systems due to the crossing inter-chain coupling.
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Other than these common features, the sublattice magnetization obtained in the TESWT is always smaller than that obtained through the CSWT for all the cases, which is similar to the situations in the study of anisotropic triangular AFM Heisenberg model. This fact confirms our previous estimation that the linear approximation within the TESWT is actually very close to the second order results. Furthermore, as the quantum fluctuation effect is suppressed by increasing the inter-chain coupling or magnetic anisotropy, the sublattice magnetization obtained within the TESWT becomes closer to the ones obtained in the CSWT, which indicates that these two spin wave theories share the same classical limit.
To summarize, the sublattice magnetization obtained within the torque equilibrium linear spin wave theory (TELSWT) is qualitatively consistent with but quantitatively smaller than the ones obtained in the LSWT. Thus the system appears to be less classical once the spin Casimir effect is taken into account. This may shed light on our understanding of the experimental fact that some edge-shared chain cuprates usually show relatively smaller sublattice magnetization than the linear spin wave prediction. Together with the more accurate ordering vector predictions, it seems that our torque equilibrium approach can provide a quite good description of the quasi one-dimensional $J_{1}$-$J_{2}$ systems within the simple linear approximation. Other than that, the TELSWT approach is much less technique-relevant compared with other sophisticated numerical and analytical methods and it has been proven to be quite accurate for the anisotropic triangular antiferromagnets. As a consequence, the TELSWT may serve as an efficient tool for experimental fitting processes of the exchange parameters, especially for the incommensurate ordered system with strong quantum fluctuation effects.
Spin-wave Spectrum
==================
Different from the static properties, the $1/S$ perturbative expansion results for the spin-wave spectrum have drastically qualitative difference from the results obtained by the linear approximation for noncollinear ordered antiferromagnets. The main reason of this fact lies in the anharmonic cubic interaction terms in the effective spin wave Hamiltonian, which are forbidden in the collinear ordered states because of the unbroken $U(1)$ symmetry. These anharmonic terms correspond to the coupling of the transverse and longitudinal fluctuations and can further lead to the zero temperature decays of magnon. This spontaneous magnon decay effect has been theoretically proposed to occur in several magnetic systems and experimentally observed in hexagonal manganites LuMnO$_{3}$ and quantum triangular antiferromagnets Ba$_{3}$CoSb$_{2}$O$_{9}$ very recently.
To get into this remarkable phenomenon, the perturbative expansion has to be performed at least at one-loop order to include the self-energies contributed from the cubic vertexes. However, the conventional spin wave expansion scheme is invalidated due to the spin Casimir effect in the incommensurate noncollinear magnetic system as explained in Ref. 31. As a consequence, the spin wave expansion has to be performed in the torque equilibrium formulism to keep the spin wave expansion procedure away from singularities and divergences. The dynamic properties of the system are expressed within the interacting normal magnon Green’s function $$G_{n}(\textbf{k},\varepsilon)=\big[\varepsilon-2S\widetilde{\varepsilon}_{\textbf{k}}-\Sigma^{tot}_{n}(\textbf{k},\varepsilon)\big]^{-1}$$ Here $\Sigma^{tot}_{n}(\textbf{k},\varepsilon)$ represents the total normal self-energy, which in the one-loop order reads $$\Sigma^{tot}_{n}(\textbf{k},\varepsilon)=\Sigma^{a}_{c}(\textbf{k})+\Sigma^{a}_{hf}(\textbf{k})+\Sigma^{a}_{3}(\textbf{k},\varepsilon)+\Sigma^{b}_{3}(\textbf{k},\varepsilon)$$ Note that $\Sigma^{a}_{c}(\textbf{k})$ only appears in the presence of the spin Casimir effect. The poles of the normal magnon Green’s function are the spin-wave spectrum, which can be obtained either by simply replacing $\varepsilon$ with linear spin-wave spectrum $2S\widetilde{\varepsilon}_{\textbf{k}}$ in the self-energies, i.e. the so-called on-sell approximation or by solving the Dyson equation self-consistently, i.e. the so-called off-shell approximation. Both the approximation schemes can manifest the features of magnon decays but with different lineshape characteristics and associated physical interpretations, as presented below.
In the subsequent subsections, we first investigate the linear spin wave spectrum within both the CSWT and TESWT and then turn to the on-shell and off-shell calculations of the renormalized spectrum. In order to perform a well-controlled calculation, all the numerical calculations are performed with the magnitude of the inter-chain coupling in both $J_{3}$- and $J_{4}$-systems fixed as 0.3$J_{1}$ and $J_{2}$=$|J_{1}|$.
Harmonic approximation
----------------------
In prior to the calculation of the $1/S$ order spin-wave spectrum, it is desirable to investigate the excitation spectrum within the harmonic approximation. In the linear spin-wave description, the magnon is a well-defined quasi-particle of the long-range ordered magnetic system with infinite long lifetime. No spin Casimir and spontaneous magnon decay effects are involved and the spin wave spectrum is simply $2S\varepsilon_{\textbf{k}}$. Thus, it seems that the linear spin wave spectrum has nothing to do with the nonlinear magnon decay effects. However, the kinematic constrains for the decay features in the one-loop on-shell spin wave spectrum are actually obtained thought the linear spin wave energy equations. In the torque equilibrium formulism, these kinematic constrains are deduced by the renormalized linear spin wave energy rather than the real harmonic one, which reads $$2S\widetilde{\varepsilon}_{\textbf{k}}=2S\sqrt{(\widetilde{J}_{\textbf{k}}-\widetilde{J}_{\textbf{Q}}+\lambda_{\textbf{Q}}+\Delta)(\widetilde{\eta}_{\textbf{k}}-\widetilde{J}_{\textbf{Q}}+\lambda_{\textbf{Q}})}$$
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Once again, we consider the two representative $J_{3}$- and $J_{4}$-systems. The resultant linear spin wave spectrum of the isotropic $J_{3}$-system is plotted in Fig. 6, in which the spectrum manifests obvious quasi-one dimensional characteristics: more curvy along the chain direction and less curvy along the vertical direction. Additionally, the three Goldstone modes at $\textbf{k}$=$\textbf{0}$ and $\textbf{k}$=$\pm\textbf{Q}$ caused by the complete breaking of the $SO(3)$ rotational symmetry in the noncollinear ground state are clearly shown. The Goldstone mode at $\textbf{k}$=$\textbf{0}$ corresponds to the in-plane sliding mode of the spiral, whereas the $\textbf{k}$=$\pm\textbf{Q}$ modes represent the out-of-plane oscillation modes related with symmetry. As a result, these three acoustic modes have only two spin wave velocities $$\begin{aligned}
\mathcal{V}_{\textbf{0}}&=&S\sqrt{2(J_{\textbf{0}}-J_{\textbf{Q}_{cl}})\nabla^{2}_{\textbf{k}}J_{\textbf{k}}\big|_{\textbf{k}=\textbf{Q}_{cl}}}\nonumber\\
\mathcal{V}_{\textbf{Q}}&=&S\sqrt{(J_{\textbf{0}}+J_{2\textbf{Q}_{cl}}-2J_{\textbf{Q}_{cl}})\nabla^{2}_{\textbf{k}}J_{\textbf{k}}\big|_{\textbf{k}=\textbf{Q}_{cl}}}\end{aligned}$$ In general, these two spin wave velocities are different, i.e. one is faster than the other. Usually, this fact actually constitutes an important decay channel from the faster Goldstone mode to the slower one, which defines the decay region as seen in such as triangular lattice antiferromagnets. However, in the TESWT, this decay channel is actually determined by the torque equilibrium results $$\begin{aligned}
\widetilde{\mathcal{V}}_{\textbf{0}}&=&S\sqrt{2(\widetilde{J}_{\textbf{0}}-\widetilde{J}_{\textbf{Q}})\nabla^{2}_{\textbf{k}}\widetilde{J}_{\textbf{k}}\big|_{\textbf{k}=\textbf{Q}}}\nonumber\\
\widetilde{\mathcal{V}}_{\textbf{Q}}&=&S\sqrt{(\widetilde{J}_{\textbf{0}}+\widetilde{J}_{2\textbf{Q}}-2\widetilde{J}_{\textbf{Q}})\nabla^{2}_{\textbf{k}}\widetilde{J}_{\textbf{k}}\big|_{\textbf{k}=\textbf{Q}}}\end{aligned}$$ with the quantum ordering vector.
Once the magnetic anisotropy is considered, the out-of-plane oscillation modes at $\textbf{k}$=$\pm\textbf{Q}$ are gapped and only the in-plane sliding mode at $\textbf{k}$=$\textbf{0}$ is present. On the other hand, the linear spin wave spectra of the corresponding $J_{4}$-systems are demonstrated in Fig. 7. The isotropic $J_{4}$-system shows three Goldstone modes and quasi-one dimensional characteristics as well. The magnetic anisotropy manifest similar effects as in the $J_{3}$-systems. However, there are some slight differences in the spectrum, e.g. the surface is concave or convex around some $\textbf{k}$ points. As a matter of fact, these seemingly insignificant differences in the linear spin wave spectrum can induce very different results in terms of the spectrum at the one-loop order, as shown below.
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On-shell approximation
----------------------
In the on-shell approximation, the self-energies are evaluated at the harmonic magnon energy in the torque equilibrium formulism, which represent strict $1/S$ corrections to the magnon energy. At the one-loop order, the spin wave spectrum $\mathcal{E}_{\textbf{k}}$ obtained within the on-shell approximation can be written as $$\begin{aligned}
\mathcal{E}_{\textbf{k}}&=&2S\widetilde{\varepsilon}_{\textbf{k}}+\Sigma^{a}_{c}(\textbf{k})+\Sigma^{a}_{hf}(\textbf{k})\nonumber\\
&&+\Sigma^{a}_{3}(\textbf{k},\widetilde{\varepsilon}_{\textbf{k}})+\Sigma^{b}_{3}(\textbf{k},\widetilde{\varepsilon}_{\textbf{k}})\end{aligned}$$ According to this expression, the one-loop spin wave spectrum $\mathcal{E}_{\textbf{k}}$ can be easily obtained by numerical integration of the self-energies.
The numerical results for the $J_{3}$- and $J_{4}$-systems are demonstrated in Fig. 8 along some representative symmetry directions in the Brillouin zone (BZ). The spin wave spectrum show prominent features of the magnon damping in the major part of the BZ with giant imaginary part of $\mathcal{E}_{\textbf{k}}$. As we have mentioned before, this remarkable magnon damping originates from the coupling between the single-particle excitations and the two-particle continuum determined by the anharmonic cubic terms. A feature of the results is that the magnon damping and the renormalization of the spin wave spectrum are stronger at large momenta. Additionally, there are many substantial singularities in both the real and imaginary parts of $\mathcal{E}_{\textbf{k}}$, which manifest themselves in the form of jump-like discontinuities and spike-like peaks. The origin of these singularities is due to the intersection of the single-magnon branch with the line of the van Hove saddle point singularities in the two-magnon continuum. The analytical properties of these singularities have deep connection with the integration dimensionality of the self-energies, which have been profoundly discussed in Ref. 25. In two-dimensional systems, they fulfill the Kramers-Kronig relations between the real and imaginary parts of the one-loop on-shell spin wave energy as: $$\mathrm{Re}(\mathcal{E}_{\textbf{k}})\simeq \mathrm{sgn}(\delta\textbf{k}),~~~~\mathrm{Im}(\mathcal{E}_{\textbf{k}})\simeq- \mathrm{ln}\Big(\frac{\Lambda}{|\delta\textbf{k}|}\Big)$$ or the other way around: $$\mathrm{Re}(\mathcal{E}_{\textbf{k}})\simeq \mathrm{ln}\Big(\frac{\Lambda}{|\delta\textbf{k}|}\Big),~~~~\mathrm{Im}(\mathcal{E}_{\textbf{k}})\simeq- \Theta(\delta\textbf{k})$$ Here $\delta\textbf{k}$=$\textbf{k}$-$\textbf{k}^{*}$ with $\textbf{k}^{*}$ represents the location of the singularities in $\textbf{k}$ space, $\Lambda$ is the cut-off parameter determined by characteristic size of the singular region, $\mathrm{sgn}(x)$ stands for the sign function and $\Theta(x)$ is the Heaviside step function. When the inter-plane coupling is considered, the self-energies integration becomes three dimensional and the associated logarithmic peaks become the square root ones.
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It is worth noting that the imaginary part of the spin wave energies is only non-vanishing in some region of the BZ, i.e. the so-called magnon decay region. The threshold boundary of this region is determined by the kinematic constrains that follow the momentum and energy conservation in the two-particle decay process $$\widetilde{\varepsilon}_{\textbf{k}}=\widetilde{\varepsilon}_{\textbf{p}}+\widetilde{\varepsilon}_{\textbf{k}-\textbf{p}}$$ and the extremum condition of the two-particle continuum, or equivalently the spin wave velocity equation $$\widetilde{\mathcal{V}}_{\textbf{p}}=\widetilde{\mathcal{V}}_{\textbf{k}-\textbf{p}}$$ These equations are expressed simply based on the harmonic approximation for the magnon energies, which can only determine the decay boundary at the one-loop order. As the anharmonic terms induced renormalization effect is considered, the kinematic constrains are modified as well. As a consequence, the threshold boundary changes as the higher order spin wave processes are taken into account. However, the decay boundary obtained within the harmonic approximation is usually considered to be immensely instructive. One of the reasons is the higher order renormalized decay region usually shows little difference from the harmonic one, as demonstrated in triangular lattice antiferromagnetic Heisenberg (TLAH) model. Another reason, may be one of the most important reasons, is that a consideration of the threshold boundary within the harmonic approximation can be carried out analytically. More than that, not only the decay region but also the location and decay channel of the singularities can be obtained.
In spite of the methodological convenience and the analytical availability, the decay boundary analysis can be difficult to carry out for higher order or in a self-consistent manner if not impossible. Additionally, the decay boundary defines the region where magnon decay is allowed, but the spin wave excitations are not strongly damped within the whole region inside the decay boundary. However, only the considerable broad peaks caused by the strong magnon decay effect can be experimentally observed because of the nonzero temperature and the finite resolution of the detector. Together with the fact that in our case, the decay region actually covers most of the area in the BZ, we then turn to the analysis of the strong damping region instead of the harmonic boundary. In order to take into account the damping rate of the spin wave excitations, it is convenient to introduce the magnon decay rate, which can be expressed as $$\mathcal{R}_{\textbf{k}}=-\mathrm{Im}\big[\Sigma^{tot}_{n}(\textbf{k},\varepsilon)\big]$$ Within the one-loop on-shell approximation, it can be written as $$\mathcal{R}_{\textbf{k}}=\frac{\pi}{2}\sum_{\textbf{p}}\widetilde{\Gamma}^{2}_{1}(\textbf{p};\textbf{k})
\cdot\delta(\widetilde{\varepsilon}_{\textbf{k}}-\widetilde{\varepsilon}_{\textbf{p}}-\widetilde{\varepsilon}_{\textbf{k}-\textbf{p}})$$ which is nothing but the imaginary part of the self-energy $\Sigma^{a}_{3}(\textbf{k},\widetilde{\varepsilon}_{\textbf{k}})$ and manifests similar form to the Fermi’s golden-rule expression.
The numerical results of the spin wave decay rate in one quarter of the BZ are shown in Fig. 9, in which the strong decay region is intuitively demonstrated. It is straightforward to verify from the decay boundary analysis that the bright line and sharp boundaries shown in Fig. 9 belong to some specific decay channels. However, not all the decay boundaries obtained from the harmonic approximation analysis have an obvious demonstration in the decay rate intensity plots. In addition, the demonstrated decay pattern of each system shows very good consistency with the corresponding on-shell spectrum. Surprisingly, the decay pattern and corresponding on-shell spin wave spectrum for isotropic $J_{3}$- and $J_{4}$-systems show drastically different features, which can also be read out from the on-shell spectrum. The isotropic $J_{4}$-system shows far more singularities than the $J_{3}$-system, e.g. there are four singularities along the $\Gamma$-X direction in the $J_{4}$-system while only one along the same path in the $J_{3}$-system.
Although it appears interesting, the pattern of the singularity distribution may not be experimentally observable because the singularities can be smoothed, considering the higher order contributions. Nevertheless, there are several remarkable differences that may survive even in the self-consistent calculation. First, the strong damping region is around the M point in the $J_{3}$-system while in the $J_{4}$-system the magnon decay is stronger around the X point. Next, there is a wide decay region around the X$'$ point in $J_{4}$-system while no decay at all near the same region in the $J_{3}$-system. This is a direct consequence of the difference in the harmonic spin wave spectrum between the $J_{3}$- and $J_{4}$-systems. The last but not the least, another striking feature is the nearly flat mode along the X-M direction of the renormalized spin wave spectrum for the $J_{4}$-system, which is very different from the results of the corresponding $J_{3}$-system and the classical $J_{4}$-system. Upon a close examination, we find that this flat mode is induced by the self-energy $\Sigma^{a}_{c}(\textbf{k})$ that is contributed from the spin Casimir effect. The conventional self-energies that describe the three- and four-magnon interactions are usually negative, which correspond to the downward renormalization of the excitation spectrum in the usual magnon decay cases. However, the self-energy $\Sigma^{a}_{c}(\textbf{k})$ has a huge positive contribution to the total self-energy around the M point in the $J_{4}$-system. As a consequence, the renormalization of the excitation spectrum turns upward compared with the TELSWT results, and then the flat mode along the X-M direction appears. Note that the total renormalization of the spin wave spectrum is still downward compared with the LSWT results, and thus the qualitative magnon decay arguments remain reliable.
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As the magnetic anisotropy is introduced, the on-shell spin wave spectra of the $J_{3}$- and $J_{4}$-systems are both drastically modified. Similar to the classical case, the Goldstone modes at $\textbf{k}$=$\pm\textbf{Q}$ are gapped and only the Goldstone mode at $\textbf{k}$=$\textbf{0}$ is present. However, the quantum fluctuation effect can modify the classical spectrum and the spin wave gap at $\textbf{k}$=$\pm\textbf{Q}$ in the $J_{3}$- and $J_{4}$-systems are both drastically downward renormalized. Furthermore, the $J_{4}$-system appears to be less sensitive to the magnetic anisotropy for the far smaller spin wave gap compared with the $J_{3}$-system with the same anisotropy energy. This fact may shed light on our understanding of the experimental results on LiCuVO$_{4}$, in which the inelastic neutron scattering data show nearly zero spin wave gap at $\textbf{k}$=$\pm\textbf{Q}$, while the electron spin resonance results indicate a 6$\%$ magnetic anisotropy in the system. Additionally, the spontaneous magnon decay region is drastically reduced due to the suppression of the quantum fluctuation effect and the kinematic condition for decays caused by the magnetic anisotropy. Surprisingly, in spite of the sharp reduction of the decay region, the decay rate is less reduced and the prominent differences in the decay pattern between the $J_{3}$- and $J_{4}$-systems still exist. At the same time, the flat mode in the isotropic $J_{4}$-system also survives in the anisotropic cases. As a consequence, the spontaneous magnon decay effects can be expected to be robust in the quasi-one dimensional $J_{1}$-$J_{2}$ magnetic systems and the decay pattern manifests different features for different types of inter-chain couplings.
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Off-shell approximation
-----------------------
As a matter of fact, the unusual singularities in the on-shell spin wave spectrum actually signify a breakdown of the standard spin wave expansion. Consequently, a self-consistent calculation has to be performed in order to obtain the actual dynamic properties of the system. In this subsection, we turn to one of the self-consistent schemes: the off-shell approximation. Within this approximation, the self-energies are evaluated in a self-consistent way by allowing the finite lifetime of the magnon at the very beginning, while the magnons created during the decay process remain stable.
The spin wave spectrum $\mathcal{D}_{\textbf{k}}$ within the off-shell approximation can be obtained by self-consistently solving the Dyson equation $$\begin{aligned}
\mathcal{D}_{\textbf{k}}&=&2S\widetilde{\varepsilon}_{\textbf{k}}+\Sigma^{a}_{c}(\textbf{k})+\Sigma^{a}_{hf}(\textbf{k})\nonumber\\
&&+\Sigma^{a}_{3}(\textbf{k},\mathcal{D}^{*}_{\textbf{k}})+\Sigma^{b}_{3}(\textbf{k},\mathcal{D}^{*}_{\textbf{k}})\end{aligned}$$ where $\mathcal{D}^{*}_{\textbf{k}}$ is the complex conjugate of $\mathcal{D}_{\textbf{k}}$ following the methodological discussion in Ref. 25 on the proper sign of the imaginary part of the decay-like self-energy. Rewriting the above equation explicitly for the real and imaginary parts, the original Dyson equation becomes the following equation sets: $$\begin{aligned}
\mathrm{Re}(\mathcal{D}_{\textbf{k}})&=&2S\widetilde{\varepsilon}_{\textbf{k}}+\Sigma^{a}_{c}(\textbf{k})+\Sigma^{a}_{hf}(\textbf{k})\nonumber\\
&&+\mathrm{Re}\big[\Sigma^{a}_{3}(\textbf{k},\mathcal{D}^{*}_{\textbf{k}})+\Sigma^{b}_{3}(\textbf{k},\mathcal{D}^{*}_{\textbf{k}})\big] \nonumber\\
\mathrm{Im}(\mathcal{D}_{\textbf{k}})&=&\mathrm{Im}\big[\Sigma^{a}_{3}(\textbf{k},\mathcal{D}^{*}_{\textbf{k}})+\Sigma^{b}_{3}(\textbf{k},\mathcal{D}^{*}_{\textbf{k}})\big]\end{aligned}$$ According to this expression, the off-shell one-loop spin wave spectrum $\mathcal{D}_{\textbf{k}}$ can be easily obtained by numerically solving the integration equations. At the same time, the magnon decay rate within the off-shell approximation can be simply expressed as $$\mathcal{R}_{\textbf{k}}=-\mathrm{Im}(\mathcal{D}_{\textbf{k}})$$ from which the self-consistent decay pattern can be directly obtained.
The off-shell spin wave spectrum for the $J_{3}$- and $J_{4}$-systems are demonstrated in Fig. 10 along the same representative symmetry directions with the on-shell results. As a self-consistent method, the spin wave spectrum and decay rate obtained within the off-shell approach contains the contributions beyond the one-loop order. Consequently, the remarkable singularities in the on-shell one-loop spectrum are regularized within the off-shell scheme. Additionally, the spin wave spectra obtained within the off-shell approximation are stretched upwards a little compared with the on-shell ones, which may be caused by the over-estimation of the energy shifts in the on-shell scheme. In spite of this upward renormalization, the off-shell spin wave gap at $\textbf{k}$=$\pm\textbf{Q}$ in the $J_{4}$-systems remains smaller compared with the $J_{3}$-systems with the same anisotropy energy as in the on-shell cases. On the other hand, the nearly flat mode along the X-M direction in $J_{4}$-systems in less significant than that in the on-shell cases but remains quite different with the $J_{3}$- and classical $J_{4}$-systems, and thus may still serve as a characteristic of the quantum $J_{4}$-systems.
The corresponding spin wave decay rate data in one quarter of the BZ are shown in Fig. 11. While the jump-like discontinuities and spike-like peaks disappear in both the real and imaginary parts of the spin wave energy, the magnon decay rate remains significant throughout BZ. The overall shape of the decay region is very different from the on-shell cases, the reason of which may lie in the sensitivity of the quasi-one dimensional systems. Nevertheless, the characteristic differences in the decay pattern between the $J_{3}$- and $J_{4}$-systems that we discussed in the on-shell case survive in the off-shell results. The strong damping region still lies around the M and X points in the $J_{3}$- and $J_{4}$-systems respectively and the wide decay region around the X$'$ point in the isotropic $J_{4}$-system still exists while the same region in the isotropic $J_{3}$-system remains no decay at all. Moreover, the effect of the magnetic anisotropy on the decay rate is further enhanced within the off-shell scheme. In particular, both the off-shell approximated decay region and decay rate are dramatically reduced compared with the corresponding on-shell results in the $J_{3}$-systems. On the other hand, the $J_{4}$-systems show less sensitivity on the area of the decay region, although the magnitude of the decay rate is much smaller than the on-shell predictions.
Different from the case in the TLAH model, the off-shell decay region is slightly different from the on-shell predictions, which may be caused by the extraordinary sensitivity of the quasi-one dimensional frustrated $J_{1}$-$J_{2}$ systems. Other than that, a more astonishing feature of the off-shell results is the appearance of non-decay area in the center of the strong decay region around the X point in the isotropic and weakly anisotropic $J_{4}$-systems. In this sudden non-decay region, the imaginary part of the off-shell spectrum vanishes and the real component manifests a flat-top peak. Although the off-shell decay pattern is not necessarily identical with the on-shell prediction, the on-shell broadening in this sudden non-decay area reaches nearly one-half of the spectrum. Consequently, one may expect that it is unlikely that this spectrum broadening can disappear in the self-consistent calculation, especially in the area isolated and surrounded by strong decay region. Moreover, this sudden non-decay region disappears once the magnetic anisotropy parameter exceeds some critical value $\Delta_{c}$ and the Dyson equation finds the solution with finite broadening as expected.
To verify this strange characteristic and further investigate the magnon decay dynamics with the off-shell approximation, we introduce the poles function $\mathcal{P}_{\textbf{k}}$, which is the inverse of the normal magnon Green’s function and expressed within the one-loop approximation as $$\begin{aligned}
\mathcal{P}_{\textbf{k}}&=&\varepsilon-2S\widetilde{\varepsilon}_{\textbf{k}}-\Sigma^{a}_{c}(\textbf{k})-\Sigma^{a}_{hf}(\textbf{k})\nonumber\\
&&-\Sigma^{a}_{3}(\textbf{k},\varepsilon)-\Sigma^{b}_{3}(\textbf{k},\varepsilon)\end{aligned}$$ The zeros of this function represent the poles of the normal magnon Green’s function, which are equivalent with the solutions of the Dyson equation. In order to analyze the sudden non-decay area, the self-energies are obtained with frequency $\varepsilon$ scans through the real axis. This scheme is very similar to the one adopted in the calculation of the spectral function, which is discussed in details in the next subsection. In order to perform a concrete calculation, we choose the representative point $\textbf{k}$=$(0.82\pi,0,0)$ which lies right in the middle of the sudden non-decay region. Additionally, we calculate another representative point $\textbf{k}$=$(\pi,0,0)$, which locates near the sudden non-decay area and lies in the decay region in the isotropic case. More than that, as a comparison, we also calculate the point $\textbf{k}$=$(0,0.2\pi,0)$, which locates far way from both the decay and the sudden non-decay region. Furthermore, to clarify the related spin wave decay dynamics, we introduce the two-magnon density of states (DOS) $$\begin{aligned}
\mathcal{M}_{\textbf{k}}&=&\sum_{\textbf{p}}\delta(\varepsilon-\widetilde{\varepsilon}_{\textbf{p}}-\widetilde{\varepsilon}_{\textbf{k}-\textbf{p}})\end{aligned}$$ which has van Hove singularities that can cross the single particle spectrum, therefore can be responsible for magnon decays.
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The results of the calculation of various functions at different representative $\textbf{k}$ points are shown in Fig. 12. The demonstrated correspondence between the step-like and spike-like singularities in $\mathrm{Re}(\mathcal{P}_{\textbf{k}})$ and $\mathrm{Im}(\mathcal{P}_{\textbf{k}})$ is very similar to that in the on-shell cases, which is also the direct consequence of the Kramers-Kronig relations. While the correspondence between the $\mathcal{P}_{\textbf{k}}$ and $\mathcal{M}_{\textbf{k}}$ is more delicate, corresponding to certain magnon decay channels, we are interested in the non-decay solutions of the Dyson equation. Here we fucus on a specific kind of crossing points between $\mathrm{Re}(\mathcal{P}_{\textbf{k}})$ and the real $\varepsilon$ axis with $\mathrm{Im}(\mathcal{P}_{\textbf{k}})$=0. Surprisingly though, such crossing point is not unique in most cases that we have investigated and there are three types.
The first type of such crossing points are the most conventional ones, which represent the true well-defined quasi-particles with zero damping and in our case only appear in Fig. 12(d). This type of crossing points locate well beneath the two-magnon DOS, and thus only exist in the real non-decay region. The second type of such crossing points are more delicate and have deep connection with the singular behavior near the bottom of the two-magnon DOS. Different with the first type, this type of solutions only appear in the decay region. Strictly speaking, this type of crossing points should not be considered as a well-defined quasi-particle state because they are caused by the singular behavior of the one-loop self-energies, thus may disappear when the one-loop singularities are regularized by the higher order approximation. However, as a matter of fact, they have very obvious demonstration in the one-loop spectral function results and are referred to as “edge” singularities in the study of the TLAH model. Contrary to the previous two types, the last type of crossing points locate well beyond the two-magnon DOS, which are actually the single-particle state pushed out of the two-magnon continuum and referred to as “antibonding” magnon states. These states exist in almost all the cases that we have investigated. Although in the usual cases only the “bonding” magnon states are considered as the right solutions of the Dyson function, the “antibonding” states also exist in some special cases but normally do not mixed up with the “bonding” ones because of the large energy gap between them. However, this conventional picture breaks down in our case of the sudden non-decay area.
Note the continues transition between the sudden non-decay region and the strong decay region shown in Fig. 10(d) and Fig. 11(d), it means that the “bonding” states are actually moving towards the upward located “antibonding” states when approaching the sudden non-decay area. As a consequence, the strange sudden non-decay area observed in the off-shell spectrum is not a true non-decay region but a region in which the “bonding” and “antibonding” magnon states are degenerate. And the flat-top peak is a direct demonstration of the “antibonding” spectrum, which is rather flat around the X point. Although this strange phenomenon appears to be interesting, the degeneration between these single-particle states is usually not robust, i.e. their appearance depend on the approximation one used in the calculation. Thus the whole sudden non-decay region may be simply an accidental product of our one-loop Dyson scheme, which can disappear once another self-consistent scheme is adopted. At last, we would like to mention that all these three types of solutions can be clearly demonstrated in the spectral function obtained within the one-loop approximation, which can be considered as another evidence of our analysis.
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Spectral function
=================
In order to further investigate the actual dynamic properties of the system we turn to the spectral function, which contain a detailed information on the spin wave energy renormalization and the magnon decay rate. The diagonal component of the spectral function is defined by the imaginary part of the normal magnon Green’s function through the expression $$\mathcal{A}(\textbf{k},\varepsilon)=-\frac{1}{\pi}\mathrm{Im}\big[G_{n}(\textbf{k},\varepsilon)\big]$$ Within the leading one-loop approximation the normal magnon Green’s function is given by Eq. (36). The spectral function $\mathcal{A}(\textbf{k},\varepsilon)$ is deeply connected with the dynamical structure factor $\mathcal{S}(\textbf{k},\varepsilon)$ which is directly measured in inelastic neutron scattering experiments. Generally, $\mathcal{S}(\textbf{k},\varepsilon)$ also contains contributions from the off-diagonal and two-particle correlations, which we leave to the next section for a detailed discussion. In this subsection, we focus on the spectral function of the system, which actually provides the major component of the dynamical structure factor but much easier to analyze.
In the classical limit, the spectral function is a $\delta$ function located at $2S\varepsilon_{\textbf{k}}$ for any momentum $\textbf{k}$ in the BZ. Similar to the spin wave spectrum, the spectral function is strongly renormalized by the quantum fluctuation as well. In the absence of the intrinsic damping, the spin wave energy is real and the quasiparticle peak in $\mathcal{A}(\textbf{k},\varepsilon)$ occurs precisely at $\mathcal{D}_{\textbf{k}}$, the off-shell spin wave spectrum obtained by solving the Dyson equation. However, when the spontaneous magnon decay occurs the spin wave energy acquires a significant imaginary part. As a consequence, the location and broaden width of the quasiparticle peak in the spectral function is different with the solution of the Dyson equation because $\mathcal{A}(\textbf{k},\varepsilon)$ is only defined on the real $\varepsilon$ axis. In additional to the single-particle peaks, the spectral function is expected to exhibit an incoherent component, which represents the contribution from the two-particle continuum due to the nonorthogonality between the single- and two-particle excitations. In the calculation of the spectral function, the normal magnon Green’s function is obtained with frequency scans through all the possible energies. Consequently, the consideration of the spectral function is also beyond the $1/S$ expansion, which is analog with the off-shell approximation and the poles function that we’ve discussed. On the other hand, the numerical integration of the self-energies is performed with an artificial broadening of $\sigma$=0.01. And the intensity plots of the spectral function for the $J_{3}$- and $J_{4}$- systems are demonstrated in Fig. 13.
A feature of the demonstrated spectral function observed in the whole BZ is the downward renormalization of the single-particle dispersion, which is consistent with our previous on-shell and off-shell results. In addition, the spontaneous magnon decay induced broadening of the spectrum can be directly observed around certain $\textbf{k}$ points in the BZ. Other than that, the characteristics differences that we expected still remain in the spectral function. As shown clearly in the results of the isotropic $J_{3}$- and $J_{4}$-systems, the strong damping region still lies around the M and X point respectively. However, this feature disappears in the anisotropic $J_{3}$-system with $\Delta$=0.95, in which the spectrum broadening is more remarkable around the X point and nearly indiscernible around the M point. This may be a consequence of the fact that the decay pattern manifested in the spectral function is different from the on-shell and off-shell results, which further indicates that the quasi-one dimensional $J_{1}$-$J_{2}$ system is very sensitive. More than that, the $J_{3}$-systems can still be concluded to be more sensitive to the magnetic anisotropy than the $J_{4}$-systems. On the other hand, the nearly flat mode along the X-M direction in $J_{4}$-systems in more significant than that in the on-shell and off-shell cases due to the remarkable broadening of the spectrum. And this salient broadening of the flat mode shrinks dramatically once the magnetic anisotropy is introduced.
Another prominent feature is the appearance of “pseudo”-quasiparticle peaks that we’ve discussed in the previous subsection. One interesting type of “pseudo”-quasiparticle peaks are the so called “edge” singularities, which usually locate near the bottom of the spectral function. They only exist in the decay region and manifest as a bright “edge” of the spectral function. More interestingly, the “edge” in the spectral function of the isotropic $J_{4}$-system goes to zero around the X point, which is consistent with the poles function results. This strange behavior may not be deeply connected with the sudden non-decay area issue because the former case own wider region than the latter one. As the magnetic anisotropy is introduced and the magnon decay get suppressed, the “edge” singularities move towards the true quasi-particle peaks and finally merge into them when the magnon decay is absent.
The most astonishing and misleading type of “pseudo”-quasiparticle peaks are the “antibonding” magnon states lie around the top of the spectral function. As a matter of fact, they are actually true single-magnon state at least within the one-loop approximation as we’ve shown in the previous subsection. However, this type of excitations only exist in certain region of the BZ and the energy scale of them are much larger than the usual spin wave excitation that we’re interested in, thus normally they are not considered as one part of the excitation spectrum. But this is not the case for the isotropic $J_{4}$-system, in which the “bonding” and “antibonding” single-magnon states are degenerate near X point as shown in Fig. 13(d). This degeneration exactly explain the appearance of the sudden non-decay area obtained within the off-shell approximation. As clearly demonstrated in the spectral function, this region is actually a strongly decay region rather than the non-decay area as the off-shell results indicated. At last, we would like to stress that this degeneration and even the “antibonding” magnon states themselves can be just an accidental product of our one-loop approximation. And to our best of knowledge, there is not yet a conclusive evidence about the existence of the “antibonding” magnon states in any real system. Nevertheless, they still exist within the one-loop scheme and can lead to very obvious phenomena in the associated off-shell excitation spectrum and the spectral function results, thus need to be clarified at some level.
Dynamic Structure Factor
========================
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The consideration of the dynamic structure factor $\mathcal{S}(\textbf{k},\varepsilon)$ in noncollinear antiferromagnets has been given for cases of the square-lattice antiferromagnets in the magnetic field, space isotropic triangular lattice antiferromagnets and more recently the anisotropic models on kagome lattice. However, the noncollinear magnetic order in all these systems is commensurate and not plagued with the divergences and singularities caused by the spin Casimir effect. For the more general incommensurate cases, the spin wave expansion has to be performed within the so called torque equilibrium approach. While the corresponding formulism of $\mathcal{S}(\textbf{k},\varepsilon)$ has not yet been established within the TESWT. In this section, we investigate the dynamic structure factor of the $J_{3}$- and $J_{4}$-systems within the torque equilibrium scheme. One of our goals of the present section is to generalize the calculation of $\mathcal{S}(\textbf{k},\varepsilon)$ to the TESWT. Our formulism is similar to the isotropic triangular lattice case, but with modifications due to the torque equilibrium condition, thus we revisit the whole derivation for comparison and completeness. And our second goal is to provide the first explicit theoretical results for the dynamic structure factor of the quasi-one dimensional $J_{1}$-$J_{2}$ models to guide experimental inelastic neutron scattering measurements in realistic materials.
The dynamic structure factor is nothing but the spin-spin correlation function which can be directly probed in inelastic neutron scattering experiments. By definition, it can be expressed as $$\mathcal{S}^{\mu\nu}(\textbf{k},\varepsilon)=\int^{\infty}_{-\infty}\frac{dt}{2\pi}e^{i\varepsilon t}\big\langle S^{\mu}_{\textbf{k}}(t)S^{\nu}_{-\textbf{k}}(0)\big\rangle$$ with $\mu,\nu\in(a,b,c)$ and $S^{\mu}_{\textbf{k}}(t)$ represents the Fourier transformed spin operator. Usually, it is convenient to define the dynamical correlation functions as $$\mathcal{G}^{\mu\nu}(\textbf{k},\varepsilon)=-i\int^{\infty}_{-\infty}\frac{dt}{2\pi}e^{i\varepsilon t}\big\langle \mathcal{T}S^{\mu}_{\textbf{k}}(t)S^{\nu}_{-\textbf{k}}(0)\big\rangle$$ with $\mathcal{T}$ represents the time-ordered operator. As a consequence, the dynamic structure factor is then connected with dynamical correlation function through the fluctuation-dissipation theorem $$\mathcal{S}^{\mu\nu}(\textbf{k},\varepsilon)=-\frac{1}{\pi}\mathrm{Im}\big[\mathcal{G}^{\mu\nu}(\textbf{k},\varepsilon)\big]$$
The inelastic neutron-scattering cross section is actually a linear combination of the diagonal components of the spin-spin correlation function with momentum dependent prefacers according to the experimental settings. For simplicity, however, we do not assume a particular experimental geometry in this section and only consider the total structure factor $$\mathcal{S}^{tot}(\textbf{k},\varepsilon)=\mathcal{S}^{aa}(\textbf{k},\varepsilon)+\mathcal{S}^{bb}(\textbf{k},\varepsilon)+\mathcal{S}^{cc}(\textbf{k},\varepsilon)$$ Rewriting each component in the twisted frame we obtain $$\begin{aligned}
\mathcal{S}^{aa}(\textbf{k},\varepsilon)&=&\frac{1}{4}\Big[\mathcal{S}^{xx}(\textbf{k}+\textbf{Q},\varepsilon)+\mathcal{S}^{xx}(\textbf{k}-\textbf{Q},\varepsilon)\nonumber\\
&&+\mathcal{S}^{zz}(\textbf{k}+\textbf{Q},\varepsilon)+\mathcal{S}^{zz}(\textbf{k}-\textbf{Q},\varepsilon)\Big] \nonumber\\
&&+\frac{i}{4}\Big[\mathcal{S}^{zx}(\textbf{k}+\textbf{Q},\varepsilon)-\mathcal{S}^{zx}(\textbf{k}-\textbf{Q},\varepsilon) \nonumber\\
&&-\mathcal{S}^{xz}(\textbf{k}+\textbf{Q},\varepsilon)+\mathcal{S}^{xz}(\textbf{k}-\textbf{Q},\varepsilon)\Big]\nonumber\\
\mathcal{S}^{bb}(\textbf{k},\varepsilon)&=&\mathcal{S}^{yy}(\textbf{k},\varepsilon),~~~~\mathcal{S}^{cc}(\textbf{k},\varepsilon)=\mathcal{S}^{aa}(\textbf{k},\varepsilon)\end{aligned}$$ Once again, we use the quantum ordering vector instead of the classical one. The total structure factor can be divided in terms of the diagonal and mixed contributions as $$\mathcal{S}^{tot}(\textbf{k},\varepsilon)=\mathcal{S}^{diag}(\textbf{k},\varepsilon)+\mathcal{S}^{mix}(\textbf{k},\varepsilon)$$ and the diagonal one can be further separated into transverse and longitudinal parts as $$\mathcal{S}^{diag}(\textbf{k},\varepsilon)=\mathcal{S}^{L}(\textbf{k},\varepsilon)+\mathcal{S}^{T}(\textbf{k},\varepsilon)$$ with $$\begin{aligned}
\mathcal{S}^{L}(\textbf{k},\varepsilon)&=&\frac{1}{2}\Big[\mathcal{S}^{zz}(\textbf{k}+\textbf{Q},\varepsilon)+\mathcal{S}^{zz}(\textbf{k}-\textbf{Q},\varepsilon)\Big] \nonumber\\
\mathcal{S}^{T}(\textbf{k},\varepsilon)&=&\frac{1}{2}\Big[\mathcal{S}^{xx}(\textbf{k}+\textbf{Q},\varepsilon)+\mathcal{S}^{xx}(\textbf{k}-\textbf{Q},\varepsilon)\Big] \nonumber\\
&&+\mathcal{S}^{yy}(\textbf{k},\varepsilon) \nonumber\\
\mathcal{S}^{mix}(\textbf{k},\varepsilon)&=&\frac{i}{2}\Big[\mathcal{S}^{zx}(\textbf{k}+\textbf{Q},\varepsilon)-\mathcal{S}^{zx}(\textbf{k}-\textbf{Q},\varepsilon) \nonumber\\
&&-\mathcal{S}^{xz}(\textbf{k}+\textbf{Q},\varepsilon)+\mathcal{S}^{xz}(\textbf{k}-\textbf{Q},\varepsilon)\Big]\end{aligned}$$
On the other hand, the relation between the dynamical correlation functions and the magnon Green’s functions can be obtained by transforming the spin operators into the Holstein-Primakoff representation and proceeding with the Bogoliubov transformation. In this rather standard procedure, the Hartree-Fock approximation has been made as we did for the quartic interaction terms and the final results about the transverse components read $$\begin{aligned}
\mathcal{G}^{xx}(\textbf{k},\varepsilon)&=&\frac{S}{2}\mathcal{C}_{x}^{2}(\widetilde{u}_{\textbf{k}}+\widetilde{v}_{\textbf{k}})^{2}
\big[G_{n}(\textbf{k},\varepsilon)+G_{n}(-\textbf{k},-\varepsilon)\nonumber\\
&&+2G_{a}(\textbf{k},\varepsilon)\big] \nonumber\\
\mathcal{G}^{yy}(\textbf{k},\varepsilon)&=&\frac{S}{2}\mathcal{C}_{y}^{2}(\widetilde{u}_{\textbf{k}}-\widetilde{v}_{\textbf{k}})^{2}
\big[G_{n}(\textbf{k},\varepsilon)+G_{n}(-\textbf{k},-\varepsilon)\nonumber\\
&&-2G_{a}(\textbf{k},\varepsilon)\big]\end{aligned}$$ with $$\begin{aligned}
\mathcal{C}_{x}&=&1-\frac{1}{4S}\sum\limits_{\textbf{p}}(2\widetilde{v}^{2}_{\textbf{p}}+\widetilde{u}_{\textbf{p}}\widetilde{v}_{\textbf{p}}) \nonumber\\
\mathcal{C}_{y}&=&1-\frac{1}{4S}\sum\limits_{\textbf{p}}(2\widetilde{v}^{2}_{\textbf{p}}-\widetilde{u}_{\textbf{p}}\widetilde{v}_{\textbf{p}})\end{aligned}$$ where $G_{a}(\textbf{k},\varepsilon)$ is the anomalous magnon Green’s functions which can be written as $$G_{a}(\textbf{k},\varepsilon)=G_{n}(\textbf{k},\varepsilon)\Sigma^{tot}_{a}(\textbf{k},\varepsilon)G_{n}(-\textbf{k},-\varepsilon)$$ Here $\Sigma^{tot}_{a}(\textbf{k},\varepsilon)$ represents the total anomalous self-energy, which in one-loop order can be expressed explicitly as $$\Sigma^{tot}_{a}(\textbf{k},\varepsilon)=\Sigma^{b}_{c}(\textbf{k})+\Sigma^{b}_{hf}(\textbf{k})+\Sigma^{c}_{3}(\textbf{k},\varepsilon)+\Sigma^{d}_{3}(\textbf{k},\varepsilon)$$ Note that the sign of the infinitely small imaginary part in the self-energies $\Sigma^{a}_{3}(\textbf{k},\varepsilon)$ and $\Sigma^{c}_{3}(\textbf{k},\varepsilon)$ have to be changed to account for the retarded self-energies and ensure the correct odd-frequency dependence of the imaginary part of the magnetic suspendibility as stressed in in the study of the TLAH model.
Similar to the calculation of the excitation spectrum and spectral function, we restrict ourselves to the leading $1/S$ order of the spin wave theory, thus several simplifications can be made following Ref. 46. Firstly, the anomalous magnon Green’s functions is of the next order in $1/S$ classification compared to the normal ones $G_{n}(\textbf{k},\varepsilon)$ due to Eq. (64), therefore can be neglected. Secondly, the $G_{n}(-\textbf{k},-\varepsilon)$ term in Eq. (62) is off-resonance compared to $G_{n}(\textbf{k},\varepsilon)$ and contains no poles for $\varepsilon$$>$0, thus has no contribution to the structure factor. All together, the approximated expressions of the transverse dynamical structure factor read $$\begin{aligned}
\mathcal{S}^{xx}(\textbf{k},\varepsilon)&=&\frac{S}{2}\mathcal{C}_{x}^{2}(\widetilde{u}_{\textbf{k}}+\widetilde{v}_{\textbf{k}})^{2}\mathcal{A}(\textbf{k},\varepsilon)\nonumber\\
\mathcal{S}^{yy}(\textbf{k},\varepsilon)&=&\frac{S}{2}\mathcal{C}_{y}^{2}(\widetilde{u}_{\textbf{k}}-\widetilde{v}_{\textbf{k}})^{2}\mathcal{A}(\textbf{k},\varepsilon)\end{aligned}$$ which are simply linear combination of the diagonal component of the spectral functions $\mathcal{A}(\textbf{k},\varepsilon)$ and $\mathcal{A}(\textbf{k}\pm\textbf{Q},\varepsilon)$.
The consideration of the longitudinal component is more delicate than the transverse ones, which is determined by the correlation function $$\mathcal{S}^{zz}(\textbf{k},t)=\langle \delta S^{z}_{\textbf{k}}(t)\delta S^{z}_{-\textbf{k}}(0)\big\rangle$$ with $$\delta S^{z}_{\textbf{k}}=-\sum\limits_{\textbf{p}}a^{\dagger}_{\textbf{p}}a_{\textbf{k}+\textbf{p}}$$ As a consequence, it is in fact an $1/S$ order smaller than the transverse correlation functions. Therefore, just linear spin wave results are adequate to account for the $1/S$ order contributions. And in the torque equilibrium formulism, the result becomes $$\begin{aligned}
\mathcal{S}^{zz}(\textbf{k},\varepsilon)&=&\frac{1}{2}\sum\limits_{\textbf{p}}\delta(\varepsilon-\widetilde{\varepsilon}_{\textbf{p}}-\widetilde{\varepsilon}_{\textbf{k}-\textbf{p}}) \cdot(\widetilde{u}_{\textbf{p}}\widetilde{v}_{\textbf{k}-\textbf{p}}\nonumber\\
&&+\widetilde{v}_{\textbf{p}}\widetilde{u}_{\textbf{k}-\textbf{p}})^{2}\end{aligned}$$ The applicability of these approximation have been examined at length in Ref. 46 for the TLAH model, in which more details can be found. At the same time, we also ignore the contributions from the mixed part of the structure factor and assume that $\mathcal{S}^{tot}(\textbf{k},\varepsilon)\approx\mathcal{S}^{diag}(\textbf{k},\varepsilon)$. And the numerical results of the total dynamical structure factor for the $J_{3}$-and $J_{4}$-systems are demonstrated in Fig. 14, which are obtained based on Eq. (66) and (69).
The dynamical structure factor for the $J_{3}$-and $J_{4}$-systems are plotted along the same representative symmetry paths with the spectral function results. The complicated view of the plot is the consequence of the superposition of three $\textbf{k}$-modulated $\mathcal{A}(\textbf{k},\varepsilon)$ terms and a background of two $\mathcal{S}^{zz}(\textbf{k},\varepsilon)$ terms with the incommensurate ordering vector $\textbf{Q}$. Similar with the spectral function results, the dynamical structure factor also shows sharp single-particle peaks as well as the substantial two-particle continuum contributions. In addition, the spontaneous magnon decay induced broadening of the quasi-particle peaks and the high energy “antibonding” single magnon states are manifested clearly. Although the strong decay region is not easy to read from such a complicated demonstration, the nearly flat mode in the $J_{4}$-systems is clearly shown. Altogether, the total structure factor exhibits a complex but consistent landscape of magnetic excitations with our previous spectrum and spectral function results.
Discussion and Conclusion
=========================
There are two main motivations for the work presented in this paper. The first one is to extend and comprehend our previously developed spin wave expansion approach to more realistic multi-parameter cases, where we can test the applicability of our theory. To accomplish this, we propose a so-called one-parameter renormalization approximation scheme, which can be considered as a good approximation for the quasi-one dimensional systems and its applicability can be further verified by the ordering vector results. Based on this scheme, a series of magnetic properties can be obtained within the TESWT and no nonphysical singularities and divergences appear in the presence of the spin Casimir effect. Not only the $1/S$ expansion results, but also the linear approximated torque equilibrium approach can indicate some important information such as the QObD effect and the magnetic anisotropy induced modification of the FM/spiral Lifshitz point. Thus, generally speaking, it seems that the TESWT can efficiently describe the magnetic dynamics in the incommensurate noncollinear long-range ordered magnetic systems where the CSWT may break down, at least qualitatively. Moreover, the TESWT can take into account part of the quantum fluctuation effects even at the linear approximation level, thus may serve as a more reliable parameter fitting tool than conventional LSWT. The reliability of the TESWT as a parameter fitting tool has been discussed in detail for Cs$_{2}$CuCl$_{4}$ in our previous work. However, for systems concerned in this article, a quantitative investigation of this issues is still impracticable due to the lacking of exact measurements of the exchange parameters.
Our second motivation is to perform a systematic investigation of the spin wave dynamics of the long-range ordered quasi-one dimensional $J_{1}$-$J_{2}$ system. This system has attracted much interest recently due to the high field spin multipolar phases observed in edge-shared chain cuprates. However, the associated spin wave expansion studies of the related zero field spiral state was lacking due to the divergent problems caused by spin Casimir effect. In this work, we perform the calculation of the excitation spectrum in both the on-shell and off-shell approximation scheme. The highlights of the anomalous features of the spectrum that should be observable in experiments are the substantial broadening of quasi-particle peaks due to the spontaneous magnon decay and strong deviations from the LSWT results, which usually serve as foundation of parameter fitting processes in experiments. Moreover, the dependence of the spin wave spectrum on the inter-chain coupling and magnetic anisotropy is uncovered, which may serve as a prob of different types of coupling. These remarkable distinct features are qualitative different decay pattern, different sensitivity to the magnetic anisotropy and the appearance of the nearly flat mode. And to highlight these differences and compare the on-shell and off-shell results in the whole BZ, we investigate the magnon decay rate in both on- and off-shell scheme.
In additional to these physical features, there are also some methodology related problems in the spectrum calculation such as the “sudden non-decay region” in the off-shell results. To clarify these issues, we introduce the poles function and two-magnon density of states and perform a detailed calculation at $\textbf{k}$ points in and outside this region. And it turns out that the “sudden non-decay region” is actually a decay region but with degenerate “bonding” and “antibonding” single-magnon states, which is consistent with the on-shell predictions. Other than that, we further investigate the spectral function as verification to the spectrum results, in which the degeneration of “bonding” and “antibonding” single-magnon states are directly demonstrated. Furthermore, we develop the analytical theory of the dynamical structure factor within the torque equilibrium formulism and present the explicit numerical results, which complete our previous analysis and can be considered as guide for the inelastic neutron scattering experiments.
The present analysis can be straightforwardly generalized to the cases where the frustrated $J_{1}$-$J_{2}$ chains are coupled in a three dimensional manner. The three dimensional coupling can have plentiful rich and varied types, all of which can suppress the quantum fluctuation effect and change the dimension of integration in the calculation of the self-energies. As a consequence, some of our results can be modified accordingly, such as the spikelike logarithmic peaks appear in the on-shell spectrum. However, we expect that the main spontaneous magnon decay related effects and the sensitivity of the magnetic dynamics to different types of inter-chain coupling can still survive due to the quasi-one dimensionality, frustrated intra-chain coupling and associated incommensurate long-range spiral order. Moreover, our investigation can also be directly applied to the large $S$ cases, where the spin wave prediction is more reliable but the over all damping will be smaller as a result of the suppression of the quantum fluctuation due to the large spin length. On the other hand, the long-range spiral order can be more robust in the compounds with large spin length. In addition, as the measurements of the lifetimes of spin expiation in the inelastic neutron and resonant inelastic x-ray scattering experiments are expected to improve dramatically in the future, this will allow for the detailed analysis of the spontaneous magnon decays.
To summarize, we have presented a systematic spin wave analysis of quasi-one dimensional $J_{1}$-$J_{2}$ systems with different types of inter-chain couplings and $XXZ$ anisotropy including various static and dynamical magnetic properties. Our results provide the first determination of the full magnetic properties for the long-range ordered quasi-one dimensional $J_{1}$-$J_{2}$ system within the framework of TESWT. In addition, these detailed calculations provide a direct analytical scheme to investigate the spin dynamics for the realistic materials as well as a guide for experimental observation of the quasi-one dimensional spontaneous magnon decay effects and inter-chain coupling dependent dynamic features. Thus this work presents the full landscape of the nonlinear spin wave dynamics in the quasi-one dimensional FM-AFM frustrated $J_{1}$-$J_{2}$ systems.
Acknowledgment
==============
The authors are indebted to A.L.Chernyshev for illuminating discussions. This work was supported by the National Natural Science Foundation of China (Grants No. 11234005 and No. 51431006), China Postdoctoral Science Foundation (2015M571729) and the National 973 Projects of China (Grant No. 2015CB654602).
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abstract: 'We show that a one-dimensional chain of trapped ions can be engineered to produce a quantum mechanical system with discrete scale invariance and fractal-like time dependence. By discrete scale invariance we mean a system that replicates itself under a rescaling of distance for some scale factor, and a time fractal is a signal that is invariant under the rescaling of time. These features are reminiscent of the Efimov effect, which has been predicted and observed in bound states of three-body systems. We demonstrate that discrete scale invariance in the trapped ion system can be controlled with two independently tunable parameters. We also discuss the extension to $n$-body states where the discrete scaling symmetry has an exotic heterogeneous structure. The results we present can be realized using currently available technologies developed for trapped ion quantum systems.'
author:
- Dean Lee
- Jacob Watkins
- Dillon Frame
- Gabriel Given
- Rongzheng He
- Ning Li
- 'Bing-Nan Lu'
- Avik Sarkar
bibliography:
- 'References.bib'
title: Time fractals and discrete scale invariance with trapped ions
---
In this work we show how to construct a one-dimensional system of trapped ions with discrete scale invariance and fractal-like time dependence. In classical systems scale invariance arises when the scale transformation acting on spatial coordinates, $r \rightarrow \lambda r,$ is a symmetry of the dynamics. This arises naturally if the Hamiltonian transforms homogeneously under rescaling. When the Hamiltonian is quantized, however, this scale invariance cannot persist for bound state solutions with discrete energy levels. Instead, the scale invariance is broken through a quantum scale anomaly. An analogous effect occurs in relativistic field theories and is responsible for the mass gap in the spectrum of non-Abelian gauge theories such as quantum chromodynamics.
While the quantum scale anomaly spoils invariance under a general scale transformation, it may preserve the symmetry associated with a discrete set of scale transformations. This was first described by Efimov for the bound state spectrum of three bosons with short-range interactions tuned to infinite scattering length [@Efimov:1971zz; @Efimov:1993a; @Bedaque:1998kg; @Bedaque:1998km]. See also Ref. [@Coon:2002sua] for a review of anomalies in quantum mechanics and the attractive $1/r^2$ potential. Efimov trimers were first observed experimentally through the loss rate of trapped ultracold cesium atoms [@Kraemer:2006Nat], and a more direct observation has been made using the Coulomb explosion of helium trimers [@Kunitski:2015qth]. As the underlying physics is of universal character, the application and generalization of the Efimov effect has been considered in various settings, including nuclear physics [@Bedaque:1999ve; @Hagen:2013jqa], bound states with more than three particles [@Platter:2004pra; @Hammer:2006ct; @vonStecher:2009a; @vonStecher:2011zz; @Carlson:2017txq], systems with reduced dimensions [@Nishida:2011ew; @Moroz:2013kf; @Happ:2019], quantum magnets [@Nishida:2012hf], molecules with spatially-varying interactions [@Nishida:2012by], and Dirac fermions in graphene [@Ovdat:2017lho].
We demonstrate that quantum scale anomalies can be produced with trapped ion quantum systems. We start with a one-dimensional chain of ions in a radio-frequency trap with qubits represented by two hyperfine “clock” states. Such systems have been investigated by the trapped ion group at the University of Maryland using $^{171}$Yb$^{+}$ ions [@Zhang:2017a; @Zhang:2017b]. Similar efforts have been pioneered by trapped ion groups at ETH Z[ü]{}rich, Freiburg, Innsbruck, Mainz, Stockholm, and the Weizmann Institute. Off-resonant laser beams are used to drive stimulated Raman transitions for all ions in the trap. This induces effective interactions between all qubits with a power-law dependence on separation distance. We define the vacuum state as the state with $\sigma^z_i=1$ for all $i$. We use interactions of the form $\sigma^x_{i}\sigma^x_{j}+\sigma^y_{i}\sigma^y_{j}$, to achieve the hopping of spin excitations. We then use a $\sigma^z_{i}\sigma^z_{j}$ interaction to produce a two-body potential felt by pairs of spin excitations, and we also consider an external one-body potential coupled to $\sigma^z_{i}$.
We can view each spin excitation with $\sigma^z_i=-1$ as a bosonic particle at site $i$ with hardcore interactions preventing multiple occupancy. In this language, the Hamiltonian we consider has the form $$\begin{aligned}
H =\frac{1}{2} \sum_{i}\sum_{j \ne i} J_{ij}[b^{\dagger}_i b_j +b^{\dagger}_j
b_i] & + \frac{1}{2}\sum_{i}\sum_{j \ne i}V^{}_{ij}b^{\dagger}_i b_i b^{\dagger}_j
b_j \nonumber\\
& + \sum_{i}U^{}_{i}b^{\dagger}_i b_i+C,\end{aligned}$$ where $b_i$ and $b^{\dagger}_i$ are annihilation and creation operators for the hardcore bosons on site $i$. See the Supplemental Materials for a derivation of this Hamiltonian. The parameter $C$ is just an overall energy constant. The hopping coefficients $J_{ij}$ have the asymptotic form $J_{ij} = J_0/|r_i-r_j|^{\alpha}$, where $r_i$ is the position of qubit $i$. For the purposes of this study, we assume $J_{ij}$ to have exactly this form for $i \ne j.$ Similarly, the two-body potential coefficients $V_{ij}$ have the asymptotic form $V_{ij} = V_0/|r_i-r_j|^{\beta}$. In this work we assume $V_{ij}$ to have exactly this form for $i \ne j$. We consider the case where the lattice of ions is uniform and large, and we start with a constant potential $U_i$ chosen so that bosons with zero momentum have zero energy. Both positive (anti-ferromagnetic) and negative (ferromagnetic) values can be realized for $J_0$ and $V_0$. The exponents $\alpha$ and $\beta$ can in principle vary in the range between $0$ and $3$. However, in practice the range between $0.5$ and $1.8$ is favored in order to enhance coherence times and reduce experimental drifts [@Zhang:2017b].
We now add to $U_i$ a deep attractive potential at some chosen site $i_0$ that traps and immobilizes one boson at that site. Without loss of generality, we take the position of that site to be the origin and add a constant to the Hamiltonian so that the energy of the trapped boson is zero. We then consider the dynamics of a second boson that feels the interactions with this fixed boson at the origin. In order to produce a Hamiltonian with classical scale invariance, we choose $\beta=\alpha-1$. Then at low energies, our low-energy Hamiltonian for the second boson has the form $$H(p,r)=2J_0\sin(\alpha\pi/2)\Gamma(1-\alpha)|p|^{\alpha-1}+
\frac{V_0}{|r|^{\alpha-1}},$$ where we omit corrections of size $O(p^{2})$. We are interested in the case where both $J_0$ and $V_0$ are negative. In that case we find an infinite tower of even parity and odd parity bound states. We label the bound state energies as $E^{(n)}_{+}$ and $E^{(n)}_{-}$, respectively, for nonnegative integers $n$. As expected, our quantized system has a quantum scale anomaly and we are left with two discrete scale symmetries, $r \rightarrow \lambda_{+}
r$ for even parity and $r \rightarrow \lambda_{-} r$ for odd parity. Correspondingly, the bound state energies follow a simple geometrical progression, $E^{(n)}_{+}=E^{(0)}_+ \lambda_{+}^{-n}$ and $E^{(n)}_{-}=E^{(0)}_-
\lambda_{-}^{-n}$. In the Supplemental Materials we provide details of the discrete scale invariance for general $\alpha$. For the special case $\alpha=2$, the scale factors are $\lambda_{\pm} = \exp (\pi/\delta_{\pm}),$ where $$\begin{aligned}
\delta_+ = \frac{V_0}{J_0\pi} \coth (\delta_+ \pi/2), \; \;
\delta_- = \frac{V_0}{J_0\pi} \tanh (\delta_- \pi/2).\end{aligned}$$
![[**Bound state wave functions.**]{} Plot of the normalized wave functions for the first twelve even-parity bound states for the case $\alpha = 2$, $\beta = 1$, $J_0 = -1$, and $V_0 = -30$. We plot the region $r>0$. All quantities are in dimensionless lattice units.[]{data-label="even_wavefunctions"}](v30_even_wavefunctions_final){width="8.5cm"}
In contrast with most other systems with a quantum scale anomaly, we note that the properties of our ion trap system can be tuned using two different adjustable parameters, $V_0/J_0$ and $\alpha$. This is convenient for probing a wide range of different phenomena exhibiting discrete scaling symmetry. In the following we will work in lattice units where physical quantities are multiplied by powers of the lattice spacing to make the combination dimensionless and have set $\hbar = 1$. As an example, consider a system with $\alpha = 2$, $\beta = 1$, $J_0 =
-1$, and $V_0
= -30$. The wave functions for the first twelve even-parity bound states are shown in Fig. \[even\_wavefunctions\]. We plot the normalized wave function for $r>0$. We see clear evidence of discrete scale invariance emerging as we approach zero energy. In Table \[energies\] we show the energies for the first fourteen even-parity and odd-parity bound states and the ratios between consecutive energies. For comparison, at the bottom we show the predictions for these ratios as we approach zero energy at infinite volume. We see that the agreement is quite good.
$n$ $E^{(n)}_+$ $E^{(n-1)}_+/E^{(n)}_+$ $E^{(n)}_-$ $E^{(n-1)}_-/E^{(n)}_+$
-------- ---------------- ---------------------------- ---------------- ----------------------------
$0$ $-27.05304149$ $-$ $-26.5188669$ $-$
$1$ $-11.93067205$ $2.267520336$ $-11.79861873$ $2.247624701$
$2$ $-6.977774689$ $1.709810446$ $-6.919891389$ $1.705029468$
$3$ $-4.553270276$ $1.5324754$ $-4.521425357$ $1.530466798$
$4$ $-3.139972298$ $1.450098869$ $-3.120231851$ $1.449067112$
$5$ $-2.233327278$ $1.405961557$ $-2.220194049$ $1.405386998$
$6$ $-1.617052389$ $1.381110033$ $-1.607920414$ $1.380786033$
$7$ $-1.182654461$ $1.367307563$ $-1.176124883$ $1.367134084$
$8$ $-0.869406941$ $1.360300229$ $-0.864656962$ $1.360221377$
$9$ $-0.640405903$ $1.357587332$ $-0.636916042$ $1.357568195$
$10$ $-0.471738446$ $1.357544438$ $-0.469161911$ $1.357561276$
$11$ $-0.347112043$ $1.359037968$ $-0.345207121$ $1.359073675$
$12$ $-0.254996818$ $1.361240684$ $-0.253589633$ $1.361282464$
$13$ $-0.187011843$ $1.363532996$ $-0.18597462$ $1.363571189$
theory – $\lambda_{+}=1.3895595319$ – $\lambda_{-}=1.3895595319$
One intriguing question is how discrete scale invariance could persist in quantum many-body systems. It has been demonstrated numerically that the Efimov effect extends beyond bosonic trimers and describes the properties of $n$-boson systems with the same discrete scaling factor [@Platter:2004pra; @Hammer:2006ct; @vonStecher:2009a; @vonStecher:2011zz; @Carlson:2017txq] . As we will see, something quite different happens in the trapped ion system. Let us start from a particular bound state of the two-body system and ask what happens when we introduce a third boson that is weakly bound and very far from the origin. The effective Hamiltonian for the third boson contains a potential energy that is doubled due to interactions of the weakly-bound third boson with the two other bosons. As a result of the stronger attractive interaction, the geometric scaling factors $\lambda_{\pm}$ for the third boson will be smaller than for the two-body system. This argument can be generalized to describe weakly-bound states for the general $n$-body system. The effective potential for the $n^{\rm th}$ boson will be a factor of $n-1$ times larger, and thus the scaling of the $n$-body energies relative to each $(n-1)$-body threshold is different from the scaling of the $k$-body bound states for each $k$ between $1$ and $n$. The properties of these exotic systems with heterogeneous discrete invariance will be investigated further in future work.
Let us now consider an initial state $|S\rangle = \sum_{n=0}^{N-1} |\psi^{(n)}_+\rangle,$ where we sum over the first $N$ even-parity two-boson bound states $|\psi^{(n)}_+\rangle$ with equal weight. We choose the even-parity states, but we could just as easily choose odd-parity states. The phase convention for each $|\psi^{(n)}_+\rangle$ is chosen so that the tail of the wave function is real and positive at large $r$. We note that the time dependent amplitude $A(t)={\rm Re} [\langle S | \exp(-iHt)
| S \rangle]$ is invariant under the rescaling $t \rightarrow \lambda^{\alpha-1}_{+}
t$, thus endowing it with the properties of a time fractal. The time fractal is particularly interesting for the case when $\lambda^{\alpha-1}_{+}$ is an integer so that each of the higher frequencies in $A(t)$ are integer multiples of the lower frequencies.
For the case $\alpha = 2$ and $J_0=-1$, we can produce the time scaling factor $\lambda^{\alpha-1}_+=\lambda_+=2$ by setting $V_0=-14.2388293$. In Fig. \[fractal\] we show the amplitude $A(t)$ ranging from $t=0$ to $80$ in the upper left, $t=0$ to $160$ in the upper right, $t=0$ to $320$ in the lower left, and $t=0$ to $640$ in the lower right. Aside from small deviations, we see that the time dependence shows fractal-like self-similarity when we zoom in or out by a scale factor very close to $2$. The best fit for the scale factor is approximately $1.9$. In the Supplemental Materials we show how a time fractal can be realized experimentally using quantum interference on a trapped ion quantum system.
The time fractals that we have discussed are closely related to the Weierstrass function $w(x)=\sum_{n=0}^\infty a^n \cos(b^n \pi x)$. Weierstrass showed that this function is continuous everywhere but differentiable nowhere when $0<a<1$, $b$ is an odd integer, and $ab > 1 + 3\pi/2$ [@Weierstrass:1886]. Hardy extended the proof to any $0<a<1<b $ and $ab\ge 1$ [@Hardy:1916]. We note that $aw(bx)$ equals $w(x)$ plus the smooth function $\cos(\pi x)$, and this suggests that the fractal dimension of the Weierstrass function should given by [@Hunt:1998] $$D = 2 + \frac{\log a}{\log b}.$$ This result for the fractal dimension is confirmed by the box-counting method for determining fractal dimensions [@Kaplan:1984].
Our initial state $|S\rangle = \sum_{n=0}^{N-1} |\psi^{(n)}_+\rangle$ produces the fractal-like amplitude $$A(t) =\sum_{n=0}^{N-1}\cos (E^{(n)}_+t) =\sum_{n=0}^{N-1}\cos (\epsilon_+ \lambda_{+}^{-n}t).$$ In the limit of large $N$, our choice of parameters corresponds to the limiting case $a \rightarrow 1$ and $b = \lambda_{+}^{}$, with $x =\epsilon_+\lambda_+^{-N+1}t/{\pi}$. Therefore, the fractal dimension for our time fractal will be $D=2$. If we instead choose the initial state to have the form $|S(a)\rangle = \sum_{n=0}^{N-1} a^{n/2}|\psi^{(n)}_+\rangle $ for $a < 1,$ then in the limit $N \rightarrow \infty$, the fractal dimension will be $$D = 2 + \frac{\log a}{\log \lambda_{+}}.$$
There are many interesting related phenomena that one can explore in connection with time fractals and the dynamics of systems with discrete scale invariance. One fascinating topic is the adiabatic evolution of a system with discrete invariance as the interactions are varied slowly. Another is the response of a system with discrete scale invariance when driven in resonance with one of its bound state energies. In this letter we have shown that the intrinsic power-law interactions of the trapped ion system make it an ideal system for exploring the physics of quantum scale anomalies, discrete scale invariance, and time fractals. There are clearly many directions that one can explore in this new area, and we look forward to working with others to develop further applications and experimental realizations of many of these concepts.
{width="8cm"} {width="8cm"} {width="8cm"} {width="8cm"}
[*We are grateful for discussions with Zohreh Davoudi, Chao Gao, Pavel Lougovski, Titus Morris, Thomas Papenbrock, and Raphael Pooser. We acknowledge financial support from the U.S. Department of Energy (DE-SC0018638 and DE-AC52-06NA25396). Computational resources were provided by the Julich Supercomputing Centre at Forschungszentrum Jülich, Oak Ridge Leadership Computing Facility, RWTH Aachen, and Michigan State University.*]{}
Supplemental Material
=====================
Trapped ion Hamiltonian {#trapped-ion-hamiltonian .unnumbered}
-----------------------
For our one-dimensional trapped ion system, the Hamiltonian we consider is $$H = T + V_2+U+C,$$ where $$\begin{aligned}
T & = \frac{1}{4}\sum_{i}\sum_{j \ne i} J_{ij}(\sigma^x_{i} \sigma^x_{j}+\sigma^y_{i}
\sigma^y_{j}), \\
V_2 & = \frac{1}{8}\sum_{i}\sum_{j \ne i}V_{ij}(1-\sigma^z_{i})( 1-\sigma^z_{j}
), \\
U & =\frac{1}{2}\sum_{i}U_{i}( 1-\sigma^z_{i}),\end{aligned}$$ and $C$ is a constant. We regard each spin configuration with $\sigma^z=-1$ as a particle excitation. Thus $T$ corresponds to the hopping of a single particle, $V_2$ is a two-particle interaction, and $U$ is a one-particle potential. In Fig. \[J\_ij\] we show a sketch of the action of the hopping coefficient $J_{ij}$. In Fig. \[V\_ij\] we show a sketch of the two-body interaction potential $V_{ij}$. Without loss of generality, we assume that both $J_{ij}$ and $V_{ij}$ are symmetric in the indices $i,j.$
![[**Hopping coefficient $J_{ij}$.**]{} This sketch shows the action of the hopping coefficient $J_{ij}$ for a single particle between sites $i$ and $j$. Particle excitations correspond with sites where $\sigma^z=-1$.[]{data-label="J_ij"}](J_ij.pdf){width="6cm"}
![[**Interaction potential $V_{ij}$.**]{} This sketch shows the two-body interaction potential $V_{ij}$ between two particles at sites $i$ and $j$. Particle excitations correspond with sites where $\sigma^z=-1.$[]{data-label="V_ij"}](V_ij.pdf){width="6cm"}
We can reorganize the $\sigma^z$ terms as $$V_2 +U=\frac{1}{8}\sum_{i}\sum_{j \ne i}V_{ij}\sigma^z_{i}\sigma^z_{j}
-\frac{1}{2}\sum_{i}U'_i\sigma^z_{i}+C',$$ where $$U'_i = U_i+\frac{1}{2}\sum_{i \ne j}V_{ij},$$ and $$C'=\frac{1}{8}\sum_{i}\sum_{j \ne i}V_{ij} +\frac{1}{2}\sum_{i}U_{i}.$$ We can view each spin excitation with $\sigma^z_i=-1$ as a bosonic particle at site $i$ with hardcore interactions preventing multiple occupancy. When expressed in terms of hardcore boson annihilation and creation operators, the Hamiltonian becomes $$H =\frac{1}{2} \sum_{i}\sum_{j \ne i} J_{ij}[b^{\dagger}_i b_j+b^{\dagger}_j
b_i] + \frac{1}{2}\sum_{i}\sum_{j \ne i}V^{}_{ij}b^{\dagger}_i b_i b^{\dagger}_j
b_j+ \sum_{i}U^{}_{i}b^{\dagger}_i b_i+C.
\label{Hamiltonian}$$
Dispersion relation {#dispersion-relation .unnumbered}
-------------------
We assume that the ions lie on a one-dimensional lattice with uniform spacing. There will be some distortion at the edges of the trap, but since our interest is in bound states with some degree of spatial localization, these edge effects can be minimized by placing the system at the middle of a trap with many ions. We work in lattice units where physical quantities are multiplied by powers of the lattice spacing to make the combination dimensionless and have also set $\hbar = 1$. We start with the case where the potential $U_i$ is set to equal $$U_{i} =- \sum_{j \ne i} J_{ij}=- \sum_{j \ne i} \frac{J_0}{|r_i-r_j|^{\alpha}}.$$ By computing the expectation value of the Hamiltonian for a single boson with momentum $p$, we find that the energy of a single boson with momentum $p$ is $$E(p) = 2J_0\sum_{n>0} \frac{\cos(pn)-1}{n^\alpha} =J_0 \left[ {\rm Li}_{\alpha}(e^{ip})+{\rm
Li}_{\alpha}(e^{-ip})-{\rm 2Li}_{\alpha}(1) \right],$$ where ${\rm Li}_{\alpha}$ is the polylogarithm function of order $\alpha$. We find that for $\alpha<3$, $$E(p)= 2J_0\sin(\alpha\pi/2)\Gamma(1-\alpha)|p|^{\alpha-1}+J_0\zeta(\alpha-2)p^{2}+O(p^4),
\label{dispersion}$$ where $\zeta$ is the Riemann zeta function. We note that the special case $\alpha=2$ corresponds with a linear dispersion relation, which has important theoretical connections to relativistic fermions as well as electrons in graphene. In Fig. \[dispersion\_relation\] we plot the dispersion relation $E(p)$ versus $p$ for $J_0=-1$ and $\alpha=1.5,2.0,2.5$.
![[**Dispersion relation.**]{} Plot of the dispersion relation $E(p)$ versus $p$ for $J_0=-1$. []{data-label="dispersion_relation"}](dispersion_relation.pdf){height="5cm"}
Two-body system {#two-body-system .unnumbered}
---------------
We now introduce a single-site deep trapping potential with large coefficient $u>0$ at some site $i_0$ that traps and immobilizes one hardcore boson at that site, $$U_{i} = - \sum_{j \ne i} \frac{J_0}{|r_i-r_j|^{\alpha}}-u\delta_{i,i_0}.$$ We also subtract a constant from the Hamiltonian so that the energy of the trapped boson is exactly zero. We then consider the dynamics of a second boson that feels the interactions with this fixed boson at $i_0$. In order to simplify our notation, let the position of the fixed boson be $r_{i_0}=0$. Let us now set $\beta=\alpha-1$. Then at low energies, our low-energy Hamiltonian for the second boson has the form $$H(p,r)=2J_0\sin(\alpha\pi/2)\Gamma(1-\alpha)|p|^{\alpha-1}+
\frac{V_0}{|r|^{\alpha-1}},$$ with corrections of size $O(p^{2})$. This follows from the result for $E(p)$ in Eq. (\[dispersion\]) and that the interaction between the particles has the form $V_{ij} = V_0/|r_i-r_j|^{\beta} = V_0/|r_i-r_j|^{\alpha-1}$. We note that this Hamiltonian has classical scale invariance.
We define by analytic continuation the Fourier transforms of the functions $|r|^\eta$, $$\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty dr e^{ipr} |r|^\eta=-\sqrt{\frac{2}{\pi}}|p|^{-1-\eta}\Gamma(1+\eta)\sin(\eta\pi/2).$$ We also compute the Fourier transforms of $\operatorname{sgn}(r)|r|^\eta$, where $\operatorname{sgn}(r)$ is the sign function,$$\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty dr e^{ipr} \operatorname{sgn}(r)|r|^\eta=i\sqrt{\frac{2}{\pi}}\operatorname{sgn}(p)|p|^{-1-\eta}\Gamma(1+\eta)\cos(\eta\pi/2).$$ In the zero energy limit, the quantum Hamiltonian $H(p,r)$ exhibits a renormalization-group limit cycle with even-parity $(+)$ and odd-parity $(-)$ wave functions at zero energy, $$\begin{aligned}
\psi_{\rm +}(r) & = \frac{1}{2}\left(|r|^{i\delta_+} + |r|^{-i\bar{\delta}_+}\right),\\
\psi_{\rm -}(r) & = \frac{1}{2}\operatorname{sgn}(r)\left(|r|^{i\delta_{-}} + |r|^{-i\bar{\delta}_{-}}\right),\end{aligned}$$ where $\delta_{\pm}$ are solutions to the constraints, $$\begin{aligned}
2J_0\delta_+\Gamma(1-\alpha)\sin(\alpha\pi/2)\Gamma(i\delta_+)\sinh(\delta_+\pi/2)&
=V_{0}
\Gamma(2-\alpha+i\delta_+)\cos((\alpha-i\delta_+)\pi/2), \\
2J_0\delta_-\Gamma(1-\alpha)\sin(\alpha\pi/2)\Gamma(i\delta_-)\cosh(\delta_-\pi/2)&
=iV_{0}
\Gamma(2-\alpha+i\delta_-)\sin((\alpha-i\delta_-)\pi/2).\end{aligned}$$ For the particular case $\alpha =2$, this constraint simplifies to $$\begin{aligned}
\delta_+ = \frac{V_0}{J_0\pi} \coth (\delta_+ \pi/2), \; \;
\delta_- = \frac{V_0}{J_0\pi} \tanh (\delta_- \pi/2).\end{aligned}$$ These solutions for the case $\alpha =2$ are real whenever $V_0/J_0$ is positive. When $V_0/J_0 \gg \pi$, these are both very well approximated by $$\delta_+ \approx \delta_- \approx \frac{V_0}{J_0\pi}.$$
In the cases where $\delta_+$ and $\delta_-$ are real, the discrete scale invariance of the renormalization-group limit cycle can be seen by writing $$\begin{aligned}
\psi_{\rm {+}}(r) = \cos[\delta_{+} \ln(|r|)], \; \; \;\psi_{\rm {-}}(r)
= \operatorname{sgn}(r)\cos[\delta_{-} \ln(|r|)].\end{aligned}$$ Under the scale transformations $r \rightarrow \lambda_{\pm} r,$ we have $$\begin{aligned}
\psi_{\rm {+}}(r) & \rightarrow \cos[\delta_{+} \ln(|r|)+\delta_{+} \ln(\lambda_{+})],
\\
\psi_{\rm {-}}(r) & \rightarrow \operatorname{sgn}(r)\cos[\delta_{-} \ln(|r|)+\delta_{-}
\ln(\lambda_{-})].\end{aligned}$$ The wave functions remain invariant up to an overall minus sign if we let$$\begin{aligned}
\lambda_{+} = \exp (\pi/\delta_{+}), \; \;
\lambda_{-} = \exp (\pi/\delta_{-}).\end{aligned}$$ The bound state energies also respect this discrete scale symmetry. Under the scale transformation $r \rightarrow \lambda_{\pm} r$, the energy scales as $E_{\pm} \rightarrow \lambda_{\pm}^{-1}E_{\pm}$. We therefore get an infinite tower of states $E^{(n)}_{+/-}$ obeying the geometric progression $$E^{(n)}_{+}=\epsilon_+ \lambda_{+}^{-n}, \; \; E^{(n)}_{-}=\epsilon_- \lambda_{-}^{-n},
\label{geometric}$$ for some negative energy constants $\epsilon_{+/-}$. We note that the case $\alpha = 2$ corresponds to a Hamiltonian of the form $$H(p,r)=-\pi J_0|p|+
\frac{V_0}{|r|},$$ which, for $J_0<0$ and $V_0<0$, is analogous to a relativistic fermion with attractive Coulomb interactions. This system is therefore directly related to the scale anomaly recently proposed in graphene for Dirac fermions and attractive Coulomb interactions [@Ovdat:2017lho].
For the cases where $\delta_+$ and $\delta_-$ are not real, the wave functions at zero energy are $$\begin{aligned}
\psi_{\rm {+}}(r) = |r|^{-{\rm Im}\,\delta_+}\cos[{\rm Re}\,\delta_{+} \ln(|r|)],
\; \; \;\psi_{\rm {-}}(r)
= |r|^{-{\rm Im}\,\delta_-}\operatorname{sgn}(r)\cos[{\rm Re}\,\delta_{-} \ln(|r|)].\end{aligned}$$ Under the scale transformations $r \rightarrow \lambda_{\pm} r$, the wave functions scale homogeneously if we let$$\begin{aligned}
\lambda_{+} = \exp (\pi/{\rm Re}\,\delta_{+}), \; \;
\lambda_{-} = \exp (\pi/{\rm Re}\,\delta_{-}).\end{aligned}$$
Time fractals {#time-fractals .unnumbered}
-------------
For the purposes of this discussion, we consider the immobile boson localized at $r=0$ as a static source and consider only the wave function of the second boson, which can occupy sites $r \ne 0$. We start with an initial state $$|S\rangle = \sum_{n=0}^{N-1} |\psi^{(n)}_+\rangle,$$ where we sum over the lowest $N$ even-parity bound states $|\psi^{(n)}_+\rangle$ with equal weight. The phase convention for each $|\psi^{(n)}_+\rangle$ is chosen so that the tail of the wave function is real and positive at large $r$. This state can be decomposed into position eigenstates $$| S \rangle = \sum_{r \ne 0} S(r)|r\rangle.$$
On a classical computer we can produce time fractals by computing the amplitude $$\begin{aligned}
A(t) &= {\rm Re} [Z(t) ],\end{aligned}$$ where $$\begin{aligned}
Z(t) &= \langle S | \exp(-iHt) | S \rangle.\end{aligned}$$ The experimental realization of time fractals on a trapped ion quantum system requires more effort. In order to compute the time evolution of the state $| S \rangle$, we define a product of single-qubit rotations $$U(\epsilon) = \prod_{r\ne 0} \exp[-i\epsilon\sigma^y_r S(r)],$$ for some infinitesmal real parameter $\epsilon$. With the immobile boson still fixed at $r=0$, let us denote the normalized state with no mobile bosons at all as $| o \rangle$. The action of $U$ on the state $| o \rangle$ produces a wave function with an indefinite number of mobile bosons. We find that $$U(\epsilon)| o \rangle = \left[1 - \frac{\epsilon^2}{2} \langle S|S\rangle
+ O(\epsilon^3)\right] |o\rangle +\epsilon
|S\rangle + O(\epsilon^2 )|X\rangle,$$As mentioned in the main text, we have subtracted a constant from the Hamiltonian so that the energy of $|o\rangle$ is zero. Hence,
$$\langle o |\exp[-iHt] |o\rangle=\langle o |o\rangle=1.$$
We now measure $$B(\epsilon,t) = |\langle o | U^\dagger(\epsilon) \exp[-iHt] U(\epsilon)|
o \rangle|^2,$$ This can be viewed as a quantum measurement of the projection operator $|o
\rangle \langle o |$ on the state $$U^\dagger(\epsilon) \exp[-iHt] U(\epsilon)|
o \rangle.$$ If we deconstruct $B(\epsilon,t)$ into powers of $\epsilon,$ we get $$B(\epsilon,t) =\left|1-\epsilon^2\langle S|S\rangle +\epsilon^2 Z(t)+O(\epsilon^3)\right|^2.$$ We then obtain $$\begin{aligned}
B(\epsilon,t) & =1-2\epsilon^2\langle S|S\rangle+\epsilon^2[Z(t)+Z^{*}(t)]+O(\epsilon^3)
\nonumber \\
&=1-2\epsilon^2\langle S|S\rangle+2\epsilon^2 A(t)+O(\epsilon^3), \end{aligned}$$ and we can thus determine the desired amplitude $A(t)$.
To our knowledge this is the first instance of the concept of time fractals appearing in the literature. However a recent preprint [@C_Gao:2019] appeared a few weeks after our preprint was posted discussing a similar concept which they called dynamical fractals.
|
---
abstract: 'We present the star formation histories, luminosities, colors, mass to light ratios, and halo masses of “galaxies” formed in a simulation of cosmological reionization. We compare these galaxies with Lyman Break Galaxies observed at high redshift. While the simulation is severely limited by the small box size, the simulated galaxies do appear as fainter cousins of the observed ones. They have similar colors, and both simulated and observed galaxies can be fitted with the same luminosity function. This is significant to the process of reionization since in the simulation the stars alone are responsible for reionization at a redshift consistent with that of the Gunn-Peterson trough seen in the absorption spectra of QSO’s. The brightest galaxies at $z = 4$ are indeed quite young in accord with observational studies. But these brightest galaxies are free riders - they contributed only about 25% to the reionization of the universe at $z > 6$. Instead, the bulk of the work was done by dimmer galaxies, those that fall within the $28 < I < 30$ magnitude range at $z\sim4$. These dimmer galaxies are not necessarily less massive than the brightest ones.'
author:
- 'A. Gayler Harford and Nickolay Y. Gnedin'
title: Lyman Break Galaxies and Reionization of the Universe
---
Introduction
============
The most dramatic global transition in the recent history of the universe is the reionization of the intergalactic medium. The recent identification of the predicted Gunn-Peterson trough in the absorption spectra of several QSO’s has placed this event at about a redshift of 6 (Djorgovski et al. 2002; Becker et al. 2001).
Ionizing radiation from newly formed stars has been suggested as the most likely source of this radiation. Bright, optically selected QSO’s appear to be too sparse during the relevant period (Fan et al. 2001). Low brightness AGN’s remain a possibility, but no other strong candidates have emerged.
It is natural to look for evidence for these ionizing stars in the accumulating observational data on high redshift ($z\la4$) galaxies (Madau et al. 1996; Madau, Pozzetti, & Dickinson 1998; Steidel et al. 1999; Ouchi et al. 2001; Ferguson, Dickinson, & Papovich 2002; Hu et al. 2002; Malhotra & Rhoads 2002; Lehnert & Bremer 2003; Kodaira et al. 2003; Iwata et al. 2003). Some studies on Lyman Break Galaxies (LBG) have questioned whether enough stars could have been formed to accomplish reionization at so early a time (Ferguson, Dickinson, & Papovich 2002; Dickinson et al. 2003).
In the popular Lambda-CDM cosmological model the seeds of galaxy formation lie in minute density fluctuations, which manifest themselves as temperature fluctuations in the Cosmic Microwave Background (CMB). As the fluctuations continue to grow into dark matter halos, they trap gas within their potential wells. That part of the gas that is able to radiate its energy efficiently eventually condenses into dense molecular clouds, that give birth to stars.
Reionization has been studied within this paradigm both by semi-analytical methods (Giroux & Shapiro 1996; Ciardi & Ferrara 1997; Haiman & Loeb 1997, 1998; Madau et al. 1999; Valageas & Silk 1999; Chiu & Ostriker 2000; Miralda-Escude, Haehnelt & Rees 2000; Barkana & Loeb 2000; Oh et al. 2001; Wyithe & Loeb 2003) and by numerical simulations (Ostriker & Gnedin 1997; Gnedin & Ostriker 1998; Gnedin 2000; Nakamoto, Umemura, & Susa 2001; Razoumov et al. 2002). The former necessarily involve broad simplifying assumptions and, therefore, have limited predictive power. The latter, although more realistic and fully self-consistent, are limited by the dynamical range that can be accommodated by existing numerical techniques on modern supercomputers. The two approaches are complementary and will have to remain so until the next generation of cosmological numerical codes become available on faster computers.
For this paper we use a simulation that was reported in Gnedin (2000). The simulation includes dark matter, gas, star formation, chemistry and ionization balance in the primordial plasma, and an approximate implementation of 3D radiative transfer. All galaxies with total masses in excess of $3\times10^8\dim{M}_\sun$ are resolved both spatially and by mass. The simulation was continued until $z=4.0$.
The overall picture for the evolution of the early universe that has emerged from this simulation is as follows:
Star formation first begins in earnest at about $200\dim{Myr}$ ($z\sim20$). The overall star formation rate increases to a maximum at about $1\dim{Gyr}$ ($z\sim5.5$) and decreases slightly thereafter.
Reionization of the intergalactic medium occurs over a prolonged period of time from about $350\dim{Myr}$ ($z\sim12$) to about $900\dim{Myr}$ ($z\sim6$), with expanding regions in the low density IGM merging at $z\sim7$. This latter moment of rapid transition is often identified as the “moment of reionization”. This moment approximately corresponds to the mass (volume) averaged neutral fraction in the low density IGM falling below 1 (0.1) percent respectively.
As a result of reionization, the gas content of halos less massive than about $3\times10^9\dim{M}_\sun$ solar masses is drastically reduced.
Since this simulation does not include AGN’s, it provides a detailed example of how stars alone might be sufficient to accomplish reionization. The addition of AGN’s to this scenario would presumably lead to even more ionizing photons, which could take up the slack in the event that the amount of star formation in the simulation were overestimated.
The simulation provides us with a detailed star formation history for each galaxy and thus allows us to make detailed photometric predictions. In this paper we present the star formation histories, luminosities, colors, mass to light ratios, and halo masses of the simulated galaxies.
Our simulation, while state of the art, is only barely sufficient for our purpose, because the computational box is so small. This inherent limitation of any simulation limits the number of bright objects that can be modeled, while the observations approach the galaxy hierarchy from the other end, detecting bright galaxies and missing the dim ones. In this sense we can think of observations and simulations as moving toward each other along the galaxy luminosity curve. Although there is not much overlap yet, the junction point has at last been reached, as we show below.
We argue that to the extent comparisons can be made, our predictions are consistent with observations of Lyman Break Galaxies. The varied star formation histories that we see provide the key to understanding why the observations appear to argue that LBG galaxies are too young to have reionized the universe.
Method
======
Simulation
----------
The specific simulation we use in this paper is fully described in Gnedin (2000). The simulation was performed using the Softened Lagrangian Hydrodynamics (SLH) code, which includes dark matter, gas, star formation, chemistry and ionization balance in the primordial plasma, and 3D radiative transfer (in an approximate implementation). All these physical ingredients are required to properly model the process of cosmological reionization.
The simulation of a representative CDM+$\Lambda$ cosmological model[^1] was performed in a comoving box with the size of $4h^{-1}\dim{Mpc}$ with the total mass resolution of $4\times10^6\dim{M}_\odot$ and the comoving spatial resolution of $1h^{-1}\dim{kpc}$. This resolution is a reasonable compromise between the need to have the box large enough to accommodate several regions and the need to have spatial resolution high enough to resolve sources of ionizing radiation. Convergence studies presented in Gnedin (2000) demonstrate that, while still limited, the resolution of this simulation is sufficient to model reionization at a semi-qualitative level (with precision of the order of 50% or so).
The simulation was stopped at $z=4$ because at this time the rms density fluctuation in the computational box is about 0.25, and at later times the box ceases to be a representative region of the universe even for the dark matter.
The simulation was designed in such a way as to approximately reproduce the measured star formation rate density at $z=4$ of about $0.1\dim{M}_\odot/\dim{yr}/\dim{Mpc}^3$ (Steidel et al. 1999).
Our simulation assumes that all cosmological reionization occurs via radiation from [*stellar*]{} sources, with star formation parameterized by the phenomenological Schmidt law as discussed in Gnedin (2000a). An alternative and complementary scenario would be one in which the bulk of reionization is produced by Active Galactic Nuclei (AGN). In light of recent high-redshift quasar counts (e.g., Fan et al. 2001), a scenario in which the universe is reionized by bright optically selected QSOs seems unlikely in any event. There remains however the possibility that low brightness AGNs contributed significantly (if not dominantly) to the reionization of the universe.
For the purpose of this paper, however, this simulation is quite suitable for the following reason. Our main goal in this paper is to investigate whether galaxies are capable of reionizing the universe and still appear as young as they do at $z\sim4$. Thus, by ignoring the AGN component, we consider the most extreme case, since we require galaxies to carry all the burden of reionizing the universe by themselves. One would then expect that if galaxies are capable of reionizing the universe [*and*]{} appear as young as they do at $z\sim4$, then it will be even easier for the models to agree with observations if some ionization is done by AGNs.
We identify galaxies in the simulation with gravitationally bound objects, which are selected with the DENMAX algorithm of Bertschinger & Gelb (1991).
Population Synthesis
--------------------
Population synthesis is carried out using the Starburst99 package (Leitherer et al. 1999). The version we use included the July 2002 update, which revised the UV spectra for O and B stars. These spectra, based on the recent work of Smith, Norris, & Crawther (2002), have decreased ionizing radiation during the early stages of star formation. These changes affect not only the amount of Lyman alpha continuum, but also the amount of Lyman alpha that can be produced if this UV is absorbed by local gas. We choose the instantaneous burst option. This is appropriate because we apply spectral synthesis to individual stellar particles formed in the simulation, which sample the stellar distribution function at discrete time steps.
We choose the lowest metallicity option (5% solar) available in Starburst99, since the simulated galaxies have comparable metallicities. Starburst99 maintains same metallicity for each subsequent generation of stars. This simplification should be sufficient for times less than about $1\dim{Gyr}$ (Leitherer et al. 1999).
Ultra-violet absorption by the IGM material along the line of sight (i.e. by the Lyman alpha forest) is taken into account using the method of Madau (1995).
Reddening corrections are made by the method of Calzetti (1997), including the recent calibration data of Leitherer et al. (2002) We find it sufficient to consider a range of reddening corresponding to a color excess of $E(B-V) = 0 - 0.3$. Higher amounts of reddening produce colors inconsistent with nearly all the observed galaxies. This range is in line with the estimate of Shapely et al. (2001) of an average color excess of $E(B-V)=0.15$ for a sample of Lyman break galaxies at a redshift of about 3. This range is also generally consistent with the results of others for high redshift galaxies.
We also allow for the partial escape of ionizing photons from the modeled galaxies by reducing the flux above $13.6\dim{eV}$ by a factor $f_{\rm ESC}$, and converting 2/3 of the removed photons into Lyman-alpha emission following the recipe of Malhotra & Rhoads (2002). The escape fraction $f_{\rm ESC}$ becomes another parameter in our modeling. We consider three values for this parameter: 0.1, 0.5, and 1.0.
The $AB$ magnitude system is used throughout. We adopt the same cosmology as used in the simulation (flat concordance cosmology with $H_0=70\dim{km/s}/\dim{Mpc}$.
For comparison purposes the $G-I$ colors for observed galaxies over a range of redshifts were corrected to a redshift of 4.0. This was done as follows. For each of a range of $z$ values, including 4.0, a series of 199 model galaxies of different ages was constructed at $5\dim{Myr}$ intervals, each having a single burst of star formation. For each observed galaxy the set of models at the nearest $z$ value was selected. The two bracketing models at this redshift plus the two corresponding models at redshift 4.0 were used to obtain a corrected color using a linear interpolation. This method was feasible because the relationship between $G-I$ and age is, with few exceptions at these intervals, monotonic. A similar correction for ${\cal R}-I$ was not possible.
[ ![\[fig0\] Calculated spectrum of a typical bright simulated galaxy at redshift 4.0. Ultra-violet absorption by the Lyman alpha forest has been taken into account. An escape fraction of 1.0 was used with no reddening correction. ](\figdir/lbg_fig0.ps "fig:"){width="\columnwidth"}]{}
Figure \[fig0\] shows an illustrative spectrum of a typical bright galaxy from our simulation. Sharp absorption edges due to intervening Lyman-$\alpha$ and Lyman-$\beta$ forests are clearly visible.
Results
=======
[ ![\[fig1\]Luminosity Function at rest frame 1600 A for simulated galaxies at $z=4.0$ in the $I$ band. Squares show the unreddened luminosity function, while triangles give the luminosity function with $E(B-V)=0.15$ reddening included. Vertical error-bars are 1-sigma Poisson errors. Diamonds show the observed luminosity function of HDF galaxies from Madau et al. (1996), while crosses and asterisks mark the observed LBG luminosity functions at $z\sim4$ and $z\sim3$ respectively from Steidel et al. (1999). The dashed line is our Schechter function fit to the combined luminosity function.](\figdir/lbg_fig1.ps "fig:"){width="\columnwidth"}]{}
Figure \[fig1\] shows a computed luminosity function for the simulated galaxies at $z=4.0$. As a measure of luminosity we use the $I$ band $AB$ magnitude, which corresponds to a rest frame wavelength of about 1600 A. Also shown for comparison is the observed luminosity function for Lyman Break Galaxies at $z\sim3 - 4$ from a variety of authors (Madau et al. 1996; Steidel et al. 1999). Because of the difficulties in observing at these redshifts, Steidel et al. (1999) have not attempted to make a Schechter fit at $z=4$, but instead have observed that the data are well described by the Schechter function fitted to Lyman Break Galaxies at a redshift of about 3. This fit, however, does not match the slope of the luminosity function of simulated galaxies at faint magnitudes, but the combined luminosity function (simulated plus observed) can be fit by a single Schechter function with a shallower slope. The parameters of both fits are listed in Table 1.
[ccc]{} Parameter & Steidel et al. fit & Our fit\
$\alpha$ & -1.60 & -1.22\
$M_*$ & -20.9 & -20.4\
$\phi_*\,(\dim{Mpc}^{-3})$ & $1.8\times10^{-2}$ & $5.7\times10^{-2}$\
Because of the small size of the simulation box, only a few of the observed points are expected to overlap the simulated ones, and these are all from the Hubble Deep Field North, which probed fainter magnitudes than did the ground based observations of Steidel et al. (1999). But it appears that both the simulation and the observations can be fitted with the same luminosity function, implying that the simulation produces galaxies similar to the real ones. The limitation of the small box size directly translates into the upper limit on galaxy luminosities (there should be at least a few galaxies of luminosity $L$ in the simulation box to determine $\phi(L)$). As one can see, our simulation has just enough volume to reach the luminosities of HDF galaxies, and is a factor of 2 too small to have a single galaxy as large as those found by Steidel et al.(1999) in the simulation volume. Thus, we are not able to make comparisons on a galaxy-by-galaxy basis yet, but future simulation will be able to model volumes comparable to those covered by observations in only a few years.
Given a Schechter function fit, we can integrate the total luminosity function to estimate the fraction of starlight in observed and simulated galaxies. We find that the observed galaxies contain about 2/3 of the total luminosity density, while the simulated ones account for about 1/3. The contribution of observed galaxies to reionization is expected to be somewhat smaller however (Gnedin 2000b). This is because these galaxies tend to be clustered (Steidel et al. 1999; Adelberger et al. 1998), and most of the time several of them sit inside the same region. When an region grows to be larger than the mean free path of ionizing photons, these photons are wasted.
These arguments imply that the redshift of reionization in the simulation is an underestimate, but not by a large factor because the star formation rate in the simulation increases rapidly with time (Gnedin 2000a). If we assume that the simulation underestimates the volume filling fraction of the ionized gas at any given time by, say, 50%, then the redshift of reionization is underestimated in the simulation by about $\Delta z\sim1.5$. As we emphasized above, our simulation is clearly insufficient to give a quantitatively accurate model of reionization, so this level of precision is more than acceptable for our purposes.
A remarkable feature of Fig. \[fig1\] is flattening of the luminosity function in the dwarf galaxy range. This effect is the result of the inhibited accretion of gas onto low mass halos because of photoionization, sometimes improperly called “photoevaporation” (Thoul & Weinberg 1996; Quinn, Katz, & Efstathiou 1996; Weinberg, Hernquist, & Katz 1997; Navarro & Steinmetz 1997; Gnedin 2000b; Chiu, Gnedin, & Ostriker 2000; Sommerville 2002; Benson et al. 2002a,b), and has been reproduced by earlier simulations (Nagamine 2002).
[ ![\[fig2\]Color-Magnitude Diagrams for Simulated Galaxies at $z=4.0$ shown as ([*a*]{}) $G-I$ bv $I$ and ([*b*]{}) ${\cal R}-I$ vs $I$. Each galaxy is represented by a region spanning the range of of reddening from $E(B-V)=0$ to $E(B-V)=0.3$ and the range of escape fractions from $f_{\rm ESC}=0.1$ to $f_{\rm ESC}=1.0$. Squares are observed LBGs from Steidel et al. (1999) in the redshift range from $z=3.5$ to $z=4.8$. A cross within the square indicates that only a minimum $G$ magnitude could be measured.](\figdir/lbg_fig2a.ps "fig:"){width="\columnwidth"}]{} [ ![\[fig2\]Color-Magnitude Diagrams for Simulated Galaxies at $z=4.0$ shown as ([*a*]{}) $G-I$ bv $I$ and ([*b*]{}) ${\cal R}-I$ vs $I$. Each galaxy is represented by a region spanning the range of of reddening from $E(B-V)=0$ to $E(B-V)=0.3$ and the range of escape fractions from $f_{\rm ESC}=0.1$ to $f_{\rm ESC}=1.0$. Squares are observed LBGs from Steidel et al. (1999) in the redshift range from $z=3.5$ to $z=4.8$. A cross within the square indicates that only a minimum $G$ magnitude could be measured.](\figdir/lbg_fig2b.ps "fig:"){width="\columnwidth"}]{}
Figure \[fig2\] shows Color-Magnitude Diagrams (CMD) for the simulated and observed galaxies. Because the simulation cannot predict the redenning correction, we plot each simulated galaxy as a dotted region (which appears almost like a single line), spanning a range in redenning corrections from $E(B-V)=0$ to $E(B-V)=0.3$, consistent with values found by Steidel et al. (1999), and a range of escape fractions from $f_{\rm ESC}=0.1$ to $f_{\rm ESC}=1.0$.
One can see that simulated galaxies have similar colors to the observed ones, which suggests that the simulation captures at least some of the physics, which is going on in these galaxies. Let us now imagine a “super-simulation”, with so large a box that it can be considered infinite. Because the colors of our brightest galaxies do not change much with magnitude, it is plausible to assume that simulated galaxies in the magnitude range $23<I<25$ would have similar colors to the observed ones. If we plausibly assume that the simulated luminosity function, when extended to higher luminosities, would match the observed one reasonably well, then the total luminosity density in such a “super-simulation” would be a factor of 3 higher than in our simulation, and, thus, the redshift of reionization would be even higher than $z=7$ - the redshift of reionization in our simulation - by about $\Delta z\sim1.5$. Such a “super-simulation” would serve as a strong counter example to claims that LBGs at $z\sim4$ are too young to reionize the universe by $z\sim 6$.
[ ![\[fig3\] Three illustrative star formation histories for simulated galaxies with different luminosities: $I=26.9$ ([*a*]{}), $I=27.2$ ([*b*]{}), $I=28.0$ ([*c*]{}). Despite the very different histories, all three have very similar $G-I$ and ${\cal R}-I$ colors. The temporal sampling of star formation histories is 10 times higher after $1.2\dim{Gyr}$, showing fine structure variability on a few Myr time scale due to formation of individual star clusters, which are resolved (by mass) in the simulation. ](\figdir/lbg_fig3a.ps "fig:"){width="\columnwidth"}]{} [ ![\[fig3\] Three illustrative star formation histories for simulated galaxies with different luminosities: $I=26.9$ ([*a*]{}), $I=27.2$ ([*b*]{}), $I=28.0$ ([*c*]{}). Despite the very different histories, all three have very similar $G-I$ and ${\cal R}-I$ colors. The temporal sampling of star formation histories is 10 times higher after $1.2\dim{Gyr}$, showing fine structure variability on a few Myr time scale due to formation of individual star clusters, which are resolved (by mass) in the simulation. ](\figdir/lbg_fig3b.ps "fig:"){width="\columnwidth"}]{} [ ![\[fig3\] Three illustrative star formation histories for simulated galaxies with different luminosities: $I=26.9$ ([*a*]{}), $I=27.2$ ([*b*]{}), $I=28.0$ ([*c*]{}). Despite the very different histories, all three have very similar $G-I$ and ${\cal R}-I$ colors. The temporal sampling of star formation histories is 10 times higher after $1.2\dim{Gyr}$, showing fine structure variability on a few Myr time scale due to formation of individual star clusters, which are resolved (by mass) in the simulation. ](\figdir/lbg_fig3c.ps "fig:"){width="\columnwidth"}]{}
Why, then, is observational determination of ages so misleading? Figure \[fig3\] illustrates the diversity of star formation histories in simulated galaxies. While our brightest galaxies ($I\la27$, such as the one shown in panel [*a*]{}) are experiencing a peak of star formation at around $z=4$, one magnitude dimmer galaxies ($I\ga28$) were more luminous in the past - indeed, at the time of reionization (at $z\sim7$ in this simulation) - and by $z\sim4$ their star formation rate has decreased by a factor of several. In fact, the galaxy shown in panel ([*c*]{}) has $1.1\times10^9\dim{M}_\sun$ in stars, whereas the galaxy in panel ([*a*]{}), while being more than a magnitude brighter, has a stellar mass of only $7\times10^8\dim{M}_\sun$. We, therefore, conclude that galaxies that reionized the universe at $z\sim7$ appear at $z\sim4$ within the magnitude range of $28\la I\la30$, while LBGs that are observed at $z\sim4$ with $23\la I \la25$ are indeed very young, experiencing their first major episode of star formation (we refrain from using the word “burst”, because the star formation history of a galaxy shown in panel [*a*]{}, while strongly rising by $z\sim4$, can still be hardly called a “burst”).
[ ![\[fig4\]Total star formation rates in galaxies that appear within a given magnitude bin at $z=4$ as a function of cosmic time. Each indicated magnitude is the central value of a bin of width 1.0 magnitudes.](\figdir/lbg_fig4.ps "fig:"){width="\columnwidth"}]{}
Figure \[fig4\] illustrates this point further. It shows total star formation rates for galaxies that appear within different magnitude bins at $z=4$. In agreement with individual examples from Fig. \[fig3\], the star formation rate at $z=4$ is dominated by our brightest galaxies, but these galaxies contribute only about 20% to the star formation at $z\sim7$. At that redshift more than half of the total star formation rate is contributed by galaxies that appear within the $28<I<30$ magnitude bin at $z\sim4$.
[ ![\[fig5\]Mean star formation time for all simulated galaxies as a function of their magnitudes at $z=4$.](\figdir/lbg_fig5.ps "fig:"){width="\columnwidth"}]{}
The large variation in possible star formation histories is further confirmed by Figure \[fig5\], which shows mean mass-weighted star formation ages for all simulated galaxies as a function of their magnitudes at $z=4$, $$t_{SF} = {1\over M_*} \int_0^{t_f} t\,\, {\rm SFR}(t)\, dt,$$ where $M_*$ is the total stellar mass. Even for our brightest galaxies the variation is significant, reaching up to a factor of 3 for the dimmest ones. A remarkable feature of Fig. \[fig5\] is a large concentration of galaxies with $40<I<45$ and little star formation after $1\dim{Gyr}$. It is tempting to identify these galaxies with dwarf spheroidals: they are old and would have luminosities of the order of $10^{5 - 6}L_\sun$ at $z=0$. A small fraction of these galaxies still have active star formation at $z\sim 4$, which may explain the diverse star formation histories of dwarf spheroidals in the Local Group (Grebel 1998; Mateo 1998). We leave this as a possible speculation, as this is not the goal of this paper, however.
Observations at longer wavelengths and/or at fainter magnitudes would be more sensitive to older populations of stars. Unfortunately, these observations are very difficult to make at a redshift of 4, and they will most likely have to wait for JWST and 30-meter class ground telescopes. But those instruments are on the horizon, and one might hope to indeed observe the culprits of reionization within the 10-year time frame.
[ ![\[fig6\] Mass-to-light ratio for simulated galaxies at $z = 4$. Squares show the luminosity function of the simulated galaxies as would be observed in the K band with $E(B - V) = 0.15$ reddening. Vertical lines through the symbols represent 1-sigma Poisson errors. Crosses show the stellar mass-to-light ratio in solar units computed with the same reddening. The vertical lines on the left show the stellar mass-to-light ranges found by Dickinson et al. (2003) assuming metallicities of $1/5$ solar (dotted) and solar (dashed). ](\figdir/lbg_fig6.ps "fig:"){width="\columnwidth"}]{}
Whatever the global history of star formation we would at least expect that the total stellar content of the universe would be monotonic with time. Thus, we have compared the stellar content of the simulated galaxies with the global history of total stellar mass as presented in Dickinson et al. (2003). In Figure \[fig6\] we show the B band rest frame luminosity function and mass-to-light ratios of the simulated galaxies, calculated from the K band magnitudes. Vertical lines give the comparison with Dickinson et al. (2003) estimates. Our results are in comfortable agreement with these estimates for all galaxies brighter than dwarf spheroidal candidates.
[ ![\[fig7\] Cumulative stellar mass and B band luminosity. Squares show the luminosity function of the simulated galaxies at $z = 4$ as would be observed in the K band. Vertical lines represent 1-sigma Poisson errors. Triangles show the total stellar mass in simulated galaxies integrated from the lowest magnitude to the plotted magnitude. Diamonds trace the luminosity density of simulated galaxies as would be observed in the K band with $E(B - V) = 0.15$ reddening. The dotted line is the mass density range observed by Dickinson et al. (2003). The dashed line is their observed luminosity density range obtained by integrating over a Schechter fit. ](\figdir/lbg_fig7.ps "fig:"){width="\columnwidth"}]{}
Finally, Figure \[fig7\] shows cumulative mass and luminosity densities of simulated galaxies as compared with Dickinson et al. (2003) estimates. Again it appears that the simulated galaxies are in a reasonable agreement with the observational data. We would like to emphasize that we are quite comfortable with a factor of 2 agreement: one should remember that the simulation can only account for about 1/3 of the total light because of the small box size. In fact, in the imaginary “super-simulation”, discussed above, the total stellar mass would be up to a factor of 3 higher (depending on whether the mass-to-light ratio goes down for brighter galaxies) and would not agree with Dickinson et al. (2003). One could hypothesize about a source of this potential discrepancy, including anything from a non-standard IMF to incorrect stellar synthesis models, but for the purpose of this paper we are content with the level of agreement we obtain, given substantial limitations of our simulation.
[ ![\[fig8\] Scatter plot of absolute magnitude at 1600 A versus total (dark matter + gas + stellar) mass of simulated galaxies at redshift 4. Reddening equivalent to a color excess of 0.15 has been applied as well as extinction due to absorption by the Lyman alpha forest. ](\figdir/lbg_fig8.ps "fig:"){width="\columnwidth"}]{}
It would also be interesting to compare the luminosities of Lyman Break Galaxies and their dark matter masses, because the relation of LBGs and their dark matter halos is still poorly understood (Somerville, Primack, & Faber 2001; Wechsler et al. 2001; Somerville 2002; Benson et al. 2002a,b). In Figure \[fig8\] we show this relation. As one can see, Lyman Break Galaxies represent a diverse population. For example, our brightest galaxies with absolute magnitudes between -18 and -17 cover about two magnitudes in the total mass: some of them are the most massive galaxies in the simulations, but some of them have significantly lower masses and are experiencing a starburst at $z=4$. There exists a general trend of less massive galaxies being dimmer, but the scatter around the mean relation is very large.
Discussion
==========
So, how well does the simulation do? While the computational box of our simulation is clearly insufficient to have even a single galaxy as luminous as those of Steidel et al.(1999), simulated galaxies do appear as fainter cousins of the observed ones. They have similar colors, and simulated and observed galaxies together can be fitted by a single luminosity function.
They are also faring well in their abilities to reionize the universe. Star formation histories of simulated galaxies are diverse, and vary systematically with magnitude. The brightest galaxies at $z=4$ are indeed quite young, in accord with conclusions of Shapely et al. (2001) and Ferguson et al. (2002). But these brightest galaxies contribute only about 25% of ionizing photons at $z>6$. The bulk of work of reionizing the universe was done by dimmer galaxies, those that fall within the $28<I<30$ magnitude range at $z\sim 4$, but which are not necessarily less massive than the bright ones.
We expect that detailed simulations will become increasingly important in sorting out the various possibilities. The general picture that our simulation produces is at least consistent with observations of Lyman Break Galaxies, an encouraging message to theorists.
The recent results from the WMAP mission (Kogut et al. 2003) indicate that the optical depth to Thompson scattering is significantly larger than in our simulation. While this measurement cannot pin down the redshift of reionization, it suggests, if correct, that there was a considerable ionizing flux at early times. Our simulation provides no support for a stellar origin of this ionizing flux. Perhaps, very massive population III stars, which we have not considered, could be responsible.
Because, as we have discussed, the photoionization feedback likely plays a significant role in the evolution of faint Lyman Break Galaxies, one might wonder how the WMAP results affect the conclusions of this paper. As has been shown by several previous studies (Gnedin 2000b and references herein), the photoionization feedback results in the substantial loss of the gas mass of objects with circular velocities below about $40\dim{km/s}$. However, this characteristic circular velocity is essentially redshift independent. So even if the universe was fully ionized throughout its entire history, the effect of the photoionization feedback at $z=4$ would be essentially at the same level as in our simulations. Our conclusion about a reasonable agreement between the simulation and the data is, in fact, independent of the redshift of reionization as long as it is above about 6.
We are grateful to Chuck Steidel for providing us with his custom filter shapes and for enlightening comments during his visit to CU. This work was supported by NSF grant AST-0134373. This work was also partially supported by National Computational Science Alliance under grant AST-960015N and utilized the SGI/CRAY Origin 2000 array at the National Center for Supercomputing Applications (NCSA).
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[^1]: With the following cosmological parameters: $\Omega_0=0.3$, $\Omega_\Lambda=0.7$, $h=0.7$, $\Omega_b=0.04$, $n=1$, $\sigma_8=0.91$, where the amplitude and the slope of the primordial spectrum are fixed by the COBE and cluster-scale normalizations.
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---
abstract: |
We propose a model of discrete time dynamic congestion games with atomic players and a single source-destination pair. The latencies of edges are composed by free-flow transit times and possible queuing time due to capacity constraints. We give a precise description of the dynamics induced by the individual strategies of players and of the corresponding costs, either when the traffic is controlled by a planner, or when players act selfishly. In parallel networks, optimal and equilibrium behavior eventually coincides, but the selfish behavior of the first players has consequences that cannot be undone and are paid by all future generations. In more general topologies, our main contributions are three-fold.
First, we show that equilibria are usually not unique. In particular, we prove that there exists a sequence of networks such that the price of anarchy is equal to $n-1$, where $n$ is the number of vertices, and the price of stability is equal to 1.
Second, we illustrate a new dynamic version of Braess’s paradox: the presence of initial queues in a network may decrease the long-run costs in equilibrium. This paradox may arise even in networks for which no Braess’s paradox was previously known.
Third, we propose an extension to model seasonalities by assuming that departure flows fluctuate periodically over time. We introduce a measure that captures the queues induced by periodicity of inflows. This measure is the increase in costs compared to uniform departures for optimal and equilibrium flows in parallel networks.
**Keywords**: Network games, dynamic flows, price of seasonality, price of anarchy, max-flow min-cut.
*OR/MS Subject Classification*: networks/graphs: multicommodity, theory; games/group decisions: noncooperative; transportation: models, network.
author:
- |
Marco Scarsini[^1]\
Dipartimento di Economia e Finanza\
LUISS\
Viale Romania 32\
00197 Roma, Italy\
`[email protected]`
- |
Marc Schröder[^2]\
Department of Quantitative Economics\
Maastricht University\
Tongersestraat 53\
6211 LM Maastricht, The Netherlands\
`[email protected]`
- |
Tristan Tomala[^3]\
HEC Paris and GREGHEC\
1 rue de la Libération\
78351 Jouy-en-Josas, France\
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title: |
Dynamic Atomic Congestion Games\
with Seasonal Flows
---
Introduction {#se:intro}
============
The analysis of transportation networks naturally leads to the consideration of congestion games, where each agent selfishly behaves as to minimize her own time on the road without regard for the effects that this behavior has on the other agents’ traveling time. The outcome of the individual selfish behavior can be compared to the outcome that a social planner would choose. A way of comparison is, for instance, the price of anarchy [see, e.g. @Rou:MIT2005; @Rou:AGT2007; @RouTar:AGT2007], namely the ratio of the worst social cost induced by selfish behavior to the optimal social cost.
Although the motivation for this theory is rooted in the study of traffic flows, most of the existing literature is actually static. The commonly adopted justification is that the static game represents the steady state of a dynamic model where the flow over the network is constant over time. Yet, for determining how the steady state is reached, a careful study of dynamic models is required. As we shall see, the behavior of agents in the transient phase may have an impact on the long-run outcome.
In this paper we study a dynamic model of congestion where the players have symmetric and unsplittable weights. This could be a high-level model for, e.g., traffic network, where each player is a car in a traffic network, or a telecommunication network, where each player is a data packet. We characterize the optimal long-run flows and latencies, i.e., the ones induced by a benevolent long-lived social planner. When the players act selfishly in order to minimize their own traveling time, without heeding the planner’s suggestion, the situation can be modeled as a noncooperative game. For some topologies of the network and when the inflow of players is uniform over time we are able to characterize the equilibria of this game. We consider the efficiency of its equilibria for various topologies and we show that some forms of Braess-type paradoxes are possible. Finally we devote our attention to the case where the inflow of players is periodic over time.
Model
-----
We analyze an atomic dynamic congestion game, based on the deterministic queuing model of @KocSku:TCS2011. Atomic models are typically more complicated to analyze than nonatomic models and have less nice properties. Nevertheless, they may be a better fit when the number of players is not huge and a nonatomic approximation is not justifiable. Atomic models have been used, for instance, in telecommunications [see, for instance, @TekLiuSouHuaAhm:IEEEACM2012]. The dynamics of the model is described as follows. Time is discrete and at each stage, a generation of finitely many players departs from the source with the goal of reaching the destination as fast as possible. We assume that each player has a unit weight and is unsplittable. We see this assumption of symmetry as a first order approximation when the size of the vehicles or of the data packet is not too dissimilar. Each player chooses a route from source to destination, knowing the choice of the previous players. Each edge of the network is endowed with a free-flow transit time and a capacity. When a player enters an edge on the chosen route, she travels on that edge at a constant speed. When reaching the head of the edge, a queue might have formed since at most the capacity number of players can exit the edge at the same time. We assume that there is a global priority among players to determine who leaves the edge first. The latency suffered by the player on an edge is thus the sum of the transit and waiting times. The total latency suffered by the player is then the sum of the latencies suffered on all the edges she uses.
Results
-------
We first study social optimality when the inflow is constant and at most the capacity of the network. We prove that optimal flows exist and show that there is an optimal flow such that, at each stage, the current flow over routes minimizes the total cost among feasible static flows, i.e., the flows that satisfy all the capacity constraints.
Then, we turn to the behavior of selfish players. In particular, we consider equilibria in which each player arrives at each intermediate vertex as fast as possible. These are called uniformly-fastest-route equilibria. In general, such equilibria are not unique. For parallel networks, in all equilibria the flow coincides with the optimal flow from some stage on, and in the worst equilibrium, all players eventually pay the highest transit cost of the network. The intuition is that the first players all choose the fastest routes and induce congestion. Eventually, all routes get so congested that all latencies become equal. This result shows the impact of the dynamic nature of the model on latencies. While optimal and equilibrium behavior eventually coincide, the selfish behavior of the first players has consequences that cannot be undone and are paid by all future generations.
In more general networks the results for equilibrium flows become more complicated. For chain-of-parallel networks, the equilibrium costs can be derived from the results for parallel networks, but the corresponding flows can be quite different and even aperiodic.
We also examine various efficiency measures of equilibria such as the price of anarchy, the price of stability and the Braess ratio. We demonstrate the following phenomena by examples. Firstly, there is a sequence of instances such that the price of anarchy is equal to $n-1$, where $n$ is the number of vertices, and the price of stability is equal to 1, illustrating the difference in long-run equilibrium costs. This is one of the main differences between the atomic and nonatomic cases, since in our atomic model there may exist multiple equilibria whose behavior can be quite different, whereas in the nonatomic model, equilibrium is unique [see @ComCorLar:ALP2011]. @KocSku:TCS2011 show that the price of anarchy increases logarithmically in the number of edges if all edge capacities are equal to $1$. As a byproduct of our results, we obtain an example of a nonatomic game, where all capacities are $1$ and the price of anarchy is linear in the number of edges. Secondly, we study Braess’s paradox [see @Bra:U1968; @Bra:TS2005], namely the decrease of the total equilibrium cost after deletion of an edge. This may happen even when the network does not contain the Wheatstone graph as a subnetwork [see @MacLarSte:TCS2013]. We also obtain a variant of Braess’s paradox: the equilibrium cost might decrease when there are initial queues in the network, or when the length of an edge increases. Thirdly, we study the Braess ratio [see @Rou:JCSS2006], namely, the largest factor by which the equilibrium cost can be improved by removal of an edge. We consider a set of networks with $n$ vertices where this ratio is $n-1$. A similar result appears in [@MacLarSte:TCS2013] in the nonatomic case.
In the last section we consider periodic inflows, we define a distance between two inflows, and we show that in parallel networks at capacity, periodicity adds the same cost in equilibrium and at the optimum, and this added cost is exactly the distance between the periodic and the uniform inflows.
Related literature {#suse:existing}
------------------
Dynamic congestion games belong to the wider class of models of flows over time. @ForFul:OR1958 [@ForFul:PUP1962] introduced these models in a discrete time setting by considering the problem of maximizing the flow from source to destination in a given finite time horizon. @Gal:MMJ1959 considered a refinement of the above problem, called earliest arrival flow, where the aim is to simultaneously maximize the flow for every time before the deadline; @Wil:OR1971 and @Min:OR1973 developed algorithms for solving it. The continuous-time versions were studied by @Phi:MOR1990 and @FleTar:ORL1998, respectively. We refer the reader to @Sku:RTCO2009 for a detailed analysis and an extensive bibliography. Equilibrium concepts in dynamic network models date back to @Vic:AER1969 in the economic literature and to @Yag:TR1971 in the transportation literature. We refer the reader to @Koc:PhD2012 for an extensive list of references on this topic. Recent mathematical formulations of the model resort to deterministic queueing theory, as introduced originally by @Vic:AER1969 and later developed by @HenKoc:TS1981. In this stream of literature @Aka:TRB2000 [@Aka:TS2001; @AkaHey:TS2003], @Mou:TRB2006 [@Mou:TS2007], @AsnUkk:AGT2009, @HoeMirRogTen:P5IWINE:2009, and especially @KocSku:TCS2011 extended some results known for static congestion games to dynamic congestion games. The latter authors use a deterministic queueing model to study dynamic flows and characterize Nash equilibria. They show the relation between dynamic and static models and they compute the price of anarchy for the dynamic model. Along these lines, @ComCorLar:ALP2011 [@ComCorLar:OR2015] studied equilibria for flows over time in the single-source single-sink deterministic queuing model and proved existence and uniqueness of equilibria when the inflow rate is piecewise constant. @KocNasSku:MMOR2011 used measure-theoretic techniques to combine continuous and discrete time models of flow over time and, among other things, extended to this general setting the classical max-flow min-cut theorem. @BhaFleAns:GEB2015 considered a Stackelberg model with a network manager acting as a leader who chooses the capacity of each edge in a way that does not exceed its physical limit. They were able to bound the price of anarchy for this model.
Among this literature, our model belongs to the class of deterministic queueing models and is close to the one developed by @KocSku:TCS2011. There are some technical differences with this literature since our model is in discrete time and with atomic players, similar to @WerHolKru:ORP2014. This can induce, for instance, a possible multiplicity of equilibria.
Importantly, our focus differs from these papers. Many of them seek to characterize equilibrium flows (e.g. @KocSku:TCS2011), prove existence results [see, e.g., @ComCorLar:ALP2011; @ComCorLar:OR2015], and provide algorithms for computing equilibria. By contrast, we emphasize how the system evolves towards a steady state. Starting from an empty network, we study how the behavior of the first users impacts the equilibrium steady state and how this steady state is reached. The transient phase that leads to the steady state is thus particularly important in our model. @ShaShi:PACMSICMMCS2010 consider the transient phase of a dynamic network before a steady state equilibrium is reached. Although their model is stochastic, some of the questions they consider are close in spirit to our model. @Com:MP2015 provides a nice survey of congestion models under uncertainty and describes a model of adaptive dynamics that gives a microfoundation for steady state traffic equilibrium models.
We now comment on some more accessory features of our work in relation to the existing literature. Our model belongs to the class of congestion games with atomic players. In his fundamental paper, @War:PICE1952 modelled the selfish behavior of a huge number of agents on a network as a nonatomic flow and introduced an equilibrium concept that has become the standard reference in the literature. @ChaCoo:TTF1961 showed the relation between Nash and Wardrop equilibria and @HauMar:N1985 proved that, under some conditions, the Wardrop equilibrium in a nonatomic model can be obtained as a limit of Nash equilibria of atomic models. The relation between atomic and nonatomic games has been recently studied by @BhaFleHua:IPCO2010. A nice survey on Wardrop equilibria can be found in @CorSti:Wiley2010. General congestion games with a finite number of players were introduced by @Ros:IJGT1973, who proved that they have pure Nash equilibria; they are actually isomorphic to potential games [see @MonSha:GEB1996]. The issue of multiplicity of equilibria in atomic congestion games was studied by @Har:TS1988 [@BhaFleHoyHua:P20AACMSIAMSDA2009]. Consistent with this literature, we find multiple equilibria for our game.
In order to obtain well defined dynamics, we use a priority order. This approach can be found in earlier works. @FarOlvVet:CJTCS2008 introduced a routing model with a general priority scheme for players on different edges. This allowed the authors to introduce a time dependence in the model and to define a cost for each player that depends on the actions of the players with a higher priority. Among the many possibilities, they considered a global priority scheme which is the same for every edge, and a time dependent priority scheme where priority is decided by who arrives first on an edge. A similar global priority scheme was exploited by @HarHeiPfe:TCS2009, who studied multicommodity flows where commodities are routed sequentially in a network. In their model demands for commodities are revealed in an online fashion and can be split along several paths. They framed the problem as an optimization problem and they studied online algorithms for its solution. In a related paper, @HarVeg:LNCS2007 considered a model in which players’ demands change over time and are released in $n$ sequential games in an online fashion. In each game, the new demands form a Nash equilibrium, and their routing remains unchanged afterwards. These three models do not explicitly take into account the dynamics of the flows of players over the network. Our model retains the idea of a priority scheme, but it is dynamic.
Finally, for measuring the efficiency of a game we use the now famous price of anarchy, i.e., the ratio between the worst Nash equilibrium latency and the socially optimal latency, and the price of stability, i.e., the ratio between the best Nash equilibrium latency and the socially optimal latency. These two measures were introduced by @KouPap:STACS1999 and @SchSti:P14SIAM2003, respectively. Their names were coined by @Pap:PACM2001 and @AnsDasKleTarWexRou:SIAMJC2008, respectively. Inefficiency of equilibria in routing games has been studied by several authors (see among others @RouTar:JACM202 [@RouTar:GEB2004], @CorSchSti:MOR2004 [@CorSchSti:GEB2008; @CorSchSti:OR2007]).
@Bra:U1968 [@Bra:TS2005] shows that removing an edge in a network can improve the equilibrium latency for all players in a static model. @Dag:TS1998 shows similar paradoxical phenomena in traffic models when queues have physical magnitude. @Rou:JCSS2006 introduces a measure, called the Braess ratio, that quantifies the extent of Braess’s paradox. A study of the network topologies for which the paradox may exists can be found in @Mil:GEB2006 for static games and in @MacLarSte:TCS2013 for dynamic games. An analysis of a dynamic Braess-type paradox in communication networks is provided by @XiaHil:IEEETCSIIEB2013.
Organization of the paper
-------------------------
The paper is organized as follows. Section \[se:model\] presents the model. Section \[se:optimum\] contains a characterization of the optimum strategy and cost for the case of constant inflows. Section \[se:topologies\] studies the case of parallel networks and chain-of-parallel networks. Section \[se:anarchy\] examines efficiency of equilibria and Braess-type phenomena. Section \[se:seasonal\] proposes an extension to model seasonalities. Section \[se:conclusion\] concludes and proposes some open problems. All proofs are relegated to the Appendix or to the online Supplementary Material.
The model {#se:model}
=========
We study a dynamic congestion game on a general directed network with a single source-destination pair, where each edge has a transit cost and a capacity. Formally, consider a directed multigraph $\mathcal{G}=(V,E)$, where $V$ is a finite set of vertices and $E$ is a finite set of edges. We then define a network $\mathcal{N}=(\mathcal{G}, (\tau_{e})_{e \in E}, (\gamma_{e})_{e \in E})$, such that for each $e\in E$, the quantities $\tau_{e}\in\mathbb{N}$ and $\gamma_{e}\in \mathbb{N}$ are the *free-flow transit cost* and the *capacity* of edge $e$, respectively.
A *path* in the network is a finite sequence of edges $(e_{1},\dots, e_{n})$ such that the head of $e_{i}$ coincides with the tail of $e_{i+1}$ for each $i=1,\dots, n-1$.
We make the following assumptions:
(a) There are two special vertices, the *source* $s$, which has only outgoing edges, and the *destination* $d$, which has only incoming edges (we use the symbol $d$ for destination, rather than the more common $t$, because we reserve $t$ for time). Source and destination are unique.
(b) For each vertex $v\in V\setminus\{s,d\}$, there exists at least a path from $s$ to $v$ and a path from $v$ to $d$.
We call *route* a path from $s$ to $d$. The set of all routes is denoted by $\mathcal{R}$. The above assumptions guarantee that any path can be extended to a route.
Time is discrete and, at each stage $t$ finitely many players enter the network at the source and choose a route from $s$ to $d$. Each player represents a unit packet of traffic. For simplicity, we assume that all players have the same size, which we normalize to 1. The dynamics of the model is the following.
- At each stage $t\in\mathbb{N}_{+}$, a finite set $G_{t}$ of players, called the *generation* at time $t$, departs from the source. For all $t \in \mathbb{N}_{+}$ define, $$\delta_{t} = {\operatorname{card}}(G_{t}) \quad\text{and}\quad \mathcal{D} = \{\delta_{t}\}_{t\in \mathbb{N}_{+}}.$$ Denote $[it]$ the $i$-th player in generation $G_{t}$ (when there is no risk of confusion, the square brackets are removed).
We thus have an infinite set of players $G:=\cup_{t} G_{t}$. We order this set (anti-lexicographically) by $\lhd$ as follows: $$[js] \lhd [it] \quad \text{iff}\quad s < t \quad\text{or}\quad (s = t \ \text{and}\ j < i).$$ This order represents priorities: if $[js] \lhd [it]$ and if these two players enter the edge $e$ at the same time, then $[js]$ exits $e$ before $[it]$. Such a global priority is a natural choice for breaking ties in congestion games, see @FarOlvVet:CJTCS2008 and @WerHolKru:ORP2014.
- Each player chooses a route in $\mathcal{R}$.
- At time $t$ player $[it]$ departs from the source $s$, takes the chosen route, and progresses with steps of size $1$ per unit of time along an edge $e$. After $\tau_e$ time units, she arrives at the head of the edge, where a queue may have formed.
- The rules for exiting the queue are the following:
- All players that entered edge $e$ before $[it]$ and those players $[js] \lhd [it]$ who entered $e$ at the same time as $[it]$ are ahead of $[it]$ in the queue. That is, there is no over-taking in queues and players are ordered first by time of arrival, then by priority.
- At most $\gamma_{e}$ players can exit $e$ simultaneously. When player $[it]$ arrives at the end of $e$, if she finds less than $\gamma_{e}$ players in the queue, then she exits immediately; otherwise, only the first $\gamma_{e}$ players exit at this stage and player $[it]$ waits for one stage. This process repeats until there remain less than $\gamma_{e}$ players in the queue ahead of player $[it]$. Then, $[it]$ exits edge $e$ and continues along the chosen route.
- The process is repeated until player $[it]$ arrives at the destination $d$ and quits the system.
These rules define a *dynamic congestion game* denoted $\Gamma(\mathcal{N}, \mathcal{D})$. Each strategy profile $\sigma \in \mathcal{R}^{G}$ induces queues on edges. We denote $\ell_{it}(\sigma)$ the *latency* suffered by player $[it]$, defined as $$\ell_{it}(\sigma)=c_{it}(\sigma)+w_{it}(\sigma),$$ where $c_{it}(\sigma):=\sum_{e\in r_{it}(\sigma)}\tau_{e}$ is the *transit cost* paid by player $[it]$ and $w_{it}(\sigma)$ is the *waiting cost* paid by player $[it]$, namely, the total number of stages that $[it]$ spends queueing, summed over the edges that she crosses. Both costs are additive, the total cost over the route is the sum of costs over the edges of the route.
We define the *total transit cost* $c_{t}$, the *total waiting cost* $w_{t}$, and the *total latency* $\ell_{t}$ at stage $t$ as follows: $$\begin{aligned}
c_{t}(\sigma) &= \sum_{[it]\in G_{t}} c_{it}(\sigma), \\
w_{t}(\sigma) &= \sum_{[it]\in G_{t}} w_{it}(\sigma), \\
\ell_{t}(\sigma) &= \sum_{[it]\in G_{t}} \ell_{it}(\sigma) =c_{t}(\sigma)+w_{t}(\sigma).\end{aligned}$$ For each integer $T$, the *average total latency* over the period $\{1,\dots, T\}$ is $$\bar{L}_{T}(\sigma)=\frac{1}{T}\sum_{t=1}^{T} \ell_{t}(\sigma).$$ If $\lim_{T \to \infty}\bar{L}_{T}(\sigma)$ exists, then it is called *asymptotic average total latency* for the strategy $\sigma$.
\[de:SO\] A strategy profile $\sigma$ is *(socially) optimal* if $$\label{eq:optimum}
\liminf_{T\to\infty}\bar{L}_{T}(\sigma')\geq\limsup_{T\to\infty}\bar{L}_{T}(\sigma) \text{ for all }\sigma' \in \mathcal{R}^{G}.$$ Call $\mathcal{O}(\mathcal{N}, \mathcal{D})$ the set of strategies $\sigma \in \mathcal{R}^G$ for which holds. Then $$\label{eq:Opt}
{\operatorname{\mathsf{Opt}}}(\mathcal{N}, \mathcal{D}):=\liminf_{T\to\infty}\bar{L}_{T}(\sigma)=\limsup_{T\to\infty}\bar{L}_{T}(\sigma) \quad\text{with }\sigma \in \mathcal{O}(\mathcal{N}, \mathcal{D})$$ is called the *optimal latency*.
\[de:SPE\] A strategy profile $\sigma$ is a *Nash equilibrium* if $$\ell_{it}(\sigma)\leq\ell_{it}(\sigma'_{it},\sigma_{-it})\text{ for all }[it]\in G, \text{ for all }\sigma'_{it}\in \mathcal{R},$$ where $\sigma_{-it}$ indicates the profile of strategies of all players different from $[it]$.
A Nash equilibrium $\sigma$ is a *uniformly fastest route (UFR) equilibrium* if for every player $[it]$ and for every vertex $v$ on the route $\sigma_{it}$, there is no alternative route $\sigma'_{it}$ that allows player $[it]$ to arrive at $v$ earlier than under $\sigma_{it}$.
There is obviously an asymmetry between Definitions \[de:SO\] and \[de:SPE\] which stems from the type of rationality driving the two concepts. An optimal strategy is the choice that a long-lived planner would like to take, in order to optimize the long-run social welfare. By contrast, an equilibrium is a strategy profile such that each finitely lived player optimizes her cost given the choices of the other players.
In non-atomic games [see @KocSku:TCS2011], the Nash-flow-over-time definition is equivalent to assuming that all particles arrive at each intermediate vertex as early as possible, which is also equivalent to requiring that no flow overtakes any other flow. For atomic games, this equivalence does not hold. For instance, take a player who is the last in her generation. The immediate successor is the first in the next generation, and therefore there is a time difference between the departures of these two subsequent players. In that case, it might be that a player does not want to arrive at each intermediate vertex as early as possible. It might also be the case that a player overtakes her predecessor in equilibrium, while arriving at the destination at the same time point. In fact, the three notions which coincides in the non-atomic case, might yield different long-run latencies in the atomic case. Example \[ex:verybadNash\] in the online Supplementary Material illustrates this phenomenon.
We denote $\mathcal{E}(\mathcal{N}, \mathcal{D})$ the set of UFR equilibria of the game $\Gamma(\mathcal{N}, \mathcal{D})$. We argue that a UFR equilibrium exists, a similar argument can be found in @WerHolKru:ORP2014. In an empty network, there is always a shortest route with the property that every intermediate vertex is reached as early as possible, since this is equivalent to the static shortest path problem. If the first player chooses such a route, then that player cannot be overtaken. Taking this choice into account, the second player chooses a route that reaches every intermediate vertex as early as possible, so that she cannot be overtaken either. Continuing this procedure iteratively yields a UFR equilibrium. We present this result as a lemma whose proof is given in the Appendix for completeness.
\[le:existence\] The set $\mathcal{E}(\mathcal{N}, \mathcal{D})$ is not empty.
The quantity $$\label{eq:WEq}
{\operatorname{\mathsf{WEq}}}(\mathcal{N}, \mathcal{D}):=\sup_{\sigma \in\mathcal{E}(\mathcal{N}, \mathcal{D})} \limsup_{T \to \infty}\bar{L}_{T}(\sigma)$$ is called the *worst equilibrium latency* and the quantity $$\label{eq:BEq}
{\operatorname{\mathsf{BEq}}}(\mathcal{N}, \mathcal{D}):=\inf_{\sigma \in\mathcal{E}(\mathcal{N}, \mathcal{D})} \limsup_{T \to \infty}\bar{L}_{T}(\sigma)$$ is called the *best equilibrium latency*.
Socially optimal strategies {#se:optimum}
===========================
In this section we characterize flows and costs generated by optimal strategies. Before stating the results, a simple observation is that the number of players entering the network over time has to be compared with the number that the network is able to absorb.
A *cut* in the network $\mathcal{N}$ is a subset of edges $C\subseteq E$ such that each route contains at least one element of $C$. The capacity of a cut $C$ is $\gamma_{C}=\sum_{e\in C}\gamma_{e}$. Call $\mathcal{C}(\mathcal{N})$ the set of all cuts in $\mathcal{N}$. A *minimum cut* is a cut $C$ such that $$\gamma_{C} \le \gamma_{C'} \quad \text{for all } C' \in \mathcal{C}(\mathcal{N}).$$ The capacity $\gamma$ of the network $\mathcal{N}$ is the capacity of any minimum cut.
Until further notice (see Section \[se:seasonal\]), we assume that the number of players in each generation is uniform over time, i.e., $\delta_{t}=\delta$ for all $t \in \mathbb{N}_{+}$ (abusing notation, the departure sequence $\mathcal{D}$ will be denoted simply by $\delta$). From the max-flow min-cut theorem of @ForFul:OR1958, if $\delta >\gamma$, then the lengths of queues on the edges of the minimum cut diverge to infinity and thus the long-run average total cost is infinite under any strategy profile. Therefore, in this section, we assume $\delta \le \gamma$.
We first recall some usual concepts of optimality for static flows over networks. A (static) *network flow* $f$ assigns a non-negative flow value $f_{e}$ to each edge $e\in E$ (in our setting, these are integers). The flow $f$ is *feasible* if it obeys the capacity constraints, i.e., $f_{e}\leq \gamma_{e}$ for each $e\in E$, and flow conservation, i.e., the outflow minus the inflow at each vertex $v\in V\setminus\{s,d\}$ is $0$. The value of a feasible network flow is the inflow at $d$, in our case this is $\delta$. A static flow may also be defined over routes. Consider a set of $\delta$ players and the routes that they choose. They induce a static flow over routes $F$ which assigns to each route $r$ an integer $F_{r}$, such that $\sum_{r\in\mathcal{R}}F_{r}=\delta$. This in turn induces a flow over edges by letting $f_e=\sum_{\{r : e\in r\}}F_{r}$.
The min-cost flow (static) optimization problem is the minimization of the total transit cost among all feasible flows with a value of $\delta$. Let $f^{*}$ be an optimal feasible solution and $F^{*}$ a corresponding optimal flow over routes. This is the optimal assignment that a planner would chose in a static framework with a single set of $\delta$ players, subject to feasibility.
Back to the dynamic problem, given a strategy profile $\sigma$ and a route $r$, denote $N^{r}_{t}(\sigma)$ the number of players who choose route $r$ at stage $t$ under the strategy profile $\sigma$.
\[th:optimum\] Consider the game $\Gamma(\mathcal{N}, \delta)$, where $\delta \le \gamma$. Let $f^{*}$ be an optimal feasible min-cost network flow with a value of $\delta$ and let $F^{*}$ be the corresponding flow over routes. Then there exists $\sigma \in\mathcal{O}(\mathcal{N}, \delta)$ such that for each stage $t\in\mathbb{N}_{+}$ and route $r\in\mathcal{R}$, $$\label{eq:optcost}
N^{r}_{t}(\sigma)=F^{*}_{r}.$$
This result says that finding the optimal long-run latency boils down to computing a min-cost static flow. This problem is well studied in the literature and algorithms for solving it efficiently are known [see, for instance, @ForFul:PUP1962; @AhuMagOrl:Prentice1993; @Sch:Springer2003A; @KorVyg:Springer2012].
The detailed proof is in the Appendix. The main insight is as follows. Consider first the case of $\delta=\gamma$. Since in this case the inflow is equal to the capacity of the network, if the planner violates the capacity constraints at some stage $t$, then this creates a queue that will remain through time. An excess of players on some edge of the min-cut, can only be compensated by a future deficit on that edge, which entails an excess on some other edge of the min-cut. Consequently, queues can never be undone and the long-run planner is better-off never creating any queue. When $\delta<\gamma$ we can consider an augmented network obtained concatenating an edge of capacity $\delta$ before the origin of the original network. The optimum of this augmented network (now at capacity $\delta$) is the optimum of the original network with a flow $\delta<\gamma$.
Note that this problem is different from finding an earliest arrival flow, where a given set of players (or particles) has to be shipped to the destination with the requirement that each particle arrives as fast as possible [see, for instance, @Gal:MMJ1959; @HopTar:MOR2000]. @JarRat:MS1982 has shown that this is equivalent to the problem of having as many players as possible to reach the destination in a prescribed amount of time, or to the problem of minimizing the average time to evacuate the system. In our setting instead, players enter the system infinitely often over time, and the goal of the planner is to minimize the average traveling time, which is achieved by not creating queues.
Equilibria for simple network topologies {#se:topologies}
========================================
This section describes the impact of the dynamic nature of the model on equilibrium latencies. We first consider parallel networks for which we are able to give sharp characterizations of equilibria. Then, we extend the results to chain-of-parallel networks.
Parallel networks {#suse:parallel}
-----------------
In a *parallel network*, each route contains a single edge (see Figure \[fi:parallelnetwork\]). For such networks, we can compute exactly the optimum and equilibrium costs for uniform departure inflow ($\delta_t=\delta$ for all $t\in\mathbb{N}_+$), with $\delta\leq\gamma$.
\(1) [$s$]{}; (2) \[right of=1\] [$d$]{};
\(1) edge \[bend right = 60\] node\[above\] [$e_{1}$]{} (2) edge \[bend right = 30\] node\[above\] [$e_{2}$]{} (2) edge node \[above\] [$e_{3}$]{} (2) edge \[bend left = 30\] node\[above\] [$e_{4}$]{} (2) edge \[bend left = 60\] node\[above\] [$e_{5}$]{} (2);
For convenience, we impose an order $\prec$ on the edges such that their lengths are weakly increasing along this order: $i < j \implies e_{i} \prec e_{j} \implies \tau_{e_{i}}\leq\tau_{e_j}$.
Observe that a parallel network admits a unique cut and thus its capacity is simply the sum of the capacities of its edges $\gamma=\sum_{e\in E}\gamma_{e}$.
For each $f \in E$, denote $f^{\prec}:=\{e \in E:e \prec f\}\quad\text{and}\quad f^{\precsim}:=f^{\prec} \cup \{f\}$, where $e_{1}^{\prec} = \varnothing$. For each $\delta \le \gamma$, there is a unique $f_{\delta}\in E$ such that $$\sum_{e \in f_{\delta}^{\prec}} \gamma_{e} < \delta \leq\sum_{e \in f_{\delta}^{\precsim}}\gamma_{e}.$$ Define the (sub-)network $\mathcal{N}_{\delta}$ with set of edges $f_{\delta}^{\precsim}$, such that each edge $e \in f_{\delta}^{\prec}$ has transit cost $\tau_{e}$ and capacity $\gamma_{e}$, and edge $f_{\delta}$ has transit cost $\tau_{f_{\delta}}$ and capacity $\delta-\sum_{e \in f_{\delta}^{\prec}} \gamma_{e}\le \gamma_{f_{\delta}}$. The total capacity of $\mathcal{N}_{\delta}$ is precisely $\delta$.
\[th:parallelunif\] Consider the game $\Gamma(\mathcal{N},\delta)$, where $\mathcal{N}$ is a parallel network and $\delta \le \gamma$. Then $$\begin{aligned}
{\operatorname{\mathsf{Opt}}}(\mathcal{N}, \delta)&=\sum_{e \in f_{\delta}^{\prec}} \gamma_{e}\tau_{e} + \left(\delta- \sum_{e \in f_{\delta}^{\prec}} \gamma_{e}\right)\tau_{f_{\delta}},\\
{\operatorname{\mathsf{WEq}}}(\mathcal{N}, \delta)&= \delta \tau_{f_{\delta}},\end{aligned}$$ and there exists a time $t_{0}$ such that for each $t\geq t_{0}$, $$N^{e}_{t}(\sigma^{{\operatorname{\mathsf{Opt}}}})=N^{e}_{t}(\sigma^{{\operatorname{\mathsf{WEq}}}})=\gamma_{e}\quad \text{for } e \prec f_{\delta} \quad \text{and}\quad N^{f_{\delta}}_{t}(\sigma^{{\operatorname{\mathsf{Opt}}}})=N^{f_{\delta}}_{t}(\sigma^{{\operatorname{\mathsf{WEq}}}})= \delta -\sum_{e \in f_{\delta}^{\prec}} \gamma_{e}.$$
The intuition for the proof is simple. First, it is clear that in a social optimum, the planner uses only the sub-network $\mathcal{N}_{\delta}$, and from Theorem \[th:optimum\], no queues are created. Thus, any optimal strategy sends exactly $\gamma_{e}$ players on each edge $e$ of $\mathcal{N}_\delta$ at each stage. Regarding equilibria, the idea is that the selfish players first fill short edges, thereby creating queues. As a result, the latencies of these edges increase for future generations, and eventually, all latencies become equal to the highest transit cost for that (sub)network. From that point on, players are basically indifferent and, as in an optimal strategies, exactly $\gamma_{e}$ players choose edge $e$ at each stage. The formal proof is in the Appendix.
Chain-of-parallel networks {#suse:chainofparallel}
--------------------------
Let $\mathcal{N}_{1}, \mathcal{N}_{2}$ be two networks with respective source-destination pairs $(s_{1},d_{1})$, $(s_{2},d_{2})$. The *series composition* of $\mathcal{N}_{1}$ and $\mathcal{N}_{2}$ is the network $\mathcal{N}=\mathcal{N}_{1}\oplus\mathcal{N}_{2}$ with source $s_{1}$, destination $d_{2}$ and where $d_{1}$ and $s_{2}$ are merged together. A *chain-of-parallel network* is obtained by composing parallel networks in series. For $h \in \{1, \dots, H\}$, let $\mathcal{N}^{(h)}=\left(E^{(h)}, (\tau_{e})_{e \in E^{(h)}}, (\gamma_{e})_{e \in E^{(h)}}\right)$ be a parallel network and consider the network $\mathcal{N}_{{\operatorname{ser}}}(H)$ obtained by composing $\mathcal{N}^{(1)}, \dots, \mathcal{N}^{(H)}$ in series. Clearly, any subnetwork $\mathcal{N}^{(h)}$ is a cut of $\mathcal{N}_{{\operatorname{ser}}}(H)$. Let $\mathcal{N}^{(*)}$ be a minimum cut of $\mathcal{N}_{{\operatorname{ser}}}(H)$ and let $\gamma^{(*)}$ be the capacity of $\mathcal{N}^{(*)}$.
We obtain the following characterization for optimal and equilibrium values.
\[th:chain\] Consider the game $\Gamma(\mathcal{N}_{{\operatorname{ser}}}(H), \gamma^{(*)})$. Then $$\begin{aligned}
{\operatorname{\mathsf{Opt}}}\left(\mathcal{N}_{{\operatorname{ser}}}(H), \gamma^{(*)}\right)&=\sum_{h=1}^{H} {\operatorname{\mathsf{Opt}}}\left(\mathcal{N}^{(h)}, \gamma^{(*)}\right),\\
{\operatorname{\mathsf{WEq}}}\left(\mathcal{N}_{{\operatorname{ser}}}(H), \gamma^{(*)}\right)&=\sum_{h=1}^{H} {\operatorname{\mathsf{WEq}}}\left(\mathcal{N}^{(h)}, \gamma^{(*)}\right).\end{aligned}$$
The insights are as follows. First, for optimal strategies the modular structure of the graph implies that each subnetwork can be analyzed separately in such a way that no queues are created. Second, modularity implies that the worst equilibrium latency has to be at least the sum of the worst equilibrium latencies of each subnetwork. The uniformly fastest route property guarantees that the latency cannot be worse.
The above result may seem straightforward. An important point to consider is that, although the number of players departing from the source is uniform over time, the number of players who exit a module may actually be non-uniform (periodic, or even aperiodic), at equilibrium. The following example illustrates this phenomenon.
\[ex:chainofparallel\] Consider the chain-of-parallel network given in Figure \[fi:chainofparallel\], where the capacity of each edge is 1 and the transit costs are indicated on the edges. The capacity $\gamma^{(*)}$ of the network is $2$.
\(1) [$s$]{}; (2) \[right of=1\] [$v$]{}; (3) \[right of=2\] [$d$]{};
\(1) edge \[bend right = 45\] node\[above\] [$\tau_{1}^{(1)}=1$]{} (2) edge node \[above\] [$\tau_{2}^{(1)}=2$]{} (2) edge \[bend left = 45\] node\[above\] [$\tau_{3}^{(1)}=2$]{} (2);
\(2) edge \[bend right = 30\] node\[above\] [$\tau_{1}^{(2)}=1$]{} (3) edge \[bend left = 30\] node\[above\] [$\tau_{2}^{(2)}=1$]{} (3);
Equilibria of this game are described in detail in the online Supplementary Material.
Consider the following strategy profile. $$\label{eq:1stequilibrium}
\sigma_{it}^{{\,\mathsf{Eq}}}=
\begin{cases}
e_{1}^{(1)} e_{1}^{(2)} &\text{ for } [it]=[11],\\
e_{1}^{(1)} e_{2}^{(2)} &\text{ for } [it]=[21],\\
e_{1}^{(1)} e_{1}^{(2)} &\text{ for } [it]=[1t] \text{ and } t\geq2,\\
e_{2}^{(1)} e_{2}^{(2)} &\text{ for } [it]=[2t] \text{ and } t\geq2.
\end{cases}$$
It is easy to check that this is an equilibrium. The first player $[11]$ takes the fastest route $e_{1}^{(1)} e_{1}^{(2)}$. The second player $[21]$ cannot pay less than a total cost of 3. She does so by taking $e_{1}^{(1)}$ first and queuing after $[11]$ (a cost of 2), then taking $e_{2}^{(2)}$. This choice of the first generation leaves a queue of size 1 on edge $e_{1}^{(1)} $ for the next generation. The next two players have to pay at least 3 each. They do so by choosing $e_{1}^{(1)} e_{1}^{(2)}$ and $e_{2}^{(1)} e_{2}^{(2)}$. The queue on edge $e_{1}^{(1)} $ is thus re-created for the next generation.
The average total latency of this equilibrium is 6. Due to the indifferences, the same average total latency can be achieved with the following periodic equilibrium strategy profile.
$$\label{eq:2ndequilibrium}
\tilde{\sigma}_{it}^{{\,\mathsf{Eq}}}=
\begin{cases}
e_{1}^{(1)} e_{1}^{(2)} &\text{ for } [it]=[1t] \text{ and } t \text{ odd},\\
e_{1}^{(1)} e_{2}^{(2)} &\text{ for } [it]=[2t] \text{ and } t \text{ odd},\\
e_{2}^{(1)} e_{1}^{(2)} &\text{ for } [it]=[1t] \text{ and } t \text{ even},\\
e_{3}^{(1)} e_{2}^{(2)} &\text{ for } [it]=[2t] \text{ and } t \text{ even}.
\end{cases}$$
Under this profile, the second player of each odd generation creates a queue on $e_{1}^{(1)}$. As both players of the even generation take a long route, none of these two players waits in a queue and thus the queue on $e_{1}^{(1)}$ disappears. Since the first player in the following odd generation uses the fast route $e_{1}^{(1)}$ again, she arrives at $v$ at the same time as the previous two players. Therefore, she waits in the queue on $e_{1}^{(2)}$. So, the first player of each odd generation waits in the queue on $e_{1}^{(2)}$, and the second player waits on $e_{1}^{(1)}$ (except for the very first player). Therefore, the strategy profile $\tilde{\sigma}^{{\,\mathsf{Eq}}}$ yields an average latency of $6$, which is the worst equilibrium latency. However, the queues vary periodically with time (one can even exploit the indifferences to construct a more complex equilibrium where queues vary with time in an aperiodic manner).
Even though such periodicities can occur in an UFR equilibrium, we prove that the worst equilibrium cost can always be obtained with stationary strategies.
Efficiency of equilibria and complex topologies {#se:anarchy}
===============================================
Efficiency of equilibria is a central issue in the theory of congestion games. Several efficiency measures have been proposed, among them, the price of anarchy, the price of stability, and the Braess ratio. Here we use these measures to show how inefficient equilibria can be for dynamic congestion games, and we look at the possible sources of inefficiencies. We first look at parallel networks. Then, we consider more complex topologies. In this section, we consider a uniform inflow in heavy traffic, i.e., $\delta=\gamma$.
Given a game $\Gamma (\mathcal{N}, \gamma)$,
(a) its *price of anarchy* is defined as $${\operatorname{\mathsf{PoA}}}(\mathcal{N}, \gamma):=\frac{{\operatorname{\mathsf{WEq}}}(\mathcal{N}, \gamma)}{{\operatorname{\mathsf{Opt}}}(\mathcal{N}, \gamma)},$$
(b) its *price of stability* is defined as $${\operatorname{\mathsf{PoS}}}(\mathcal{N}, \gamma):=\frac{{\operatorname{\mathsf{BEq}}}(\mathcal{N}, \gamma)}{{\operatorname{\mathsf{Opt}}}(\mathcal{N}, \gamma)},$$
(c) its *Braess ratio* ${\,\mathsf{BR}}(\mathcal{N}, \gamma)$ is defined as the largest factor by which the removal of one or more edges can improve the latency of traffic in an equilibrium flow.
Parallel networks {#parallel-networks}
-----------------
A direct consequence of Theorem \[th:parallelunif\] is a computation of the price of anarchy for parallel networks.
\[co:poaunif\] Consider the game $\Gamma (\mathcal{N}, \gamma)$, where $\mathcal{N}$ is a parallel network. Then $${\operatorname{\mathsf{PoA}}}(\mathcal{N}, \gamma)\le\frac{\max_{e}\tau_{e}}{\min_{e}\tau_{e}}.$$
The inequality is straightforward and shows that the price of anarchy admits an upper bound the does not depend on capacities but only on the relative lengths of edges. To see that the bound is tight, consider a parallel network with two parallel edges such that the first is short and wide, $\tau_{1}=1$, $\gamma_{1}=N^{p}$, where $p\in\mathbb{N}_+$, and the second is long and narrow, $\tau_{2}=N$, $\gamma_{2}=1$. The number of players per stage is the capacity of the network $N^{p}+1$. This instance is similar to the classical example of Pigou where in equilibrium, the congestion on the fast edge creates a latency which matches the latency of the slow edge. The price of anarchy for this network is $(N^{p}+1)/(N^{p-1}+1)$, which is roughly $N= \max_{e}\tau_{e}/\min_{e}\tau_{e}$ for $p$ sufficiently large.
Series-parallel networks {#sususe:seriesparallel}
------------------------
Let $\mathcal{N}_{1}, \mathcal{N}_{2}$ be two networks with respective source-destination pairs $(s_{1},d_{1})$, $(s_{2},d_{2})$. The *parallel composition* of $\mathcal{N}_{1}$ and $\mathcal{N}_{2}$ is the network $\mathcal{N}=\mathcal{N}_{1}\vee\mathcal{N}_{2}$ where the sources (resp. destinations) of $\mathcal{N}_{1}$ and $\mathcal{N}_{2}$ are merged together and the set of edges is the disjoint union of $E_{1}$ and $E_{2}$. A *series-parallel network* is a network which can be obtained by iterated parallel and series compositions of networks, starting with a network containing only one edge.
The well-known paradox due to @Bra:U1968 [@Bra:TS2005] arises when adding a new edge to a network increases the worst equilibrium latency. In static games, this paradox can only occur if the networks contains a Wheatstone subnetwork (see Figure \[fi:Wheatstone\]), or in other words is not series-parallel [@Mil:GEB2006]. @MacLarSte:TCS2013 noticed that in non-atomic dynamic congestion games, Braess’s paradox can arise in networks that are series-parallel.
In dynamic congestion games a different sort of Braess’s paradox can arise: the presence of initial queues in the network, or increasing the transit costs of an edge may decrease the worst equilibrium latency. The following example is an adjustment of one of the networks considered in @MacLarSte:TCS2013 .
\[ex:strictineq\] Consider the series-parallel network in Figure \[fi:seriesparallel\] where the associated free-flow transit costs and capacities are given. The network has two minimum cuts $\{e_{1},e_{4}\}$ and $\{e_{2},e_{3}, e_{4}\}$ with a capacity of 3, and each edge is part of one cut. Deleting one edge would cause the total cost to explode, thus Braess’s paradox cannot happen in its usual form.
(source) at (0,0) [$s$]{}; (main) at (4,0) [$v$]{}; (dest) at (8,0) [$d$]{}; (source)\[out=60,in=120\] to node\[above\][$\tau_{4}=1$]{} node\[below\][$\gamma_{4}=1$]{} (dest); (source) to node\[above\][$\tau_{1}=0$]{} node\[below\][$\gamma_{1}=2$]{} (main); (main)\[out=30,in=150\] to node\[above\][$\tau_{3}=1$]{} node\[below\][$\gamma_{3}=1$]{} (dest); (main)\[out=-30,in=-150\] to node\[above\][$\tau_{2}=0$]{} node\[below\][$\gamma_{2}=1$]{} (dest);
Consider the following equilibrium strategy. $$\label{eq:equilMacko}
\sigma_{it}^{{\,\mathsf{Eq}}}=
\begin{cases}
e_{1} e_{2} &\text{ for } [it]=[11],\\
e_{1} e_{3} &\text{ for } [it]=[21],\\
e_{1} e_{2} &\text{ for } [it]=[31],\\
e_{1} e_{2} &\text{ for } [it]=[1t], t\geq2,\\
e_{4} &\text{ for } [it]=[2t], t\geq2,\\
e_{1} e_{3} &\text{ for } [it]=[3t], t\geq2.
\end{cases}$$
To verify that this is indeed an equilibrium, the reader is referred to the online Supplementary Material. The strategy $\sigma^{{\,\mathsf{Eq}}}$ yields a latency of 4, which means that the last player of each generation pays more than the maximum free flow transit costs.
If we have an initial queue of length one on $e_{2}$, or if we increase the transit cost of $e_{2}$ from $0$ to $1$, then in equilibrium one player of each generation chooses $e_{1}e_{2}$, one player chooses $e_{1}e_{3}$ and one player chooses $e_{4}$. This equilibrium is efficient and the latency is $1$ for every player: a paradox! Recall that this paradox occurs in a network for which no Braess’s paradox is possible in its classical form. Related paradoxical phenomena have been studied by @Dag:TS1998, who showed that, in a model with physical queues, decreasing the capacity of an edge can improve the equilibrium flow of a network. Our example shows that this paradox can occur also with point queues that do not spill over preceding edges.
Notice that Figure \[fi:seriesparallel\] corresponds to a series-parallel network considered in @MacLarSte:TCS2013, where the capacity of edge $e_{3}$ is decreased. So this form of Braess’ paradox may occur in all series-parallel networks considered in their work.
Wheatstone networks {#sususe:braessnetwork}
-------------------
One open question is the impact of the multiplicity of equilibria. The following example will show that for the Wheatstone network, the worst equilibrium has latency costs that are three times higher than the latency costs of the best equilibrium.
\[ex:Wheatstone\] Consider the Wheatstone network in Figure \[fi:Wheatstone\] with associated free-flow transit costs and capacity equal to $1$ for all edges. The capacity of the network is $2$.
\(1) at (0,3) [$s$]{}; (2) at (-2.5,0) [$v$]{}; (3) at (0,-3) [$d$]{}; (4) at (2.5,0) [$w$]{};
\(1) edge node \[right\] [$\tau_{2}=1$]{} (4) edge node\[left,color=black\] [$\tau_{1}=0$]{} (2) (2) edge node \[left\] [$\tau_{4}=1$]{} (3) edge node [$\tau_{3}=0$]{} (4) (4) edge node \[right,color=black\] [$\tau_{5}=0$]{} (3);
Consider the following UFR equilibrium strategy $$\label{eq:Wheatstoneequil}
\sigma_{it}^{{\,\mathsf{Eq}}}=\begin{cases}
e_{1} e_{3} e_{5} &\text{ for } [it]=[i1],i=1,2,\\
e_{1} e_{3} e_{5} &\text{ for } [it]=[12],\\
e_{2} e_{5} &\text{ for } [it]=[22],\\
e_{1} e_{3} e_{5} &\text{ for } [it]=[13],\\
e_{1} e_{4} &\text{ for } [it]=[23],\\
e_{2} e_{5} &\text{ for } [it]=[14],\\
e_{1} e_{3} e_{5} &\text{ for } [it]=[24],\\
e_{1} e_{4} &\text{ for } [it]=[1t],t\geq5,\\
e_{2} e_{5} &\text{ for } [it]=[2t],t\geq5.
\end{cases}$$ The strategy $\sigma^{{\,\mathsf{Eq}}}$ yields a latency of 6. If we remove edge $e_{3}$ from the network, then the worst equilibrium latency improves by a factor of three (from $6$ to $2$) and is equal to the optimum latency of the network. This appears even more paradoxical than in a static model, since the edge $e_{3}$ is not used in equilibrium in steady state.
We can generalize the results of the Wheatstone network by considering Braess’s graphs as defined by @Rou:JCSS2006. Using these graphs, we can construct an example where multiple equilibria exist, some of them efficient and some others unboundedly bad. Details can be found in the online Supplementary Material.
\[pr:pospoa\] For every even integer $n$, there exists a network $\mathcal{N}=(\mathcal{G},(\tau_{e})_{e\in E},(\gamma_{e})_{e\in E})$ in which $\mathcal{G}$ has $n$ vertices such that $$\begin{aligned}
{\operatorname{\mathsf{PoA}}}(\mathcal{N},\gamma)&={\,\mathsf{BR}}(\mathcal{N},\gamma)=n-1,\\
{\operatorname{\mathsf{PoS}}}(\mathcal{N},\gamma)&=1.\end{aligned}$$
Seasonal inflows on parallel network {#se:seasonal}
====================================
In this section, we consider an inflow sequence which is a periodic function of time: there exists an integer $K$ such that $\delta_{t+K}=\delta_t$ for all $t$. Considering parallel networks once more, we provide a characterization of optimum and equilibrium costs with periodic inflow.
From the max-flow min-cut theorem of @ForFul:OR1958, if the average number of players $(\sum_{k=1}^K\delta_{k})/K$ exceeds the capacity $\gamma$ of the network, then the lengths of queues on the edges of the minimum cut diverge to infinity and thus the long-run average total cost is infinite under any strategy profile. We assume from now on that $\gamma=\frac{1}{K}\sum_{k=1}^K\delta_{k}$.
Let $\mathbb{N}_{K}(\gamma)$ be the set of $K$-dimensional integer vectors $\boldsymbol{\delta}=(\delta_{1},\dots, \delta_{K})$ such that $\sum_{k=1}^{K} \delta_{k} = \gamma K$. A $K$-periodic inflow sequence will be identified with a vector $\boldsymbol{\delta}\in \mathbb{N}_{K}(\gamma)$. We denote $\Gamma(\mathcal{N}, K, \boldsymbol{\delta})$ the game with $K$-periodic inflow sequence given by $\boldsymbol{\delta}$.
For each integer $p$, the total latency over the period $\{pK+1,\dots, (p+1)K\}$ is $$L_{p}(\sigma)=\sum_{t=pK+1}^{(p+1)K}\ell_{t}(\sigma).$$
The *average total latency* over $P$ periods is $$\widetilde{L}_{P}(\sigma)=\frac1P\sum_{p=1}^P L_{p}(\sigma).$$ If $\lim_{P \to \infty}\widetilde{L}_{P}(\sigma)$ exists, then it is called *asymptotic average total latency* for the strategy $\sigma$.\
We want to capture how optimum and equilibrium costs are affected by seasonality. The main point is that at peak hours, the inflow exceeds the capacity, and therefore queues build up, even if all the flow is controlled by the planner. This is exemplified as follows.
\[ex:period600\] Consider a parallel network having two edges $e_{1}, e_{2}$ each connecting the source to the destination. We assume $\gamma_{e_{1}}=\gamma_{e_{2}}=1$, $\tau_{e_{1}}=1$ and $\tau_{e_{2}}=2$. The capacity of the network is thus $2$.
Consider the $3$-periodic sequence of departures $\boldsymbol{\delta}= (6,0,0)$. Then, the following strategy profile that allocates three player per period to each edge is optimal. $$\sigma^{{\operatorname{\mathsf{Opt}}}}_{it}=
\begin{cases}
e_{1} & \text{for } i \text{ odd,}\\
e_{2} & \text{for } i \text{ even.}
\end{cases}$$ To see it, consider the first two players and send the first one to $e_{1}$, the second one to $e_{2}$. Then, the next two players have to queue at least for one period, so it is as if they had departed one period later and it is optimal to send one of them to $e_{1}$ and the other to $e_{2}$. Now, the remaining two players have to queue at least two periods, so it is as if they had departed two periods later, and it is again optimal to send one of them to $e_{1}$ and the other to $e_{2}$.
The total latency over a period of time $\{1,2,3\}$ (modulo 3) is $15$, that is, $3$ times the single-period optimal total latency that we would have if departures were uniform $(2,2,2)$ plus the added cost of $6$ induced by the waiting times: two players pay an extra cost of $1$ and two players pay an extra cost of $2$.
Consider now the following equilibrium strategy $\sigma^{{\,\mathsf{Eq}}}$. For $t = 1$ we let $$\sigma^{{\,\mathsf{Eq}}}_{it}=
\begin{cases}
e_{1} & \text{for $i=1$ or $i$ even,} \\
e_{2} & \text{for $i>2$, odd,}
\end{cases}$$ therefore the latencies for the first six players are $$\begin{aligned}
\ell_{11}(\sigma^{{\,\mathsf{Eq}}})&=1, \ \ell_{21}(\sigma^{{\,\mathsf{Eq}}})=2, \ \ell_{31}(\sigma^{{\,\mathsf{Eq}}})=2, \\ \ell_{41}(\sigma^{{\,\mathsf{Eq}}})&=3,\ \ell_{51}(\sigma^{{\,\mathsf{Eq}}})=3, \ \ell_{61}(\sigma^{{\,\mathsf{Eq}}})=4.\end{aligned}$$ For $t \ge 4$ and $t=1 \mod 3$, $$\sigma^{{\,\mathsf{Eq}}}_{it}=
\begin{cases}
e_{1} & \text{for $i$ odd,}\\
e_{2} & \text{for $i$ even,}
\end{cases}$$ and $$\begin{aligned}
\ell_{1t}(\sigma^{{\,\mathsf{Eq}}})&=2, \ \ell_{2t}(\sigma^{{\,\mathsf{Eq}}})=2, \ \ell_{3t}(\sigma^{{\,\mathsf{Eq}}})=3, \\ \ell_{4t}(\sigma^{{\,\mathsf{Eq}}})&=3,\ \ell_{5t}(\sigma^{{\,\mathsf{Eq}}})=4, \ \ell_{6t}(\sigma^{{\,\mathsf{Eq}}})=4.\end{aligned}$$ It is easy to check that this is an equilibrium. It is constructed in such a way that each player chooses $e_1$ when he is indifferent between the two edges. This choice makes it the worst equilibrium. In the steady state, the total equilibrium payoff over a $3$-period is $18$, that is $3$ times the single-period equilibrium total latency when departures are uniform $(2,2,2)$ plus the added cost of $6$ induced by the waiting times.
We define now a quantity that measures the non-uniformity of the inflow. Define the following binary relation on $\mathbb{N}_{K}(\gamma)$.
For any two elements $\boldsymbol{\delta}, \boldsymbol{\delta}' \in \mathbb{N}_{K}(\gamma)$, we say that *$\boldsymbol{\delta}'$ is obtained from $\boldsymbol{\delta}$ by an elementary operation* (denote it $\boldsymbol{\delta}\to \boldsymbol{\delta}'$), if there exists a stage $t$ such that $$\begin{aligned}
\delta_{t} &> \gamma,\\
\delta'_{t}&=\delta_{t}-1, \\
\delta'_{t+1}&=\delta_{t+1}+1, \\
\delta'_{k}&=\delta_{k} \text{ for $k\notin\{t,t+1\}$},\end{aligned}$$ where indices are considered modulo $K$.
Denote $\boldsymbol{\gamma}_{K} = (\gamma, \dots, \gamma) \in \mathbb{N}_{K}(\gamma)$ the uniform vector. Consider the directed graph representing the above binary relation $\to$ and denote $D(\boldsymbol{\delta})$ the distance in this graph from $\boldsymbol{\delta}$ to $\boldsymbol{\gamma}_{K}$. An elementary operation $\boldsymbol{\delta}\to \boldsymbol{\delta}'$ consists in moving one unit from a slot where the capacity is over-filled, to the next slot. Note that indices are considered modulo $K$, so this definition is invariant under circular permutation. Any $\boldsymbol{\delta}\neq \boldsymbol{\gamma}_{K}$ has at least one successor in the graph and $\boldsymbol{\gamma}_{K}$ is the only element with no successor. Then, $D(\boldsymbol{\delta})$ is the minimum number of elementary operations needed to transform $\boldsymbol{\delta}$ into $\boldsymbol{\gamma}_{K}$. See Figure \[fi:600to222\].
=\[ball color=black!10,circle,minimum size=17pt,inner sep=0pt\] =\[ball color=red,circle,minimum size=17pt,inner sep=0pt\] =\[circle,draw=red!75,minimum size=17pt,inner sep=0pt\] =\[minimum size=17pt,inner sep=0pt\]
/in [1/1, 2/2, 3/3]{} (T-) at (, 0) [$\name$]{};
/in [1/1, 2/2, 3/3, 4/4, 5/5, 6/6]{} (G-) at (1, ) ;
/in [1/1, 2/2, 3/3]{} (T-) at (, 0) [$\name$]{};
/in [1/1, 2/2, 3/3, 4/4, 5/5]{} (G-) at (1, ) ; /in [6/6]{} (G-) at (1, ) ; /in [1/1]{} (F-) at (2, ) ; (G-6) .. controls +(-30:1cm) .. (F-1);
/in [1/1, 2/2, 3/3]{} (T-) at (, 0) [$\name$]{};
/in [1/1, 2/2, 3/3, 4/4]{} (G-) at (1, ) ; /in [5/5]{} (G-) at (1, ) ; /in [1/1]{} (F-) at (2, ) ; /in [2/2]{} (F-) at (2, ) ; (G-5) .. controls +(-30:1cm) .. (F-2);
/in [1/1, 2/2, 3/3]{} (T-) at (, 0) [$\name$]{};
/in [1/1, 2/2, 3/3]{} (G-) at (1, ) ; /in [4/4]{} (G-) at (1, ) ; /in [1/1, 2/2]{} (F-) at (2, ) ; /in [3/3]{} (F-) at (2, ) ; (G-4) .. controls +(-30:1cm) .. (F-3);
/in [1/1, 2/2, 3/3]{} (T-) at (, 0) [$\name$]{};
/in [1/1, 2/2, 3/3]{} (G-) at (1, ) ; /in [1/1, 2/2]{} (F-) at (2, ) ; /in [3/3]{} (F-) at (2, ) ; /in [1/1]{} (H-) at (3, ) ; (F-3) .. controls +(-30:1cm) .. (H-1);
/in [1/1, 2/2, 3/3]{} (T-) at (, 0) [$\name$]{};
/in [1/1, 2/2]{} (G-) at (1, ) ; /in [3/3]{} (G-) at (1, ) ; /in [1/1, 2/2]{} (F-) at (2, ) ; /in [3/3]{} (F-) at (2, ) ; /in [1/1]{} (H-) at (3, ) ; (G-3) – (F-3);
/in [1/1, 2/2, 3/3]{} (T-) at (, 0) [$\name$]{};
/in [1/1, 2/2]{} (G-) at (1, ) ; /in [1/1, 2/2]{} (F-) at (2, ) ; /in [3/3]{} (F-) at (2, ) ; /in [1/1]{} (H-) at (3, ) ; /in [2/2]{} (H-) at (3, ) ; (F-3) .. controls +(-30:1cm) .. (H-2);
/in [1/1, 2/2, 3/3]{} (T-) at (, 0) [$\name$]{};
/in [1/1, 2/2]{} (G-) at (1, ) ; /in [1/1, 2/2]{} (F-) at (2, ) ; /in [1/1, 2/2]{} (H-) at (3, ) ;
Proposition \[pr:parallelperiod\] below states that the quantity $D(\boldsymbol{\delta})$ measures the total waiting time incurred at the optimum and at the worst equilibrium by the players due to non-uniform departures.
\[pr:parallelperiod\] Consider the game $\Gamma(\mathcal{N},K, \boldsymbol{\delta})$, where $\mathcal{N}$ is a parallel network and $\boldsymbol{\delta}\in\mathbb{N}_{K}(\gamma)$. Then $$\begin{aligned}
{\operatorname{\mathsf{Opt}}}(\mathcal{N},K, \boldsymbol{\delta})&=K\sum_{e\in E}\gamma_{e}\tau_{e}+
D(\boldsymbol{\delta}),\\
{\operatorname{\mathsf{WEq}}}(\mathcal{N}, K, \boldsymbol{\delta})&=K\gamma\max_{e\in E}\tau_{e}+D(\boldsymbol{\delta}).\end{aligned}$$
The main idea of the proof is that if $\boldsymbol{\delta}'$ is obtained from $\boldsymbol{\delta}$ by an elementary operation, then the optimum and the worst equilibrium under $\boldsymbol{\delta}'$ are obtained from the optimum and the worst equilibrium under $\boldsymbol{\delta}$ by letting a player postpone her departure by one unit of time. In other words, since a player departing over capacity has to queue anyhow, it would save one unit of total cost if this player’s departure were postponed to the next unit of time. The formal proof is in the online Supplementary Material.
Regarding the impact on efficiency, it is easy to see that the ratio ${\operatorname{\mathsf{WEq}}}(\mathcal{N}, K, \boldsymbol{\delta})/{\operatorname{\mathsf{Opt}}}(\mathcal{N},K, \boldsymbol{\delta})$ is decreasing in $D(\boldsymbol{\delta})$. Intuitively, when seasonality is high, the planner has to create queues, and thus the optimum tends to resemble the equilibrium.
Conclusions and open problems {#se:conclusion}
=============================
In this paper, we have considered dynamic congestion games with atomic players and common source-destination pair. We have shown that when the inflow rate is uniform over time and does not exceed the capacity of the network, an optimal dynamic flow never create queues. This result is independent of the topology of the network. For special topologies such as parallel network, we have provided exact computation of optimum and equilibrium costs. An important insight is that optimum and equilibrium flows eventually coincide, but the transient phase before reaching the steady state induces an important difference in costs.
We have studied efficiency of equilibria and have shown that the price of anarchy is unbounded, even for parallel networks. We also found that there exist networks that admit efficient equilibria, but for which both the price of anarchy and the Braess ratio are arbitrarily large. This shows that multiplicity of equilibria in atomic games may have a significant impact.
We have shown that several Braess-type paradoxes can occur in atomic dynamic network games. First of all, we have the usual Braess’s paradox according to which removing an edge from a network can improve the equilibrium cost. Unlike what happens in static games, this paradox can occur also in networks that do not include a Wheatstone subnetwork. Moreover, we can have another paradoxical phenomenon for which initial queues in the network reduce the equilibrium cost. Alternatively, increasing the transit cost of an edge may reduce the equilibrium cost.
Finally, we have studied the impact of seasonality of inflow by considering parallel networks and periodic inflow sequences. The main result is that the optimum cost and the equilibrium cost are shifted upwards by the same amount which is interpreted as a measure of seasonality.
We think of this work as a first attempt to understand atomic dynamic congestion games. Several problems remain open, among them:
(a) Are queues always bounded in equilibrium? If yes, how much worse can the equilibrium costs be, compared to the socially optimal costs, and what would be a characterization of this cost for a given network?
(b) We found a new kind of paradoxical phenomenon: the presence of an initial queue improves the equilibrium latency. In which networks does such paradox exist? And by how much can the latency improve?
(c) Based on many examples that we have solved numerically, we conjecture that $D(\boldsymbol{\delta})$ is always an upper bound of the extra equilibrium cost due to seasonality. Is this true?
Acknowledgments {#acknowledgments .unnumbered}
---------------
We thank three referees, an Associate Editor, and the Area Editor Professor Asuman Ozdaglar, whose insightful observations allowed us to expand the scope of our analysis. We are indebted to Ludovic Renou, who provided helpful comments and to Thomas Rivera whose suggestions helped us improve the exposition. We thank Roberto Cominetti and José Correa for some insightful conversations.
Proofs
======
Proofs of Section \[se:model\] {#proofs-of-sectionsemodel .unnumbered}
------------------------------
We prove the existence of a uniformly fastest route equilibrium of the game $\Gamma(\mathcal{N}, \mathcal{D})$. Define the strategy profile $\sigma\in\mathcal{R}^G$ as follows. In an empty network there is always a shortest route with the property that every intermediate vertex is reached as early as possible, since that case is equivalent to the static shortest path problem. Let player $[11]$ choose a route with that property. We define the strategy for each other player $[it]$ iteratively. Given the choices of players $[js] \lhd [it]$, let player $[it]$ choose a route such that each intermediate vertex is reached as early as possible. A slight modification of Dijkstra’s algorithm can be used to compute such path. Let us argue that the above strategy profile $\sigma$ is a UFR equilibrium.
By definition of $\sigma$, a player $[js]$ does not influence the costs of a player $[it]$ with $[it] \lhd [js]$, since $[js]$ does not overtake $[it]$. Hence the latency of a player $[it]$ does not depend on a player $[js]$ with $[it] \lhd [js]$. So player $[it]$ has the same latency value she had when she chose her route. Since she chose a shortest route with the property that every intermediate vertex is reached as early as possible, the strategy profile is a UFR equilibrium.
Proofs of Section \[se:optimum\] {#proofs-of-sectionseoptimum .unnumbered}
--------------------------------
We start proving the theorem for the case $\delta=\gamma$, i.e., when the inflow is at capacity. The proof starts with two lemmas. The first lemma actually holds for any $\delta \le \gamma$.
For edge $e$ and each stage $t$, denote $y_{t}^{e}(\sigma)$ the number of players who *enter* edge $e$ at stage $t$ under strategy $\sigma$ and $$\bar{y}_{T}^{e}(\sigma) = \frac{1}{T} \sum_{t\leq T}y_{t}^{e}(\sigma).$$
\[le:limsupLbar\] Let $C$ be a minimum cut and $c\in C$. If a strategy $\sigma$ is such that $\limsup_{T}\bar{y}_{T}^{c}(\sigma)>\gamma_{c}$, then $\limsup_{T}\bar{L}_{T}(\sigma)=+\infty$.
Take a strategy $\sigma$, a minimum cut $C$ and an edge $c\in C$ such that $\limsup_{T}\bar{y}_{T}^{c}(\sigma)>\gamma_{c}$. Then, there exists $\alpha>0$ and a subsequence $\{T_{k}\}$ such that along this subsequence we have $\bar{y}_{T_{k}}^{c}(\sigma)\geq \gamma_{c}+\alpha$. Since at most $\gamma_{c}$ players can exit edge $c$ at any given time, this implies that there exists a player who has a waiting time of $w=\lfloor \alpha T_{k}/\gamma_c\rfloor$. This in turn implies that for each integer $s< w$, there exist $\gamma_c$ players who have waiting time $s$. Thus, the total waiting time adds up to at least $$\gamma_c\cdot(1+\dots+w-1)=\frac{\gamma_c\cdot(w-1)\cdot w}{2}$$ and the average waiting time at stage $T_{k}$ is such that $$\bar{w}_{T_{k}}(\sigma)\geq \frac{(\lfloor \alpha T_{k}/\gamma_c\rfloor-1)\cdot\lfloor \alpha T_{k}/\gamma_c\rfloor}{2T_{k}}.$$ The r.h.s. diverges as $k\to\infty$, which concludes the proof.
For each stage $T$ and edge $e$, denote $x_{T}^{e}(\sigma)$ the number of players who *exit* edge $e$ at stage $T$ under strategy $\sigma$ and $$\bar{x}_{T}^{e}(\sigma) = \frac{1}{T} \sum_{t\leq T}x_{t}^{e}(\sigma).$$
\[co:FF\] If a strategy $\sigma$ is such that $\limsup_{T}\bar{L}_{T}(\sigma)<+\infty$, then, for any minimum cut $C$ and $c\in C$, we have $$\lim_{T}\bar{y}_{T}^{c}(\sigma)=\lim_{T}\bar x_{T}^{c}(\sigma)=\gamma_{c}.$$
Consider such a strategy $\sigma$ and a minimum cut $C$. Thanks to Lemma \[le:limsupLbar\], for each edge $c\in C$, $\limsup_{T}\bar{y}_{T}^{c}(\sigma)\leq \gamma_{c}$. If there exists an edge $c\in C$ such that $\liminf_{T}\bar{y}_{T}^{c}(\sigma) < \gamma_{c}$, then there must be another edge $c'\in C$ such that $\limsup_{T}\bar{y}_{T}^{c'}(\sigma) > \gamma_{c}$. This is a consequence of the Ford and Fulkerson theorem. If for $\alpha>0$, $\bar{y}_{T_{k}}^{c}(\sigma)\leq \gamma_{c}-\alpha$ along a subsequence $\{T_{k}\}$, then there is a deficit of players on edge $c$ which has to be compensated by an an excess of players on another edge $c'$ of the minimum cut $C$. As a consequence of Lemma \[le:limsupLbar\], this results in an unbounded average cost.
Finally, if the inflows satisfy $$\lim_{T}\bar{y}_{T}^{c}(\sigma)=\gamma_{c},\quad \forall c\in C,$$ then the outflows $\bar{x}_{T}^{c}(\sigma), c\in C$, satisfy it as well.
We may now conclude the proof of Theorem \[th:optimum\] when $\delta=\gamma$. Lemmas \[le:limsupLbar\] and \[co:FF\] imply that, to guarantee $\limsup_{T}\bar{L}_{T}(\sigma)<+\infty$, flows have to match capacities on every minimal cut. Thus, in order to minimize the asymptotic average latency, there should remain no queues, i.e., at the optimum the total waiting time is zero. A simple way to achieve that is to repeat a static flow with no queues at each stage. By construction, the min-cost flow $f^{*}$ has a value $\gamma$, the capacity of the network, and satisfies $f^{*}_e\leq\gamma_e$ for each edge $e$. Thus, repeating the assignment $F^{*}$ at each stage yields an asymptotic average latency $L^{*}$, which is the value of the min-cost static flow problem. This is clearly the best that can be achieved without creating queues and therefore this is the optimal asymptotic average latency. Notice that under this assignment *all* edges are queue free, not just on the edges of the minimum cut.
Now, the case $\delta < \gamma$ can be treated by augmenting the network with a fictitious edge $f$ of capacity $\delta$ and length $0$, whose tail is the new source and whose head is the old source. This new edge $f$ is clearly the unique minimum cut of the new network. This way, we obtain a game where the inflow is at capacity, and so we can apply the first part of the proof. Since the output of $f$ is the input of the original network, and is constantly $\delta$, the result follows.
Proofs of Section \[se:topologies\] {#proofs-of-sectionsetopologies .unnumbered}
-----------------------------------
First, by Theorem \[th:optimum\], the optimal latency can be computed by sending the capacity number of players on each edge of the subnetwork $\mathcal{N}_{\delta}$. Hence the result follows.
Second, we show that there exists an equilibrium such that each player pays the transit cost of $\tau_{f_{\delta}}$. To simplify notation, the proof below assumes that $\delta=\gamma$. A similar proof can be given if $\delta<\gamma$. Call $n$ the number of edges in $E$.
We start by defining several times. Let $T_0=0$ and, for all $j\in\{1,\ldots,n-1\}$, $$T_j=\sum_{k=1}^j\left(\frac{\sum_{i=1}^k\gamma_i}{\delta-\sum_{i=1}^k\gamma_i}\cdot(\tau_{k+1}-\tau_k)\right).$$ Denote $\underline{T}_j=\left\lfloor T_j\right\rfloor$ for all $j\in\{1,\ldots,n-1\}$.
Define a strategy profile $\sigma\in\mathcal{R}^G$ for $\Gamma(\mathcal{N},\delta)$ as follows. For all $[it]\in G$, choose $e\in E$ with minimum latency, and if there are multiple edges with minimum latency, then choose among these the first one in the order $\prec$.
We divide the proof into three parts: (i) stages $t$ with $t\leq \underline{T}_1$, (ii) stages $t$ with $\underline{T}_{j-1}<t\leq \underline{T}_j$ for $j\in\{2,\ldots,n-1\}$ and (iii) stages $t$ with $t>\underline{T}_{n-1}$. Note that part (ii) is redundant if $n=2$.
\(i) Each player who sees a queue of size $\gamma_1\cdot(\tau_2-\tau_1)=(\delta-\gamma_1)\cdot T_1$ on $e_1$, faces a waiting cost of $\tau_2-\tau_1$, and consequently is indifferent between $e_1$ and $e_2$. If all $[it]$ with $t<\underline{T}_1$ choose $e_1$, then the queue on $e_1$ contains $(\delta-\gamma_1)\cdot \underline{T}_1$ players at the start of stage $\underline{T}_1$. Define $$\alpha_1=(\delta-\gamma_1)\cdot(T_1- \underline{T}_1).$$ This is the number of players of $G_{\underline{T}_1}$ needed before a player is indifferent between $e_1$ and $e_2$. Since $0\leq\alpha_1<(\delta-\gamma_1)$, we know that player $[\alpha_1+1,\underline{T}_1]$ sees a queue of size $(\delta-\gamma_1)\cdot T_1\cdot$ on $e_1$ (and is indifferent between $e_1$ and $e_2$), and that player $[\alpha_1+\gamma_1+1,\underline{T}_1]$ is the first player to choose $e_2$. In other words, at all stages $t$ with $t< \underline{T}_1$ all players choose $e_1$.
\(ii) For $j\in\{2,\ldots,n-1\}$, the following analysis holds true iteratively. Consider the sum of the queues on $e_1, \ldots, e_j$ that have grown starting from the first player who was indifferent between $e_1, \ldots, e_j$ onwards. We call this the joint queue of $e_1, \ldots, e_j$. The joint queue of $e_1, \ldots, e_j$ contains $\max\{0,\delta-\sum_{i=1}^j\gamma_i-\alpha_{j-1}\}$ players at the start of stage $\underline{T}_{j-1}+1$.
Each player who sees a queue of size $\sum_{i=1}^j\gamma_i\cdot(\tau_{j+1}-\tau_j)=(\delta-\sum_{i=1}^j\gamma_i)\cdot(T_j-T_{j-1})$ on the joint queue of $e_1, \ldots, e_j$, is indifferent between $e_1$, …, $e_j$ and $e_{j+1}$. If all $[it]$ with $\underline{T}_{j-1}<t<\underline{T}_j$ choose one edge in $\{e_1, \ldots, e_j\}$, then the joint queue of $e_1, \ldots, e_j$ contains $\max\{0,\delta-\sum_{i=1}^j\gamma_i-\alpha_{j-1}\}+(\delta-\sum_{i=1}^j\gamma_i)\cdot(\underline{T}_j-\underline{T}_{j-1}-1)$ players at the start of stage $\underline{T}_j$. Define $$\alpha_j=\left(\delta-\sum_{i=1}^j\gamma_i\right)\cdot(T_j- \underline{T}_j+\underline{T}_{j-1}+1-T_{j-1})-\max\left\{0,\delta-\sum_{i=1}^j\gamma_i-\alpha_{j-1}\right\}.$$ Thus $\alpha_j$ is the number of players needed in $G_{\underline{T}_j}$ before a player is indifferent between $e_1$, …, $e_j$ and $e_{j+1}$.
We have $0\leq\alpha_j<\delta-\gamma_1$ for all $j\in\{2,\ldots,n-1\}$.
First, we show that $\alpha_j\geq 0$ for all $j\in\{2,\ldots,n-1\}$. Notice that if $\delta-\sum_{i=1}^j\gamma_i-\alpha_{j-1}\leq0$, then the result follows. So assume that $\delta-\sum_{i=1}^j\gamma_i-\alpha_{j-1}>0$. We prove by induction that $\delta-\sum_{i=1}^j\gamma_i-\alpha_{j-1}\leq(\delta-\sum_{i=1}^j\gamma_i)\cdot(\underline{T}_{j-1}+1-T_{j-1})$ for all $j\in\{2,\ldots,n-1\}$.
For $j=2$, $$\begin{aligned}
\delta-\sum_{i=1}^2\gamma_i-\alpha_1&=\left(\delta-\sum_{i=1}^2\gamma_i\right)\cdot(\underline{T}_1+1-T_1)-\gamma_2\cdot(T_1-\underline{T}_1)\\
&\leq\left(\delta-\sum_{i=1}^2\gamma_i\right)\cdot(\underline{T}_1+1-T_1).\end{aligned}$$
Suppose the inequality holds true for $j\in\{2,\ldots,n-2\}$. Then $$\begin{aligned}
\delta-\sum_{i=1}^{j+1}\gamma_i-\alpha_j&=\left(\delta-\sum_{i=1}^{j+1}\gamma_i\right)\cdot(\underline{T}_j-T_j+T_{j-1}-\underline{T}_{j-1})\\
&\quad-\gamma_{j+1}\cdot(T_j-\underline{T}_j+\underline{T}_{j-1}+1-T_{j-1})+\max\left\{0,\delta-\sum_{i=1}^j\gamma_i-\alpha_{j-1}\right\}\\
&\leq\left(\delta-\sum_{i=1}^{j+1}\gamma_i\right)\cdot(\underline{T}_j+1-T_j)-\gamma_{j+1}\cdot(T_j-\underline{T}_j)\\
&\leq\left(\delta-\sum_{i=1}^{j+1}\gamma_i\right)\cdot(\underline{T}_j+1-T_j),\end{aligned}$$ where the first inequality follows from the induction hypothesis.
The above result implies $$\alpha_j\geq\left(\delta-\sum_{i=1}^{j+1}\gamma_i\right)\cdot(T_j-\underline{T}_j)\geq 0.$$
Second, we show that $\alpha_j<\delta-\gamma_1$ for all $j\in\{2,\ldots,n-1\}$. We prove by induction that $\delta-\sum_{i=1}^j\gamma_i-\alpha_{j-1}\geq(\delta-\sum_{i=1}^j\gamma_i)\cdot(\underline{T}_{j-1}+1-T_{j-1})-\sum_{i=2}^j\gamma_i\cdot(T_{i-1}-\underline{T}_{i-1})$ for all $j\in\{2,\ldots,n-1\}$.
For $j=2$, $$\begin{aligned}
\delta-\sum_{i=1}^2\gamma_i-\alpha_1&=\left(\delta-\sum_{i=1}^2\gamma_i\right)\cdot(\underline{T}_1+1-T_1)-\gamma_2\cdot(T_1-\underline{T}_1).\end{aligned}$$
Suppose the inequality holds true for $j\in\{2,\ldots,n-2\}$. Then $$\begin{aligned}
\delta-\sum_{i=1}^{j+1}\gamma_i-\alpha_j&=\left(\delta-\sum_{i=1}^{j+1}\gamma_i\right)\cdot(\underline{T}_j-T_j+T_{j-1}-\underline{T}_{j-1})\\
&\quad-\gamma_{j+1}\cdot(T_j-\underline{T}_j+\underline{T}_{j-1}+1-T_{j-1})+\max\left\{0,\delta-\sum_{i=1}^j\gamma_i-\alpha_{j-1}\right\}\\
&\geq\left(\delta-\sum_{i=1}^{j+1}\gamma_i\right)\cdot(\underline{T}_j+1-T_j)-\sum_{i=2}^{j+1}\gamma_i\cdot(T_{i-1}-\underline{T}_{i-1}),\end{aligned}$$ where the inequality follows from the induction hypothesis.
The above result implies $$\alpha_j\leq\left(\delta-\sum_{i=1}^{j+1}\gamma_i\right)\cdot(T_j-\underline{T}_j)+\sum_{i=2}^{j+1}\gamma_i\cdot(T_{i-1}-\underline{T}_{i-1})<\delta-\gamma_1.$$ This concludes the proof of the claim.
Since $0\leq\alpha_j<\delta-\gamma_1$, we know that player $[\alpha_j+1,\underline{T}_j]$ sees a queue of size $\sum_{i=j+1}^n\gamma_i\cdot(T_j-T_{j-1})\cdot$ on the joint queue of $e_1, \ldots, e_j$ (and is indifferent between $e_1, \ldots, e_j$ and $e_{j+1}$). So, if player $[\alpha_j+\sum_{i=1}^{j}\gamma_i+1,\underline{T}_j]$ exists, then she is the first player to choose $e_{j+1}$, and if player $[\alpha_j+\sum_{i=1}^{j}\gamma_i+1,\underline{T}_j]$ does not exist, then player $[\sum_{i=1}^j\gamma_i+1,\underline{T}_j+1]$ is the first player to choose $e_{j+1}$.
\(iii) For all stages $t$ with $t>\underline{T}_{n-1}$, player $[1t]$ faces a latency of $\tau_n$ on each $e\in E$ and therefore is indifferent between $e_1$, …, and $e_n$. So the first $\gamma_1$ players choose $e_1$, the second $\gamma_2$ players choose $e_2$, …, and the last $\gamma_n$ players choose $e_n$. This implies no additional queue is created during this stage. Since at most $\gamma$ players arrive at each stage, individual costs cannot become higher than $\tau_n$.
Third, notice that both in the socially optimal strategy and the worst equilibrium flows on each edge of the subnetwork $\mathcal{N}_{\delta}$ are eventually equal to capacity.
Supplementary material
======================
Section \[se:topologies\] {#sectionsetopologies .unnumbered}
-------------------------
*Optimum*. It is clear that a minimum cut of $\mathcal{N}_{{\operatorname{ser}}}(H)$ is a subnetwork $\mathcal{N}^{(h)}$ with minimum capacity. Denote $\mathcal{N}^{(*)}$ such a minimum cut of $\mathcal{N}_{{\operatorname{ser}}}(H)$ and $\gamma^{(*)}$ its capacity which is also the capacity of the whole network $\mathcal{N}_{{\operatorname{ser}}}(H)$. If the size of each generation is $\delta=\gamma^{(*)}$, then each subnetwork $\mathcal{N}^{(h)}$ has a capacity at least $\delta$. Thus, the planner can choose the global flow in order to minimize the cost on each subnetwork separately, which is clearly the best achievable total cost.
*Equilibrium*. Consider again $\mathcal{N}^{(*)}$, a minimum cut of $\mathcal{N}_{{\operatorname{ser}}}(H)$ with capacity $\gamma^{(*)}$. First, we show that there is an equilibrium with corresponding latency equal to the sum of the worst latencies of each module. Consider the subnetwork $\mathcal{N}^{(1)}$. The worst equilibrium cost on $\mathcal{N}^{(1)}$ with corresponding strategy profile is given by Theorem \[th:parallelunif\]. Now, from the structure of this equilibrium, from some time onwards, there are $\gamma^{(*)}$ players outgoing from $\mathcal{N}^{(1)}$ at each stage. Since the output of $\mathcal{N}^{(1)}$ is at most $\gamma^{(*)}$ in earlier stages, the long-run worst equilibrium cost for the next modules is the same as under a constant inflow of $\gamma^{(*)}$. Hence the latency of this equilibrium is the sum of the worst latencies of each module.
Second, we show that the sum of the worst latencies is the worst equilibrium latency for this network. If the inflow of each module is constant from some point onwards, then the above equilibrium is the worst equilibrium. The UFR property assures that within each module an equilibrium is played. Therefore, on each (sub)module, costs are at most the maximum transit costs.
However, a module $\mathcal{N}^{(h)}$ with a capacity larger than $\gamma^{(*)}$ is able to produce a non-uniform outflow. As long as this outflow is below $\gamma^{(*)}$, the latency of the following module $\mathcal{N}^{(h+1)}$ is at most the worst latency of $\mathcal{N}^{(h+1)}$. Let $t^{*}$ be the first period in which the outflow of $\mathcal{N}^{(h)}$ is above $\gamma^{(*)}$. Each player that departs after $\gamma^{(*)}$ players already departed (potentially) faces an additional queue in $\mathcal{N}^{(h+1)}$. However, in order to obtain an outflow above $\gamma^{(*)}$, players from two different generations must leave at the same moment. This implies that all players from the second generation have a latency which is one unit below the latency of the first generation. So the additional queue that will be created in $\mathcal{N}^{(h+1)}$ is offset by the decrease in latency in $\mathcal{N}^{(h)}$. A similar idea applies to subsequent periods in which this additional queue is maintained. Hence overall the equilibrium latency cannot be worse than the sum of the worst latencies.
*Details of Example \[ex:chainofparallel\].* Consider the chain-of-parallel network given in Figure \[fi:chainofparallelSM\], where the capacity of each edge is 1 and the transit costs are indicated on the edges. The capacity $\gamma^{(*)}$ of the network is $2$.
\(1) [$s$]{}; (2) \[right of=1\] [$v$]{}; (3) \[right of=2\] [$d$]{};
\(1) edge \[bend right = 45\] node\[above\] [$\tau_{1}^{(1)}=1$]{} (2) edge node \[above\] [$\tau_{2}^{(1)}=2$]{} (2) edge \[bend left = 45\] node\[above\] [$\tau_{3}^{(1)}=2$]{} (2);
\(2) edge \[bend right = 30\] node\[above\] [$\tau_{1}^{(2)}=1$]{} (3) edge \[bend left = 30\] node\[above\] [$\tau_{2}^{(2)}=1$]{} (3);
=\[circle,fill=blue!20,draw,minimum size=25pt,font=**\] =\[circle,fill=green!20,draw,minimum size=25pt,font=**\] =\[circle,fill=red!20,draw,minimum size=25pt,font=**\] =\[circle,minimum size=22pt\] =\[blue,fill=white,font=**\] =\[red,fill=white,font=**\] =\[black,fill=white,font=**\] =\[font=**\]**************
(source) [$s$]{}; (main) \[right of=source\] [$v$]{}; (dest) \[right of=main\] [$d$]{};
(source)\[red, out=-30,in=-150\] to node\[above\] node\[below\] (main); (main)\[red, out=-30,in=-150\] to node\[above\] node\[below\] (dest); at (12,0) [$e_{1}^{(1)},e_{1}^{(2)}$]{};
(source) [$s$]{}; (main) \[right of=source\] [$v$]{}; (dest) \[right of=main\] [$d$]{};
(source)\[blue, out=-30,in=-150\] to node\[above\] node\[below\] (main); (main)\[blue, out=30,in=150\] to node\[above\] node\[below\] (dest); at (12,0) [$e_{1}^{(1)},e_{2}^{(2)}$]{};
(source) [$s$]{}; (main) \[right of=source\] [$v$]{}; (dest) \[right of=main\] [$d$]{};
(source)\[green\] to node\[above\] node\[below\] (main); (main)\[green, out=-30,in=-150\] to node\[above\] node\[below\] (dest); at (12,0) [$e_{2}^{(1)},e_{1}^{(2)}$]{};
(source) [$s$]{}; (main) \[right of=source\] [$v$]{}; (dest) \[right of=main\] [$d$]{};
(source)\[brown\] to node\[above\] node\[below\] (main); (main)\[brown, out=30,in=150\] to node\[above\] node\[below\] (dest); at (12,0) [$e_{2}^{(1)},e_{2}^{(2)}$]{};
(source) [$s$]{}; (main) \[right of=source\] [$v$]{}; (dest) \[right of=main\] [$d$]{};
(source)\[out=30,in=150\] to node\[above\] node\[below\] (main); (main)\[out=30,in=150\] to node\[above\] node\[below\] (dest); at (12,0) [$e_{3}^{(1)},e_{2}^{(2)}$]{};
Consider the following strategy profile. $$\label{eq:1stequilibriumSM}
\sigma_{it}^{{\,\mathsf{Eq}}}=
\begin{cases}
e_{1}^{(1)} e_{1}^{(2)} &\text{ for } [it]=[11],\\
e_{1}^{(1)} e_{2}^{(2)} &\text{ for } [it]=[21],\\
e_{1}^{(1)} e_{1}^{(2)} &\text{ for } [it]=[1t] \text{ and } t\geq2,\\
e_{2}^{(1)} e_{2}^{(2)} &\text{ for } [it]=[2t] \text{ and } t\geq2.
\end{cases}$$
=\[circle,fill=blue!20,draw,minimum size=25pt,font=**\] =\[circle,fill=green!20,draw,minimum size=25pt,font=**\] =\[circle,fill=red!20,draw,minimum size=25pt,font=**\] =\[circle,minimum size=22pt\] =\[blue,fill=white,font=**\] =\[red,fill=white,font=**\] =\[green,fill=white,font=**\] =\[black,fill=white,font=**\] =\[brown,fill=white,font=**\] =\[font=**\]******************
\(1) [$s$]{}; (2) \[right of=1\] [$v$]{}; (3) \[right of=2\] [$d$]{};
(source)\[\] to node\[above\] node\[below\] (main); (source)\[out=40,in=140\] to node\[above\] node\[below\] (main); (source)\[out=-40,in=-140\] to node\[above\] node\[below\] (main);
(main)\[out=40,in=140\] to node\[above\] node\[below\] (dest); (main)\[out=-40,in=-140\] to node\[above\] node\[below\] (dest);
at (5.3,-0.6) [${}_{[11]}$]{}; at (3.6,-0.6) [${}_{[21]}$]{};
at (12,0) [$t=2$]{};
\(1) [$s$]{}; (2) \[right of=1\] [$v$]{}; (3) \[right of=2\] [$d$]{};
(source)\[\] to node\[above\] node\[below\] (main); (source)\[out=40,in=140\] to node\[above\] node\[below\] (main); (source)\[out=-40,in=-140\] to node\[above\] node\[below\] (main);
(main)\[out=40,in=140\] to node\[above\] node\[below\] (dest); (main)\[out=-40,in=-140\] to node\[above\] node\[below\] (dest);
at (10,0) [${}_{[11]}$]{}; at (5.3,0.7) [${}_{[21]}$]{}; at (3.6,-0.6) [${}_{[12]}$]{}; at (2.3,0) [${}_{[22]}$]{};
at (12,0) [$t=3$]{};
\(1) [$s$]{}; (2) \[right of=1\] [$v$]{}; (3) \[right of=2\] [$d$]{};
(source)\[\] to node\[above\] node\[below\] (main); (source)\[out=40,in=140\] to node\[above\] node\[below\] (main); (source)\[out=-40,in=-140\] to node\[above\] node\[below\] (main);
(main)\[out=40,in=140\] to node\[above\] node\[below\] (dest); (main)\[out=-40,in=-140\] to node\[above\] node\[below\] (dest);
at (10,0) [${}_{[21]}$]{}; at (5.3,-0.6) [${}_{[12]}$]{}; at (5.3,0.6) [${}_{[22]}$]{}; at (3.6,-0.6) [${}_{[13]}$]{}; at (2.3,0) [${}_{[23]}$]{};
at (12,0) [$t=4$]{};
\(1) [$s$]{}; (2) \[right of=1\] [$v$]{}; (3) \[right of=2\] [$d$]{};
(source)\[\] to node\[above\] node\[below\] (main); (source)\[out=40,in=140\] to node\[above\] node\[below\] (main); (source)\[out=-40,in=-140\] to node\[above\] node\[below\] (main);
(main)\[out=40,in=140\] to node\[above\] node\[below\] (dest); (main)\[out=-40,in=-140\] to node\[above\] node\[below\] (dest);
at (10.3,0) [${}_{[22]\textcolor{red}{[12]}}$]{}; at (5.3,-0.6) [${}_{[13]}$]{}; at (5.3,0.6) [${}_{[23]}$]{}; at (3.6,-0.6) [${}_{[14]}$]{}; at (2.3,0) [${}_{[24]}$]{};
at (12,0) [$t=5$]{};
\(1) [$s$]{}; (2) \[right of=1\] [$v$]{}; (3) \[right of=2\] [$d$]{};
(source)\[\] to node\[above\] node\[below\] (main); (source)\[out=40,in=140\] to node\[above\] node\[below\] (main); (source)\[out=-40,in=-140\] to node\[above\] node\[below\] (main);
(main)\[out=40,in=140\] to node\[above\] node\[below\] (dest); (main)\[out=-40,in=-140\] to node\[above\] node\[below\] (dest);
at (10.3,0) [${}_{[23]\textcolor{red}{[13]}}$]{}; at (5.3,-0.6) [${}_{[14]}$]{}; at (5.3,0.6) [${}_{[24]}$]{}; at (3.6,-0.6) [${}_{[15]}$]{}; at (2.3,0) [${}_{[25]}$]{};
at (12,0) [$t=6$]{};
Figure \[fi:chainofparallel1steq\] shows that this is an equilibrium. The first player $[11]$ takes the fastest route $e_{1}^{(1)} e_{1}^{(2)}$. The second player $[21]$ cannot pay less than a total cost of 3. She does so by taking $e_{1}^{(1)}$ first and queuing after $[11]$ (a cost of 2), then taking $e_{2}^{(2)}$. This choice of the first generation leaves a queue of size 1 on edge $e_{1}^{(1)} $ for the next generation. The next two players have to pay at least 3 each. They do so by choosing $e_{1}^{(1)} e_{1}^{(2)}$ and $e_{2}^{(1)} e_{2}^{(2)}$. The queue on edge $e_{1}^{(1)} $ is thus re-created for the next generation.
The same average total latency of 6 can be achieved with the following periodic equilibrium strategy profile (see Figure \[fi:chainofparallel2ndeq\]).
$$\label{eq:2ndequilibriumSM}
\tilde{\sigma}_{it}^{{\,\mathsf{Eq}}}=
\begin{cases}
e_{1}^{(1)} e_{1}^{(2)} &\text{ for } [it]=[1t] \text{ and } t \text{ odd},\\
e_{1}^{(1)} e_{2}^{(2)} &\text{ for } [it]=[2t] \text{ and } t \text{ odd},\\
e_{2}^{(1)} e_{1}^{(2)} &\text{ for } [it]=[1t] \text{ and } t \text{ even},\\
e_{3}^{(1)} e_{2}^{(2)} &\text{ for } [it]=[2t] \text{ and } t \text{ even}.
\end{cases}$$
=\[circle,fill=blue!20,draw,minimum size=25pt,font=**\] =\[circle,fill=green!20,draw,minimum size=25pt,font=**\] =\[circle,fill=red!20,draw,minimum size=25pt,font=**\] =\[circle,minimum size=22pt\] =\[blue,fill=white,font=**\] =\[red,fill=white,font=**\] =\[green,fill=white,font=**\] =\[black,fill=white,font=**\] =\[font=**\]****************
\(1) [$s$]{}; (2) \[right of=1\] [$v$]{}; (3) \[right of=2\] [$d$]{};
(source)\[\] to node\[above\] node\[below\] (main); (source)\[out=40,in=140\] to node\[above\] node\[below\] (main); (source)\[out=-40,in=-140\] to node\[above\] node\[below\] (main);
(main)\[out=40,in=140\] to node\[above\] node\[below\] (dest); (main)\[out=-40,in=-140\] to node\[above\] node\[below\] (dest);
at (5.3,-0.6) [${}_{[11]}$]{}; at (3.6,-0.6) [${}_{[21]}$]{};
at (12,0) [$t=2$]{};
\(1) [$s$]{}; (2) \[right of=1\] [$v$]{}; (3) \[right of=2\] [$d$]{};
(source)\[\] to node\[above\] node\[below\] (main); (source)\[out=40,in=140\] to node\[above\] node\[below\] (main); (source)\[out=-40,in=-140\] to node\[above\] node\[below\] (main);
(main)\[out=40,in=140\] to node\[above\] node\[below\] (dest); (main)\[out=-40,in=-140\] to node\[above\] node\[below\] (dest);
at (10,0) [${}_{[11]}$]{}; at (5.3,0.7) [${}_{[21]}$]{}; at (2.3,0) [${}_{[12]}$]{}; at (2.3,1) [${}_{[22]}$]{};
at (12,0) [$t=3$]{};
\(1) [$s$]{}; (2) \[right of=1\] [$v$]{}; (3) \[right of=2\] [$d$]{};
(source)\[\] to node\[above\] node\[below\] (main); (source)\[out=40,in=140\] to node\[above\] node\[below\] (main); (source)\[out=-40,in=-140\] to node\[above\] node\[below\] (main);
(main)\[out=40,in=140\] to node\[above\] node\[below\] (dest); (main)\[out=-40,in=-140\] to node\[above\] node\[below\] (dest);
at (5.5,-0.6) [${}_{\textcolor{red}{[13]}[12]}$]{}; at (10,0) [${}_{[21]}$]{}; at (5.3,0.7) [${}_{[22]}$]{}; at (3.6,-0.6) [${}_{[13]}$]{}; at (3.6,-0.6) [${}_{[23]}$]{};
at (12,0) [$t=4$]{};
\(1) [$s$]{}; (2) \[right of=1\] [$v$]{}; (3) \[right of=2\] [$d$]{};
(source)\[\] to node\[above\] node\[below\] (main); (source)\[out=40,in=140\] to node\[above\] node\[below\] (main); (source)\[out=-40,in=-140\] to node\[above\] node\[below\] (main);
(main)\[out=40,in=140\] to node\[above\] node\[below\] (dest); (main)\[out=-40,in=-140\] to node\[above\] node\[below\] (dest);
at (10.3,0) [${}_{[22]\textcolor{green}{[12]}}$]{}; at (8.1,-0.7) [${}_{[13]}$]{}; at (5.3,0.7) [${}_{[23]}$]{}; at (2.3,0) [${}_{[14]}$]{}; at (2.3,1) [${}_{[24]}$]{};
at (12,0) [$t=5$]{};
\(1) [$s$]{}; (2) \[right of=1\] [$v$]{}; (3) \[right of=2\] [$d$]{};
(source)\[\] to node\[above\] node\[below\] (main); (source)\[out=40,in=140\] to node\[above\] node\[below\] (main); (source)\[out=-40,in=-140\] to node\[above\] node\[below\] (main);
(main)\[out=40,in=140\] to node\[above\] node\[below\] (dest); (main)\[out=-40,in=-140\] to node\[above\] node\[below\] (dest);
at (10.3,0) [${}_{[23]\textcolor{red}{[13]}}$]{}; at (5.5,-0.6) [${}_{\textcolor{red}{[15]}[14]}$]{}; at (5.3,0.7) [${}_{[24]}$]{}; at (3.6,-0.6) [${}_{[25]}$]{};
at (12,0) [$t=6$]{};
\(1) [$s$]{}; (2) \[right of=1\] [$v$]{}; (3) \[right of=2\] [$d$]{};
(source)\[\] to node\[above\] node\[below\] (main); (source)\[out=40,in=140\] to node\[above\] node\[below\] (main); (source)\[out=-40,in=-140\] to node\[above\] node\[below\] (main);
(main)\[out=40,in=140\] to node\[above\] node\[below\] (dest); (main)\[out=-40,in=-140\] to node\[above\] node\[below\] (dest);
at (10.3,0) [${}_{[24]\textcolor{green}{[14]}}$]{}; at (8.1,-0.7) [${}_{[15]}$]{}; at (5.3,0.7) [${}_{[25]}$]{}; at (2.3,0) [${}_{[16]}$]{}; at (2.3,1) [${}_{[26]}$]{};
at (12,0) [$t=7$]{};
Under this profile, the second player of each odd generation creates a queue on $e_{1}^{(1)}$. As both players of the even generation take a long route, none of these two players waits in a queue and thus the queue on $e_{1}^{(1)}$ disappears. Since the first player in the following odd generation uses the fast route $e_{1}^{(1)}$ again, she arrives at $v$ at the same time as the previous two players. Therefore, she waits in the queue on $e_{1}^{(2)}$. So, the first player of each odd generation waits in the queue on $e_{1}^{(2)}$, and the second player waits on $e_{1}^{(1)}$ (except for the very first player).
Section \[se:anarchy\] {#sectionseanarchy .unnumbered}
----------------------
*Details of Example \[ex:strictineq\].* Consider the series-parallel network in Figure \[fi:seriesparallelSM\] where the associated free-flow transit costs and capacities are given. The network has two minimum cuts $\{e_{1},e_{4}\}$ and $\{e_{2},e_{3}, e_{4}\}$ with a capacity of 3, and each edge is part of one cut.
(source) at (0,0) [$s$]{}; (main) at (4,0) [$v$]{}; (dest) at (8,0) [$d$]{}; (source)\[out=60,in=120\] to node\[above\][$\tau_{4}=1$]{} node\[below\][$\gamma_{4}=1$]{} (dest); (source) to node\[above\][$\tau_{1}=0$]{} node\[below\][$\gamma_{1}=2$]{} (main); (main)\[out=30,in=150\] to node\[above\][$\tau_{3}=1$]{} node\[below\][$\gamma_{3}=1$]{} (dest); (main)\[out=-30,in=-150\] to node\[above\][$\tau_{2}=0$]{} node\[below\][$\gamma_{2}=1$]{} (dest);
Consider the following equilibrium strategy. $$\label{eq:equilMackoSM}
\sigma_{it}^{{\,\mathsf{Eq}}}=
\begin{cases}
e_{1} e_{2} &\text{ for } [it]=[11],\\
e_{1} e_{3} &\text{ for } [it]=[21],\\
e_{1} e_{2} &\text{ for } [it]=[31],\\
e_{1} e_{2} &\text{ for } [it]=[1t], t\geq2,\\
e_{4} &\text{ for } [it]=[2t], t\geq2,\\
e_{1} e_{3} &\text{ for } [it]=[3t], t\geq2.
\end{cases}$$ To verify that this is indeed an equilibrium, the reader is referred to Figures \[fi:seriesparallelcolorcode\] and \[fi:Macko\].
=\[circle,fill=blue!20,draw,minimum size=25pt,font=**\] =\[circle,fill=green!20,draw,minimum size=25pt,font=**\] =\[circle,fill=red!20,draw,minimum size=25pt,font=**\] =\[circle,minimum size=22pt\] =\[blue,fill=white,font=**\] =\[red,fill=white,font=**\] =\[black,fill=white,font=**\] =\[font=**\]**************
(source) at (0,0) [$s$]{}; (main) at (4,0) [$v$]{}; (dest) at (8,0) [$d$]{}; (source plus) at (0,0.1) ; (source minus) at (0,-0.1) ; (main plus) at (4,0.1) ; (main minus) at (4,-0.1) ;
(source)\[red\] to node\[above\] node\[below\] (main); (main)\[red, out=-30,in=-150\] to node\[above\] node\[below\] (dest); at (12,0) [$e_{1},e_{2}$]{};
(source) at (0,0) [$s$]{}; (main) at (4,0) [$v$]{}; (dest) at (8,0) [$d$]{}; (source plus) at (0,0.1) ; (source minus) at (0,-0.1) ; (main plus) at (4,0.1) ; (main minus) at (4,-0.1) ;
(source)\[blue\] to node\[above\] node\[below\] (main);
(main)\[blue, out=30,in=150\] to node\[above\] node\[below\] (dest);
at (12,0) [$e_{1},e_{3}$]{};
(source) at (0,0) [$s$]{}; (main) at (4,0) [$v$]{}; (dest) at (8,0) [$d$]{}; (source plus) at (0,0.1) ; (source minus) at (0,-0.1) ; (main plus) at (4,0.1) ; (main minus) at (4,-0.1) ;
(source)\[out=60,in=120\] to node\[above\] node\[below\] (dest);
at (12,0) [$e_{4}$]{};
=\[circle,fill=blue!20,draw,minimum size=25pt,font=**\] =\[circle,fill=green!20,draw,minimum size=25pt,font=**\] =\[circle,fill=red!20,draw,minimum size=25pt,font=**\] =\[circle,minimum size=22pt\] =\[blue,fill=white,font=**\] =\[red,fill=white,font=**\] =\[black,fill=white,font=**\] =\[font=**\]**************
(source) at (0,0) [$s$]{}; (main) at (4,0) [$v$]{}; (dest) at (8,0) [$d$]{}; (source)\[out=60,in=120\] to node\[above\] node\[below\] (dest); (source) to node\[above\] node\[below\] (main); (main)\[out=30,in=150\] to node\[above\] node\[below\] (dest); (main)\[out=-30,in=-150\] to node\[above\] node\[below\] (dest); at (3,0) [${}_{[31]}$]{}; at (4.8,.5) [${}_{[21]}$]{}; at (9,0) [${}_{[11]}$]{}; at (12,0) [$t=1$]{};
(source) at (0,0) [$s$]{}; (main) at (4,0) [$v$]{}; (dest) at (8,0) [$d$]{}; (source)\[out=60,in=120\] to node\[above\] node\[below\] (dest); (source) to node\[above\] node\[below\] (main); (main)\[out=30,in=150\] to node\[above\] node\[below\] (dest); (main)\[out=-30,in=-150\] to node\[above\] node\[below\] (dest); at (3,0) [${}_{[32]}$]{}; at (0.7,1) [${}_{[22]}$]{}; at (7,-.5) [${}_{[12]}$]{}; at (9.2,0) [${}_{\textcolor{red}{[31]},\textcolor{blue}{[21]}}$]{}; at (12,0) [$t=2$]{};
(source) at (0,0) [$s$]{}; (main) at (4,0) [$v$]{}; (dest) at (8,0) [$d$]{}; (source)\[out=60,in=120\] to node\[above\] node\[below\] (dest); (source) to node\[above\] node\[below\] (main); (main)\[out=30,in=150\] to node\[above\] node\[below\] (dest); (main)\[out=-30,in=-150\] to node\[above\] node\[below\] (dest); at (3,0) [${}_{[33]}$]{}; at (0.7,1) [${}_{[23]}$]{}; at (7,-.5) [${}_{[13]}$]{}; at (5,.5) [${}_{[32]}$]{}; at (9.2,0) [${}_{[22],\textcolor{red}{[12]}}$]{}; at (12,0) [$t=3$]{};
(source) at (0,0) [$s$]{}; (main) at (4,0) [$v$]{}; (dest) at (8,0) [$d$]{}; (source)\[out=60,in=120\] to node\[above\] node\[below\] (dest); (source) to node\[above\] node\[below\] (main); (main)\[out=30,in=150\] to node\[above\] node\[below\] (dest); (main)\[out=-30,in=-150\] to node\[above\] node\[below\] (dest); at (3,0) [${}_{[34]}$]{}; at (0.7,1) [${}_{[24]}$]{}; at (7,-.5) [${}_{[14]}$]{}; at (5,.5) [${}_{[33]}$]{}; at (9.5,0) [${}_{[23],\textcolor{red}{[13]},\textcolor{blue}{[32]}}$]{}; at (12,0) [$t=4$]{};
(source) at (0,0) [$s$]{}; (main) at (4,0) [$v$]{}; (dest) at (8,0) [$d$]{}; (source)\[out=60,in=120\] to node\[above\] node\[below\] (dest); (source) to node\[above\] node\[below\] (main); (main)\[out=30,in=150\] to node\[above\] node\[below\] (dest); (main)\[out=-30,in=-150\] to node\[above\] node\[below\] (dest); at (3,0) [${}_{[35]}$]{}; at (0.7,1) [${}_{[25]}$]{}; at (7,-.5) [${}_{[15]}$]{}; at (5,.5) [${}_{[34]}$]{}; at (9.5,0) [${}_{[24],\textcolor{red}{[14]},\textcolor{blue}{[33]}}$]{}; at (12,0) [$t=5$]{};
*Details of Example \[ex:Wheatstone\].* Consider the Wheatstone network in Figure \[fi:WheatstoneSM\] with associated free-flow transit costs and capacity equal to $1$ for all edges. The capacity of the network is $2$.
\(1) at (0,3) [$s$]{}; (2) at (-2.5,0) [$v$]{}; (3) at (0,-3) [$d$]{}; (4) at (2.5,0) [$w$]{};
\(1) edge node \[right\] [$\tau_{2}=1$]{} (4) edge node\[left,color=black\] [$\tau_{1}=0$]{} (2) (2) edge node \[left\] [$\tau_{4}=1$]{} (3) edge node [$\tau_{3}=0$]{} (4) (4) edge node \[right,color=black\] [$\tau_{5}=0$]{} (3);
=\[circle,fill=blue!20,draw,minimum size=25pt,font=**\] =\[circle,fill=green!20,draw,minimum size=25pt,font=**\] =\[circle,fill=red!20,draw,minimum size=25pt,font=**\] =\[circle,minimum size=22pt\] =\[blue,fill=white,font=**\] =\[red,fill=white,font=**\] =\[black,fill=white,font=**\] =\[font=**\]**************
\(1) at (0,3) [$s$]{}; (2) at (-2.5,0) [$v$]{}; (3) at (0,-3) [$d$]{}; (4) at (2.5,0) [$w$]{}; (5) at (-.2,3.1) ; (6) at (0,3) ; (7) at (-2.7,0.1) ; (8) at (-2.5,0) ; (9) at (-.1,-3) ; (10) at (.1,-3.1) ; (11) at (2.6,-.1) ; (12) at (2.4,0) ;
(1)\[\] to (2);
(2)\[\] to (3);
at (0,-4.5) [$e_{1},e_{4}$]{}; at (0,-5) ;
\(1) at (0,3) [$s$]{}; (2) at (-2.5,0) [$v$]{}; (3) at (0,-3) [$d$]{}; (4) at (2.5,0) [$w$]{}; (5) at (-.2,3.1) ; (6) at (0,3) ; (7) at (-2.7,0.1) ; (8) at (-2.5,0) ; (9) at (-.1,-3) ; (10) at (.1,-3.1) ; (11) at (2.6,-.1) ; (12) at (2.4,0) ;
(1)\[red\] to (2); (2)\[red\] to (4); (4)\[red\] to (3);
at (0,-4.5) [$e_{1},e_{3},e_{5}$]{}; at (0,-5) ;
\(1) at (0,3) [$s$]{}; (2) at (-2.5,0) [$v$]{}; (3) at (0,-3) [$d$]{}; (4) at (2.5,0) [$w$]{}; (5) at (-.2,3.1) ; (6) at (0,3) ; (7) at (-2.7,0.1) ; (8) at (-2.5,0) ; (9) at (-.1,-3) ; (10) at (.1,-3.1) ; (11) at (2.6,-.1) ; (12) at (2.4,0) ;
(1)\[blue\] to (4); (4)\[blue\] to (3);
at (0,-4.5) [$e_{2},e_{5}$]{}; at (0,-5) ;
Consider the following equilibrium strategy $$\label{eq:WheatstoneequilSM}
\sigma_{it}^{{\,\mathsf{Eq}}}=\begin{cases}
e_{1} e_{3} e_{5} &\text{ for } [it]=[i1],i=1,2,\\
e_{1} e_{3} e_{5} &\text{ for } [it]=[12],\\
e_{2} e_{5} &\text{ for } [it]=[22],\\
e_{1} e_{3} e_{5} &\text{ for } [it]=[13],\\
e_{1} e_{4} &\text{ for } [it]=[23],\\
e_{2} e_{5} &\text{ for } [it]=[14],\\
e_{1} e_{3} e_{5} &\text{ for } [it]=[24],\\
e_{1} e_{4} &\text{ for } [it]=[1t],t\geq5,\\
e_{2} e_{5} &\text{ for } [it]=[2t],t\geq5.
\end{cases}$$ We refer to Figure \[fi:Wheatstoneequilibrium\] to check that this is indeed an equilibrium.
=\[circle,fill=blue!20,draw,minimum size=25pt,font=**\] =\[circle,fill=green!20,draw,minimum size=25pt,font=**\] =\[circle,fill=red!20,draw,minimum size=25pt,font=**\] =\[circle,minimum size=22pt\] =\[blue,fill=white,font=**\] =\[red,fill=white,font=**\] =\[black,fill=white,font=**\] =\[font=**\]**************
\(1) at (0,3) [$s$]{}; (2) at (-2.5,0) [$v$]{}; (3) at (0,-3) [$d$]{}; (4) at (2.5,0) [$w$]{};
\(1) edge node \[right\] (4) edge node\[left,color=black\] (2) (2) edge node \[left\] (3) edge node (4) (4) edge node \[right,color=black\] (3);
at (0,-4) [${}_{[11]}$]{}; at (-1.9,.9) [${}_{[21]}$]{}; at (0,-4.5) [$t=1$]{}; at (0,-5) ;
\(1) at (0,3) [$s$]{}; (2) at (-2.5,0) [$v$]{}; (3) at (0,-3) [$d$]{}; (4) at (2.5,0) [$w$]{};
\(1) edge node \[right\] (4) edge node\[left,color=black\] (2) (2) edge node \[left\] (3) edge node (4) (4) edge node \[right,color=black\] (3);
at (.8,2.1) [${}_{[22]}$]{}; at (-1.9,.9) [${}_{[12]}$]{}; at (0,-4) [${}_{[21]}$]{}; at (0,-4.5) [$t=2$]{}; at (0,-5) ;
\(1) at (0,3) [$s$]{}; (2) at (-2.5,0) [$v$]{}; (3) at (0,-3) [$d$]{}; (4) at (2.5,0) [$w$]{};
\(1) edge node \[right\] (4) edge node\[left,color=black\] (2) (2) edge node \[left\] (3) edge node (4) (4) edge node \[right,color=black\] (3);
at (-1.9,.9) [${}_{[23],\textcolor{red}{[13]}}$]{}; at (.7,-2.1) [${}_{[22]}$]{}; at (0,-4) [${}_{[12]}$]{}; at (0,-4.5) [$t=3$]{}; at (0,-5) ;
\(1) at (0,3) [$s$]{}; (2) at (-2.5,0) [$v$]{}; (3) at (0,-3) [$d$]{}; (4) at (2.5,0) [$w$]{};
\(1) edge node \[right\] (4) edge node\[left,color=black\] (2) (2) edge node \[left\] (3) edge node (4) (4) edge node \[right,color=black\] (3);
at (.8,2.1) [${}_{[14]}$]{}; at (-1.9,.9) [${}_{[24],\textcolor{black}{[23]}}$]{}; at (.7,-2.1) [${}_{[13]}$]{}; at (0,-4) [${}_{[22]}$]{}; at (0,-4.5) [$t=4$]{}; at (0,-5) ;
\(1) at (0,3) [$s$]{}; (2) at (-2.5,0) [$v$]{}; (3) at (0,-3) [$d$]{}; (4) at (2.5,0) [$w$]{};
\(1) edge node \[right\] (4) edge node\[left,color=black\] (2) (2) edge node \[left\] (3) edge node (4) (4) edge node \[right,color=black\] (3);
at (.8,2.1) [${}_{[25]}$]{}; at (-1.9,.9) [${}_{[15],\textcolor{red}{[24]}}$]{}; at (-1.9,-.9) [${}_{[23]}$]{}; at (.7,-2.1) [${}_{[14]}$]{}; at (0,-4) [${}_{[13]}$]{}; at (0,-4.5) [$t=5$]{}; at (0,-5) ;
\(1) at (0,3) [$s$]{}; (2) at (-2.5,0) [$v$]{}; (3) at (0,-3) [$d$]{}; (4) at (2.5,0) [$w$]{};
\(1) edge node \[right\] (4) edge node\[left,color=black\] (2) (2) edge node \[left\] (3) edge node (4) (4) edge node \[right,color=black\] (3);
at (.8,2.1) [${}_{[26]}$]{}; at (-1.9,.9) [${}_{[16],[15]}$]{}; at (.8,-2) [${}_{[25],\textcolor{red}{[24]}}$]{}; at (0,-4) [${}_{\textcolor{blue}{[14]},[23]}$]{}; at (0,-4.5) [$t=6$]{}; at (0,-5) ;
\(1) at (0,3) [$s$]{}; (2) at (-2.5,0) [$v$]{}; (3) at (0,-3) [$d$]{}; (4) at (2.5,0) [$w$]{};
\(1) edge node \[right\] (4) edge node\[left,color=black\] (2) (2) edge node \[left\] (3) edge node (4) (4) edge node \[right,color=black\] (3);
at (.8,2.1) [${}_{[27]}$]{}; at (-1.9,.9) [${}_{[17],[16]}$]{}; at (.8,-2) [${}_{[26],[25]}$]{}; at (-1.9,-.9) [${}_{[15]}$]{}; at (0,-4) [${}_{\textcolor{red}{[24]}}$]{}; at (0,-4.5) [$t=7$]{}; at (0,-5) ;
\(1) at (0,3) [$s$]{}; (2) at (-2.5,0) [$v$]{}; (3) at (0,-3) [$d$]{}; (4) at (2.5,0) [$w$]{};
\(1) edge node \[right\] (4) edge node\[left,color=black\] (2) (2) edge node \[left\] (3) edge node (4) (4) edge node \[right,color=black\] (3);
at (.8,2.1) [${}_{[28]}$]{}; at (-1.9,.9) [${}_{[18],[17]}$]{}; at (.8,-2) [${}_{[27],[26]}$]{}; at (-1.9,-.9) [${}_{[16]}$]{}; at (0,-4) [${}_{\textcolor{blue}{[25]},[15]}$]{}; at (0,-4.5) [$t=8$]{}; at (0,-5) ;
\(1) at (0,3) [$s$]{}; (2) at (-2.5,0) [$v$]{}; (3) at (0,-3) [$d$]{}; (4) at (2.5,0) [$w$]{};
\(1) edge node \[right\] (4) edge node\[left,color=black\] (2) (2) edge node \[left\] (3) edge node (4) (4) edge node \[right,color=black\] (3);
at (.8,2.1) [${}_{[29]}$]{}; at (-1.9,.9) [${}_{[19],[18]}$]{}; at (.8,-2) [${}_{[28],[27]}$]{}; at (-1.9,-.9) [${}_{[17]}$]{}; at (0,-4) [${}_{\textcolor{blue}{[26]},[16]}$]{}; at (0,-4.5) [$t=9$]{}; at (0,-5) ;
For $k\in\mathbb{N}_{+}$, define *Braess’s $k$-th graph* as follows. Since this is just a graph and not a multigraph, edges are uniquely identified by their tail and head. Let $$V^{k}=\{s,v_{1},\ldots,v_{k}, w_{1},\ldots,w_{k},d\}$$ be the set of $2k+2$ vertices and $$E^{k}=\{(s,v_{i}),(v_{i},w_{i}),(w_{i},d)\mid1\leq i\leq k\}\cup \{(v_{i},w_{i-1})\mid2\leq i\leq k\}\cup \{(v_{1},d)\}\cup \{(s,w_{k})\}$$ the set of edges. See Figure \[fi:braess1\].
(source) at (0,4) [$s$]{}; (left) at (-5,0) [$v_{1}$]{}; (upl) at (-2,1) [$v_{2}$]{}; (upr) at (2,1) [$v_{k}$]{}; (right) at (5,0) [$w_{k}$]{}; (downl) at (-2,-1) [$w_{1}$]{}; (downr) at (2,-1) [$w_{k-1}$]{}; (sink) at (0,-4) [$d$]{}; at (0,1) […]{}; at (0,-1) […]{}; (source) to (left); (source) to (right); (source) to (upl); (source) to (upr); (left) to (downl); (left) to (sink); (right) to (sink); (upl) to (downl); (upr) to (downr); (upr) to (right); (upl) to (-.5,-.7); (.5,.7) to (downr); (downl) to (sink); (downr) to (sink);
Let $\gamma_{e}=1$ for all $e\in E^{k}$ and $$\begin{aligned}
\tau_{e}&=\begin{cases}
1 &\text{ if }e=(v_{1},d)\text{, }e=(s,w_{k})\text{ or }e=(v_{i},w_{i-1})\text{ for }2\leq i\leq k,\\
0 &\text{ otherwise,}
\end{cases}\end{aligned}$$ Notice that Braess’s $k$-th graph has a capacity $k+1$.
For $i=1,\ldots,k$, let $P_{i}$ denote the path $(s,v_{i})(v_{i},w_{i})(w_{i},d)$. Let $Q_{1}$ denote the path $(s,v_{1})(v_{1},d)$, for $i=2,\ldots,k$ let $Q_{i}$ denote the path $(s,v_{i})(v_{i},w_{i-1})(w_{i-1},d)$ and let $Q_{k+1}$ denote the path $(s,w_{k})(w_{k},d)$.
The optimal latency is achieved by the strategy profile in which each player of every generation chooses a different path $Q_{i}$ for $i=1,\ldots,k+1$. Hence $${\operatorname{\mathsf{Opt}}}(\mathcal{N},\gamma)=k+1.$$
Consider the subnetwork $\mathcal{N}'$ obtained from $\mathcal{N}$ by deleting each edge $(v_{i},w_{i})$ for $i=1,\ldots,k$. Observe that $\mathcal{N}'$ is a parallel network and the unique equilibrium latency is achieved by the strategy profile in which each player of every generation chooses a different path $Q_{i}$ for $i=1,\ldots,k+1$. Hence $${\operatorname{\mathsf{WEq}}}(\mathcal{N}',\gamma)=k+1.$$
Now, on $\mathcal{N}$ the best equilibrium latency is achieved by the following strategy profile.
(i) In the $j$-th period, where $1\leq j\leq k$, the first $k+1-j$ players choose path $P_{i}$ for $i=1,\ldots,k$ in increasing order and players $k+2-j$, …, $k+1$ choose path $Q_{k+2-j}$, …, $Q_{k+1}$, respectively.
(ii) From period $k+1$ onwards, each player chooses a path $Q_{i}$ for $i=1,\ldots,k+1$ in increasing order.
Since no queue is created in any of the periods, $${\operatorname{\mathsf{BEq}}}(\mathcal{N},\gamma)=k+1.$$
The worst equilibrium latency is achieved by the following strategy profile. For all $[it]\in G$, choose a path $p_{it}$ with minimum latency that has no possibility of overtaking, according to the following preference relation over paths $P_{1}\succ\ldots\succ P_{k}\succ Q_{1}\succ Q_{k+1}\succ Q_{2}\succ\ldots\succ Q_{k}$. The idea is that the players in the transient states create queues on each $P_{i}$ for $i=1,\ldots,k$, in such a way that in the steady state, in each generation exactly one player chooses the path $Q_{i}$ for $i=1,\ldots,k+1$, with a latency of $2k+1$. For $k=1$, this strategy profile is illustrated in Example \[ex:Wheatstone\]. For $k\geq2$, queues grow as follows.
(i) In the first $k$ periods, each player chooses a path $P_{i}$ for $i=1,\ldots,k$ in increasing order such that after $k$ periods, each edge $(s,v_{i})$ for $i=1,\ldots,k$ has a waiting cost of one.
(ii) Partition the following $(k-1)\cdot2k$ periods into $k-1$ sets of $2k$ periods. Each set of $2k$ periods consists of two subsets of $k$ periods such that players in the second $k$ periods choose the same routes as players in the first $k$ periods. During the $j$-th set of $2k$ periods, where $1\leq j\leq k-1$, all players in the first $k-j$ periods choose a path $P_{i}$ for $i=1,\ldots,k$ in increasing order and create a queue on $(s,v_{i})$ for $i=1,\ldots,k-j$. In the next $j$ periods, the UFR property implies that a path $P_{i}$ is replaced by a path $Q_{i+1}$ for $i=k,\ldots,k+1-j$ such that a queue grows on $(w_{i},d)$ instead of on $(s,v_{i})$.
So during the $j$-th set of $2k$ periods, where $1\leq j\leq k-1$, the queue on each edge $(s,v_{i})$ for $i=1,\ldots,k-j$ has increased by two, and the queue on $(w_{i},d)$ for $i=k,\ldots,k+1-j$ has increased by two.
(iii) After $(k-1)\cdot2k+k$ periods, the length of the queue on each edge $(s,v_{i})$ for $i=1,\ldots,k$ is $2\cdot(k-i)+1$, and the length of the queue on each edge $(w_{i},d)$ for $i=2,\ldots,k$ is $2\cdot(i-1)$. The UFR property implies that in the following $k$ periods a queue grows on each edge $(w_{i},d)$ for $i=1,\ldots,k$.
(iv) The subsequent $k+1$ periods are summarized as follows. First, a queue grows on $(s,v_{1})$, then a queue grows on $(s,v_{i})$ and $(w_{i-1},d)$ for $i=2,\ldots,k$, finally a queue grows on $(w_{k},d)$. Summarizing, each path $P_{i}$ for $i=1,\ldots,k$ has a latency of $2k+2$ and each path $Q_{i}$ for $i=1,\ldots,k+1$ has a latency of $2k+1$.
(v) In all of the upcoming periods, each player chooses a different path $Q_{i}$ for $i=1,\ldots,k+1$. The UFR property guarantees that queues cannot grow any further.
Hence $${\operatorname{\mathsf{WEq}}}(\mathcal{N},\gamma)=(k+1)\cdot(2k+1).$$
Concluding, we found that $$\begin{aligned}
{\operatorname{\mathsf{PoS}}}(\mathcal{N},\gamma)&=\frac{k+1}{k+1}=1,\\
{\operatorname{\mathsf{PoA}}}(\mathcal{N},\gamma)&={\,\mathsf{BR}}(\mathcal{N},\gamma)=\frac{(k+1)\cdot(2k+1)}{(k+1)}=2k+1=n-1. \qedhere\end{aligned}$$
For the $k$-th Braess’s graph, the Nash latency of the non-atomic game is achieved by the following strategy profile. First, a queue of length one is created on each $(s,v_{i})$ for $i=1,\ldots,k$. Then congestion occurs on each $(s,v_{i})$ for $i=1,\ldots,k-1$ and $(w_{k},d)$ until there is an additional queue of one. This process, where more queues grow on $(w_{i},d)$ instead of on $(s,v_{i})$ for $i=1,\ldots,k$, continues until each path has the same latency equal to $k+1$. Hence the non-atomic game has a price of anarchy of $n/2$.
Section \[se:seasonal\] {#sectionseseasonal .unnumbered}
-----------------------
We first prove the formula for the optimum cost, $${\operatorname{\mathsf{Opt}}}(\mathcal{N},K, \boldsymbol{\delta})=K\sum_{e\in E}\gamma_{e}\tau_{e}+
D(\boldsymbol{\delta}).$$ We prove this result by induction on $D( \boldsymbol{\delta})$. For $D( \boldsymbol{\delta})=0$, note that $ \boldsymbol{\delta}= \boldsymbol{\gamma}$ and the result is obvious.
Suppose the result is true for $ \boldsymbol{\delta}'$ with $D( \boldsymbol{\delta}')\in\mathbb{N}$ and let $ \boldsymbol{\delta}\rightarrow \boldsymbol{\delta}'$. Since $ \boldsymbol{\delta}\rightarrow \boldsymbol{\delta}'$, there is some $k\in\{1,\ldots,K\}$ such that $\delta_k>\gamma$, $\delta'_k=\delta_k-1$, $\delta'_{k+1}=\delta_{k+1}+1$ and $\delta'_{\ell}=\delta_{\ell}$ for all $\ell\notin\{k,k+1\}$, where $k+1$ is considered modulo $K$.
At each stage $t_k$ such that $t=k \mod K$, players depart above capacity under $\boldsymbol{\delta}$. This implies that there is at least one player $[jt_k]$ who sees a queue on his route, and thus who adds one unit of waiting time to the total cost. Denote $[j^*t_k]$ such a player with the highest index, i.e., the player with the lowest priority. Consider the relaxed optimization problem where the planner postpones the departure of this player by one stage and let her depart as the first player of the next generation, that is, to transform $\boldsymbol{\delta}$ into $\boldsymbol{\delta}'$.
By the choice of $j^*$ (the last one in the generation who sees a queue), the postponing of this player does not affect the costs nor the choices of the other players. This is clear for those who have higher priority. For those who have lower priority, this player will be ahead of them in the queue in both cases. So the choice of strategy for player $[j^*t_k]$ that has to be made by the social planner is the same in $\boldsymbol{\delta}$ as in $\boldsymbol{\delta}'$. Hence all players, including player $[j^*t_k]$, choose the same strategy, and thus in each period, one unit of waiting cost is saved by postponing the departure of player $[j^*t_k]$. This concludes the proof.
We now turn to the proof of the formula for the equilibrium. We start by showing some preliminary results. The first claim shows that for computing equilibrium costs, without loss of generality all capacities can be assumed to be $1$.
Let $\mathcal{N}=(\mathcal{G}, (\tau_e)_{e\in E}, (\gamma_e)_{e\in E})$ be a network. Given $e\in E$, let $\mathcal{N}^{e}$ be the network obtained from $\mathcal{N}$ by replacing the edge $e$ of capacity $\gamma_e$, by a set $E(e)$ of $\gamma_e$ parallel edges of capacity 1, with the same head and tail, and same length as $e$.
\[cl:cap1\] Every equilibrium of $\mathcal{N}$ (resp. $\mathcal{N}^{e}$) can be mapped to an equilibrium of $\mathcal{N}^{e}$ (resp. $\mathcal{N}$) with the same total cost.
We index the $\gamma_e$ edges $E(e)$ by the integers $\{1,\dots,\gamma_e\}$. We consider an equilibrium $\sigma$ of $\mathcal{N}$ and construct an equilibrium of $\mathcal{N}^{e}$ with the same total cost.
Suppose that under $\sigma$, at generation $t$, $n$ players numbered $[i_1t],\dots,[i_nt]$ enter edge $e$.
First assume that there is no initial queue on $e$ at the beginning of stage $t$. For each $k=1,\dots, n$, assign player $[i_kt]$ to the $q$-th edge if $k=q\mod\gamma_e$. In words, take those $n$ players and assign them to the edges according to their priority, following the numbering of edges: player $[i_1t]$ is assigned to edge 1, …, player $[i_kt]$ is assigned to edge $k$ for $k\leq \gamma_e$. If $n>\gamma_e$, then player $[i_{\gamma_{e}+1}t]$ is assigned to edge 1, and so on. By construction, both ways, player $[i_kt]$ will queue on $e$ if $k>\gamma_e$, and if $k=w\gamma_e+r$ ($w, r$ integers, $r<\gamma_e$), player $[i_kt]$ will queue for $w-1$ units of time. Therefore the total cost paid by this player on $e$ is the same in both cases. This defines an equilibrium for those players: there is no point in deviating to a route feasible in $\mathcal{N}$, as it would imply a profitable deviation from $\sigma$. By construction, each player is assigned to an edge of $E(e)$ which is fastest, given the priorities.
With this construction, the queues left by this generation to the next one has the following structure: there exist $w$ and $q^*\leq \gamma_e$ such that all edges of $E(e)$ numbered $1,\dots, q^*$ have a queue of length $w$, and edges numbered $q^*+1,\dots, \gamma_e$ have a queue of length $w-1$ (if $q^*=\gamma_e$, all queues have the same length).
Second, suppose again that at generation $t$, $n$ players numbered $[i_1t],\dots,[i_nt]$ enter edge $e$ under $\sigma$, but that on $E(e)$, they see queues with the above structure. If $q^*<\gamma_e$, then let the first $\gamma_e-q^*$ players fill the edges numbered $q^*+1,\dots,\gamma_e$ in an orderly fashion, according to priorities. The remaining $n-(\gamma_e-q^*)$ choose edges as in the previous case: the first player chooses the first edge, and so on.
As in the previous case, since the choice of edges in $E(e)$ respects the priorities, the waiting time is the same for each player on both networks. Also for the same reason as in the previous case, this is an equilibrium choice. Note that the above structure of queues is preserved from one generation to the next, so that the analysis can be iterated. We have thus constructed an equilibrium of $\mathcal{N}^{e}$ with the same total cost as $\sigma$.
Conversely, take an equilibrium $\sigma^{e}$ of $\mathcal{N}^{e}$. For each route in $\mathcal{N}^{e}$ that uses an edge $f\in E(e)$, there is a unique corresponding route in $\mathcal{N}$ which uses edge $e$. This maps uniquely the strategy profile $\sigma^{e}$ in a strategy profile $\sigma$ on $\mathcal{N}$. Then $\sigma$ has to be an equilibrium. Actually, a deviation in $\mathcal{N}$ is also feasible in $\mathcal{N}^{e}$, so a profitable deviation from $\sigma$ would imply a profitable deviation from $\sigma^{e}$.
From the fact that $\sigma^{e}$ is an equilibrium of $\mathcal{N}^{e}$, the queues on the edges of $E(e)$ (if at all) must have a structure as above, there is an integer $w$ such that each edge in an non-empty subset of $E(e)$ has a queue of length $w$, and all other edges in $E(e)$ have a queue of length $w-1$. Therefore, the waiting time of a player on edge $e$ is the same as on $E(e)$.
Applying this result iteratively we can transform any network into another where all capacities are one.
\[le:worsteq\] Let $\mathcal{N}$ be a parallel network. In a worst equilibrium of $\Gamma(\mathcal{N},\mathcal{D})$, whenever a player is indifferent between several edges, she chooses one where there is a queue, if there is one.
Using Claim \[cl:cap1\], we assume that all capacities are one. For a parallel network, arriving at intermediary nodes is not an issue. Therefore, each generation of players chooses an equilibrium as in a game where no subsequent generations exist. More precisely, consider a parallel network with edges $e\in E$ and lengths $(\tau_e)_{e\in E}$. Consider the game where at stage 1 a generation of $\delta$ players enter the network and where there are no subsequent players. Denote $W(\delta,(\tau_e)_{e\in E})$ the worst equilibrium total cost of this game.
\[cl:monotonic\] $W(\delta,(\tau_e)_{e\in E})$ is weakly increasing in free-flow transit costs. That is, if, for all $e\in E$, $\tau_e\leq\tau_e'$, then $W(\delta,(\tau_e)_{e\in E})\leq W(\delta,(\tau'_e)_{e\in E})$.
Since we are dealing with just one generation, we denote the players $1,\dots, \delta$. Take an equilibrium $\sigma$ and let $\ell_i(\sigma)$ be the equilibrium latency of player $i$. A simple remark is that $\ell_i(\sigma)$ weakly increases with $i$ and that from one player to the next, it can only increase by one unit. Precisely, there exist integers $1\leq k_1<k_2<\cdots<k_n\leq\delta$ such that
- whenever $1\leq i\leq k_1$, we have $\ell_i(\sigma)=\min_e\tau_e$,
- if $1\leq m<n$, then, whenever $k_m<i\leq k_{m+1}$, we have $\ell_i(\sigma)=\min_e\tau_e+m$.
To see this, note first that if $i<j$, then $\ell_i(\sigma)\leq\ell_j(\sigma)$. Otherwise, player $i$ who has priority over $j$ could profitably imitate $j$. Second, $\ell_{i+1}(\sigma)\leq \ell_i(\sigma)+1$. Otherwise, player $i+1$ could profitably imitate $i$ and pay $\ell_i(\sigma)+1$.
If follows directly that if we increase $\min_e\tau_e$, then the equilibrium costs of all players are pushed (weakly) upwards. Suppose now that we increase by one unit the length of an edge which is used in equilibrium. That is, take an edge $f$ with length $\tau_f=\min_e\tau_e+m$ for some $m$, with $1\leq m<n$ as above, and replace it by an edge with length $\tau_f+1$. In this new situation, we have the same number of players who pay $\tau_f-1$ at most. Among the players who paid $\tau_f$ in the old situation, one has to pay now $\tau_f+1$ (the one with the lowest priority; whether she chooses edge $f$ or another one with the same total cost). Subsequent players have to pay weakly more. So, all costs are weakly pushed upwards.
Consider now a worst equilibrium of $\Gamma(\mathcal{N},\mathcal{D})$. The first generation chooses an equilibrium for the network with lengths $(\tau_e)_{e\in E}$. Let $n_1(e)$ be the number of players of the first generation who choose edge $e$. If $n_1(e)\leq 1$, then the first generation leaves no queue on $e$ for the next one. If $n_1(e)>1$, then the first player in the second generation meets a queue of $r_1(e)=n_1(e)-1$ on edge $e$. Iteratively, let $n_t(e)$ denote the number of players of generation $t$ who choose edge $e$ and $r_t(e)$ the queue that the first player in generation $t+1$ meets on $e$. We have the following recursion for $t>1$, $$r_t(e)=(r_{t-1}(e)+n_t(e)-1)_+,$$ where $x_+=\max\{x,0\}$.
Then, generation $t+1$ chooses an equilibrium for the network $(\tau ^{t+1}_e)_{e\in E}$ with $\tau_e^{t+1}:=\tau_e+r_t(e)$.
Now, suppose that there is a generation $t$, a player $[it]$ and two edges $e,f$ such that player $[it]$ is indifferent between $e$ and $f$ and, there is a queue on $e$ but not on $f$. There are two equilibrium scenarios. In the best scenario (BS) player $[it]$ chooses $f$, in the worst scenario (WS) player $[it]$ chooses $e$. We argue that the queues left for future generations are all weakly higher in WS than in BS.
Consider first the case where player $[it]$ is the last in generation $t$. Then by choosing $f$, she leaves no queue on $f$ for the next generation and the queue on $e$ decreases by one unit. If she chooses $e$, she recreates the queue on $e$, there is still no queue on $f$. For all other edges, the queue is the same under both scenarios.
The second case is when player $[it]$ is not the last in her generation. Let $p=\delta_t-i+1$ be the number of players who come weakly after player $i$ in generation $t$, and let $q$ be the number of edges that have the same total cost as $e$ for player $[it]$. If $p>q$, then one player must choose $e$ and another one must choose $f$, no matter what player $[it]$ does, so the queues are the same under both scenarios. If $p\leq q$, then at most one player will choose $f$ (so no queue is created there) and no player chooses the same edge as $[it]$. Therefore, if she chooses $e$ she maintains the queue there, whereas she creates no queue by choosing $f$.
We conclude that whenever a player is indifferent between queuing or not, choosing the edge with the queue weakly increases all queues for the next generation. From the recursion $r_t(e)=(r_{t-1}(e)+n_t(e)-1)_+$, this weakly increases queues for all future generations. From Claim \[cl:monotonic\], the conclusion follows.
We now turn to the proof of the formula for the equilibrium, $${\operatorname{\mathsf{WEq}}}(\mathcal{N}, K, \boldsymbol{\delta})=K\gamma\max_{e\in E}\tau_{e}+D(\boldsymbol{\delta}).$$
We prove it by induction on $D(\boldsymbol{\delta})$, the result being obvious for $D(\boldsymbol{\delta})=0$. We assume that the result is true for $\boldsymbol{\delta}'$ with $D(\boldsymbol{\delta}')\in\mathbb{N}$ and let $\boldsymbol{\delta}\rightarrow\boldsymbol{\delta}'$. We take $k\in\{1,\ldots,K\}$ such that $\delta_k>\gamma$, $\delta'_k=\delta_k-1$, $\delta'_{k+1}=\delta_{k+1}+1$ and $\delta'_{\ell}=\delta_{\ell}$ for all $\ell\notin\{k,k+1\}$, where $k+1$ is considered modulo $K$.
First, we show that there is an equilibrium for $\Gamma(\mathcal{N},K,\boldsymbol{\delta})$ with costs equal to ${\operatorname{\mathsf{WEq}}}(\mathcal{N},K,\boldsymbol{\delta}')+1$. This implies that ${\operatorname{\mathsf{WEq}}}(\mathcal{N},K,\boldsymbol{\delta})\geq {\operatorname{\mathsf{WEq}}}(\mathcal{N},K,\boldsymbol{\delta}')+1$.
Let $\sigma'$ be the strategy profile as defined for $\Gamma(\mathcal{N},K,\boldsymbol{\delta}')$ yielding the worst equilibrium latency. We construct a strategy profile $\sigma$ for $\Gamma(\mathcal{N},K,\boldsymbol{\delta})$ corresponding to $\sigma'$ such that the same queues are created. The definition of the strategy is iterative. We indicate below how the construction works for one period and how the iteration proceeds to the next.
(I) \[it:algo-I\] Let $k<K$ and $t\in\mathbb{N}$. If $t<k$, then let each player $[it]$ choose the same edge as in $\sigma'$. If $t=k$, then let each player $[it]$ with $i<\delta_k$ choose the same edge as in $\sigma'$.
(1) \[it:algoI-1\] If the edge chosen by player $[1,k+1]$ in $\sigma'$ has minimum latency and waiting costs for player $[\delta_k k]$, then let $[\delta_k k]$ choose this edge and let each player $[it]$ with $t<k+K$ choose the same edge as in $\sigma'$. In this case, bringing forward a player does not affect the choice of the other players, because for them there is no difference whether the player waits a stage in a queue or whether the player waits a stage to depart. From stage $k+K$ onwards, go to \[it:algo-I\] and iterate.
(2) \[it:algoI-2\] If the edge chosen by player $[1,k+1]$ in $\sigma'$ has either no minimum latency or no waiting costs for player $[\delta_k k]$, then let $[\delta_k k]$ choose an edge with minimum latency and no waiting costs (observe that in the former case, the edge has waiting costs and thus there must be a different edge with minimum latency but no waiting costs) and let each player $[i,k+1]$ with $i\leq\delta_{k+1}$ choose the same edge as in $\sigma'$.
(a) \[it:algoI2-a\] Either there is a first generation $G_s$ with $k+1\leq s<k+K$ which has a last player with no waiting costs in $\sigma'$. Let each player $[it]$ with $k+1<t\leq s$ choose the same edge as the player departing before $[it]$ in $\sigma'$ and let each player $[it]$ with $s<t<k+K$ choose the same edge as in $\sigma'$. In this case, $[\delta'_s s]$ does not affect the other players in $\sigma'$. So it is no problem if he does not depart. From stage $k+K$ onwards, go to \[it:algo-I\].
(b) \[it:algoI2-b\] Or all generations $G_t$ with $k+1\leq t<k+K$ have a last player with waiting costs in $\sigma'$, then let each player $[it]$ with $k+1<t\leq k+K$ choose the same edge as the player departing before $[it]$ in $\sigma'$. From stage $k+K+1$ onwards, go to \[it:algo-II\].
(II) \[it:algo-II\] Let $k=K$ and $t\in\mathbb{N}$. Let each player $[i1]$ with $i\leq\delta_k$ choose the same edge as in $\sigma'$.
(1) \[it:algoII-1\] Either, there is a first generation $G_s$ with $1\leq s<K$ which has a last player with no waiting costs in $\sigma'$. Let each player $[it]$ with $1<t\leq s$ choose the same edge as the player departing before $[it]$ in $\sigma'$ and let each player $[it]$ with $s<t<K$ choose the same edge as in $\sigma'$. In this case, $[\delta'_ss]$ does not affect the other players in $\sigma'$. So it is no problem if she does not depart. From stage $K$ onwards, go to \[it:algo-I\].
(2) \[it:algoII-2\] Or, all generations $G_t$ with $1\leq t<K$ have a last player with waiting costs in $\sigma'$, then let each player $[it]$ with $1<t\leq K$ choose the same edge as the player departing before $[it]$ in $\sigma'$. From stage $K+1$ onwards, go to \[it:algo-II\].
Notice that $\sigma$ is defined in such a way that queues have the same length as under $\sigma'$, either at the beginning or at end of stage $k+K$ . Queues have the same length at the beginning of stage $k+K$ in cases where the algorithm goes to \[it:algo-I\], and queues have the same length at the end of stage $k+K$ in cases where the algorithm goes to \[it:algo-II\].
Now, in order to compute the long-run latency, let us focus on the steady state. We know that with uniform departures there is a $t_0$ such that for all generations $t\geq t_0$, queues are such that $\gamma_e$ players choose edge $e$ for all $e\in E$. Recall that $\delta_k>\gamma$. By construction, for each generation $t=k\mod{K}$, where $t\geq t_0+K$, and for all edges $e$ at least $\gamma_e$ players choose $e$. This implies that player $[\delta_k t]$ must wait for at least one period. So the waiting costs for $[\delta_k t]$ increases by one unit compared to $\sigma'$.
Second, we show that there is an equilibrium of $\Gamma(\mathcal{N},K,\boldsymbol{\delta}')$ with costs equal to ${\operatorname{\mathsf{WEq}}}(\mathcal{N},K,\boldsymbol{\delta})-1$. This implies that ${\operatorname{\mathsf{WEq}}}(\mathcal{N},K,\boldsymbol{\delta})\leq {\operatorname{\mathsf{WEq}}}(\mathcal{N},K,\boldsymbol{\delta}')+1$.
Fix a worst equilibrium $\sigma$ of $\Gamma(\mathcal{N}, K, \boldsymbol{\delta})$ and consider stage $k$. Since $\delta_k>\gamma$, queues must be created on some edges. Let $i^{*}$ be the maximal index such that player $[i^{*}k]$ meets a queue under $\sigma$. Then, by the choice of $i^{*}$, it must be that each subsequent player meets no queue. Further, each such player pays the same cost as $[i^{*}k]$. Indeed, if for $j>i^{*}$, player $[jk]$ pays less than $[i^{*}k]$, then $[i^{*}k]$ has a profitable deviation by imitating $[jk]$. If $[jk]$ pays more than $[i^{*}k]$, then she would meet a queue. Recall from Lemma \[le:worsteq\] that in a worst equilibrium, in case of indifference, players choose an edge with a queue over an edge without a queue. So $[jk]$ would imitate $[i^{*}k]$, paying the cost of $[i^{*}k]$ plus $1$. This contradicts the definition of $i^{*}$.
Now, consider the game where $[i^{*}k]$ is postponed by one stage, starting as the first player of the next generation. In this game, if the postponed player chooses the exact same strategy, she pays one unit less, since queues have decreased by one. She cannot pay less than that, since that would have offered a profitable deviation for $[i^{*}k]$ in the original game. So it is an equilibrium. Since the two situations are identical for all other players, this reasoning can be iterated at each period.
Hence, combining the previous two results yields $$\begin{aligned}
{\operatorname{\mathsf{WEq}}}(\mathcal{N},K,\boldsymbol{\delta})&={\operatorname{\mathsf{WEq}}}(\mathcal{N},K,\boldsymbol{\delta}')+1,\\
&={\operatorname{\mathsf{WEq}}}(\mathcal{N},K,\gamma)+D(\boldsymbol{\delta}')+1,\\
&={\operatorname{\mathsf{WEq}}}(\mathcal{N},K,\gamma)+D(\boldsymbol{\delta}),\end{aligned}$$ where the second equality follows from the induction hypothesis.
Nash equilibria {#nash-equilibria .unnumbered}
---------------
The following example shows that there are Nash (but not UFR) equilibria for chain-of-parallel networks, where players may end up paying strictly more than the cost of the costlier route. The reason is that in a Nash equilibrium players need not arrive at intermediate vertices as early as possible and this may create additional queues.
\[ex:verybadNash\] Consider the chain-of-parallel network in Figure \[fi:chainofparallel2SM\] with associated free-flow transit costs and capacity equal to $1$ for all edges. The capacity $\gamma^{(*)}$ of the network is $2$.
\(1) [$s$]{}; (2) \[right of=1\] [$v$]{}; (3) \[right of=2\] [$d$]{};
\(1) edge \[bend right = 30\] node\[above\] [$\tau_{1}^{(1)}=1$]{} (2) edge \[bend left = 30\] node\[above\] [$\tau_{2}^{(1)}=2$]{} (2);
\(2) edge \[bend right = 30\] node\[above\] [$\tau_{1}^{(2)}=1$]{} (3) edge \[bend left = 30\] node\[above\] [$\tau_{2}^{(2)}=2$]{} (3);
Consider the following equilibrium strategy profile $$\begin{aligned}
\sigma_{it}^{{\,\mathsf{Eq}}}=\begin{cases}
e_{1}^{(1)} e_{1}^{(2)} &\text{ for }[it]=[1t], t\leq2,\\
e_{1}^{(1)} e_{1}^{(2)} &\text{ for }[it]=[2t], t\leq2,\\
e_{2}^{(1)} e_{1}^{(2)} &\text{ for }[it]=[13],\\
e_{1}^{(1)} e_{1}^{(2)} &\text{ for }[it]=[23],\\
e_{2}^{(1)} e_{2}^{(2)} &\text{ for }[it]=[1t], t\geq4,\\
e_{1}^{(1)} e_{1}^{(2)} &\text{ for }[it]=[2t], t\geq4,\\
\end{cases}\end{aligned}$$ In the first two periods a queue of length two is created on $e_{1}^{(1)}$. Note that player $[22]$ cannot be overtaken as the next player departs in the following period. In period three, a new queue starts on $e_{1}^{(2)}$ and in stage four we reach the steady state.
The latency of this strategy profile equals $9$, which means that the last player of each generation pays more than the maximum free-flow transit costs.
[^1]: This author is a member of GNAMPA-INdAM. The support of PRIN 20103S5RN3 and MOE2013-T2-1-158 is gratefully acknowledged.
[^2]: The support of GSBE and MOE2013-T2-1-158 is gratefully acknowledged.
[^3]: The support of the HEC foundation and of the Agence Nationale de la Recherche under grant ANR JEUDY, ANR-10-BLAN 0112 is gratefully acknowledged.
|
---
abstract: 'A recent experiment reported the first violation of a Bell correlation witness in a many-body system \[Science 352, 441 (2016)\]. Following discussions in this paper, we address here the question of the statistics required to witness Bell correlated states, i.e. states violating a Bell inequality, in such experiments. We start by deriving multipartite Bell inequalities involving an arbitrary number of measurement settings, two outcomes per party and one- and two-body correlators only. Based on these inequalities, we then build up improved witnesses able to detect Bell-correlated states in many-body systems using two collective measurements only. These witnesses can potentially detect Bell correlations in states with an arbitrarily low amount of spin squeezing. We then establish an upper bound on the statistics needed to convincingly conclude that a measured state is Bell-correlated.'
author:
- Sebastian Wagner
- Roman Schmied
- Matteo Fadel
- Philipp Treutlein
- Nicolas Sangouard
- 'Jean-Daniel Bancal'
title: 'Bell correlations in a many-body system with finite statistics'
---
#### Introduction –
Physics research fundamentally relies on the proper analysis of finite experimental data. In this exercise, assumptions play a subtle but crucial role. On the one hand, they are needed in order to reach a conclusion; even device-independent assessments rely on assumptions [@Scarani12]. On the other hand, they open the door for undesirable effects ranging from a reduction of the conclusion’s scope, when more assumptions are used than strictly needed, to biased results when relying on unmet assumptions. Contrary to popular belief, such cases are frequent in science, even for common assumptions [@Bailey16; @Youden72]. Relying on fewer hypotheses, when possible, is thus desirable to obtain more general, accurate and trustworthy conclusions [@diew1; @diew2].
Bell nonlocality, as revealed by the violation of a Bell inequality, constitutes one of the strongest forms of non-classicality known today. However, its demonstration has long been restricted to systems involving few particles [@Lanyon14; @Eibl02; @Zhao03; @Lavoie09; @BellExp]. Recently, the discovery of multipartite Bell inequalities that only rely on one- and two-body correlators opened up new possibilities [@Tura14]. Although these inequalities have not yet lead to the realization of a multipartite Bell test, they can be used to derive witnesses able to detect Bell correlated states, i.e. states capable of violating a Bell inequality.
Using such a witness, an experiment recently detected the presence of Bell correlations in a many-body system under the assumption of gaussian statistics [@Schmied16]. While this demonstration uses spin squeezed states, the detection of Bell correlations in other systems was also recently investigated [@Pelisson16]. The witness used in Ref. [@Schmied16] involves one- and two-body correlation functions and takes the form $\mathcal{W} \geq 0$, where the inequality is satisfied by measurements on states that are not Bell-correlated. Observation of a negative value for $\mathcal{W}$ then leads to the conclusion that the measured system is Bell-correlated. However, due to the statistics loophole [@Gill12; @Zhang11], reaching such a conclusion in the presence of finite statistics requires special care. In particular, an assessment of the probability with which a non-Bell-correlated state could be responsible for the observed data is required before concluding about the presence of Bell correlations without further assumptions.
Concretely, the witness of Refs. [@Schmied16] has the property of admitting a quantum violation lower-bounded by a constant $\mathcal{W}_\text{opt} < 0$, while the largest possible value $\mathcal{W}_\text{max} > 0$ is achievable by a product state and increases linearly with the size of the system $N$. These properties imply that a small number of measurements on a state of the form $$\label{eq:counterExample}
\rho=(1-q){|\psi\rangle\!\!\!\;\langle\psi|} + q({|{\uparrow}\rangle\!\!\!\;\langle{\uparrow}|})^{\otimes N},$$ where $\mathcal{W}(\ket{\psi})=\mathcal{W}_\text{opt}$, $\mathcal{W}(\ket{\uparrow}^{\otimes N})=\mathcal{W}_\text{max}$ and $q$ is small, is likely to produce a negative estimate of $\mathcal{W}$, even though the state is not detected by the witness in the limit of infinitely many measurement rounds [@Schmied16]. This state thus imposes a lower bound on the number of measurements required to exclude, through such witnesses, all non-Bell-correlated states with high confidence. Contrary to other assessments, this lower bound increases with the number of particles involved in the many-body system. Therefore, it is not captured by the standard deviation of one- and two-body correlation functions (which on the contrary decreases as the number of particles increases).
It is worth noting that states of the form put similar bounds on the number of measurements required to perform any hypothesis tests in a many-body system satisfying the conditions above. This includes in particular tests of entanglement [@Gross10; @Riedel10; @Bernd14; @Pezze] based on the entanglement witnesses of Ref. [@Sorensen01a; @Toth09; @Hyllus12].
In this article, we address this statistical problem in the case of Bell correlation detection by providing a number of measurement rounds sufficient to exclude non-Bell-correlated states from an observed witness violation. Let us mention that in Ref. [@Schmied16], the statistics loophole is circumvented by the addition of an assumption on the set of local states being tested. This has the effect of reducing the scope of the conclusion: the data reported in Ref. [@Schmied16], are only able to exclude a subset of all non-Bell-correlated states (as pointed out in the reference). Here, we show that such additional assumptions are not required in experiments on many-body systems, and thus argue that they should be avoided in the future.
In order to minimize the amount of statistics required to reach our conclusion, we start by investingating improved Bell correlation witnesses. For this, we first derive Bell inequalities with two-body correlators and an arbitrary number of settings. This allows us to obtain Bell-correlation witnesses that are more resistant to noise compared to the one known to date [@Schmied16]. We then analyse the statistical properties of these witnesses and provide an upper bound on the number of measurement rounds needed to rule out all local states in a many-body system. We show that this upper bound is linear in the number of particles, hence making the detection of Bell correlations free of the statistical loophole possible in systems with a large number of particles.
#### Symmetric two-body correlator Bell inequalities with an arbitrary number of settings –
Multipartite Bell inequalities that are symmetric under exchange of parties and which involve only one- and two-body correlators have been proposed in scenarios where each party uses two measurement settings and receives an outcome among two possible results [@Tura14]. Similar inequalities were also obtained for translationally invariant systems [@Tura14a], or based on Hamiltonians [@Tura16]. Here, we derive a similar family of Bell inequalities that is invariant under arbitrary permutations of parties but allows for an arbitrary number of measurement settings per party.
Let us consider a scenario in which $N$ parties can each perform one of $m$ possible measurements $M_k^{(i)}$ ($k=0,...,m-1$; $i=1,...,N$) with binary outcomes $\pm 1$. We write the following inequality: $$\begin{aligned}
I_{N,m} = \sum_{k=0}^{m-1} \alpha_k S_k + \frac12 \sum_{k,l} S_{kl} \geq -\beta_c \, ,
\label{eq:symBell}\end{aligned}$$ where $\alpha_k=m-2k-1$, $\beta_c$ is the local bound, and the symmetrized correlators are defined as $$\begin{aligned}
S_k := \sum_{i=1}^N \langle M_k^{(i)} \rangle \, , \quad S_{kl}:= \sum_{i \neq j} \langle M_k^{(i)} M_l^{(j)}\rangle \, .
\label{eq:correlators}\end{aligned}$$ Let us show that is a valid Bell inequality for $\beta_c=\left\lfloor \frac{m^2 N}{2}\right\rfloor$, where $\lfloor x\rfloor$ is the largest integer smaller or equal to $x$. Below, we assume that $m$ is even; see Appendix A for the case of odd $m$.
Since $I_{N,m}$ is linear in the probabilities and local behaviors can be decomposed as a convex combination of deterministic local strategies, the local bound of Eq. can be reached by a deterministic local strategy [@Brunner14]. We thus restrict our attention to these strategies and write $$\begin{aligned}
\langle M_k^{(i)} \rangle = x_k^i = \pm 1 \quad \Rightarrow S_{kl}=S_k S_l -\sum_{i=1}^N x_k^i x_l^i \, ,
\label{eq:determinism}\end{aligned}$$ where $x_k^i$ is the (deterministic) outcome party $i$ produces when asked question $k$. This directly leads to the following decomposition: $$\begin{aligned}
I_{N,m} = \sum_{k=0}^{\frac{m}{2}-1}\alpha_k (S_k-S_{m-k-1}) +\frac12 B^2 - \frac12 C \geq -\beta_c\, ,
\label{eq:BellEven}\end{aligned}$$ with $B:= \sum_{k=0}^{m-1}S_k$ and $C:=\sum_{i=1}^N\left(\sum_{k=0}^{m-1}x_k^i\right)^2$. Due to the symmetry under exchange of parties of this Bell expression, it is convenient to introduce, following [@Tura14], variables counting the number of parties that use a specific deterministic strategy: $$\begin{aligned}
a_{j_1<...<j_n} :&=\#\{i \in \{1,...,N\} \vert x_k^i = -1 \text{ iff } k \in \{j_1,...,j_n\}\} \nonumber \\
\bar{a}_{j_1<...<j_n} :&=\#\{i \in \{1,...,N\} \vert x_k^i = +1 \text{ iff } k \in \{j_1,...,j_n\}\}\nonumber\\
n &\leq \frac{m}{2} \, , \quad \bar{a}_{j_1,...,j_{\frac{m}{2}}} \equiv 0 \, ,
\label{eq:strategyvar}\end{aligned}$$ where $\#$ denotes the set cardinality. Since each party has to choose a strategy, the variables sum up to $N$: $$\begin{aligned}
\sum_{\text{all variables}}= \sum_{n=0}^{\frac{m}{2}}\sum_{j_1<...<j_n}\left(a_{j_1...j_n}+\bar{a}_{j_1...j_n}\right)=N \, .
\label{eq:SumN}\end{aligned}$$ The correlators can now be expressed as $$\begin{aligned}
S_k = \sum_{n=0}^{\frac{m}{2}}\sum_{j_1<...<j_n}\left(a_{j_1...j_n}-\bar{a}_{j_1...j_n}\right)y_k^{j_1...j_n} \, ,
\label{eq:Sky}\end{aligned}$$ with $y_k^{j_1...j_n}= -1$ if $k\in\{j_1,...,j_n\}$, and $+1$ otherwise.
The first term of concerns the difference between two correlators. Let us see how this term decomposes as a function of the number of indices present in its variables. From Eq. , it is clear that a variable with $n$ indices only appears in the difference $S_k-S_l$ if $y_k^{j_1...j_n}\neq y_l^{j_1...j_n}$. But the corresponding strategy only has $n$ differing outcomes and each correlator in this term only appears once, so a variable with $n$ indices appears in at most $n$ of these differences. Moreover, if it appears, it does so with a factor $\pm 2$. The coefficient in front of a variable with $n$ indices in the first sum of thus cannot be smaller than $-2\sum_{k=0}^{n-1}\alpha_k=2n(n-m)$.
The second term of can be bounded as $B^2\geq 0$, while the third one can be expressed as $$\begin{aligned}
C=\sum_{n=0}^{\frac{m}{2}}\sum_{j_1<...<j_n}\left(a_{j_1...j_n}+\bar{a}_{j_1...j_n}\right) (m-2n)^2 \, .\end{aligned}$$ Putting everything together and using property , we arrive at $$\begin{aligned}
I_{N,m}&\geq \sum_{k=0}^{\frac{m}{2}-1}\alpha_k (S_k-S_{m-k-1}) - \frac12 C \nonumber\\
&\geq -\frac{m^2}{2}\sum_{\text{all variables}}=-\frac{m^2 N}{2} = -\beta_c \, ,\end{aligned}$$ which concludes the proof.
Note that this bound is achieved for $a_{01...\frac{m}{2}-1}=N$, i.e. when for each party exactly the first half of the measurements yields result $-1$. Note also that the Bell inequality does not reduce to Ineq. (6) of Ref. [@Tura14] when $m=2$. Indeed, while none of these inequalities is a facet of the local polytope, the latter one is a facet of the symmetrized 2-body correlator local polytope [@Tura14; @symm10].
#### From Bell inequalities to Bell-correlation witnesses –
Let us now derive a set of Bell-correlation witnesses assuming a certain form for the measurement operators. Here, no assumptions are made on the measured state.
Following Ref. [@Schmied16], we start from inequality and introduce spin measurements along the axes $\vec{d}_k$, $k=0,...,m-1$, as well as the collective spin observables $\hat{S}_k$: $$\begin{aligned}
M_k^{(i)}= \vec{d}_k \cdot \vec{\sigma}^{(i)} \, , \quad \hat{S}_k = \frac12 \sum_{i=1}^N M_k^{(i)} \, ,
\label{eq:SpinMeas}\end{aligned}$$ where $\vec \sigma$ is the Pauli vector acting on a spin-$\frac12$ system. The correlators can be expressed in terms of these total spin observables and the measurement directions [@Tura14]: $$\begin{aligned}
S_k &= 2\langle \hat{S}_k \rangle \nonumber\\
S_{kl} &= 2\left[\left\langle\hat{S}_k\hat{S}_l\right\rangle+\left\langle\hat{S}_l\hat{S}_k\right\rangle\right] - N \vec{d}_k\cdot\vec{d}_l\, .\end{aligned}$$ This defines the Bell operators $$\begin{aligned}
\hat{W}_{N,m} := 2\!\sum_{k=0}^{m-1}\! \alpha_k \hat{S}_k + 2\!\sum_{k,l} \hat{S}_k\hat{S}_l -\frac{N}{2}\! \sum_{k,l} \vec{d}_k\!\cdot\!\vec{d}_l + \left\lfloor\!\frac{m^2N}{2}\!\right\rfloor \, , \end{aligned}$$ whose expectation values are positive for states that are not Bell-correlated. Note that the expectation value of these operators need not be negative for all Bell-correlated states and every choice of mesurement, though. A negative value may only be achieved for specific choices of states and measurement settings.
We now consider measurement directions $\vec{d}_k = \vec{a}\cos(\vartheta_k)+\vec{b}\sin(\vartheta_k)$ lying in a plane spanned by two orthonormal vectors $\vec{a}$ and $\vec{b}$, with the antisymmetric angle distribution $\vartheta_{m-k-1}=-\vartheta_k$. Note that the coefficients $\alpha_k$ share the same antisymmetry. Defining $\mathcal{W}_m := \left\langle \frac{\hat{W}_{N,m}}{2 \hat{N}} \right\rangle $ for even $m$, we arrive at the following family of witnesses: $$\begin{aligned}
\mathcal{W}_{m} = {\mathcal{C}}_b {\sum_{k=0}^{\frac{m}{2}-1} \alpha_k \sin(\vartheta_k)} \,{-} {\left(1-\zeta_a^2\right)} {\left[\sum_{k=0}^{\frac{m}{2}-1}\cos(\vartheta_k)\right]^2}\! {+}\, \frac{m^2}{4} \, ,
\label{eq:witness}\end{aligned}$$ with $\mathcal{W}_m\geq 0$ for states that are not Bell correlated. These Bell correlation witnesses depend on $\frac{m}{2}$ angles $\vartheta_k$ and involve just two quantities to be measured: the scaled collective spin ${\mathcal{C}}_b:=\left\langle \frac{\hat{S}_{\vec{b}}}{\hat{N}/2}\right\rangle$ and the scaled second moment $\zeta_a^2 :=\left\langle \frac{\hat{S}_{\vec{a}}^2}{\hat{N}/4}\right\rangle$.
The tightest constraints on ${\mathcal{C}}_b$ and $\zeta_a^2$ that allow for a violation of $\mathcal{W}_m\geq0$ are obtained by minimizing $\mathcal{W}_m$ over the angles $\vartheta_k$. Solving $\frac{\partial \mathcal{W}_m}{\partial \vartheta_k}=0$ yields (see Appendix B): $$\begin{aligned}
\vartheta_k = -\arctan[\lambda_m (m-2k-1)] \, ,
\label{eq:OptAng} \\
\frac{{\mathcal{C}}_b}{2\lambda_m(1-\zeta_a^2)}=\sum_{k=0}^{\frac{m}{2}-1}\cos(\vartheta_k)\, . \label{eq:Selfcons} \end{aligned}$$ Equation is a self-consistency equation for $\lambda_m$ that has to be satisfied in order to minimize $\mathcal{W}_m$.
Using these parameters, we can rewrite our witness in terms of the physical parameters ${\mathcal{C}}_b$ and $\zeta_a^2$ only. For two measurement directions ($m=2$), we find that states which are not Bell-correlated satisfy $$\begin{aligned}
\zeta_a^2\geq Z_2({\mathcal{C}}_b)= \frac12\left(1-\sqrt{1-{\mathcal{C}}_b^2}\right) \, .
\label{eq:Z2}\end{aligned}$$ This recovers the bound obtained from a different inequality in [@Schmied16]. Note that in the present case, the argument is more direct since it does not involve ${\mathcal{C}}_a$, the first moment of the spin operator in the $a$ direction.
![Plots of the critical lines $Z_2$, $Z_4$ and $Z_{\infty}$. The witness obtained from the Bell inequality with $4$ settings already provides a significant improvement over the case of $2$ settings. The black point in the inset shows the data point from [@Schmied16], with $N = 476\pm 21$.[]{data-label="fig:Z2,4,Inf"}](ContourPlotInset.png){width="0.9\linewidth"}
Increasing the number of measurement directions allows for the detection of Bell correlations in additional states. In the limit $m\to\infty$, we find (see Appendix B): $$\begin{aligned}
\zeta_a^2\geq Z_{\infty}({\mathcal{C}}_b) = 1-\frac{{\mathcal{C}}_b}{\text{artanh}\left({\mathcal{C}}_b\right)} \, .
\label{eq:ZInf} \end{aligned}$$
Figure \[fig:Z2,4,Inf\] shows the two witnesses and together with the one obtained similarly for $m=4$ settings in the ${\mathcal{C}}_b$-$\zeta_a^2$ plane. The curve $Z_\infty$ reaches the point ${\mathcal{C}}_b=\zeta_a^2=1$, therefore allowing in principle for the detection of Bell correlations in presence of arbitrarily low squeezing. It is known, however, that some values of ${\mathcal{C}}_b$ and $\zeta_a^2$ can only be reached in the limit of a large number of spins [@Sorensen01]. For any fixed $N$, a finite amount of squeezing is thus necessary in order to allow for the violation of our witness (see Appendix C). The corresponding upper bound on $\zeta_a^2$ is shown in Figure \[fig:zetaUpper\].
![Upper bound on the value of $\zeta_a^2$ required to see a violation of the Bell correlation witness . The bound depends on the number of particles $N$.[]{data-label="fig:zetaUpper"}](minSqueezing.pdf){width="0.95\linewidth"}
Points below the curve $Z_m$ in Fig. \[fig:Z2,4,Inf\] indicate a violation of the witness $\mathcal{W}_{m} \geq 0$ obtained from the corresponding $m$-settings Bell inequality. Violation of any such bound reveals the presence of a Bell-correlated state. However, as discussed in the introduction, conclusions in the presence of finite statistics have to be examined carefully, since in practice, one can never conclude from the violation of a witness that the measured state is Bell correlated with $100\%$ confidence. The point shown in the inset of Fig. \[fig:Z2,4,Inf\] corresponds to the data reported in Ref. [@Schmied16] from measurements on a spin-squeezed Bose-Einstein condensate. This point clearly violates the witnesses for $m=2,4,\infty$ by several standard deviations, although the number of measurement rounds is too small to guarantee that the measured state is Bell correlated without further assumptions [@Schmied16].
#### Finite Statistics –
In this section, we put a bound on the number of experimental runs needed to exclude with a given confidence that a measured state is not Bell-correlated. Note that such a conclusion does not follow straightforwardly from the violation of the witness by a fixed number of standard deviations. Indeed, standard deviations inform on the precision of a violation, but fail at excluding arbitrary local models [@Zhang11], including e.g. models which may showing non-gaussian statistics with rare events. We thus look here for a number of experimental runs that is sufficient to guarantee a p-value lower than a given threshold for the null hypothesis ‘The measured state is not Bell-correlated’. Since we are concerned with the characterization of physical systems in the absence of an adversary, we assume that the same state is prepared in each round (i.i.d. assumption).
For this statistical analysis, let us consider a different Bell correlation witness than . Indeed, we derived this inequality in order to maximize the amount of violation for given data, but here we rather wish to maximize the statistical evidence of a violation. For this, we take and consider the representation of the angles given in Eq. , but without taking Eq. into account. In the limit of infinitely-many measurement settings, we find (see Appendix B) $$\begin{aligned}
\mathcal{W_\text{stat}} =-{\mathcal{C}}_b \Delta_\nu &-(1-\zeta_a^2)\Lambda_\nu^2+\frac{1}{4} \geq 0 \, \text{, with} \label{eq:finalW}\\
\Delta_\nu=\frac{\sqrt{1+\nu^2}}{4\nu}-&\frac{\mathrm{arsinh}(\nu)}{4\nu^2} \, ,\quad \Lambda_\nu= \frac{\mathrm{arsinh}(\nu)}{2\nu}\, , \label{eq:DeltaLambda}\end{aligned}$$ where $\nu=\lim\limits_{m\to\infty}\lambda_m\cdot m$ is a free parameter that fully specifies the set of measurement angles.
In order to model the experimental evaluation of $\mathcal{W_\text{stat}}$, we introduce the following estimator: $$\begin{split}
\mathcal{T}=& \frac{\chi(Z=0)}{q}X + \frac{\chi(Z=1)}{1-q}Y + (\frac14-\Delta_\nu-\Lambda_\nu^2) \, .
\end{split}$$ Here, $\chi$ denotes the indicator function and the binary random variable $Z$ accounts for the choice between the measurement of either ${\mathcal{C}}_b$ or $\zeta_a$. Each measurement allows for the evaluation of the corresponding random variables $X=\Delta_\nu(1-{\mathcal{C}}_b)$ and $Y=\Lambda_\nu^2\zeta_a^2$. Assuming that $Z$ is independent of $X$ and $Y$ and choosing $q=P[Z=0]$ guarantees that $\mathcal{T}$ is a proper estimator of $\mathcal{W}_\text{stat}$, i.e. $\langle\mathcal{T}\rangle=\mathcal{W}$. $q$ then corresponds to the probability of performing a measurement along the $b$ axis. We choose $q=\left(1+\frac{\Lambda_\nu^2 N}{2\Delta_\nu}\right)^{-1}$ so that the contributions of both measurements to $\mathcal{T}$ have the same magnitude, i.e. the maximum values of $X/q$ and $Y/(1-q)$ are equal within the domain $|{\mathcal{C}}_b|\leq1$ and $\zeta_a^2\in[0,N]$. This also guarantees that the spectrum of $\mathcal{T}$ matches that of $\mathcal{W}_\text{stat}$.
Suppose the measured state is non-Bell-correlated, i.e. that its mean value $\mu=\langle \mathcal{T}\rangle= \mathcal{W}_\text{stat} \geq 0$. We are now interested in the probability that after $M$ experimental runs the estimated value $T=\frac1M \sum_{i=1}^M \mathcal{T}_i$ of the witness $\mathcal{W}_\text{stat}$ falls below a certain value $t_0<0$, with $\mathcal{T}_i$ being the value of the estimator in the $i^\text{th}$ run.
In statistics, concentration inequalities deal with exactly this issue. In Appendix D, we compare four of these inequalities, namely the Chernoff, Bernstein, Uspensky and Berry-Esseen ones and show explicitly that in the regime of interest the tightest and therefore preferred bound results from the Bernstein inequality: $$\begin{aligned}
P[T \leq t_0]\leq \exp\left(-\frac{(\mu-t_0)^2 M}{2\sigma_0^2+\frac23 (t_u-t_l) (\mu-t_0)}\right) \leq \varepsilon\, .
\label{eq:Bern}\end{aligned}$$ Here, $t_0$ is the experimentally observed value of $T$ after $M$ measurement rounds, $t_l=\frac14-\Delta_\nu-\Lambda_\nu^2$ and $t_u=\frac14+\Delta_\nu+\Lambda_\nu^2(N+1)$ are lower and upper bounds on the random variable $\mathcal{T}$ respectively, and $\sigma_0^2$ is its variance for a local state.
We show in Appendix D that the largest p-value is obtained by setting $\mu=0$ and $\sigma_0^2=-t_l t_u$. A number of measurements sufficient to exclude the null hypothesis with a probability larger than $1-\varepsilon$ is then given by: $$\begin{aligned}
M\geq \frac{-2t_l t_u -\frac23(t_u-t_l)t_0}{t_0^2}\ln\left(\frac1\varepsilon\right) \, .
\label{eq:MBern}\end{aligned}$$ This quantity can be minimized by choosing the free parameter $\nu$ appropriately. As shown in Appendix D, optimizing $\nu$ at this stage allows us to reduce the number of measurement by $\sim\!\!30$%. It is thus clearly advantageous not to consider the witness when evaluating statistical significance.
The number of runs in depends linearly on $t_l$ and therefore also linearly on $N$. The ratio $\frac{M}{N}$ thus tends to a constant for large $N$ (see Appendix D for more details). This implies that a number of measurements growing linearly with the system size is both necessary and sufficient to close the statistics loophole [@Schmied16].
Figure \[fig:MNvsCb\] depicts the required number of experimental runs per spin as a function of the scaled collective spin ${\mathcal{C}}_b$ and of the scaled second moment $\zeta_a^2$. For a confidence level of $1-\epsilon=99\%$, between 20 and 500 measurement runs per spins are required in the considered parameter region.
![Number of experimental runs per spins required to rule out non-Bell-correlated states with a confidence of $1-\varepsilon$ as a function of ${\mathcal{C}}_b$ and $\zeta_a$. For ${\mathcal{C}}_b=0.98$ and $\zeta_a^2=0.272$ (as reported in [@Schmied16]), approximately $17\cdot\ln(100)\simeq80$ runs per spin are sufficient to reach a confidence level of $99\%$.[]{data-label="fig:MNvsCb"}](MNvsCbJoined2.png){width="\linewidth"}
#### Conclusion –
In this paper, we introduce a class of multipartite Bell inequalities involving two-body correlators and an arbitrary number of measurement settings. Assuming collective spin measurements, these inequalities give rise to the witness , which can be used to determine whether Bell correlations can be detected in a many-body system. This criterion detects states that were not detected by the previously-known witness [@Schmied16].
We then discuss the statistics loophole arising in experiments involving many-body systems, i.e. the difficulty of ruling out, without further assumptions, non-Bell-correlated states in the presence of finite statistics. We provide a bound, Eq. , on the number of measurement rounds that allows one to close this loophole. This bound shows that all non-Bell-correlated states can be convincingly ruled out at the cost of performing a number of measurements that grows linearly with the system size. This opens the way for a demonstration of Bell-correlations in a many-body system free of the statistics loophole.
#### Acknowledgements –
We thank Baptiste Allard, Remik Augusiak and Valerio Scarani for helpful discussions. This work was supported by the Swiss National Science Foundation (SNSF) through grants PP00P2-150579, 20020-169591 and NCCR QSIT. NS acknowledges the Army Research Laboratory Center for Distributed Quantum Information via the project SciNet.
Proof of the Bell inequalities
==============================
In this appendix, we expand on the proof of Ineq. given in the main text, and cover the case of odd numbers of measurements.
Symmetric Bell inequality for $m$ measurements
----------------------------------------------
We consider local measurements on $N$ parties. For each party, one can choose between $m$ measurements $M_k^{(i)}$, where $k \in \{0,1,...,m-1\}$ and $i \in \{1,...,N\}$. Each measurement has the two possible outcomes $\pm 1$. We are interested in Bell inequalities, i.e. inequalities every local theory has to obey [@Brunner14]. We only consider one- and two-body mean values, so the general form of such an inequality is $$\begin{aligned}
I_{N,m} &= \sum_{k=0}^{m-1}\sum_{i=1}^N \alpha_k^i \langle M_k^{(i)} \rangle + \sum_{k,l} \sum_{i < j} \beta_{kl}^{ij} \langle M_k^{(i)} M_l^{(j)} \rangle \geq -\beta_c \, ,
\label{eq:generalI}\end{aligned}$$ where $\langle M_k^{(i)} \rangle=\sum_{a\in\{-1,1\}}a\, \text{Prob}(M_k^{(i)}=a)$ and $\langle M_k^{(i)} M_l^{(j)} \rangle=\sum_{a,b\in\{-1,1\}}ab\, \text{Prob}(M_k^{(i)}=a,M_l^{(j)}=b)$.
We now restrict ourselves to Bell inequalities which are symmetric under exchange of parties, i.e. $\alpha_k^i=\alpha_k$ and $\beta_{kl}^{ij}=\beta_{kl}$. After defining the symmetrized correlators $$\begin{aligned}
S_k = \sum_{i=1}^N \langle M_k^{(i)} \rangle \quad , \quad S_{kl} = \sum_{i\neq j} \langle M_k^{(i)}M_l^{(j)}\rangle \, ,
\label{eq:Cor}\end{aligned}$$ symmetric inequalities can be expressed as $$\begin{aligned}
I_{N,m} = \sum_{k=0}^{m-1}\alpha_k S_k +\frac{1}{2}\sum_{k,l}^{m-1}\beta_{kl} S_{kl} \geq -\beta_c \, .\end{aligned}$$
We are interested in cases for which the coefficients are $\alpha_k=m-2k-1$ ($ k=0,...,m-1$) and $\beta_{k,l}=1$. We note that $\alpha_{m-k-1}=-\alpha_k$ and claim that local theories have to fulfill the Bell inequalities $$\begin{aligned}
I_{N,m} &= \sum_{k=0}^{m-1}(m-2k-1)S_k +\frac{1}{2}\sum_{k,l}^{m-1}S_{kl} \geq -\left\lfloor\frac{m^2N}{2}\right\rfloor = -\beta_c \, ,
\label{eq:final_I}\end{aligned}$$ where $\lfloor x\rfloor$ is the largest integer smaller or equal to $x$.
Computation of the local bound
------------------------------
In this section we prove the claim above. One of the most important properties of a local theory is its equivalence to a mixture of deterministic local theory. That is why, by considering only deterministic theories, there is no loss of generality. We can therefore assume that a measurement $M_k^{(i)}$ will lead to an outcome $x_k^i=\pm 1$ with probability $1$, i.e. $\langle M_k^{(i)} \rangle = x_k^i$. The two-body correlators $S_{kl}$ can thus be expressed as $S_{kl}=S_k S_l-\sum_{i=1}^N x_k^i x_l^i$. By also taking the antisymmetry of $\alpha_k$ into account, and introducing the quantities $$\begin{aligned}
A=\sum_{k=0}^{\left\lfloor\frac{m}{2}\right\rfloor -1}(m-2k-1) (S_k-S_{m-k-1})\ ,\quad
B=\sum_{k=0}^{m-1} S_k\ ,\quad
C=\sum_{i=1}^N \left[\sum_{k=0}^{m-1}x_k^i\right]^2 \, .
\label{eq:detI}\end{aligned}$$ we arrive at $$\begin{aligned}
I_{N,m} = A + \frac12 B^2 - \frac12 C\, .\end{aligned}$$
### Strategy variables
We want to rewrite $I_{N,m}$ further. Therefore we introduce variables counting the strategies chosen by the parties. Because there are $m$ measurements with binary outcomes, the number of possible strategies per party is $2^m$. We define the following $2^m$ variables: $$\begin{aligned}
a_{j_1<...<j_n} :&=\#\{i \in \{1,...,N\} \vert x_k^i = -1 \text{ iff } k \in \{j_1,...,j_n\}\} \quad\text{ for } n\leq \left\lfloor\frac{m}{2}\right\rfloor \, , \nonumber \\
\bar{a}_{j_1<...<j_n} :&=\#\{i \in \{1,...,N\} \vert x_k^i = +1 \text{ iff } k \in \{j_1,...,j_n\}\} \quad \text{ for } n\leq \left\lfloor\!\frac{m-1}{2}\!\right\rfloor \, , \nonumber\\
\bar{a}_{j_1,...,j_{\left\lceil\!\frac{m}{2}\!\right\rceil}} &\equiv 0 \, ,
\label{eq:Strategy}\end{aligned}$$ where $\#$ denotes the set cardinality. For example, $a_j$ counts the parties $k$ whose outcomes are $x_k^{j'}=1-2\delta_{jj'}$. $\bar{a}_j$ is the number of parties following the opposite strategy. Variables with $n$ indices thus correspond to a strategy for which either exactly $n$ of the $m$ outcomes are $+1$ or exactly $n$ of the outcomes are $-1$, i.e. $n$ outcomes differ from the rest. Note that the conjugate variables in the case of $\frac{m}{2}$ indices are set to zero for the case of even $m$ in order to prevent strategies from being counted twice. Since every party has to choose one strategy, the variables sum up to $N$, i.e. $$\begin{aligned}
a+\bar{a}+\sum_{j=0}^{m-1}(a_j+\bar{a}_j) + ... =\sum_{n=0}^{\left\lfloor\frac{m}{2}\right\rfloor}\sum_{j_1<...<j_n} (a_{j_1...j_n}+\bar{a}_{j_1...j_n})=N \, \text{.}
\label{eq:SumStrategies}\end{aligned}$$
Note that $S_k$ can be expressed in terms of the strategy variables as follows: $$\begin{aligned}
&S_k = \sum_{n=0}^{\left\lfloor\frac{m}{2}\right\rfloor}\sum_{j_1<...<j_n}\left(a_{j_1...j_n}-\bar{a}_{j_1...j_n}\right)y_k^{j_1...j_n} \text{ , with} \label{eq:S_k}\\
&y_k^{j_1...j_n} =\begin{cases}-1 \quad \text{if }k\in\{j_1,...,j_n\} \\+1 \quad \text{else}\end{cases} .\label{eq:outcomes}\end{aligned}$$
### Decomposition of $A$ and $C$ in terms of the strategy variables
Equation results in the following representation of $S_k-S_l$: $$\begin{aligned}
S_k-S_l = \sum_{n=0}^{\left\lfloor\frac{m}{2}\right\rfloor}\sum_{j_1<...<j_n}\left(a_{j_1...j_n}-\bar{a}_{j_1...j_n}\right)\left(y_k^{j_1...j_n}-y_l^{j_1...j_n}\right) \text{ .}\end{aligned}$$ Clearly, a variable only appears in this expression if $y_k \neq y_l$, i.e. if the strategy is such that the outcome of the $k^{th}$ measurement differs from the $l^{th}$. This, for example, cannot be the case if the number of indices is zero, i.e. if all measurement outcomes are the same. So $a$ and $\bar{a}$ do not show up in $S_k-S_l$. With the help of the introduced strategy variables, we can express $A$ as $$\begin{aligned}
A=\sum_{k=0}^{\left\lfloor\frac{m}{2}\right\rfloor -1}(m-2k-1) \sum_{n=0}^{\left\lfloor\frac{m}{2}\right\rfloor}\sum_{j_1<...<j_n}\left(a_{j_1...j_n}-\bar{a}_{j_1...j_n}\right)\left(y_k^{j_1...j_n}-y_{m-k-1}^{j_1...j_n}\right) \text{ .}\end{aligned}$$ and $C$ as $$\begin{aligned}
C &= m^2(a+\bar{a}) + (m-2)^2\sum_{j=0}^{m-1}(a_j+\bar{a}_j)+(m-4)^2\sum_{j_1<j_2}(a_{j_1j_2}+\bar{a}_{j_1j_2})+... \nonumber \\
&= \sum_{n=0}^{\left\lfloor\frac{m}{2}\right\rfloor}(m-2n)^2\sum_{j_1<...<j_n}(a_{j_1...j_n}+\bar{a}_{j_1...j_n}) \, .\label{eq:C}\end{aligned}$$ In other words, we notice that if a variable has $n$ indices it contributes to $C$ with a factor $(m-2n)^2$.
### A bound independent of the number of indices
In this section, we study the contributions of $A$ and $C$ to $I_{N,m}$. For this, we make use of the following theorem:
A strategy with $n$ equal outcomes satisfies the inequality $$\begin{aligned}
\sum_{k=0}^{\left\lfloor\frac{m}{2}\right\rfloor-1}\left\vert y_k^{j_1...j_n}-y_{m-k-1}^{j_1...j_n}\right\vert \leq 2n \text{ .}\end{aligned}$$ \[Th1\]
First we note that the summation is such that no $y_k^{j_1...j_n}$ appears twice. Also, we know that since we consider binary outcomes, $\left\vert y_k-y_{m-k-1}\right\vert$ is either $0$ or $2$. We thus have $$\begin{aligned}
\sum_{k=0}^{\left\lfloor\frac{m}{2}\right\rfloor-1}\left\vert y_k^{j_1...j_n}-y_{m-k-1}^{j_1...j_n}\right\vert = 2l \, ,\quad l \in \mathbb{N} \, .\end{aligned}$$ Assume now that the above inequality is violated, i.e. $l>n$.
$\Leftrightarrow y_k^{j_1...j_n} \neq y_{m-k-1}^{j_1...j_n}$ for $l>n$ values of $k$.
$\Leftrightarrow$ The strategy $(j_1,...,j_n)$ has $l$ differing outcomes.
This is a contradiction to the definition of the strategy. Therefore the assumption must be wrong and the inequality holds for all strategies.
A function $f(k)$ which is monotonically decreasing with $k$, satisfies $$\begin{aligned}
\sum_{k=0}^{\left\lfloor\frac{m}{2}\right\rfloor-1} f(k)\left\vert y_k^{j_1...j_n}-y_{m-k-1}^{j_1...j_n}\right\vert \leq 2\sum_{k=0}^{n-1} f(k) \, .\end{aligned}$$ \[Co1\]
Theorem \[Th1\] implies that $y_k \neq y_{m-k-1}$ for at most $n$ values of $k$. Taking into account that $f(k)$ is monotonically decreasing, we find that $$\begin{aligned}
\sum_{k=0}^{\left\lfloor\frac{m}{2}\right\rfloor-1} f(k)\left\vert y_k^{j_1...j_n}-y_{m-k-1}^{j_1...j_n}\right\vert \leq \sum_{\substack{\text{n values} \\ \text{of k}}} f(k)\cdot 2 \leq 2 \sum_{k=0}^{n-1} f(k) \, .\end{aligned}$$
With the help of Corollary \[Co1\], we rewrite the quantity $A$ as $$\begin{aligned}
A &=\sum_{n=0}^{\left\lfloor\frac{m}{2}\right\rfloor}\sum_{j_1<...<j_n}\left(a_{j_1...j_n}-\bar{a}_{j_1...j_n}\right)\sum_{k=0}^{\left\lfloor\frac{m}{2}\right\rfloor-1}\alpha_k \left(y_k^{j_1...j_n}-y_{m-k-1}^{j_1...j_n}\right)\nonumber\\
&\geq \sum_{n=0}^{\left\lfloor\frac{m}{2}\right\rfloor}\sum_{j_1<...<j_n}\left(a_{j_1...j_n}-\bar{a}_{j_1...j_n}\right)(-2)\sum_{k=0}^{n-1} \alpha_k
= -2\sum_{n=0}^{\left\lfloor\frac{m}{2}\right\rfloor}n(m-n)\sum_{j_1<...<j_n}\left(a_{j_1...j_n}-\bar{a}_{j_1...j_n}\right) \, .\end{aligned}$$ Making use of Eq. and , we then find that $$\begin{aligned}
A-\frac12C &\geq \sum_{n=0}^{\left\lfloor\frac{m}{2}\right\rfloor}\left[-2n(m-n)-\frac{(m-2n)^2}{2}\right]\sum_{j_1<...<j_n}\left(a_{j_1...j_n}-\bar{a}_{j_1...j_n}\right) \nonumber\\
&= -\frac{m^2}{2}\sum_{n=0}^{\left\lfloor\frac{m}{2}\right\rfloor}\sum_{j_1<...<j_n}\left(a_{j_1...j_n}-\bar{a}_{j_1...j_n}\right) = -\frac{m^2}{2}N\end{aligned}$$
### Putting the pieces together
In order to conclude the proof, we now only miss the contribution of the term $B$. For this, we look at the case of even and odd $m$ separately. When $m$ is even, $B=\sum_k S_k$ is also even. We thus find that $B^2 \geq 0$. This means that $$\begin{aligned}
I_{N,m} \geq A-\frac12 C \geq -\frac{m^2 N}{2} \, .\end{aligned}$$ If $m$ is odd, $B$ shares the parity of $N$. That is why we have $B^2\geq 0$ for even $N$, and $B^2\geq 1$ for odd $N$, resulting in $$\begin{aligned}
I_{N,m} \geq \begin{cases} -\frac{m^2 N}{2} &\text{ for even N}\\ -\frac{m^2 N}{2}+\frac{1}{2} \quad &\text{ for odd N} \end{cases} \text{ .}\end{aligned}$$
In general, the classical bound is thus $\beta_c=\left\lfloor \frac{m^2 N}{2} \right\rfloor$.
Optimization of the witnesses
=============================
In this appendix, we optimize the witnesses $\mathcal{W}_m$ as given in Eq. of the main text over the measurement angles. Let us remind the form of $\mathcal{W}_m$: $$\begin{aligned}
\mathcal{W}_{m} = {\mathcal{C}}_b {\sum_{k=0}^{\frac{m}{2}-1} \alpha_k \sin(\vartheta_k)} \,{-} {\left(1-\zeta_a^2\right)} {\left[\sum_{k=0}^{\frac{m}{2}-1}\cos(\vartheta_k)\right]^2}\! {+}\, \frac{m^2}{4} \, .
\label{eq:witness.A}\end{aligned}$$ We do this optimization by searching for those angles leading to the minimum of $\mathcal{W}_m$. This is equivalent to solving the system of equations arising from $\frac{\partial \mathcal{W}_m}{\partial \vartheta_k}=0$: $$\begin{aligned}
\frac{\partial\mathcal{W}_m}{\partial\vartheta_k}=(m-2k-1){\mathcal{C}}_b\cos(\vartheta_k)+2\sin(\vartheta_k)\left(1-\zeta_a^2\right)\sum_{l=0}^{\frac{m}{2}-1}\cos(\vartheta_l) = 0 \, . \label{eq:Partial1}\end{aligned}$$ We eventually want to find angles such that $\mathcal{W}_m$ is negative. To achieve this, the last term of Eq. must be compensated. Since $\zeta_a^2 \geq 0$, the second term of Eq. is bounded by $-\frac{m^2}{4}$ and thus cannot be sufficient for a negative $\mathcal{W}_m$. On the other hand, the first term is bounded by $-\frac{m^2}{4}+1$ due to the fact that $\vert {\mathcal{C}}_b\vert\leq 1$. So we find that in order to reach $\mathcal{W}_m<0$, we need $\sin(\vartheta_k)$, $\cos(\vartheta_k)$ and ${\mathcal{C}}_b$ to differ from zero and $\zeta_a^2<1$. In the following studies, we assume these necessary constraints, allowing us to rewrite Eq. as $$\begin{aligned}
\frac{2(1-\zeta_a^2)}{{\mathcal{C}}_b}\sum_{l=0}^{\frac{m}{2}-1}\cos(\vartheta_l) = -(m-2k-1)\frac{\cos(\vartheta_k)}{\sin(\vartheta_k)} \quad \forall k \, . \label{eq:Partial2}\end{aligned}$$ Since the left side of Eq. does not explicitly depend on $k$, this can only be achieved if both sides are equal to a constant. The assumptions about $\zeta_a^2$, ${\mathcal{C}}_b$ and the angles, as reasoned above, allow us to write $$\begin{aligned}
\frac{2(1-\zeta_a^2)}{{\mathcal{C}}_b}\sum_{l=0}^{\frac{m}{2}-1}\cos(\vartheta_l)=\frac{1}{\lambda_m} = -(m-2k-1)\frac{\cos(\vartheta_k)}{\sin(\vartheta_k)}
\, , \label{eq:Partial3}\end{aligned}$$ where $\lambda_m$ is a constant depending for given ${\mathcal{C}}_b$ and $\zeta_a^2$ only on $m$. We find the optimal angles $$\begin{aligned}
\vartheta_k = -\arctan\left(\lambda_m(m-2k-1)\right) \, .\label{eq:atan} \end{aligned}$$ For a minimal $\mathcal{W}_m$, the constants $\lambda_m$ have to fulfill the self-consistency equations $$\begin{aligned}
\frac{{\mathcal{C}}_b}{2\lambda_m(1-\zeta_a^2)}=\sum_{l=0}^{\frac{m}{2}-1}\cos(\vartheta_l)=\sum_{l=0}^{\frac{m}{2}-1}\frac{1}{\sqrt{1+\lambda_m^2(m-2l-1)^2}}\, . \label{eq:selfc}\end{aligned}$$ Here, we used the fact that $\cos(\arctan(x))=\frac{1}{\sqrt{1+x^2}}$. For further steps, we note that $\sin(\arctan(x))=\frac{x}{\sqrt{1+x^2}}$ and define the following functions: $$\begin{aligned}
\Lambda_m(\lambda_m) :&= \sum_{k=0}^{\frac{m}{2}-1}\frac{1}{\sqrt{1+\lambda_m^2(m-2k-1)^2}}
= \,\sum_{k=1}^{\frac{m}{2}}\frac{1}{\sqrt{1+\lambda_m^2(2k-1)^2}} , \label{eq:Lambda.A} \\
\Delta_m(\lambda_m) :&= \sum_{k=0}^{\frac{m}{2}-1}\frac{\lambda_m(m-2k-1)^2}{\sqrt{1+\lambda_m^2(m-2k-1)^2}}
=\sum_{k=1}^{\frac{m}{2}}\frac{\lambda_m(2k-1)^2}{\sqrt{1+\lambda_m^2(2k-1)^2}} \label{eq:Delta.A} \, .\end{aligned}$$ If we assume the representation of the angles given in Eq. , the witnesses can be expressed as $$\begin{aligned}
\mathcal{W}_m = - {\mathcal{C}}_b\Delta_m(\lambda_m)-\left(1-\zeta_a^2\right)\Lambda_m^2(\lambda_m) +\frac{m^2}{4} \geq 0\, , \label{eq:finalW.A} \end{aligned}$$ which holds for non-Bell-correlated states. Note that in this expression, we only assume the $arctan$-angle-distribution, without taking the self-consistency equations into account, i.e. without optimizing the actual differences between angles.
For the case of $m\to\infty$, we need to rewrite the above witnesses, since $\mathcal{W}_m$ diverges in this limit. We define $$\begin{aligned}
\mathcal{W}_m':= \frac{\mathcal{W}_m}{m^2} = -{\mathcal{C}}_b\frac{\Delta_m(\lambda_m)}{m^2}-\left(1-\zeta_a^2\right)\left(\frac{\Lambda_m(\lambda_m)}{m}\right)^2 +\frac{1}{4} \geq 0 \, .
\label{eq:witness_inf}\end{aligned}$$ We also have to rewrite the constant $\lambda_m$. We define $\lambda_m=\frac{\nu_m}{m}$ and rewrite Eq. as $$\begin{aligned}
\frac{{\mathcal{C}}_b}{2\nu_m(1-\zeta_a^2)}=\frac{\Lambda_m\left(\frac{\nu_m}{m}\right)}{m} \, . \label{eq:selfc_nu}\end{aligned}$$ If we define $s_k=\frac{2k-1}{m}$, we see that $\frac1m$ can be expressed as $\frac{s_{k+1}-s_k}{2}$. Note that for $m\to\infty$, $s_1\to 0$ and $s_{m/2}\to 1$. Using the convention $\nu_\infty=\nu$, we find $$\begin{aligned}
\Lambda_\nu :=\lim_{m \to \infty} \frac{\Lambda_m(\nu_m/m)}{m} &= \lim_{m \to \infty} \sum_{k=1}^{\frac{m}{2}}\frac{1}{\sqrt{1+\nu_m^2\frac{(2k-1)^2}{m^2}}} \cdot\frac{1}{m} =\lim_{m \to \infty}\sum_{k=1}^{\frac{m}{2}}\frac{1}{\sqrt{1+\nu_m^2 s_k^2}}\frac{s_{k+1}-s_k}{2} \nonumber\\
&= \frac12\int_0^1 \frac{1}{\sqrt{1+\nu^2 s^2}}ds =\frac{\mathrm{arsinh}(\nu)}{2\nu} \label{eq:LambdaInf}\\
\Delta_\nu:= \lim_{m \to \infty} \frac{\Delta_m(\nu_m/m)}{m^2} &= \lim_{m \to \infty}\sum_{k=1}^{\frac{m}{2}}\frac{\nu_m \frac{(2k-1)^2}{m^2}}{\sqrt{1+\nu_m^2\frac{(2k-1)^2}{m^2}}}\cdot\frac{1}{m} = \lim_{m \to \infty}\sum_{k=1}^{\frac{m}{2}} \frac{\nu_m s_k^2}{\sqrt{1+\nu_m^2 s_k^2}}\frac{s_{k+1}-s_k}2 \nonumber \\
&= \frac12 \int_0^1 \frac{\nu s^2}{\sqrt{1+\nu^2s^2}}ds = \frac{\sqrt{1+\nu^2}}{4\nu}-\frac{\mathrm{arsinh}(\nu)}{4\nu^2} \label{eq:DeltaInf} \, .\end{aligned}$$ This yields the witness $$\begin{aligned}
\mathcal{W}_\infty'&=-{\mathcal{C}}_b\Delta_\nu - (1-\zeta_a^2)\Lambda_\nu+\frac14\\
&= -{\mathcal{C}}_b\left(\frac{\sqrt{1+\nu^2}}{4\nu}-\frac{\mathrm{arsinh}(\nu)}{4\nu^2}\right) -(1-\zeta_a^2)\frac{\mathrm{arsinh}^2(\nu)}{4\nu^2} +\frac14 \geq 0 \, .
\label{eq:WInf}\end{aligned}$$
We now search for those points in the ${\mathcal{C}}_b$-$\zeta_a^2$-plane that allow for a violation of the correlation witnesses of Ineq. and . For this purpose, we assume the optimal angles given in Eq. and respectively. We define $Z_m({\mathcal{C}}_b)$ to be the scaled second moment, as a function of the scaled collective spin, such that $\mathcal{W}_m$ vanishes.
### $m=2$
In the case of $m=2$ measurement settings, we have to solve the following system of equations in order to find $Z_2$: $$\begin{aligned}
0 =\mathcal{W}_2 &= -{\mathcal{C}}_b \frac{\lambda_2}{\sqrt{1+\lambda_2^2}}-(1-\zeta_a^2)\frac{1}{1+\lambda_2^2}+1 \, ,\\
\frac{{\mathcal{C}}_b}{2\lambda_2(1-\zeta_a^2)} &= \frac{1}{\sqrt{1+\lambda_2^2}} \, .\end{aligned}$$ From these, we find the critical line $Z_2$ and therefore the following condition, satisfied by every non-Bell-correlated state: $$\begin{aligned}
\zeta_a^2 \geq Z_2({\mathcal{C}}_b) = \frac12\left(1-\sqrt{1-{\mathcal{C}}_b^2}\right) \, . \label{eq:Z2.A}\end{aligned}$$
### Limit $m\to \infty$
The critical line $Z_\infty$ is determined by solving the following equation for $\zeta_a^2$ $$\begin{aligned}
0 = \mathcal{W}_\infty' = -{\mathcal{C}}_b\left(\frac{\sqrt{1+\nu^2}}{4\nu}-\frac{\mathrm{arsinh}(\nu)}{4\nu^2}\right) -(1-\zeta_a^2)\frac{\mathrm{arsinh}^2(\nu)}{4\nu^2} +\frac14 \, , \quad\text{where } \nu=\sinh\left(\frac{{\mathcal{C}}_b}{1-\zeta_a^2}\right) \, .
\label{eq:bestNu}\end{aligned}$$ We find that any non-Bell-correlated state satisfies $$\begin{aligned}
\zeta_a^2\geq Z_\infty({\mathcal{C}}_b)=1-\frac{{\mathcal{C}}_b}{\mathrm{artanh}({\mathcal{C}}_b)} \, . \label{eq:ZInf.A}\end{aligned}$$
Squeezing requirement
=====================
Here, we find a bound on the amount of squeezing that is needed as a function of the number of spins $N$ in order to violate the Bell correlation witness described in the main text. Due to the structure of spin systems, the first moment ${\mathcal{C}}_b$, the second moment $\zeta_a^2$ and the number of spins $N$ satisfy the following constraints [@Sorensen01]: $$\zeta_a^2 \geq 1-\frac{N}{2}\left[\sqrt{(1-{\mathcal{C}}_b^2)\left[\left(1+\frac{2}{N}\right)^2 - {\mathcal{C}}_b^2\right]} + {\mathcal{C}}_b^2 - 1\right]$$ (notice that there is an error in the expression given in the reference). For any number of spins $N$, equating the right-hand side of this constraint with the right-hand side of Eq. gives the maximum value of ${\mathcal{C}}_b$ under which a violation of the witness is possible. The corresponding maximum value of $\zeta_a^2$ is plotted as a function of the number of spins $N$ in Fig. \[fig:zetaUpper\]. For large $N$, this function can be expanded as $$Z_N^* = 1-\frac{1}{\omega}-\frac1{2\omega^3}-\frac3{4\omega^4}-O(\omega^{-5})$$ where $\omega=W_{-1}\left(-\frac{1}{2\sqrt{N+1}}\right)$ and $W_{-1}$ is the lower branch of the Lambert $W$ function.
Finite statistics
=================
In this appendix, we introduce four concentration inequalities and determine their bound on the p-value for generic non-Bell-correlated states. We also compare these p-values and choose the optimal one to estimate a number of experimental runs sufficient to exclude non-Bell-correlated states with a confidence $1-\varepsilon$. Eventually we minimize this number of runs by optimizing the measurement angles.
Concentration inequalities
--------------------------
In statistics, concentration inequalities bound the probability that a random variable $X$ exceeds or falls below a certain value. In what follows, we recall the definitions of some of these inequalities. We then discuss some of their properties in view of our problem in the following section.
### Chernoff bound
The following version of the Chernoff bound was proven by Van Vu at the University of California, San Diego [@Chernoff]. It was done for discrete, independent random variables. However, the bound also applies in the case of continuous random variables.
Let $X_1,...,X_M$ be independent random variables with $\vert X_i \vert \leq 1$ and expectation values $E[X_i]=0$ for all $i$. Let $X=\sum\limits_{i=1}^M X_i$ and $\sigma^2$ be the variance of $X$. Then $$\begin{aligned}
&P[X \leq - \lambda \sigma] \leq \exp\left(-\frac{\lambda^2}{4}\right) \, , &\text{for } 0 \leq \lambda \leq 2\sigma \, ,\quad\\
&P[X\leq -x_0] \leq \exp\left(-\frac{x_0^2}{4\sigma^2}\right) \, , &\text{for } 0 \leq x_0 \leq 2\sigma^2 \, .\,\end{aligned}$$ \[th:Chern\]
Let $X_1,...,X_M$ be independent random variables with $E[X_i]=\mu$ and $a\leq X_i\leq b$ for all $i$. Let $X=\frac1M \sum\limits_{i=1}^M X_i$, $\sigma_i^2=\mathrm{Var}[X_i]$ and $\sigma_0^2=\max\limits_i\left\{\sigma_i^2\right\}$. Then $$\begin{aligned}
&P\left[X\leq x_0\right] \leq \exp\left( -\frac{(\mu-x_0)^2 M}{4 \sigma_0^2}\right) \, , &\text{for } \mu \geq x_0 \geq \mu-\frac{\sum_i \sigma_i^2}{M(b-a)} \, .\end{aligned}$$ \[cor:Chern\]
### Bernstein inequality
The following expression known as Bernstein inequality, was proven by Bernstein in 1927, but we refer to the work of George Bennett [@Bernstein62]. The inequality is valid under certain restrictions for the absolute moments. Since our random variables are bounded, we can be sure these restrictions to be fulfilled.
Let $X_1,...,X_M$ be independent random variables with $E[X_i]=0$ and $\vert X_i \vert \leq \xi$ for all $i$. Also let $X=\frac{1}{M}\sum\limits_{i=1}^M X_i$ and $\sigma^2=\frac{1}{M}\sum\limits_{i=1}^M \mathrm{Var}[X_i]$. Then $$\begin{aligned}
&P[X \geq x_0] \leq \exp\left(-\frac{x_0^2 M}{2\sigma^2+\frac23 \xi x_0}\right)\, , &\forall x_0> 0 \, , \\
&P[X \leq -x_0] \leq \exp\left(-\frac{x_0^2 M}{2\sigma^2+\frac23 \xi x_0}\right)\, , &\forall x_0> 0 \, .\end{aligned}$$ \[th:bern\]
Let $X_1,...,X_M$ be independent random variables with $E[X_i]=\mu$, $\sigma_i^2=\mathrm{Var}[X_i]$ and $a\leq X_i \leq b$ for all $i$. Also let $X=\frac{1}{M}\sum\limits_{i=1}^M X_i$ and $\sigma^2=\frac{1}{M}\sum\limits_{i=1}^M \sigma_i^2\leq \sigma_0^2 = \max\limits_i\{\sigma_i^2\}$. Then $$\begin{aligned}
P[X \leq x_0] &\leq \exp\left(-\frac{(\mu-x_0)^2 M}{2\sigma^2+\frac23 (b-a) (\mu-x_0)}\right) \\
&\leq \exp\left(-\frac{(\mu-x_0)^2 M}{2\sigma_0^2+\frac23 (b-a) (\mu-x_0)}\right)\, , &\forall x_0< \mu \, .\end{aligned}$$ \[cor:Bern\]
### Uspensky inequality
Uspensky stated in [@Usp] an inequality for a stochastic variable:
Let $X$ be a random variable with mean value $E[X]=0$ and $a\leq X\leq b$. Additionally $b\geq \vert a \vert$. Let $\sigma^2=\mathrm{Var}[X]$. Then for $x_0\leq 0$ $$\begin{aligned}
P[X\leq x_0] \leq \frac{\sigma^2}{\sigma^2+x_0^2} \, . \end{aligned}$$
Let $X_1,...,X_M$ be independent random variables with mean values $E[X_i]=\mu$ and $a\leq X_i\leq b$ for all $i$. Additionally $b\geq \vert a \vert$. Let $\sigma_i^2= \mathrm{Var}[X_i]$ and $\sigma_0^2=\max\limits_i\left\{\sigma_i^2\right\}$. Then $\sigma^2=\mathrm{Var}[X]=\mathrm{Var}\left[\frac1M \sum\limits_{i=1}^M X_i\right] = \frac{1}{M^2}\sum\limits_{i=1}^M \sigma_i^2\leq \frac1M \sigma_0^2$ so that for $x_0\leq \mu$ $$\begin{aligned}
P[X\leq x_0]\leq \frac{\sigma_0^2}{\sigma_0^2+(x_0-\mu)^2 M} \, .\end{aligned}$$ \[cor:Usp\]
### Berry-Esseen inequality
Andrew C. Berry and Carl-Gustav Esseen proved the following theorem [@Berry]:
Let $X_1,...,X_M$ be independent identically distributed random variables with $E[X_i]=\mu$ for all $i$ and with $X=\frac1M \sum\limits_{i=1}^M X_i$. Additionally, the variance $\sigma^2=\mathrm{Var}[X_i]=E[(X_i-\mu)^2]$ and the third absolute moment $\rho=E\left[\vert X_i-\mu\vert^3\right]$ are finite. Then there exists a constant $C$ such that for all $x$ $$\begin{aligned}
\vert F_M(x)-\Phi(x)\vert \leq \frac{C \rho}{\sigma^3 \sqrt{M}} \, , \end{aligned}$$ where $F_M(x)=P\left[\sqrt{\frac{M}{\sigma^2}}(X-\mu)\leq x\right]$ and $\Phi(x)=\frac12\left(1+\mathrm{erf}(x)\right)$.
Esseen also proved that the constant $C$ has to fulfill $C\geq\frac{\sqrt{10}+3}{6\sqrt{2\pi}}$. A good estimate for $C$ follows from $$\begin{aligned}
\vert F_M(x)-\Phi(x)\vert \leq \frac{0.33554(\rho+0.415\sigma^3)}{\sigma^3 \sqrt{M}} \, . \end{aligned}$$ \[th:berry\]
A direct consequence of Theorem \[th:berry\] is $$\begin{aligned}
P\left[X\leq x_0\right]&=P\left[\sqrt{\frac{M}{\sigma^2}}(X-\mu)\leq \sqrt{\frac{M}{\sigma^2}}(x_0-\mu)\right] \\
&\leq \Phi\left(\sqrt{\frac{M}{\sigma^2}}(x_0-\mu)\right) + \frac{C\rho}{\sigma^3 \sqrt{M}} \, .\end{aligned}$$ \[cor:Berry\]
Largest p-value of the concentration inequalities
-------------------------------------------------
In this section we determine the largest p-value for the concentration inequalities listed above, under the assumption that $x_0<0$ and $\mu\geq 0$. We denote with $X$ the random variable $\frac1M\sum_{i=1}^M X_i$, with $x_l\leq X_i \leq x_u$ and $x_l<0$.
The crucial point here is that we have no information about the random variable apart from its non-negative mean value. This lack of information directly disqualifies the Berry-Esseen inequality of Corollary \[cor:Berry\] as a potential tight bound. Indeed, the Berry-Esseen bound does not result in a tighter restriction than the trivial bound $P[X\leq x_0] \leq 1$. To see this, we consider the following distribution with three peaks:
Consider the case, for which all $X_i$ satisfy the following probability distribution: $$\begin{aligned}
P[X_i=x]=
\begin{cases}
p_l \quad &\text{ for }x=x_l \\
p_u \quad &\text{ for }x=x_u \\
1-p_l-p_u \quad &\text{ for }x=0 \\
0 \quad &\text{ else}
\end{cases} \, ,\end{aligned}$$ where $p_l, p_u<1$. Additionally, we demand $E[X_i]=0$ leading to the condition $-p_l x_l =p_u x_u$. We thus arrive at the variance and third absolute moment: $$\begin{aligned}
\sigma^2 &= p_u x_u(x_u-x_l)\, , \\
\rho &= p_u x_u(x_u^2+x_l^2) \\
\Rightarrow \frac{\rho}{\sigma^3} &= \frac{x_u^2+x_l^2}{\sqrt{p_u x_u(x_u-x_l)^3}} \, . \label{eq:tri}\end{aligned}$$ In this expression, $p_u$ remains as a parameter scaling the weight on the edges compared to the weight at $x=0$. Note that for $p_u\to \frac{-x_l}{x_u-x_l}$, we arrive at a binomial distribution while for $p_u\to 0$ we have a delta distribution. From Eq. , we see that in the limit $p_u\to 0$, $\frac{\rho}{\sigma^3} \to \infty$. Thus, we can write for the Berry-Esseen bound $$\begin{aligned}
P[X\leq x_0] &\leq \Phi\left(\sqrt{\frac{M}{\sigma^2}}(x_0-\mu)\right) + \frac{C\rho}{\sigma^3 \sqrt{M}} \leq \frac{C\rho}{\sigma^3 \sqrt{M}} = \frac{C}{\sqrt{M}}\frac{x_u^2+x_l^2}{\sqrt{p_u x_u(x_u-x_l)^3}} \xrightarrow[p_u \to 0]{} \infty \, .\end{aligned}$$ Since we have no information on the actual probability distribution, the largest p-value of the Berry-Esseen bound is $1$. We thus restrict our interests to the Chernoff, Bernstein and Uspensky bounds.
The inequalities of Corollaries \[cor:Chern\], \[cor:Bern\] and \[cor:Usp\] have certain properties in common: They all depend on the variance $\sigma_0^2$ and the mean value $\mu$. More explicitly, the dependence on $\sigma_0^2$ in all three cases is such that if one increases the variance, the bounds are also increased. We therefore use the following strategy to determine the largest p-values: We increase the variance of an arbitrary random variable in a way leaving the mean value unaffected. We eventually arrive at an easy-to-handle probability distribution with a maximal variance. The bounds resulting from this distribution then serve as upper bounds for all distributions of the same mean value. The bounds given by the concentration inequalities will then only depend on the mean value, so that we can optimize over $\mu$.
Eventually, we show that the largest p-value results from the binomial distribution centered around $\mu=0$.
Let $X$ be a random variable in the interval $[a,b]$ with an arbitrary probability distribution and with $E[X]=\mu$. Let $X_{bi}$ be a binomially distributed random variable with peaks at the edges $a$ and $b$. Furthermore, $X_{bi}$ has the same mean value as $X$, i.e. $P[X_{bi}=a]=\frac{\mu-b}{a-b}$, $P[X_{bi}=b]=\frac{a-\mu}{a-b}$ and $P[X_{bi}=x]=0$ otherwise. Additionally, let $a\leq 0$. Then $$\begin{aligned}
\mathrm{Var}[X_{bi}]\geq \mathrm{Var}[X] \, .\end{aligned}$$ \[th:Var\]
We define a third random variable $Y$ satisfying $$\begin{aligned}
P[Y=x]=
\begin{cases} P[X=x] &\text{if } x \notin dx_0\cup\{a,b\} \\
0 & \text{if } x \in dx_0 \\
P[X=a]+qP[X\in dx_0] &\text{if } x=a \\
P[X=b]+(1-q)P[X\in dx_0] &\text{if } x=b
\end{cases} \, .\end{aligned}$$ Here, $dx_0$ is an infinitesimal set around $x_0\in ]a,b[$. So in other words, the probability distribution function of $Y$ is almost the same as the one of $X$. The only difference is that the set $dx_0$ is “cut out” and the probabilities at $a$ and $b$ are increased (see Fig. \[fig:Var\]). They are increased in a way which leaves the mean value unaffected, i.e. $q$ is chosen such that $E[Y]=E[X]=\mu$: $$\begin{aligned}
E[Y] &= E[X]-P[X\in dx_0]x_0+qP[X\in dx_0]a+(1-q)P[X\in dx_0]b \nonumber\\
&= \mu + P[X\in dx_0][-x_0+q(a-b)+b] = \mu \nonumber\\
&\Leftrightarrow \quad q=\frac{x_0-b}{a-b} \, .\end{aligned}$$ With this and $a\leq 0$, we show that $\mathrm{Var}[Y]\geq \mathrm{Var}[X]$: $$\begin{aligned}
\mathrm{Var}[Y] &= \mathrm{Var}[X]+P[X\in dx_0]\left[-x_0^2+q(a^2-b^2)+b^2\right] = \mathrm{Var}[X]+P[X\in dx_0]\left[-x_0^2+x_0(a+b)-ab\right]\nonumber\\
&\geq \mathrm{Var}[X]+P[X\in dx_0]\left[-x_0^2+x_0(a+x_0)-a x_0\right] \geq \mathrm{Var}[X]+P[X\in dx_0]\left[-x_0^2+x_0^2\right] = \mathrm{Var}[X] \, .\end{aligned}$$ So we find that $\mathrm{Var}[Y]\geq \mathrm{Var}[X]$, with the equal sign only if $P[X\in dx_0]=0$. By induction, one can gradually “cut out” all the other points in $]a,b[$. During this process, the variance is constantly increased while the mean value remains unchanged. Eventually, one arrives at the random variable $X_{bi}$ with its binomial distribution.
![Sketch of the probability distribution function of $Y$ in Theorem \[th:Var\].[]{data-label="fig:Var"}](proof2.pdf){width="80.00000%"}
By applying the different bounds and using Theorem \[th:Var\], we find $$\begin{aligned}
P[X\leq x_0]\leq
\begin{cases}
\exp\left(-\frac{(\mu-x_0)^2 M}{4\sigma_{bi}^2(\mu)}\right) =:p_C(\mu,M) &\text{ Chernoff} \\
\\
\exp\left(-\frac{(\mu-x_0)^2 M}{2\sigma_{bi}^2(\mu)+\frac23(x_u-x_l)(\mu-x_0)}\right) =:p_B(\mu,M) &\text{ Bernstein}\\
\\
\frac{\sigma_{bi}^2(\mu)}{\sigma_{bi}^2(\mu)+(x_0-\mu)^2 M} =:p_U(\mu,M) &\text{ Uspensky}
\end{cases} \, ,\end{aligned}$$ where $\sigma_{bi}^2(\mu)=(x_u-\mu)(\mu-x_l)$. As stated above, $\mu\in[0,x_u]$. Restricted to this interval, the three functions $p_C$, $p_B$ and $p_U$ are strictly monotonous decreasing with $\mu$ and therefore take their maximal values at $\mu=0$. This allows us to write $$\begin{aligned}
P[X\leq x_0]\leq
\begin{cases}
\exp\left(-\frac{x_0^2 M}{4\sigma_{bi}^2}\right) =:p_C(M) &\text{ Chernoff} \\
\\
\exp\left(-\frac{x_0^2 M}{2\sigma_{bi}^2-\frac23(x_u-x_l)x_0}\right) =:p_B(M) &\text{ Bernstein}\\
\\
\frac{\sigma_{bi}^2}{\sigma_{bi}^2+x_0^2 M} =:p_U(M) &\text{ Uspensky}
\end{cases} \, ,\end{aligned}$$ with $\sigma_{bi}^2:=\sigma_{bi}^2(0)=-x_l x_u$.
We identify $M$ with the number of experimental runs, and determine the minimum number of runs required to have $P[X\leq x_0]\leq \varepsilon$. This corresponds to solving $p_i(M)\leq \varepsilon $ to $M$, where $i=C,B,U$. We find $$\begin{aligned}
M\geq
\begin{cases}
M_C := \frac{4\sigma_{bi}^2}{x_0^2}\ln\left(\frac{1}{\varepsilon}\right) &\text{ Chernoff} \\
\\
M_B := \frac{2\sigma_{bi}^2-\frac23(x_u-x_l)x_0}{x_0^2}\ln\left(\frac{1}{\varepsilon}\right) &\text{ Bernstein}\\
\\
M_C := \frac{\sigma_{bi}^2(1-\varepsilon)}{\varepsilon x_0^2} &\text{ Uspensky}
\end{cases} \, .
\label{eq:bounds}\end{aligned}$$
Comparison of the bounds
------------------------
We now compare the three bounds stated in Ineq. and show that in the regime of interest, the Bernstein bound is the tightest bound. By comparing $M_B$ to $M_C$, one finds $$\begin{aligned}
M_B\cdot\frac{x_0^2}{\ln(1/\varepsilon)}&=2\sigma_{bi}^2-\frac23(x_u-x_l)x_0<2\sigma_{bi}^2-(x_u-x_l)x_0 \nonumber\\
&<-2x_l x_u-x_u x_l-x_l x_u =-4x_l x_u = 4\sigma_{bi}^2= M_C\cdot\frac{x_0^2}{\ln(1/\varepsilon)} \nonumber\\
\Rightarrow \, M_B&<M_C \, .\end{aligned}$$ This means the bound resulting from Bernstein’s inequality is tighter than the Chernoff bound for all values of $x_l$, $x_u$ and $x_0$.
Now we compare $p_B$ to $p_U$. Plotting both functions reveals that Uspensky’s inequality is better for small numbers of experimental runs, whereas Bernstein’s is better for larger numbers of runs. We now want to estimate for which $\varepsilon$ both approximations require the same number of runs $M$. We therefore set $M_B=M_U$: $$\begin{aligned}
\frac{2\sigma_{bi}^2-\frac23(x_u-x_l)x_0}{x_0^2}\ln\left(\frac{1}{\varepsilon}\right) =\frac{\sigma_{bi}^2(1-\varepsilon)}{\varepsilon x_0^2} \,\Leftrightarrow\, 2+\frac23 \frac{(x_u-x_l)(-x_0)}{\sigma_{bi}^2} = \frac{1-\varepsilon}{\varepsilon\ln\left(\frac{1}{\varepsilon}\right)} \, . \label{eq:intersect}\end{aligned}$$ The function on the right side of Eq. is strictly monotonous decreasing with $\varepsilon$. The left side is just a number. So the minimal $\varepsilon$ possible is achieved if the left side is maximized. We thus make the following estimation: $$\frac{(x_u-x_l)\vert x_0\vert}{\sigma_{bi}^2}=\frac{(x_u+\vert x_l\vert)\vert x_0\vert}{\vert x_l\vert x_u} \leq \frac{x_u\vert x_l\vert + \vert x_l\vert x_u}{\vert x_l\vert x_u} = 2 \, .$$ For this value, Eq. yields $\varepsilon\approx 0.127$. This means that the Uspensky bound can only be better than the Bernstein bound for $\varepsilon\geq 0.127$. Since we are interested in probabilities $\varepsilon\leq 0.1$, the Bernstein bound remains the preferred one.
Statistical optimization {#sec:exp}
------------------------
Let us now present the result of numerical studies on the number of measurement runs allowing one to rule out non-Bell-correlated states. Here we rely on the Bernstein bound presented in Ineq. .
### Choice of settings
First, we study the choice of measurement settings, i.e. the choice of $\nu$ in Eq. , which allow for the strongest statistical claim. For this, we minimize the number of experimental runs sufficient to conclude about the presence of a Bell-correlated state with given confidence $1-\varepsilon$ over the choices of $\nu$. We then compare this number of measurements $M$ to the one obtained when choosing the $\nu=\text{sinh}\left(\frac{{\mathcal{C}}_b}{1-\zeta_a^2}\right)$, i.e. for the settings which maximize the witness value (see Eq. ).
The result of this comparison is presented in Figure \[fig:VarNu\] for a particular choice of ${\mathcal{C}}_b$ and $\zeta_a^2$. Clearly, it is advantageous to reoptimize the measurement setting in order to maximize the statistical evidence, and thus a different witness should be considerd in this case. In the rest of this appendix, measurement settings are optimized in order to maximize statistical evidence.
![Plots of the required number of experimental runs $M$ per spin for a confidence of $\varepsilon$ in units of $\ln(\frac1\varepsilon)$, as a function of the number of spins $N$. The plots are for $m=2,4,\infty$, with $\zeta_a^2=0.272$ and ${\mathcal{C}}_b=0.98$. The ratios tend to constants for larger systems.[]{data-label="fig:RatioMN"}](OptimalNu.pdf){width="95.00000%"}
![Plots of the required number of experimental runs $M$ per spin for a confidence of $\varepsilon$ in units of $\ln(\frac1\varepsilon)$, as a function of the number of spins $N$. The plots are for $m=2,4,\infty$, with $\zeta_a^2=0.272$ and ${\mathcal{C}}_b=0.98$. The ratios tend to constants for larger systems.[]{data-label="fig:RatioMN"}](RatioMN.pdf){width="0.95\linewidth"}
### Linear relation between the number of measurement runs and the number of spins
Figure \[fig:RatioMN\] depicts the number of measurement runs per spin needed in order to reach a confidence of $1-\varepsilon$. The plots are for the values of ${\mathcal{C}}_b$ and $\zeta_a^2$ stated in [@Schmied16], and for $m=2,4,\infty$ settings. As explained in the main text, we observe that the ratio $\frac{M}{N}$ tends to a constant as $N$ increases. This plot also illustrates the gain obtained in using the newly derived Bell inequality with $m$ settings.
### Minimum squeezing requirement for finite $M$
As discussed in the main text, violation of our witness requires a finite amount of squeezing for systems of finite size $N$. Here we study this requirement when finite number of measurement runs are taken into account. In Fig. \[fig:Ca2vsN\] we show the upper bound on $\zeta_a^2$ as a function of the number of spins $N$ and the number of measurement runs $M$ for the case ${\mathcal{C}}_b=0.98$, $\varepsilon=0.1$. Although Fig. 2 in the main text shows that a violation with $N=1000$ spins is possible when $\zeta_a^2 \lesssim 0.8$, we see here that a squeezing of $\zeta_a^2 \lesssim 0.5$ is needed if the number of measurement is limited to $10^6$.
![Plots of the required scaled second moment $\zeta_a^2$ as a function of the number of atoms $N$, with ${\mathcal{C}}_b=0.98$ and $\varepsilon=0.1$. The different curves represent different numbers of experimental runs ($M=10^4,10^5,10^6$).[]{data-label="fig:Ca2vsN"}](VariousM.pdf){width="43.00000%"}
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abstract: 'Submillimeter observations are a key for answering many of the big questions in modern-day astrophysics, such as how stars and planets form, how galaxies evolve, and how material cycles through stars and the interstellar medium. With the upcoming large submillimeter facilities ALMA and Herschel a new window will open to study these questions. ARTIST is a project funded in context of the European ASTRONET program with the aim of developing a next generation model suite for comprehensive multi-dimensional radiative transfer calculations of the dust and line emission, as well as their polarization, to help interpret observations with these groundbreaking facilities.'
---
Introduction
============
The Atacama Large Millimeter Array is the largest ground based project in astronomy. It will be the world’s most powerful instrument for millimeter and submillimeter astronomy, providing enormous improvements in sensitivity, resolution and imaging fidelity in these wavelength bands. The main focus of the early use of ALMA, compared to present-day facilities, will be its high angular resolution and sensitivity. For studies of star formation, for example, ALMA will zoom in to AU scales in circumstellar disks in nearby star forming regions and thereby address some of the key questions in disk formation. It will also open up new possibilities in the study of planet formation, resolving the effects of planets on the disks around young stars and thus providing direct observational constraints on planet formation models. At the same time, the Herschel Space Observatory is providing high spatial and spectral resolution observations at wavelengths unobservable from the ground, in particular, of lines of H$_{2}$O and high excitation transitions of molecules tracing the dense warm gas, e.g., in the innermost regions of protostars or in shocks.
The simultaneous observation of dust and of a multitude of spectral lines opens new horizons for the physical and chemical analysis of the objects, e.g., to determine the excitation conditions (density, temperature, radiation), the chemical abundances and chemical network, and through polarization measurements, the magnetic field. It will be necessary to take all these observational constraints into account for a realistic quantitative description. ALMA will offer a new chance to study magnetic fields: due to receiver constraints, full polarization calibration and imaging will be the norm rather than the exception, making it essential that a self-consistent polarization modeling tool is available to the ALMA users.
With the novelty of these observations it will be critical to have an efficient, flexible and state-of-the-art modeling package that can provide a direct link between the theoretical predictions and the quantitative constraints from the submillimeter observations. The new observational opportunities require a new generation of modeling tools that can model the full multi-dimensional structure of, e.g., a low-mass protostar, including its envelope, disk, outflow and magnetic field, and their time evolution. Current tools are inadequate for the modeling of such complex structures because of their speed and inaccessibility, while tools to model polarization are completely lacking. Both ALMA and Herschel will provide us with large samples of sources, observed homogeneously as part of large key and legacy programs. This makes it prudent to have easily accessible and efficient tools, which with high convergence speed incorporate all observational constraints, for large source samples, and in a systematic fashion. It is the aim of this program to provide such an innovative suite of model tools and test it with existing submillimeter data and with new data from ALMA and Herschel.
Objectives
==========
The goal of this project is to deliver a next generation radiative transfer modeling package that provides a self-consistent model for the emission of a multi-dimensional source observed at submillimeter wavelengths. For this project we are specifically motivated, without any loss of more general applicability, by low-mass star formation. Our modeling package shall be able to provide a self-consistent modeling tool for the line and continuum as well as polarization emission from, e.g., a young stellar object, incorporating an infalling large-scale envelope, rotationally decoupled protoplanetary disk, outflow cavity, and magnetic field - with no restrictions of the intrinsic geometry. With such an innovative model tool it will be possible to provide quantitative constraints on the relation between the large-scale angular momentum in the core and the disk evolution, on the direct impact of the outflows and their launching close to the disk surface, and on the importance of the magnetic field. Two important issues need to be addressed in this particular example: $(i)$ young stellar objects are characterized by structure on a wide range of spatial scales and with complex geometries, but current radiative transfer tools are locked to fixed linear or logarithmic grids and can therefore model multi-dimensional source structures only with great computational time expense; $(ii)$ while most current observations indicate that magnetic fields play an important role in various stages of the star formation process, their relation to the physical source structure is yet poorly constrained. ALMA observations will study the magnetic fields through resolved line and continuum polarization observations that can not be properly analyzed with current models. The modeling tool will also be useful for tackling scientific questions relating to ALMA observations of, e.g., evolved stars, planetary nebulae or extragalactic starbursts. Providing radiative transfer tools that take complex source structures into account will also be of great help to interpret Herschel observations, e.g., to understand the origin of H$_{2}$O and high excitation CO lines in the interface between protostellar cores and outflows or jets. There is a high degree of coupling between different modeling aspects: e.g., to understand the polarization of a given molecule’s emission it is necessary to understand the physical conditions and chemical network that leads to the molecule’s formation and excitation. For complex source structures it is necessary to develop an approach to time-dependent multi-dimensional modeling, e.g., through libraries of theoretical model prescriptions that can readily be incorporated and expanded for comparison to the data.
Tools
=====
The planned model suite will have the following three components: $(a)$ an innovative radiative transfer code using adaptive gridding that allows simulations of sources with arbitrary multi-dimensional and time-dependent structures ensuring a rapid convergence and thus allowing an exploration of parameters; $(b)$ unique tools for modeling the polarization of the line and dust emission, information that will come with standard ALMA observations; $(c)$ a comprehensive Python-based interface connecting these packages, thus with direct link to, e.g., ALMA data reduction software (CASA). A schematic overview of the program is shown in Fig. \[schematic\].\
ALMA’s high resolution data will produce the need to model phenomena with non-symmetric structures, such as spiral-waves, proto-planet resonances in evolving circumstellar disks, close protostellar binaries etc. Conventional radiative transfer tools use simple linear or logarithmic spatial grids. To model complex source structures in higher dimensions thus requires increasingly finer grid scaling, which becomes very difficult to handle computationally. As an alternative, the SimpleX algorithm ([@RitzerveldIcke06 Ritzerveld & Icke 2006]) uses a Poisson method to define a grid based on the density distribution. The cornerstone of ARTIST is the 3D line radiative transfer code, LIME (Brinch & Hogerheijde 2010), which utilizes the SimpleX gridding algorithm in a 3D extension to the RATRAN radiative transfer code ([@HogerheijdevanderTak00 Hogerheijde & van der Tak 2000]). LIME is currently being applied to for example modeling new H$_{2}$O observations of protostars and disks from the Herschel Space Observatory. For these models the adaptive gridding method ensures rapid convergence, for lines from molecules such as H$_{2}$O that are far from LTE excitation.\
Various theoretical studies predict that magnetic fields, turbulence or/and magneto-hydrodynamic waves may be the main agents controlling both the evolution of molecular clouds and the star-formation process (e.g., [@BertoldiMcKee92 Bertoldi & McKee 1992]; [@MacLowKlessen04 Mac Low & Klessen 2004]; [@Mouschovias06 Mouschovias et al. 2006]; [@vanLoo07 van Loo et al. 2007]). Unfortunately, the magnetic field is the least-known observable in star formation, due to the inherent difficulty to measure it with present telescopes (mainly through polarimetric observations of dust and molecular emission; e.g., [@Girartetal06 Girart, Rao & Marrone 2006]). This situation will change dramatically with ALMA, which will provide such an improvement in sensitivity that polarization observations of dust and molecular lines can be done in many sources at a very good angular resolution. However, the interpretation of the polarized data is difficult and appropriate tools are needed to scientifically harvest the wealth of polarization data expected from ALMA. Note that the ALMA band-7 (275 - 373 GHz) receivers will require full polarization calibration and imaging even for projects with no primary interest in polarization. The ARTIST package consists of a set of modeling tools for polarization and magnetic fields in different molecular and dust environments (e.g., low- and high-mass protostars, envelopes around evolved stars).\
An important component of ARTIST is a common interface for the codes used for radiative transfer modeling of typical data produced by submillimeter telescopes such as ALMA and Herschel. The model interface will include: $(i)$ a library of standard input models, e.g., for collapsing protostars, circumstellar disks and evolved stars; $(ii)$ a wrapper package that links the existing dust and line radiative transfer codes; $(iii)$ tools to analyze model output, e.g., to extract information about molecular excitation and its deviation from LTE, or optical depth surfaces, and to import this in existing visualization packages; $(iv)$ a ray tracing backend that readily provides data cubes that can be used in data reduction packages.
Current Status
==============
The aim of the ARTIST project is to supply the community with the described tools in one coherent modeling package. The tools will be made publicly available as they are finished: we expect LIME to be released medio-2010 with the remaining components of the package to follow. The tools will be distributed and supported through the ALMA regional center nodes in Bonn and Leiden as well as the Danish initiative for Far-infrared and Submillimeter Astronomy (DFSA; Copenhagen, Denmark). For more information see [`http://www.astro.uni-bonn.de/ARC/artist`]{}.
1992, *ApJ*, 395, 140
2010, *A&A*, 523, 25
2006 *Science*, 313, 812
2000, *A&A*, 362, 697
2004, *RvMP*, 76, 125
2006, *ApJ*, 646, 1043
2006, *PhRvE*, 74, 26704
2007 *A&A*, 471, 213
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abstract: 'Named Entity Recognition (NER) is a fundamental Natural Language Processing (NLP) task to extract entities from unstructured data. The previous methods for NER were based on machine learning or deep learning. Recently, pre-training models have significantly improved performance on multiple NLP tasks. In this paper, firstly, we introduce the architecture and pre-training tasks of four common pre-training models: BERT, ERNIE, ERNIE2.0-tiny, and RoBERTa. Then, we apply these pre-training models to a NER task by fine-tuning, and compare the effects of the different model architecture and pre-training tasks on the NER task. The experiment results showed that RoBERTa achieved state-of-the-art results on the MSRA-2006 dataset.'
author:
-
title: 'Application of Pre-training Models in Named Entity Recognition'
---
named entity recognition; pre-training model; BERT; ERNIE; ERNIE2.0-tiny; RoBERTa;
Introduction
============
Named Entity Recognition (NER) is a basic and important task in Natural Language Processing (NLP). It aims to recognize and classify named entities, such as person names and location names[@R; @IEEE; @ACCESS]. Extracting named entities from unstructured data can benefit many NLP tasks, for example Knowledge Graph (KG), Decision-making Support System (DSS), and Question Answering system. Researchers used rule-based and machine learning methods for the NER in the early years[@R; @M.; @Song][@R; @Y.; @Zhao]. Recently, with the development of deep learning, deep neural networks have improved the performance of NER tasks[@R; @Z.; @Huang][@R; @Y.; @Xia]. However, it may still be inefficient to use deep neural networks because the performance of these methods depends on the quality of labeled data in training sets while creating annotations for unstructured data is especially difficult[@R; @J.; @Lee]. Therefore, researchers hope to find an efficient method to extract semantic and syntactic knowledge from a large amount of unstructured data, which is also unlabeled. Then, apply the semantic and syntactic knowledge to improve the performance of NLP task effectively.
Recent theoretical developments have revealed that word embeddings have shown to be effective for improving many NLP tasks. The Word2Vec and Glove models represent a word as a word embedding, where similar words have similar word embeddings[@R; @word; @embedding]. However, the Word2Vec and Glove models can not solve the problem of polysemy. Researchers have proposed some pre-training models, such as BERT, ERNIE, and RoBERTa, to learn contextualized word embeddings from unstructured text corpus[@R; @BERT][@R; @ERNIE][@R; @RoBERTa]. These models not only solve the problem of polysemy but also obtain more accurate word representations. Therefore, researchers pay more attention to how to apply these pre-training models to improve the performance of NLP tasks.
The purpose of this paper is to introduce the structure and pre-training tasks of four common pre-trained models (BERT, ERNIE, ERNIE2.0-tiny, RoBERTa), and how to apply these models to a NER task by fine-tuning. Moreover, we also conduct experiments on the MSRA-2006 dataset to test the effects of different pre-training models on the NER task, and discuss the reasons for these results from the model architecture and pre-training tasks respectively.
Related work
============
Named Entity Recognition
------------------------
Named entity recognition (NER) is the basic task of the NLP, such as information extraction and data mining. The main goal of the NER is to extract entities (persons, places, organizations and so on) from unstructured documents. Researchers have used rule-based and dictionary-based methods for the NER[@R; @M.; @Song]. Because these methods have poor generalization properties, researchers have proposed machine learning methods, such as Hidden Markov Model (HMM) and Conditional Random Field (CRF)[@R; @Y.; @Zhao][@R; @D.; @Li]. But machine learning methods require a lot of artificial features and can not avoid costly feature engineering. In recent years, deep learning, which is driven by artificial intelligence and cognitive computing, has been widely used in multiple NLP fields. Huang $et$ $al$. [@R; @Z.; @Huang] proposed a model that combine the Bidirectional Long Short-Term Memory (BiLSTM) with the CRF. It can use both forward and backward input features to improve the performance of the NER task. Ma and Hovy [@R; @X.; @Ma] used a combination of the Convolutional Neural Networks (CNN) and the LSTM-CRF to recognize entities. Chiu and Nichols [@R; @J.; @Chiu] improved the BiLSTM-CNN model and tested it on the CoNLL-2003 corpus.
Pre-training model
------------------
As mentioned above, the performance of deep learning methods depends on the quality of labeled training sets. Therefore, researchers have proposed pre-training models to improve the performance of the NLP tasks through a large number of unlabeled data. Recent research on pre-training models has mainly focused on BERT. For example, R. Qiao $et$ $al$. and N. Li $et$ $al$. [@R; @R.; @Qiao][@R; @N.; @Li] used BERT and ELMO respectively to improve the performance of entity recognition in chinese clinical records. E. Alsentzer $et$ $al$. , L. Yao $et$ $al$. and K. Huang $et$ $al$. [@R; @E.; @Alsentzer][@R; @L.; @Yao][@R; @K.; @Huang] used domain-specific corpus to train BERT(the model structure and pre-training tasks are unchanged), and used this model for a domain-specific task, obtaining the result of SOTA.
Methods
=======
In this section, we first introduce the four pre-trained models (BERT, ERNIE, ERNIE 2.0-tiny, RoBERTa), including their model structures and pre-training tasks. Then we introduce how to use them for the NER task through fine-tuning.
BERT
----
BERT is a pre-training model that learns the features of words from a large amount of corpus through unsupervised learning[[@R; @BERT]]{}.
There are different kinds of structures of BERT models. We chose the **BERT-base** model structure. **BERT-base**’s architecture is a multi-layer bidirectional Transformer[[@R; @Attention]]{}. The number of layers is $L=12$, the hidden size is $H=768$, and the number of self-attention heads is $A=12$[[@R; @BERT]]{}.
Unlike ELMO, BERT’s pre-training tasks are not some kind of N-gram language model prediction tasks, but the “Masked LM (MLM)” and “Next Sentence Prediction (NSP)” tasks. For MLM, like a $Cloze$ task, the model mask 15% of all tokens in each input sequence at random, and predict the masked token. For NSP, the input sequences are sentence pairs segmented with `[SEQ]`. Among them, only 50% of the sentence pairs are positive samples.
ERNIE
-----
ERNIE is also a pre-training language model. In addition to a basic-level masking strategy, unlike BERT, ERNIE using entity-level and phrase-level masking strategies to obtain the language representations enhanced by knowledge [[@R; @ERNIE]]{}.
ERNIE has the same model structure as **BERT-base**, which uses 12 Transformer encoder layers, 768 hidden units and 12 attention heads.
As mentioned above, ERNIE using three masking strategies: basic-level masking, phrase-level masking, and entity-level masking. the basic-level making is to mask a character and train the model to predict it. Phrase-level and entity-level masking are to mask a phrase or an entity and predict the masking part. In addition, ERNIE also performs the “Dialogue Language Model (DLM)” task to judge whether a multi-turn conversation is real or fake [[@R; @ERNIE]]{}.
ERNIE2.0-tiny
-------------
ERNIE2.0 is a continual pre-training framework. It could incrementally build and train a large variety of pre-training tasks through continual multi-task learning [@R; @ERNIE2.0].
ERNIE2.0-tiny compresses ERNIE 2.0 through the method of structure compression and model distillation. The number of Transformer layers is reduced from 12 to 3, and the number of hidden units is increased from 768 to 1024.
ERNIE2.0-tiny’s pre-training task is called continual pre-training. The process of continual pre-training including continually constructing unsupervised pre-training tasks with big data and updating the model via multi-task learning. These tasks include word-aware tasks, structure-aware tasks, and semantic-aware tasks.
RoBERTa
-------
RoBERTa is similar to BERT, except that it changes the masking strategy and removes the NSP task[@R; @RoBERTa].
Like ERNIE, RoBERTa has the same model structure as BERT, with 12 Transformer layers, 768 hidden units, and 12 self-attention heads.
RoBERTa removes the NSP task in BERT and changes the masking strategy from static to dynamic[[@R; @RoBERTa]]{}. BERT performs masking once during data processing, resulting in a single static mask. However, RoBoERTa changes masking position in every epoch. Therefore, the pre-training model will gradually adapt to different masking strategies and learn different language representations.
Applying Pre-training Models
----------------------------
After the pre-training process, pre-training models obtain abundant semantic knowledge from unlabeled pre-training corpus through unsupervised learning. Then, we use the fine-tuning approach to apply pre-training models in downstream tasks. As shown in Figure 1, we add the Fully Connection (FC) layer and the CRF layer after the output of pre-training models. The vectors output by pre-training models can be regarded as the representations of input sentences. Therefore, we use a fully connection layer to obtain the higher level and more abstract representations. The tags of the output sequence have strong restrictions and dependencies. For example, “I-PER” must appear after “B-PER”. Conditional Random Field, as an undirected graphical model, can obtain dependencies between tags. We add the CRF layer to ensure the output order of tags.
Experiments and Results
=======================
We conducted experiments on Chinese NER datasets to demonstrate the effectiveness of the pre-training models specified in section III. For the dataset, we used the MSRA-2006 published by Microsoft Research Asia.
The experiments were conducted on the AI Studio platform launched by the Baidu. This platform has a build-in deep learning framework PaddlePaddle and is equipped with a V100 GPU. The pre-training models mentioned above were downloaded by PaddleHub, which is a pre-training model management toolkit. It is also launched by the Baidu. For hyper-parameter configuration, we adjusted them according to the performance on development sets. In this article, the number of the epoch is 2, the learning rate is 5e-5, and the batch size is 16.
--------------------- ------------- ----------- -----------
\[-6pt\] Models Precision/% Recall/% F1/%
\[-6pt\] Baseline 92.54 88.20 90.32
\[-6pt\] BERT-base 92.68 94.18 93.30
\[-6pt\] ERNIE 92.92 94.07 93.37
\[-6pt\] ERNIE-tiny 83.89 89.88 86.52
\[-6pt\] RoBERTa **93.64** **94.93** **94.17**
--------------------- ------------- ----------- -----------
: the results of NER using different pre-training models[]{data-label="table1"}
The BiGRU+CRF model was used as the baseline model. Table I shows that the baseline model has already achieved an F1 value of 90.32. However, using the pre-training models can significantly increase F1 values by 1 to 2 percentage points except for ERNIE-tiny model. Among the pre-training models, the RoBERTa model achieves the highest F1 value of 94.17, while the value of ERNIE-tiny is relatively low, even 4 percentage points lower than the baseline model.
Discussion
==========
This section discusses the experimental results in detail. We will analyze the different model structures and pre-training tasks on the effect of the NER task.
First of all, it is shown that the deeper the layer, the better the performance. All pre-training models have 12 Transformer layers, except ERNIE2.0-tiny. Although Ernie2.0-tiny increases the number of hidden units and improves the pre-training task with continual pre-training, 3 Transformer layers can not extract semantic knowledge well. The F1 value of ERNIE-2.0-tiny is even lower than the baseline model.
Secondly, for pre-training models with the same model structure, RoBERTa obtained the result of SOTA. BERT and ERNIE retain the sentence pre-training tasks of NSP and DLM respectively, while RoBERTa removes the sentence-level pre-training task because Liu $et$ $al$. [@R; @RoBERTa] hypothesizes the model can not learn long-range dependencies. The results confirm the above hypothesis. For the NER task, sentence-level pre-training tasks do not improve performance. In contrast, RoBERTa removes the NSP task and improves the performance of entity recognition. As described by Liu $et$ $al$. [@R; @RoBERTa], the NSP and the MLP are designed to improve the performance on specific downstream tasks, such as the SQuAD 1.1, which requires reasoning about the relationships between pairs of sentences. However, the results show that the NER task does not rely on sentence-level knowledge, and using sentence-level pre-training tasks hurts performance because the pre-training models may not able to learn long-range dependencies.
Moreover, as mentioned before, RoBERTa could adapt to different masking strategies and acquires richer semantic representations with the dynamic masking strategy. In contrast, BERT and ERNIE use the static masking strategy in every epoch. In addition, the results in this paper show that the F1 value of ERNIE is slightly lower than BERT. We infer that ERNIE may introduce segmentation errors when performing entity-level and phrase-level masking.
Conclusion
==========
In this paper, we exploit four pre-training models (BERT, ERNIE, ERNIE2.0-tiny, RoBERTa) for the NER task. Firstly, we introduce the architecture and pre-training tasks of these pre-training models. Then, we apply the pre-training models to the target task through a fine-tuning approach. During fine-tuning, we add a fully connection layer and a CRF layer after the output of pre-training models. Results showed that using the pre-training models significantly improved the performance of recognition. Moreover, results provided a basis that the structure and pre-training tasks in RoBERTa model are more suitable for NER tasks.
In future work, investigating the model structure of different downstream tasks might prove important.
Acknowledgment {#acknowledgment .unnumbered}
==============
This research was funded by the major special project of Anhui Science and Technology Department (Grant: 18030801133) and Science and Technology Service Network Initiative (Grant: KFJ-STS-ZDTP-079).
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abstract: 'It is shown that the penetration of an oscillating electric field in a semi-infinite classical plasma obeys the standard exponential attenuation law $e^{-x/\lambda_{e}}$ (besides oscillations), where $x$ is the distance from the wall and $\lambda_{e}$ is the extinction length (penetration depth, attenuation length). The penetration depth is computed here explicitly; it is shown that it is of the order $\lambda_{e}\simeq[\mid\varepsilon\mid/(1-\varepsilon)]^{1/3}v_{th}/\omega$, where $\varepsilon$ is the dielectric function, $\omega$ is the frequency of the field and $v_{th}=\sqrt{T/m}$ is the thermal velocity ($T$ being the temperature and $m$ the particle (electron) mass). The result is obtained by including explicitly the contribution of the surface term.'
author:
- |
[M. Apostol ]{}\
[Department of Theoretical Physics, Institute of Atomic Physics, ]{}\
[Magurele-Bucharest MG-6, POBox MG-35, Romania ]{}\
[email: [email protected]]{}
title: 'Penetration depth of an electric field in a semi-infinite classical plasma'
---
*Key words: Landau damping, semi-infinite plasma; electric field; penetration depth*
Introduction
============
It is well known that there exists a mechanism of energy transfer between collective modes and individual particles in collisionless classical plasmas, governed by the Landau damping.[@key-1] The origin of this mechanism is the causal character of the response of the plasmas to external excitations. The Landau damping received much interest, due to its application to heating plasmas by radiofrequency electric fields.[@key-2]-[@key-6] Also, the Landau damping enjoyed controversies along the years, as a consequence of the counter-intuitive character of an energy loss in collisioness plasmas.[@key-7]-[@key-19] Apart from theoretical and experimental investigations, numerical-analysis[@key-2]-[@key-5] and mathematical studies are devoted to the phenomenon,[@key-20]-[@key-23] which show both the complexity of the concept and difficulties related to its understanding at the fundamental level.
In semi-infinite plasmas the Landau damping appears as attenuated spatial oscillations (vibrations). This phenomenon, with its characteristic penetration depth, has a particular relevance for surface effects. Specifically, the Landau damping in semi-infinite plasmas implies an attenuated electric field, with spatial and temporal oscillations, besides a uniform component, as a response to a uniform oscillating external electric field perpendicularly applied to the plasma surface. The calculation of the exact form of this response is complicate, due, on one hand, to the difficulties related to the Landau damping, and, on the other hand, as a consequence of the presence of the surface. The latter point is particularly interesting, because the response is discontinuous at the surface, and the usual Fourier or Laplace techniques may not include properly this discontinuity. In addition, the surface boundary conditions may bring further complications. These difficulties have been analyzed recently in a clear formulation in Ref. [@key-26]. In various approximations (see, for instance, Refs. [@key-24]-[@key-27]), including the original calculation in Ref. [@key-1], the asymptotically attenuated field is presented as being proportional to $x^{2/3}e^{-\frac{3}{4}(\omega x/v_{th})^{2/3}}$, where $x$ is the distance from the wall, $\omega$ is the frequency of the field and $v_{th}=\sqrt{T/m}$ is the thermal velocity, $T$ being the temperature and $m$ being the particle mass (electrons); sometimes, an exponential attenutation $\sim e^{-\omega_{0}x/v_{th}}$ is included, where $\omega_{0}$ is the plasma frequency. A non-linear $x$-dependence ($\sim x^{2/3})$ is related to model assumptions made upon the surface and an asymptotic treatment of the Landau damping for the Boltzmann kinetic equation (see, for instance, Ref. [@key-26]). We show here that, when the surface condition (surface term) is included explicitly, the attenuated field obeys the standard exponential attenuation law $e^{-x/\lambda_{e}}$ (apart from factors oscillating in space), where $\lambda_{e}$ is an extinction length (penetration depth, attenuation length) which is computed here explicitly; up to immaterial numerical factors, it is of the order $\lambda_{e}\simeq[\mid\varepsilon\mid/(1-\varepsilon)]^{1/3}v_{th}/\omega$, where $\varepsilon$ is the dielectric function.
Semi-infinite plasma
====================
We consider a classical plasma at thermal equlibrium consisting of mobile charges $q$ with mass $m$ and concentration $n$ (electrons) moving in a rigid neutralizing background. We confine this plasma to a semi-infinite space (half-space) $x>0$, bounded by a plane surface $x=0$. The plasma is subject to a uniform oscillating external electric field $E_{0}e^{-i\omega t}$, where $E_{0}$ is directed along the $x$-direction (capacitively coupled plasma). The plasma is governed by the Maxwell distribution. The mean thermal velocity is sufficiently small to consider plasma unmagnetized. Since the field is directed along the $x$-direction we may integrate over the transverse velocities and use for the Maxwell distribution $F=n(\beta m/2\pi)^{1/2}e^{-\frac{1}{2}\beta mv^{2}}$, where $v$ is the velocity along the $x$-direction and $\beta=1/T$ is the reciprocal temperature. In the collisionless regime the change $f(x,v)e^{-i\omega t}$ in the Maxwell distribution is governed by the Boltzmann (Vlasov) equation $$-i\omega f+v\frac{\partial f}{\partial x}+\frac{q}{m}(E_{0}+E+E_{1})\frac{\partial F}{\partial v}=0\,\,\,,\label{1}$$ where $E$ is a uniform internal electric field and $E_{1}$ is another internal electric field, which may vary in space; these fields are generated by internal charges and currents. The uniform reaction field $E$ occurs in an infinite space too, *i.e.* a space bounded by surfaces at infinity (it is a bulk reaction field), while the non-uniform field $E_{1}$ is due to the presence of the surface (it is a surface field). We seek the solution of equation (\[1\]) as $f(x,v)=f_{0}(v)+f_{1}(x,v)$, where $$-i\omega f_{0}+\frac{q}{m}(E_{0}+E)\frac{\partial F}{\partial v}=0\label{2}$$ and $$-i\omega f_{1}+v\frac{\partial f_{1}}{\partial x}+\frac{q}{m}E_{1}\frac{\partial F}{\partial v}=0\:\:.\label{3}$$
The uniform part $f_{0}$ of the solution does not generate charge density in plasma; it generates a current density. Therefore, it should satisfy the equation $$i\omega E=4\pi q\int dv\cdot vf_{0}\,\,;\label{4}$$ it is easy to see that this equation arises from the general equation $\partial\mathbf{E}/\partial t+4\pi\mathbf{j}=0$, where $\mathbf{j}$ is the current density; this equation ensures the vanishing of the (internal) magnetic field, as expected. The non-uniform part $f_{1}$ of the solution generates a charge density in plasma; it satisfies the equation $$\frac{\partial E_{1}}{\partial x}=4\pi q\int dvf_{1}\,\,.\label{5}$$
The solution of equations (\[2\]) and (\[4\]) is $$f_{0}=-\frac{iq\omega E_{0}}{m(\omega^{2}-\omega_{0}^{2})}\frac{\partial F}{\partial v}\label{6}$$ and $$E=\frac{\omega_{0}^{2}}{\omega^{2}-\omega_{0}^{2}}E_{0}\,\,,\,\,E_{t}=E_{0}+E=\frac{\omega^{2}}{\omega^{2}-\omega_{0}^{2}}E_{0}\,\,\:,\label{7}$$ where $\omega_{0}=(4\pi nq^{2}/m)^{1/2}$ is the plasma frequency; we recognize here the response of a boundless plasma to an electric field (restricted to $x>0$), where $\varepsilon=1-\omega_{0}^{2}/\omega^{2}$ is the dielectric function and $E_{t}$ is the total field in plasma ($P=\chi E_{t}$ is the polarization and $\chi=(\varepsilon-1)/4\pi=-nq^{2}/m\omega^{2}$ is the electric susceptibility).
In order to deal conveniently with the boundary condition at the surface we multiply equation (\[3\]) by the step function $\theta(x)$ ($\theta(x)=1$ for $x>0$, $\theta(x)=0$ for $x<0$) and restrict ourselves to the solution for $x>0$; equation (\[3\]) becomes $$-i\omega f_{1}+v\frac{\partial f_{1}}{\partial x}+\frac{q}{m}E_{1}\frac{\partial F}{\partial v}=vf_{s}\delta(x)\,\,\,,\label{8}$$ where $f_{s}=f_{s}(v)=f_{1}(x=0,v)$; we can check directly this surface term by integrating equation (\[8\]) along a small distance perpendicular to the surface $x=0$. Similarly, equation (\[5\]) becomes $$\frac{\partial E_{1}}{\partial x}-E_{1s}\delta(x)=4\pi q\int dvf_{1}\,\,\,,.\label{9}$$ where $E_{1s}=E_{1}(x=0)$. The inclusion of the surface $\delta$-terms in equations (\[8\]) and (\[9\]) is the main point of this paper.
In equations (\[8\]) and (\[9\]) we use the Fourier transforms with respect to the coordinate $x$ (and restrict ourselves to $x>0$); we get $$f_{1}(k,v)=\frac{i}{\omega-vk+i\gamma}\left[vf_{s}(v)-\frac{q}{m}\frac{\partial F}{\partial v}E_{1}(k)\right]\label{10}$$ and $$E_{1}(k)=\frac{4\pi q\int dv\frac{vf_{s}(v)}{\omega-vk+i0^{+}}-iE_{s1}}{k+\frac{4\pi q^{2}}{m}\int dv\frac{\partial F/\partial v}{\omega-vk+i0^{+}}}\,\,\:,\label{11}$$ where $\gamma\rightarrow0^{+}$. It is worth noting that in the Fourier transforms we replace $\omega$ by $\omega+i\gamma$, $\gamma\rightarrow0^{+}$, in order to ensure the causal behaviour (*i.e.*, zero response for time $t<0$, which requires a pole in the lower $\omega$-half-plane). This procedure gives a pole in the upper $k$-half-plane (this is the connection between the Landau damping and the spatial decay). At the same time, in the integrals with respect to $v$ we may take the limit $\gamma\rightarrow0^{+}$, which avoids the singularity $\omega=vk$; the insertion of the parameter $\gamma$ produces the Landau damping. We denote by $A$ the denominator in equation (\[11\]); it can be estimated as $$\begin{array}{c}
A=k+\frac{4\pi q^{2}}{m}\int dv\frac{\partial F/\partial v}{\omega-vk+i0^{+}}=k+\frac{4\pi q^{2}}{m}P\int dv\frac{\partial F/\partial v}{\omega-vk}-i\frac{4\pi^{2}q^{2}}{mk}\frac{\partial F}{\partial v}\mid_{v=\omega/k}\simeq\\
\\
\simeq k(1-\omega_{0}^{2}/\omega^{2})-i\frac{4\pi^{2}q^{2}}{mk}\frac{\partial F}{\partial v}\mid_{v=\omega/k}\,\,;
\end{array}\label{12}$$ we can see that the zeroes of $A$ give the damped collective eigenmodes $\omega=\pm\omega_{0}-i\Gamma$ (plasma frequency), where $\Gamma$ is given by the imaginary part in equation (\[12\])\
($\Gamma\simeq-2\pi^{2}q^{2}\omega_{0}/mk^{2})(\partial F/\partial v)\mid_{v=\omega_{0}/k}$); this is the Landau damping.
Penetrating electric field
==========================
In order to estimate the field $E_{1}(x)$ we need the zeroes of $A$ with respect to $k$ in equation (\[11\]). It is convenient to introduce the variable $\xi=\sqrt{\beta m/2}\omega/k$. We can see easily that the zeroes of $A$ are given by $\xi^{2}\mid\xi\mid e^{-\xi^{2}}=-i\alpha$, where $\alpha=\mid\varepsilon\mid/2\sqrt{\pi}(1-\varepsilon)$; we consider the case $\omega<\omega_{0}$ ($\varepsilon<0$; the rather unrealistic case $\omega>\omega_{0}$ can be treated similarly, by using the equation $\xi^{2}\mid\xi\mid e^{-\xi^{2}}=i\alpha$). For small values of $\alpha$ we get two roots of the equation $A=0$, given by $k_{1,2}\simeq\pm\frac{1}{2\alpha^{1/3}}\sqrt{\beta m}\omega(1+i)$; only $k_{1}$ (placed in the upper half-plane) contributes to the $k$-integration for $x>0$. In estimating the integral in the numerator of equation (\[11\]) we may leave aside the contribution of the principal value. For $k$ near $k_{1}$ the field $E_{1}(k)$ has the form $$\begin{array}{c}
E_{1}(k)\simeq\frac{B}{k-k_{1}+\frac{i}{5}(k-k_{1})^{*}}\,\,,\\
\\
B=\frac{8\sqrt{2}\pi q\alpha^{2/3}v_{th}^{2}}{5\omega\mid\varepsilon\mid}(1+i)f_{s}\left(\alpha^{1/3}v_{th}(1-i)\right)+\frac{2i}{5\mid\varepsilon\mid}E_{1s}\,\,.
\end{array}\label{13}$$ The reverse Fourier transformation leads to $$\begin{array}{c}
E_{1}(x)=E_{1s}e^{(i-1)\omega x/2\alpha^{1/3}v_{th}}\end{array}\label{14}$$ with the relationship $$E_{1s}=-\frac{8\sqrt{2}\pi q\alpha^{2/3}v_{th}^{2}}{(2+5\mid\varepsilon\mid)\omega}(1-i)f_{s}\left(\alpha^{1/3}v_{th}(1-i)\right)\label{15}$$ (or $E_{1s}=iB$). The final result is given by $E_{1}(t,x)=Re\left[E_{1}(x)e^{-i\omega t}\right]$. We can see that an additional, non-uniform, electric field $E_{1}(x)$ appears as a result of the presence of the surface. This field oscillates in space and is attenuated with an attenuation length (penetration depth, extinction length) $\lambda_{e}\simeq(1/\pi)^{1/6}[\mid\varepsilon\mid/(1-\varepsilon)]^{1/3}v_{th}/\omega$. It is worth noting that the penetration depth and the wavelength of the spatial oscillations have the same order of magnitude.
Discussion and conclusions
==========================
Making use of $E_{1}(k)$ given by equations (\[11\]) and (\[13\]) we can calculate the change $f_{1}(x,v)$ in the distribution function (equation (\[10\])); if we limit ourselves to slow spatial oscillations, we get $$f_{1}(x,v)\simeq-\frac{iq}{m\omega}sgn(v)\frac{\partial F}{\partial v}E_{1}(x)\label{16}$$ (compare with equations (\[6\]) and (\[7\])). Within this approximation $f_{s}(v)=-(iq/m\omega)sgn(v)(\partial F/\partial v)E_{1s}$ and the polarization charge and current densities are zero (as expected for slow oscillations).
The amplitude of the field $E_{1}(x)$ depends on the parameter $E_{1s}$, which accounts for the boundary condition at $x=0$. It is related to $f_{s}(v)=\frac{1}{2\pi}\int dkf_{1}(k,v)$ by equation (\[15\]), where $f_{1}(k,v)$ is given by equation (\[10\]); it is easy to see that the integration of the first term in equation (\[10\]) gives $f_{s}$, while, making use of equations (\[13\]), the integration with respect to $k$ of the term which includes $E_{1}(k)$ is zero.
Within the kinetic approach we may estimate the local change in temperature by $\delta T=2T\overline{f/F}$, where the overbar implies an integration over velocities (thermal average). We can see that only $f_{1}$ contributes to this integration. Making use of equation (\[16\]) we get $\delta T=0$. However, if we keep the contribution of the fast oscillations, we get a surface change of temperature $$\delta T\simeq\frac{2iT}{n\omega}\int dv\cdot vf_{s}(v)\cdot\delta(x)+...\:\:.\label{17}$$ (*i.e.*, $Re\left(\delta Te^{-i\omega t}\right)$). The $\delta$-type contribution in equation (\[17\]) corresponds to the surface sheath in plasma heating models.[@key-6; @key-25]
Similar calculations of the penetration depth can be made for a plasma confined between two plane-parallel walls (or other geometries); the result depends on the boundary conditions incorporated in parameters like $f_{s}$.[@key-24] The boundary parameter $f_{s}$ is a model parameter; we may take $f_{0}+f_{s}=0$ ($f(x=0,v)=0$) as a natural assumption, an equation which provides the parameter $f_{s}$. For $f_{s}=-f_{0}$ the field $E_{1}$ at the surface (maximum value) is of the order $E_{1}\simeq E/\mid\varepsilon\mid$, where $E$ is the internal uniform field given by equation (\[7\]). The surface change in temperature (equation (\[17\])) can be written in this case as $$\delta T=\frac{1}{2\pi}\left(\frac{E_{0}}{q/a^{2}}\right)T\cdot a\delta(x)\label{18}$$ (for $\omega\ll\omega_{0}$), where $a$ is the mean separation distance between the particles ($a=n^{-1/3}$); $q/a^{2}\gg E_{0}$ is an electric field of the order of the microscopic (inter-particle) field.
In conclusion, it is shown in this paper that the penetration of an oscillating electric field in a semi-infinite classical plasma obeys the standard exponential penetration law $e^{-x/\lambda_{e}}$ (beside a uniform component), which may exhibit spatial oscillations, the extinction length $\lambda_{e}$ (penetration depth, attenuation length) being of the order $\lambda_{e}\simeq[\mid\varepsilon\mid/(1-\varepsilon)]^{1/3}v_{th}/\omega$; ($\varepsilon$ is the dielectric function, $\omega$ is the frequency of the field and $v_{th}=\sqrt{T/m}$ is the thermal velocity). The surface term is included explicitly in these calculations.
**Acknowledgments.** The author is indebted to the members of the Laboratory of Theoretical Physics at Magurele-Bucharest for many fruitful discussions. This work has been supported by the Scientific Research Agency of the Romanian Government through Grants 04-ELI/2016 (Program 5/5.1/ELI-RO), PN 16 42 01 01/2016 and PN (ELI) 16 42 01 05/2016.
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|
---
author:
- Julius Ranninger
title: '**Introduction to polaron physics: Basic Concepts and Models**'
---
Introduction
============
Putting an electron into a crystalline lattice, it will interact with the dynamical deformations of that latter. Depending on the type of material we are dealing with—such as ionic polar crystals, covalent materials or simple metals—the form of the electron-lattice coupling as well as the relevant lattice modes being involved in this coupling will be different. The mutual interaction between the electron and the lattice deformations, which in general is of a highly complex dynamical nature, results in composite entities of electrons surrounded by clouds of virtual phonons which, in a dynamical way, correlate the position of the electron and the associated local lattice deformation. Such entities are referred to as polarons, which can be either spatially quite extended “large polaron”, or rather constraint in space “small polarons”. Large polarons generally are itinerant entities, while small polarons in real materials, have a tendency to self-trap themselves in form of localized states. By far the most spectacular manifestation of such electron-lattice interactions in the weak coupling large polaron limit is the classical phonon mediated superconductivity.
The earliest indications for polarons came from F centers—electrons trapped in negative ion vacancy positions in alkali halides—which led Landau to the concept of localized strong coupling polarons [@LandauPZS33]. Subsequently, the field was developed mainly in connection with scenarios where the materials could be characterized by a continuous elastic medium and specifically for:\
(i) ionic crystals [@FrohlichAP54], where positively and negatively charged ions oscillate out of phase around their equilibrium positions and thus give rise to large electric polarization fields in form of longitudinal optical modes, which then strongly couple back onto the conduction band electrons,\
(ii) covalent materials [@BardeenPR50], where local dilations of the material couple the electrons to corresponding deformation potentials and\
(iii) simple metals [@Scalapino69], characterized by high frequency plasma modes which, because of strong charge density fluctuations, renormalize into acoustic modes.
With the arrival of new materials since the late sixties, the necessity to consider the microscopic lattice structure for the polaron formation became more and more crucial and initiated the theoretical work on the transition between large and small polarons and the question of polaron localization.
After reviewing the different kinds of electron-lattice coupling leading to polaron formation we shall discuss the fundamental issues of the cross-over between large and small polarons, the question of continuous versus discontinuous transition and the difference of the polaron self-trapping with respect to localization in systems with attractive interaction potentials. We then present examples of a few decisive experiments on polarons which, early on, shed some light on their dynamical formation and disintegration.
The Fröhlich Large Polaron {#FLP}
==========================
Polarons in ionic crystals
--------------------------
Let us to begin with consider the case of an ionic crystal where longitudinal optical phonons strongly couple to the electrons. A single electron in such a dielectricum induces via its charge an electric displacement field $${\bf D}({\bf r},{\bf r}_{\rm el})=-{\rm grad}{e\over |{\bf r} - {\bf r}_{\rm el}|}\, ,
\label{eq1}$$ at a spatial coordinate ${\bf r}$ with ${\bf r}_{\rm el}$ denoting the position of the electron. This displacement field couples to the dynamics of the lattice which can be described in terms of a polarization field ${\bf P}({\bf r})$ of the dielectric medium and which (via Poisson’s equation $div \; {\bf D}({\bf r})$ = $4\pi e\delta({\bf r} - {\bf r}_{\rm el})$) can be expressed in terms of a polarization potential $\Phi({\bf r})$ such as: $${\bf P}({\bf r})= \frac{1}{4\pi} {\rm grad} \; \Phi({\bf r}), \quad
\Phi({\bf r_{\rm el}}) = - \frac{1}{e}\int {\rm d}^3{\bf r} \; {\bf D}({\bf r},{\bf r}_{\rm el}) \cdot
{\bf P}({\bf r})\, .
\label{eq2}$$ We assume for simplicity the absence of any shear and vorticity in the elastic medium forming the dielectricum.
The standard way to describe the physics of polarons is to introduce measurable material quantities such as the frequency dependent dielectric constant $\varepsilon(\omega)$, defined by ${\bf D} = \varepsilon(\omega) {\bf E}$, in order to account for the dynamics of the dielectric medium via the frequency dependence of the polarization field, related to ${\bf E}$ via $4\pi{\bf P}({\bf r})$ = ${\bf D}({\bf r})-{\bf E}({\bf r})$. ${\bf D}({\bf r})$ being exclusively determined by the point charge is independent on $\varepsilon(\omega)$. In a crude sense one can consider the response of the dielectric medium as being given by a superposition of two contributions: (i) a high frequency contribution in the ultraviolet regime $\varepsilon_{\infty}$, arising from the electron clouds oscillating around the ionic positions and (ii) a low frequency contribution $\varepsilon_0$ in the infrared regime, arising from the oscillations of the positive and negative ions against each other. This separates the polarization field ${\bf P}({\bf r})$ into two contributions ${\bf P}({\bf r}) = {\bf P}_0({\bf r}) +
{\bf P}_{\infty}({\bf r})$ which in the low and high frequency limits are determined by: $${\bf P}_0({\bf r}) +{\bf P}_{\infty}({\bf r}) =
\frac{1}{4\pi}\left(1 - \frac{1}{\varepsilon_0}\right) {\bf D}({\bf r})\, ,
\label{eq3}$$ $${\bf P}_{\infty}({\bf r}) = \frac{1}{4\pi}\left(1 -
\frac {1}{\varepsilon_{\infty}}\right){\bf D}({\bf r})\, ,
\label{eq4}$$ considering the fact that the low frequency contribution of the polarization field remains unaffected by high frequency perturbations. The determinant contribution to the polaron dynamics, arising from the low frequency polarization field is hence given by $${\bf P}_0({\bf r}) = \frac{1}{4\pi} \left( \frac{1}{\varepsilon_{\infty}}
-\frac{1}{\varepsilon_0} \right){\bf D}({\bf r})
\equiv \frac{1}{4\pi\tilde{\varepsilon}}{\bf D}({\bf r})\, .
\label{eq5}$$
Before entering into a detailed discussion of the intricate nature of the dynamics of this problem let us consider certain limiting cases and start with the picture of an electron inside a dielectric continuous medium being constraint to a finite small volume in a sphere of a certain radius $R_1$ to be determined.
We consider an electron, moving inside such a sphere of radius $R_1$ with a velocity $v$, to be fast compared to the characteristic time of the atomic oscillations $(2\pi/\omega_0)$ (given by the longitudinal optical lattice modes). The polarization field induced in the medium by the motion of the charge carrier can then be considered as static for distances much greater than $2 \pi v/\omega_0$ and being described by a static Coulomb potential. Inside this sphere the test charge can be considered as being uniformly distributed and hence the potential as being constant. We thus have: $$E_{\rm pot} = -e^2 / \tilde\varepsilon R_1\;\;
(r < R_1), \quad
E_{\rm pot} = -e^2 / \tilde\varepsilon r\;\; (r \geq R_1)\, .
\label{eq6}$$
Determining the size of the polaron in a purely static fashion, considering the electron kinetic energy inside this sphere as being given by $E_{\rm kin} \simeq (2\pi \hbar)^2/2m R^2_1)$, we find by minimizing the total energy with respect to $R_1$: $$R_1 = (2\pi \hbar)^2 \tilde\varepsilon / m e^2\, .
\label{eq7}$$
Determining the size of the polaron radius in a dynamical fashion we require that the characteristic wavelength $2 \pi \hbar/mv$ of the electron must be smaller than $2 \pi v/\omega_0$. This implies that the typical radius $R_2$ of the trapping potential is determined by the distance for which $\hbar/mv \simeq v/\omega_0$, or in other words: $$R_2 = 2 \pi (\hbar /m\omega_0)^{\frac{1}{2}}\, .
\label{eq8}$$ >From these simple arguments it follows that the interaction between the displacement field and the polarization field, arising from the dynamics of the lattice, (the polar coupling between the electron and the longitudinal optical phonons)) is of the order of $E_{\rm pot}$ = $-e^2/\tilde{\varepsilon}R_2$.
The potential energies for the two limiting cases where the dielectric medium is considered (i) as static and (ii) as dynamic, are given by $$-{U_1 \over \hbar \omega_0} = \left({U_2 \over \hbar \omega_0}\right)^2 =
{1 \over 2 \pi^2} \alpha^2_{\rm Fr}\, ,
\label{eq9}$$ where $\alpha_{\rm Fr}$ is a dimensionless coupling constant introduced by Fröhlich $$\alpha_{\rm Fr}= {e^2 \over \tilde{\varepsilon} \sqrt2}
\sqrt{{m \over \hbar^3 \omega_0}}\, .
\label{eq10}$$ The value for $\alpha_{\rm Fr}$ can vary over one decade in the interval $\sim[0.1, 5]$ depending on the material. Large values of $\alpha$ favor the static picture, while smaller ones favor the dynamical one. For most standard materials (polar and ionic crystals) the typical phonon frequency is of the order of $10^{-14}$ seconds and the electrons can be assumed to be essentially free electrons. This results in a value for $R_2$ which is much bigger than the lattice constant and hence justifies the continuum approach described above permitting to treat the dielectric constant as wave vector independent. Within such an approach Fröhlich first formulated this polaron problem in a field theoretical form and proposed a corresponding Hamiltonian for it, designed to describe the so called large polarons.
Let us now briefly sketch the derivation of the corresponding Hamiltonian [@FrohlichPM50]. To within a first approximation the dynamics of the polarization field can be described by harmonic oscillators driven by the electric displacement fields. Dealing with the polaron problem on the basis of the continuum model of a dielectricum one is restricted to consider waves of the polarization field with wave length much bigger than the inter-atomic distance and moreover to limit oneself in this problem to frequencies well below the optical excitations which would result from the deformations of the ions rather than from their motion. Under those conditions one can limit the discussion of the dynamics of the polarization field to the infrared longitudinal component of it, being driven by a source term due to its coupling with the electric displacement field. Using a simple Ansatz in form of a harmonic motion of the polarization field one has $$\left({{\rm d}^2 \over {\rm d}t^2} + \omega^2_0 \right) {\bf P}_0({\bf r}) =
{1 \over \gamma}{\bf D}({\bf r},{\bf r}_{\rm el})\, .
\label{eq11}$$ The coefficient ${1\over \gamma}$ = $\equiv{\omega_0^2 \over 4\pi}{1\over \tilde\varepsilon}$ of ${\bf D}({\bf r},{\bf r}_{\rm el})$ on the rhs of this equation is determined from its static limit, following eqs. (\[eq3\], \[eq4\]). To this equation of motion for the polarization field one has to add the term which describes the motion of the electron and which is controlled by the interaction energy $e\Phi({\bf r})$, given in eq. (\[eq2\]), i.e., $$m{{\rm d}^2 \over {\rm d}t^2} {\bf r}_{\rm el} = - e \; {\rm grad} \; \Phi({\bf r}_{\rm el})\, .
\label{eq12}$$ Considering $\gamma{\bf P}_0({\bf r})$, $m{\bf r}_{\rm el}$ and $\gamma\dot{\bf P}_0({\bf r})$, ${\bf p}_{\rm el}$ = $m\dot{\bf r}_{\rm el}$ as a set of generalized coordinates and conjugate momenta $\{Q,\partial L\partial \dot Q\}$ the corresponding Hamiltonian is given by $$H = \frac{1}{2}\gamma\int{\rm d}^3{\bf r}\left[\dot{\bf P}^2_0({\bf r})+\omega_0^2{\bf P}^2_0({\bf r})\right]-\int{\rm d}^3{\bf r}{\bf D}({\bf r},{\bf r}_{\rm el}){\bf P}_0({\bf r})+\frac{1}{2}m \dot{\bf r}_{\rm el}^2\, .
\label{eq13}$$ In order to obtain a quantum field theoretical formulation of this problem one introduces habitually the vector field representation for the polarization fields $${\bf B}^{\pm}({\bf r})=\sqrt{{\gamma \omega_0 \over 2\hbar}}\left({\bf P}_0({\bf r})\pm {i \over \omega_0}\dot{\bf P}_0({\bf r}) \right)={1 \over N} \sum_{{\bf q}\lambda}{{\bf q} \over q}\left(\begin{array}{c}
e^{-iqr}a^+_{{\bf q}\lambda}\\
e^{+iqr}a_{{\bf q}\lambda}\end{array}\right) \, ,
\label{eq14}$$ with the phonon annihilation (creation) operators $a^{(+)}_{{\bf q}\lambda}$ with $[a_{{\bf q}\lambda},a^+_{{\bf q}'\lambda'}]$ = $\delta_{{\bf q},{\bf q}'}\delta_{\lambda,\lambda'}$.
For ionic polar crystals the dominant contribution of the polarization field comes from longitudinal optical phonons. We shall in the following restrict ourselves to those modes only. Introducing the quantized form of the electron momentum ${\bf p}_{\rm el} \equiv {\hbar\over i}{\partial\over\partial {\bf r}_{\rm el}}$ with $[{\bf p}_{\rm el},{\bf r}_{\rm el}]$ = $-i\hbar$, we rewrite the Hamiltonian, eq. (\[eq13\]) in terms of the phonon creation and annihilation operators and the electron coordinates and conjugate momenta as $$H ={p^2_{\rm el} \over 2m} + \hbar \omega_0 \sum_{\bf q} \left(a^+_{\bf q} a_{\bf q}+\frac{1}{2}\right) + \sum_{\bf q} V^{\rm opt}_{\bf q}(a^+_{\bf q}e^{-i{\bf q} \cdot {\bf r}_{\rm el}} - a_{\bf q}e^{i{\bf q} \dot{\bf r}_{\rm el}})\, ,
\label{eq15}$$ where the effective electron lattice coupling constant is $$V^{\rm opt}_{\bf q} = i \sqrt{4 \pi \alpha}
(\hbar/2 \omega_0 m)^{\frac{1}{4}}\hbar \omega_0 (1/q)\, .
\label{eq16}$$ Upon introducing dimensionless electron coordinates as well as phonon wave vectors $\bar{\bf r}_{\rm el}$ = $(2m\omega_0 / \hbar)^{\frac{1}{2}}{\bf r}_{\rm el}$ and $\bar {\bf q}$ = $(2m\omega_0 / \hbar)^{-\frac{1}{2}}{\bf q}$, $H/\hbar\omega_0$ turns out to be exclusively controlled by the unique parameter $\alpha_{\rm Fr}$.
Replacing in the expression for the Hamiltonian, eq. (\[eq15\]), the Fourier transform $e^{i{\bf q}\cdot {\bf r}}$ of the electron density $\rho({\bf r})$ = $\sum_{{\bf r}_{\rm el}} \delta({\bf r} -{\bf r}_{\rm el})$ by its second quantization form $\sum_{\bf k\sigma}c^+_{\bf k+q\sigma}c_{\bf k\sigma}$ we finally obtain what is generally called the Fröhlich Hamiltonian $$H =\sum_{{\bf k}\sigma} \varepsilon_{\bf k}c^+_{{\bf k}\sigma}c_{{\bf k}\sigma}+\sum_{\bf q}\hbar \omega_0(a^+_{\bf q}a_{\bf q}+\frac{1}{2})+\sum_{{\bf q}{\bf k} \sigma}V^{\rm opt}_{\bf q}c^+_{{\bf k+q}\sigma}c_{{\bf k}\sigma}(a_{\bf q}-a^+_{-{\bf q}})\, .
\label{eq17}$$ $\varepsilon_{\bf k}$ = $D-t\gamma_{\bf k}$, $\gamma_{\bf k}$ = $\frac{1}{z}\sum_{\delta} \exp(i{\bf k} \cdot {\delta})$ denotes the bare electron dispersion (with $\delta$ being the lattice vectors linking nearest neighboring sites) and $D=zt$ is the band half-width and $z$ the coordination number.
Polarons in non-polar covalent materials
----------------------------------------
A structurally very similar Hamiltonian is obtained for covalent materials when the density of charge carriers is very low and screening effects can be neglected. The electron lattice interaction then can be derived within the so-called [*deformation potential*]{} method [@BardeenPR50] where it is assumed that the electron dispersion of the rigid lattice $\varepsilon^0_{\bf k}$ gets modified due to some elastic strain which, in the simplest case, amounts to a dilation. This then results in a shift of the electron dispersion $\varepsilon_{\bf k}$ = $\varepsilon^0_{\bf k}+C\Delta$ with $$\Delta(x)={\partial u^{\nu} \over \partial x^{\nu}} = i {1 \over \sqrt N}\sum_{\bf q} {q \over \sqrt{2 M \omega_{\bf q}}} (a_{\bf q} e^{i {\bf q} \cdot {\bf x}} -
a^+_{\bf q} e^{-i {\bf q} \cdot {\bf x}})\, .
\label{eq18}$$ The dilation, being exclusively related to longitudinal acoustic phonons ($a_{\bf q}, a^+_{\bf q}$) with a phonon frequency $\omega_{\bf q}$ and $M$ denoting the ion mass, the constant $C$ can be estimated form pressure measurements and is typically of the order of $10$ eV. The effective electron-lattice interaction in that case is then similar to that derived above for ionic crystals and is described by the Hamiltonian eq. \[eq17\] upon replacing $V^{\rm opt}_{\bf q}$ by $$V^{\rm ac}_{\bf q} = i C {q \over \sqrt{\frac{1}{2} M \omega_{\bf q}}}\, .
\label{eq19}$$
Polarons in metals
------------------
A yet very different approach is required to treat the electron-lattice coupling in metals. The electrons at the spatial coordinate ${\bf x}$ experience a pseudo potential $V({\bf x}-{\bf R}_i)$ exerted on them by the positively charged ions situated at lattice sites ${\bf R}_i$. The major effect giving rise to the coupling of the electrons to the lattice now comes from the dynamical displacements ${\bf u}_i = {\bf R}_i - {\bf R}^0_i$ of the ions from their equilibrium positions ${\bf R}^0_i$. The dynamics of the lattice arises primarily from the ion-ion interaction $\tilde{V}({\bf R}_i-{\bf R}_j)$, which can being assumed in form of a Coulomb potential. But since those ions are imbeded in a kind of electron jellium, they sense the charge density fluctuations of the electrons via the electron-lattice interaction. Considering plane waves $\psi({\bf x})$ for the electrons one has the following Hamiltonian to consider $$\begin{aligned}
H &=& \int {\rm d}{\bf x} \psi^+({\bf x}) \left[ - {\nabla^2 \over 2m} +
\sum_i V({\bf x}-{\bf R}_i)\right] \psi({\bf x})
+ \sum_i {P^2_i \over 2M}
+ {1 \over 2} \sum_{i,j} \tilde{V}({\bf R}_i-{\bf R}_j) \nonumber \\
&&+ {1 \over 2}
\int {\rm d}{\bf x} {\rm d}{\bf x}' \psi^+({\bf x}) \psi^+({\bf x}')
{e^2 \over |{\bf x} - {\bf x}'|}\psi({\bf x}')\psi({\bf x})\, ,
\label{eq20}\end{aligned}$$ where $P^2_i/2M$ denotes the kinetic energy of the ions at site $i$. Using the standard expansion of the lattice displacements ${\bf u}_i$ in terms of the phonon operators $${\bf u}_i = {1 \over \sqrt N } \sum_{{\bf q}\lambda} {\bf e}_{{\bf q}\lambda}
e^{i {\bf q} \cdot {\bf R}^0_i} \left(\hbar \over 2 M
\Omega_{{\bf q}\lambda} \right)^\frac{1}{2}(a_{{\bf q}\lambda} +
a^+_{-{\bf q}\lambda})\, .
\label{eq21}$$ (${\bf e}_{{\bf q}\lambda}$ denotes the polarization vectors and $\Omega_{{\bf q}\lambda}$ the eigen-frequencies of the various phonon branches) one can write the above Hamiltonian in the form $$\begin{aligned}
H &=& \sum_{{\bf k}\sigma} \varepsilon_{\bf k} c^+_{{\bf k}\sigma}
c_{{\bf k}\sigma} + \sum_{{\bf q}\lambda} \hbar \Omega_{{\bf q}\lambda}
(a^+_{{\bf q}\lambda}a_{{\bf q}\lambda}+\frac{1}{2})
+ \sum_{\bf q}{4 \pi e^2 \over q^2}\rho_{\bf q}
\rho_{-{\bf q}}\nonumber\\
&&+ \sum_{{\bf q}\lambda}V^{\rm met}_{{\bf q}\lambda}\rho_{\bf q}
(a_{{\bf q}\lambda} + a^+_{-{\bf q}\lambda})\, ,
\label{eq22}\end{aligned}$$ where $\rho_{{\bf q}} = {1 \over N}\sum_{{\bf k}\sigma}
c^+_{{\bf k}+ {\bf q}\sigma} c_{{\bf k}\sigma}$ is the charge density fluctuation operator for the electrons and $V^{\rm met}_{{\bf q}\lambda}$ = $-i(N/2M \hbar\Omega_{{\bf q}\lambda})^\frac{1}{2}({\bf q} \cdot {\bf e}_{{\bf q}\lambda}) V({\bf q})$. The relevant phonon modes arising from the ion-ion interaction are the practically dispersion-less ionic plasma modes with frequencies $\Omega_{{\bf q}\lambda}$. However, these modes being coupled to the electron charge fluctuations, this results in a strong renormalization leading to: (i) a longitudinal acoustic branch with a correspondingly phonon frequency $\omega^{ac}_{\bf q}=\simeq q \sqrt{m/3M} v_F$ ($v_F$ denoting the Fermi velocity) and (ii) to a corresponding dressed electron-phonon coupling given by [@Scalapino69] $$\tilde{v}_{{\bf q}\lambda} = -i \sqrt {N \over 2M\omega^{ac}_{{\bf q}\lambda}}
({\bf q} \cdot {\bf e}_{{\bf q}\lambda}) {\Lambda({\bf q})V({\bf q}) \over
\varepsilon({\bf q})Z_c}\, .
\label{eq23}$$ $\varepsilon({\bf q})$ denotes a wave vector dependent dielectic constant, $\Lambda({\bf q})$ the Coulomb screened renormalization of the bare ion potential and $Z_c$ the spectral weight renormalization constant of the electron quasi-particle spectrum.
In spite of the diversity of the physical systems considered above, the various Hamiltonians describing them are of rather general form. The main message which they contain is that, because of the electron-lattice interaction $V_{\bf q}$, the electrons will be accompanied by a lattice deformation. In the weak coupling limit, this is restricted to simply a single phonon accompanying the electron. Within a lowest order perturbative approach this leads to states of the form $$c^+_{{\bf k}\sigma}|0\rangle + \sum_{\bf q} V_{\bf q} c^+_{{\bf k}- {\bf q}\sigma}a^+_{\bf q}|0\rangle
{1 \over (\hbar \omega_{\bf q} +\varepsilon_{{\bf k}- {\bf q}} -
\varepsilon_{\bf k})}\, ,
\label{eq24}$$ where $|0\rangle$ denotes the vacuum state for the electrons as well as of the phonons. This expression clearly indicates that electrons carry with them an electrical polarization field or, in other terms, the electrons are accompanied by a phonon. This feature reflects the fact that the total wave-vector $${\bf K} = \sum_{{\bf k}\sigma} {\bf k} c^+_{{\bf k}\sigma}c_{{\bf k}\sigma} +
\sum_{{\bf q}\lambda}{\bf q}a^+_{{\bf q}\lambda}a_{{\bf q}\lambda}\, ,
\label{eq25}$$ is a conserved quantity. The most spectacular consequence of this lies in the formation of Cooper pairs [@CooperPR56], as evidenced by the isotope effect of their binding energy [@FrohlichPPSA50] and the ultimately resulting phonon mediated BCS theory of superconductivity [@BardeenPR57]. Electron pairing then can be understood by a process in which the passage of a first electron polarizes the lattice and where subsequently a second electron reabsorbes that polarization. The effective interaction Hamiltonian for this pairing is given by second order perturbation theory: $$H_{\rm el-el} = \sum_{\bf kk'q}|V_{\bf q}|^2
c^+_{{\bf k+q}\uparrow}c^+_{{\bf k'-q}\downarrow}c_{{\bf k'}\downarrow}
c_{{\bf k}\uparrow}{\hbar \omega_{\bf q} \over (\varepsilon_{\bf k} -
\varepsilon_{\bf k-q})^2 -(\hbar \omega_{\bf q})^2 }\, .
\label{eq26}$$ This effective electron-electron interaction shows that within a small region around the Fermi surface i.e., $|\varepsilon_{\bf k} -
\varepsilon_{{\bf k} \pm {\bf q}}| \leq \omega_{\bf q}$ this interaction is attractive and therefore leads to an instability of the Fermi surface resulting in a superconducting ground state via Cooper pair formation in ${\bf k}$-space.
This present discussion of the continuum approach to the coupling between the electrons and the lattice-deformations has shown interaction Hamiltonians which, depending on the underlying materials, show either a coupling between the phonon coordinates $(a^{\phantom +}_{\bf q} + a^+_{-{\bf q}})/\sqrt 2$ and the electron charge density $\rho_{\bf q}$ = $\frac{1}{N}\sum_{\bf k}c^+_{{\bf k}+ \frac{\bf q}{2}}
c^{\phantom +}_{{\bf k}+ \frac{\bf q}{2}}$ (metals) or between the conjugate phonon momenta $(a^{\phantom +}_{\bf q} -a^+_{-{\bf q}})/i\sqrt 2$ and the charge current density ${\bf p}_{\bf q} = \frac{1}{N}\sum_{\bf k}{\bf q}c^+_{{\bf k}+ \frac{\bf q}{2}}c^{\phantom +}_{{\bf k}+ \frac{\bf q}{2}}$ (polar and covalent materials). Formally these interaction terms can be written in a unifying way by rotating the phonon coordinate into the phonon momenta and vice versa by a suitable unitary transformation $U =\exp(-i\frac{\pi}{2}a^+_{\bf q}a^{\phantom +}_{\bf q})$.
The Holstein small polaron {#Hsp}
==========================
The systems discussed in the previous section are characterized by long range electron-lattice coupling which show up in form of (i) a moderate mass renormalization of charge carriers in their band states, (ii) an equally moderate reduction in their mobility due to the scattering of the electrons on the lattice vibrations and (iii) the emergence of phonon sidebands in optical absorption spectra.
The fundamental theoretical question which posed itself in the context of polaron physics ever since Landau [@LandauPZS33] proposed self-trapped localized polarons was to establish whether in such systems one would have a local lattice instability upon increasing the coupling constant $\alpha_{\rm Fr}$, passing from large mobile Fröhlich polarons for weak coupling to localized polarons when the coupling strength exceeds a certain critical value. This question could not be addressed within the continuum approach since it requires a physics which is related to the dynamics of the local lattice deformations on the scale of the unit cell. On the experimental side, more and more new materials where synthesized in the mean time which clearly showed polaronic effects on such short length scales. It was for these reasons that a scenario was introduced which could describe such lattice polarons.
The generic model to capture such a situation, generally referred to as the Holstein molecular crystal model [@HolsteinAP59] treats the problem consequently in real space. Its corresponding Hamiltonian is given by $$\begin{aligned}
H &=& D\sum_{{\bf i}\sigma}n_{{\bf i}\sigma} -
t\sum_{{\bf i}\neq{\bf j}\sigma}(c^+_{{\bf i}\sigma}c^{\phantom{\dag}}_{{\bf j}\sigma}+
h.c.)
-\lambda\sum_{\bf i}n_{\bf i}u_{\bf i}\nonumber \\
& & + \sum_{\bf i}\frac{M}{2}(\dot{u}^2_{\bf i}+\omega^2_0u^2_{\bf i})
+ \sum_{\bf i}Un_{{\bf i}\uparrow}n_{{\bf i}\downarrow}, \label{H}\,
\label{eq27}\end{aligned}$$ where $n_{{\bf i}\sigma} = c^+_{{\bf i}\sigma}c^{\phantom{\dag}}_{{\bf i}\sigma}$ ($n_{\bf i} = \sum_{\sigma}n_{{\bf i}\sigma}$) denotes the density of charge carriers having spin $\sigma$ at molecular sites ${\bf i}$. The electrons are assumed to be coupled to the intra-molecular deformations $u_{\bf i}$ via charge density fluctuations and the coupling constant is denoted by $\lambda$. The dynamics of the lattice is treated purely locally with Einstein oscillators describing the intra-molecular oscillations with frequency $\omega_0$. $M$ denotes the mass of the atoms making up the diatomic molecular units. The additional Hubbard $U$ intra-molecular repulsion is introduced sometimes in order to account for possible correlation effects in conjunction with the purely polaronic features. We shall not consider the effect of this term in this present discussion, but several specific lectures in this school will be devoted to it.
This model is capable of describing the self-trapping of charge carriers which arises from a competition between the energy gain coming from the itinerancy of the electrons and that coming from the potential energy due to the induced local deformations of the molecular units. In order to see that let us rewrite this Hamiltonian in a form which makes more explicit such a self-trapped localized picture: $$\begin{aligned}
H_0 &=& (D-\varepsilon_p)\sum_{\bf i}n_{\bf i} - t\sum_{{\bf i}\neq
j\sigma}(c^+_{{\bf i}\sigma}c^{\phantom{+}}_{{\bf j}\sigma}+
h.c.)+
\sum_{\bf i}(U-2\varepsilon_p)n_{{\bf i}\uparrow}n_{{\bf i}\downarrow} \nonumber \\
&&+ \sum_{\bf i}\frac{M}{2}\left[\left(\dot{u_{\bf i}}-\frac{\lambda \dot{n_{\bf i}}}
{M\omega_0^2}\right)^2+
\omega^2_0\left(u_{\bf i}-\frac{\lambda n_{\bf i}}{M\omega_0^2}\right)^2\right]
- \frac{M}{2}\left[\left(\frac{\lambda \dot n_{\bf i}}{M \omega_0^2}\right)^2-
2\dot u_{\bf i}\frac{\lambda \dot n_{\bf i}}{M \omega_0^2}\right].
\label{eq28}\end{aligned}$$ The major features which evolve out of such a representation are:
- The energy of the electrons is lowered by an amount $\varepsilon_p=\lambda^2/2M\omega^2_0$ which corresponds to the ionization energy of the polaron
- A lattice induced intra-molecular attraction of strength $2\varepsilon_p$ between the electrons on different sites of the diatomic molecule units which can partially or totally compensate their intra-molecular Coulomb repulsion $U$.
- The intra-molecular distance of the Einstein oscillators are shifted by a time dependent quantity $u^0_{\bf i}$ =$\lambda n_{\bf i}/M\omega_0^2$ which follows the time evolution of the charge redistribution on site ${\bf i}$. The frequency of the oscillators is modified as a consequence.
Since in this school we shall primarily be concerned with systems which can be described by the Holstein model, let us now focus in more detail on the basic physics inherent in this scenario. To begin with, we consider this problem in a semi-classical fashion and in the adiabatic approach.
In the limit of small coupling $\varepsilon_p\ll D$, the form of the Hamiltonian, given in eq. (\[eq28\]) is reminiscent of one which describes itinerant electrons in a static potential, given by $V^{\rm wc}(\{u_i\})$ = $-\lambda \langle u_i\rangle c^+_{{\bf i}\sigma} c^{\phantom +}_{{\bf i}\sigma} d^+_{\bf i}d^{\phantom +}_{\bf i}$ with $\langle u_{\bf i}\rangle$ = $\lambda/M\omega^2_0$ and $d^+_{\bf i}d^{\phantom +}_{\bf i}$ representing the density of some fictitious particles. Solving the eigenvalue problem for the ground state energy $E_0$, i.e., $${\lambda^2 \over M \omega^2_0}\sum_{\bf k} {1 \over \varepsilon_{\bf k} - E_0} = 1
\label{eq29}$$ predicts a splitting off of the ground state energy from the bottom of the free itinerant band ($E_0<0$), resulting in a localization of the charge carriers.
In the limit of strong coupling, for large values for $\lambda$ such that $\varepsilon_p \geq D$, the adiabatic approach tells us to ignore the electron itinerancy and assume the electron to be fixed at a particular lattice site. This then results in an adiabatic potential for the electron given by $V^{\rm sc}(\{u_{\bf i}\})$ = $-\varepsilon_{\rm p}n_{\bf i}+\frac{M}{2}\omega^2_0\left(u_{\bf i}-\frac{\lambda n_{\bf i}}{M\omega_0^2}\right)^2$, which follows directly from the form of the Hamiltonian given in eq. (\[eq28\]). If one requires that this potential is deep enough to bind the electron in the first place, the selfconsistency of the adiabatic approach is guaranteed and leads to localized states in this strong coupling limit where the time derivatives of the local deformations can be neglected. As $\lambda$ is varied, these potentials for the weak and strong coupling limits are expected to join up smoothly. Depending on the strength of the coupling $\lambda$, this small exercise shows that one can have a situation where the energies of the two configurations are degenerate. In this intermediary coupling regime one should expect a coexistence of small localized polarons and weakly bound electrons. It has become customary in the theory of small polarons to introduce two dimensionless parameters: $\alpha\equiv \sqrt{\varepsilon_p/\hbar \omega_0}$ measuring the strength of the interaction and $\gamma$ = $t / \hbar \omega_0$ the adiabaticity ratio ($\alpha$ has nothing to do with the dimensionless coupling constant $\alpha_{\rm Fr}$ used in the theory of large polarons and introduced in section \[FLP\]). We then can rewrite the potential $V^{\rm sc}(\{u_i\})$ in a compact form in terms of those parameters and a dimensionless local lattice deformation $\bar{u}_{\bf i}$ = $u_{\bf i} \sqrt{M\omega_0/2\hbar}$. Its variation with $\alpha$ and $\gamma$ as a function of $\bar{u}$ illustrated in fig. \[V\].
![The variation of $V^{\rm sc}(\tilde{u})$ (full lines) felt by an electron with wave vector $k$=0 in the presence of a static continuous deformation of the intramolecular distance $\bar{u}$ for various values of $\alpha$ and $\gamma$. The dotted line describes the energy of the electron with homogeneously deformed oscillators uncoupled to the electron. The crossing points (dots) of these curves indicate a change from itinerant to localized behavior.[]{data-label="V"}](fig1.eps){width="6.5cm"}
This picture of the polaron physics, widely used and appreciated until the nineteen sixties (see ref. [@polarons_excitons63]), although intuitively very appealing is qualitatively incorrect - as we shall see in the next section. There we shall show that a polaron, although self-trapped, remains delocalized for any dimension.
A formal and correct treatment of this problem was first initiated by Holstein and collaborators [@HolsteinAP59; @EminAP69] and in a very elegant and efficient way in terms of a unitary transformation by Firsov and collaborators [@LangJETP63], given by $$\tilde H = e^S H e^{-S}, \qquad S = \alpha \sum_{{\bf i}\sigma}
n_{{\bf i}\sigma}(a_{\bf i} - a^+_{\bf i})\, ,
\label{eq30}$$ which transform the electron operators $c^{(+)}_{{\bf i}\sigma}$ into operators which describe charge carriers rigidly tied to local lattice deformations $$\tilde c^+_{{\bf i}\sigma} = c^+_{{\bf i}\sigma} X^+_{\bf i}, \qquad \tilde c_{{\bf i}\sigma} =
c_{{\bf i}\sigma} X^-_{\bf i}, \qquad
X^{\pm}_{\bf i} = e^{\pm \alpha(a_{\bf i} - a^+_{\bf i})}
\label{eq31}$$ and which correspond to localized polaronic self-trapped states $$X^+_{\bf i}|0)_{\bf i} = \sum_n e^{-\frac{1}{2}\alpha^2}{(\alpha)^n \over \sqrt{n!}} |n)_{\bf i}\, .
\label{eq32}$$ $|n)_{\bf i}$ denotes the n-th excited oscillator states at a molecular sites ${\bf i}$ and $X^+_{\bf i}|0)_i$ signifies an oscillator ground state whose equilibrium position is shifted by an amount $\lambda/M \omega^2_0$. The corresponding transformed Hamiltonian is consequently given by $$\begin{aligned}
\tilde H &=& \sum_{{\bf i}\sigma} (D - \varepsilon_p) c^+_{{\bf i}\sigma} c_{{\bf i}\sigma} -
t\sum_{{\bf i} \neq {\bf j}\sigma} (c^+_{{\bf i}\sigma} c_{{\bf j}\sigma} X^+_{\bf i}X^-_{\bf j} + H.c.)\nonumber\\
&&+ \sum_{\bf i}(U - 2 \varepsilon_p)c^+_{{\bf i}\uparrow} c^+_{{\bf i}\downarrow} c_{{\bf i}\downarrow}c_{{\bf i}\uparrow} + \hbar \omega_0 \sum_{\bf i}(a^+_{\bf i} a_{\bf i} + \frac{1}{2})\,.
\label{eq33}\end{aligned}$$
Such an approach to the polaron problem is particularly useful in the limit of large $\alpha$ where $\varepsilon_p \geq D$ and for a small adiabaticity ratio $\gamma$. In the strong coupling limit and anti-adiabaticity ($\gamma < 1$), the term $X^+_{\bf i}X^-_{\bf j}$ in the transformed Hamiltonian can be averaged over the bare phonon states which, for zero temperature, results in an effective polaron hopping integral $t^* = t e^{-\alpha^2}$. This is justified a posteriori since then $t^*\ll\omega_0$, where the local lattice deformations can be considered to adapt themselves quasi instantaneously to the slowly in time varying positions of the electrons. Such an approximation implies that the number of phonons in the phonon clouds surrounding each charge carrier remains largely unchanged during the transfer of a charge carrier from one site to the next, while processes where the number of phonons in the clouds change give rise to a polaron damping. Neglecting such damping effects, one obtains well defined Bloch states for the polaronic charge carriers (defined by $c^+_{{\bf i}\sigma} X^+_{\bf i}$) , albeit with a much reduced hopping integral $t^*$, while at the same time the electrons loose practically all of their coherence and thus their quasi-particle features. In order to illustrate that, let us consider the Green’s function for a single localized polaron, respectively for a single localized electron by putting $t^* = 0$. For the polaron retarded Green’s function we have $$\begin{aligned}
G_{\rm p}^{\rm ret}(t) &=& -\theta(t)(0|X^-_{\bf i}[ e^{{\bf i} \tilde H t} \tilde{c}^+_{{\bf i}\sigma}(0)
e^{-{\bf i} \tilde H t}\tilde{c}_{{\bf i}\sigma}(0)] X^+_{\bf i}|0)\nonumber\\
&=& -\theta(t)\sum_n e^{{\bf i}( \varepsilon_{\rm p} - n\hbar \omega_0)t}
|(0|X^-_{\bf i} \tilde{c}^+_{{\bf i}\sigma}(0) |n)|^2 \delta_{n,0}=-\theta(t)
e^{{\bf i}\varepsilon_{\rm p}t}|(0|c^+_{{\bf i}\sigma}(0) |0)|^2
\label{eq34}\end{aligned}$$ which after Fourier transforming becomes $$G_{\rm p}^{\rm ret}(\omega) = \lim_{\delta \rightarrow 0}
{1 \over \omega + i\delta + \varepsilon_{\rm p}}
\label{eq35}$$ and displays a spectrum which consists of exclusively a coherent contribution. On the contrary, the electron retarded Green’s function $$\begin{aligned}
G_{\rm el}^{\rm ret}(t) &=& -\theta(t)(0|X^-_{\bf i}[ e^{i \tilde H t} c^+_{\bf i}(0)
e^{- i \tilde H t}c_{{\bf i}\sigma}(0)] X^+_{\bf i}|0)\nonumber\\
&=& \sum^{\infty}_{n=0} e^{i( \varepsilon_{\rm p} - n\hbar \omega_0)t}
|(0|X^-_{\bf i} c^+_{{\bf i}\sigma}(0) |n)|^2
\label{eq36}\end{aligned}$$ displays a spectrum given by its Fourier transform $$G_{\rm el}^{\rm ret}(\omega)\simeq \lim_{\delta \rightarrow 0}\sum^{\infty}_{n=0}
{e^{-\alpha^2}\alpha^{2n}
\over n!} {1 \over \omega + i\delta + \varepsilon_{\rm p} - n \hbar \omega_0}\, .
\label{eq37}$$ The spectral weight of the coherent part is now reduced to $\exp(-\alpha^2)$, corresponding to the term $n$ = 0 in eq. (\[eq37\]), while the major part of the spectrum is made up by the incoherent contributions which track the structure of the composite nature of the polaron.
Generalizing these results by including the itinerancy of the charge carriers leads to very similar results [@RanningerPRB93] for the electron Green’s function for a many polaron system, which in this strong coupling anti-adiabatic limit reduces to a system of small polarons in band states and where the Green’s function for the electrons with wavevectors ${\bf k}$ is given by
and where $\varepsilon^*_{\bf k}$ = $e^{-\alpha^2}\varepsilon_{\bf k}$. These characteristic spectral properties of small polarons can be tested by photo-emission spectroscopy and present crucial tests which permit to distinguishing between different mechanisms leading to heavily dressed composite quasi-particles. The Poisson type phonon distribution was particularly well demonstrated early on in such experiments on simple molecules such as $H_2$ molecules [@AsbrinkCPL70] and carbon rings [@HandschuhPRL95], on localized polaron states in the manganites by neutron spectroscopy [@LoucaPRB99] and in the cuprate superconductors by infrared absorption measurements [@BiPRL93].
The localized, or almost localized, nature of such small polarons is apparent in their electron occupation number distribution illustrated in fig. \[n-k\]
![Occupation number $n_{{\bf k}\sigma}$ for a system of itinerant small polarons for $\alpha\gg 1$. The dashed and dotted curves indicate the case $n = 1$ for $\alpha \simeq 1$ and $\alpha\ll 1$ respectively (after ref. [@RanningerPRB93]).[]{data-label="n-k"}](figpolaron1.eps){width="6.5cm"}
showing an almost flat distribution covering the whole Brillouin zone, i.e., $$n_{{\bf k}\sigma} = \langle c^+_{{\bf k}\sigma}c^{\phantom +}_{{\bf k}\sigma}\rangle =
(1 -e^{-\alpha^2})n_{\sigma} + e^{-\alpha^2}n_{\sigma}(\varepsilon^*_{\bf k})\, .
\label{eq39}$$
There is a qualitative difference between the weak coupling and the strong coupling limit discussed here. We have seen that in the weak coupling limit of the Fröhlich Hamiltonian the electrons are accompanied by phonons with which they continuously exchange the momentum such as to keep the total momentum of electron plus phonon constant. In the strong coupling limit the electrons are surrounded by real lattice deformations which in terms of phonons means clouds of phonons and which in the limit of extreme strong coupling, where the electrons can be considered as localized on a given site, have a Poisson distribution. The new feature now is that this deformation corresponds to a local mass inhomogeneity capable of carrying true momentum rather than the pseudo-momentum as is the case for weak coupling. Again, as we shall see below, the momentum between the electron and this deformation is perpetually exchanged as the polaronic charge carrier moves through the lattice, leading to a space-time dependent interaction between the charge carrier and the deformation associated to it. This effect will be particularly important in the cross-over between the weak coupling adiabatic and the strong coupling anti-adiabatic regime, discussed in the next section.
A key point which occupied the field of polaron theory for several decades was to try to establish if a large delocalized polaron changes discontinuously or continuously—although abruptly— into a small self-trapped polaron as the electron-lattice coupling is increased beyond a certain critical value. We shall review this issue in the following.
Self-trapping
=============
We shall in this section discuss the meaning of self-trapping. For that purpose let us go back for a moment to the semi-classical approach of the polaron problem (discussed in section \[Hsp\] above) and the adiabatic lattice potential which goes with it. On the basis of such an approach, amounting to treat the polaron problem as a potential problem, it is tempting to conclude localization of charge carriers as their coupling to the lattice degrees of freedom increases. As we shall see below, the polaron problem can not be treated as a potential problem. Its intrinsic dynamics of the local lattice deformations is determinant in correlating the dynamics of the charge and the lattice degrees of freedom, thus resulting in itinerant delocalized states of electrons surrounded by clouds of phonons whose density varies and increases as we go from the adiabatic weak coupling to the anti-adiabatic strong coupling limit. For that reason the lattice degrees of freedom must be treated in a quantized version.
Let us start with the weak coupling limit for which we can assume a polaron state in the form $$|\Psi\rangle_{\bf k} = {1 \over \sqrt N} \sum_{\bf ij}e^{{\bf ik} \cdot {\bf r}_{\bf i}}
[c^+_{{\bf i}\sigma}\delta_{\bf ij} + f_{\bf ij}^{\bf k}c^+_{{\bf j}\sigma} a^+_{\bf i}]|0\rangle\, ,
\label{eq40}$$ which is the real space version of the state previously discussed, i.e., eq. (\[eq24\]). Diagonalizing the Hamiltonian within such a subspace of zero and respectively one phonon present, leads to the following selfconsistent equations for the eigenvalues and parameters $f_{ij}^{\bf k}$. $$\begin{aligned}
E_{\bf k} &=& \varepsilon_{\bf k} - (\alpha \hbar \omega_0)^2 \frac{1}{N}\sum_{{\bf k}'}
{1 \over \varepsilon_{{\bf k} - {\bf k}'} - E_{\bf k} + \hbar \omega_0}\,,\label{eq41} \\
f^{\bf k}_{ij} &=& \alpha \hbar \omega_0 \frac{1}{N}\sum_{{\bf k}'}
{e^{i {\bf k}' \cdot ({\bf r}_i - {\bf r}_j)} \over
\varepsilon_{{\bf k} - {\bf k}'} -E_{\bf k} + \hbar \omega_0}\, .\label{eq42}\end{aligned}$$
Considering the solution for the ground state (${\bf k}=0$) of this problem, one notices that its corresponding eigenvalue $E_0$ falls below the bottom of the free electron band ($E_0 \leq 0$). This signals a bound state which is characterized by an exponential drop-off of the spatial correlation between the electron and the accompanying phonon, which for a 3D system is given by: $$f^0_{\bf ij} = v\frac{\alpha \omega_0 m}{2 \pi \hbar}\frac{1}
{|{\bf r}_{\bf i}- {\bf r}_{\bf j}|}e^{-\sqrt{2m(-E_0 +\hbar \omega_0)}
\frac{|{\bf r}_{\bf i}- {\bf r}_{\bf j}|}{\hbar}}
\label{eq43}$$ ($v$ denoting the unit cell volume) and which indicates that the dynamically (not thermally!) excited phonons accompanying the electron remain in its immediate vicinity. The exponential drop-off does not depend on any dimensionality (in contrast to the semi-classical approach outlined in section \[Hsp\]) and gives us a first indication for the intricate correlation between the inherent dynamics of lattice fluctuations and the charge dynamics. Hence, already in this weak coupling limit, the polaron problem can not be reduced to an effective (adiabatic) potential problem.
Let us next turn to the strong coupling limit, where on the basis of such an adiabatic approach one would expect self-trapped lowest energy eigenstates of the form $$|\Psi\rangle_{\bf k} = {1 \over \sqrt N}\sum_{\bf i} e^{i{\bf k}\cdot{\bf r}_i}c^+_{\bf k}
|\Phi_{\bf k}(\{u_j\})\rangle\, ,
\label{eq44}$$ where $\Phi_{\bf k}(\{u_{\bf j}\})$ denotes a function of the ensemble of oscillator coordinates ${u_{\bf j}}$ situated at sites ${\bf j}$ in the vicinity of site ${\bf i}$ where the electron is located. Guided by the semi-classical picture of the adiabatic potential, resulting in the strong coupling limit, one can try an intuitively appealing Ansatz for the lattice wavefunction in form of a series of displaced oscillators around the site where the electron is situated and which, for the ground state, can be assumed to be of the form $$|\Phi_{{\bf k}=0}(\{u_{\bf j}\})\rangle= e^{-\alpha
\sum_{\bf j}(f^*_{\bf ij} a^+_{\bf j}- f_{\bf ij} a_{\bf j})}|0\rangle \, .
\label{eq45}$$ Determining the parameters $f_{\bf ij}$ variationally [@ToyozawaPTP61] leads to solutions which change discontinuously in the intermediary coupling regime. Associated with that is a discontinuous change of the mass of the corresponding quasi-particles amounting to a change-over from a practically free band dispersion with a electron mass $m_{\rm el}$ to an effective polaron mass $m_{\rm p}$ given by $m_{\rm p}/m_{\rm el}=e^{\alpha^2}$ (see fig. \[Shore1\]).
It has been a matter of dispute for many decades and up to the 1960-ties whether this discontinuity is a real effect or is an artifact of the theory. From the experimental side, small polarons where found to be localized and have ever since, for the presently available materials, shown a mobility via hopping rather than Bloch like band motion and metallic conductivity. The theoretical results obtained for the cross-over regime between large and small polarons on the basis of the semi-classical description in terms of an adiabatic lattice potential with two minima of comparable energy (see the discussion in section \[Hsp\]), suggest already that in this regime one should have particularly strong fluctuations of the local lattice displacements which could possibly be modeled by some effective lattice wave function in terms of a superposition of two sorts of oscillator states: one practically undisplaced and one displaced oscillator state on the site housing the electron, as hypothesized early on in a slightly different context [@EaglesPSSB71]. These ideas were subsequently followed up by a variety of different approaches such as: exact diagonalization studies [@ShorePRB73] on small clusters, upon assuming the charge induced deformation to be constraint to the site where the electron sits [@ChoJPSJ71] (when the problem can be solved analytically) and Quantum Monte Carlo studies [@deRaedtPRB83].
All these approaches converge to a lattice wave function for the ground state which can be approximated by $$|\Phi_{\bf p}(\{u_{\bf j}\})\rangle=
[e^{-\alpha\sum_{\bf j}(f^*_{\bf ij}a^+_{\bf j}- f_{\bf ij}a_{\bf j})} +
\eta e^{-\alpha
\sum_{\bf j}(g^*_{\bf ij}a^+_{\bf j}- g_{\bf ij} a_{\bf j})}|]0\rangle \, .
\label{eq46}$$ This results (after variationally determining the parameters $f$, $g$ and $\eta$) in a smooth, but nevertheless very abrupt, cross-over between the weak and strong coupling features and a change-over from a quasi-free electron band mass to a strongly enhanced one (see fig. \[Shore1\]). The form of eq. (\[eq46\]) represents for any electron situated on a given site to be associated with an oscillator wave function on this site, illustrated in fig. \[Shore2\]. It indicates an essentially undisplaced oscillator for weak coupling, a displaced one for strong coupling and a superposition of such two oscillator states for the cross-over regime between those two limits. This latter has led to the suggestion of strong retardation effects between the dynamics of the electron and that of the local lattice deformation which could possibly result in a dynamically disordered system and subsequent localization, a scenario which we shall return to in the lecture “From Cooper-pairs to resonating Bipolarons”.
Let us here consider this point in more detail and investigate this cross-over behavior in terms of the strong dynamical correlation which act between the charge and the lattice degrees of freedom. We shall demonstrate that on hand of a polaron toy problem, such as a two-site system involving two adjacent diatomic molecules whose individual oscillations are uncorrelated with each other and an electron with spin $\sigma$ hopping between those two molecular units. The Hamiltonian, eq. (\[eq27\]), for that then reduces to $$\begin{aligned}
H &=& t(n_{1\sigma} +n_{2\sigma})-
t(c^{+}_{1\sigma}c^{\phantom{+}}_{2\sigma} +
c^{+}_{2\sigma}c^{\phantom{+}}_{1\sigma})
-\lambda(n_{1}u_1+n_{2}u_2) \nonumber\\
&&+ \frac{M}{2} \left[(\dot{u}^2_1+ \dot{u}^2_2) + \omega^2_0(u^2_1+u^2_2)\right]\, .
\label{eq47}\end{aligned}$$
![The variation of the oscillator wave function on a given site (after ref. [@ShorePRB73]) as a function of $ \{u,\,W_0\}$ (= $\{ \alpha^2(\gamma 2z)^{-1}),\,\alpha^2 \}$ in our notation) for oscillator trial wave functions $\Psi^{II}$ such as given by eq. (\[eq46\]) and with $x = \frac{1}{\sqrt 2}\langle a^+_i + a_i\rangle$.[]{data-label="Shore2"}](figShore1.eps){width="5.5cm"}
![The variation of the oscillator wave function on a given site (after ref. [@ShorePRB73]) as a function of $ \{u,\,W_0\}$ (= $\{ \alpha^2(\gamma 2z)^{-1}),\,\alpha^2 \}$ in our notation) for oscillator trial wave functions $\Psi^{II}$ such as given by eq. (\[eq46\]) and with $x = \frac{1}{\sqrt 2}\langle a^+_i + a_i\rangle$.[]{data-label="Shore2"}](figShore2.eps){width="7.5cm"}
Upon introducing the variables $$X ={u_1 - u_2 \over \sqrt{2}},\qquad Y ={u_1 + u_2 \over \sqrt{2}}
\label{eq48}$$ this Hamiltonian separates into two contributions $H_X$ and $H_Y$ depending respectively on lattice coordinates $X$ and $Y$. $Y$ couples exclusively to the total number of charge carriers in the system, ie. $(n_{1\sigma} + n_{2\sigma})$. It presents an in-phase oscillation of the two molecules and has hence no effect on the dynamics of the polaron and will be disregarded for the present considerations. The lattice coordinate $X$, on the contrary, presents an out of phase oscillation of the two molecules and couples to the relative charge distribution $(n_{1\sigma} - n_{2\sigma})$. The Hamiltonian for this subspace which describes the intertwined dynamics of the charge and the lattice deformations is $$\quad H_X = t(n_{1\sigma} + n_{2\sigma})-
t(c^{+}_{1\sigma}c^{\phantom{+}}_{2\sigma} +
c^{+}_{2\sigma}c^{\phantom{+}}_{1\sigma})
-{\lambda \over \sqrt{2}} (n_{1\sigma} - n_{2\sigma}) X
+ \frac{M}{2} \left[\dot{X}^2 + \omega^2_0 X^2\right]\, .
\label{eq49}$$ The term proportional to $\lambda$ in this Hamiltonian clearly indicates how the charge fluctuations between the two molecules induces a coupling between the oscillators of those two molecules and which ultimately couples back to the dynamics of the charge transfer. This generates, as we shall see, a dynamical lattice deformation mode which accompanies the transfer of charge from one to the other molecule in a slow and smooth continuous fashion. Upon introducing the quantification of the out of phase lattice coordinate $X=(a^{\phantom{+}}+a^+)/\sqrt{2M\omega_0/\hbar}$ in terms of corresponding phonon operators $a^+, a^{\phantom{+}}$ and diagonalizing the above Hamiltonian in a truncated Hilbert space (keeping the number of phonon states finite but large) [@RanningerPRB92] we obtain for the ground state of this system with a single electron with spin $\sigma$: $$|GS\rangle_{\sigma} = \frac{1}{\sqrt 2}\left[c^+_{1\sigma} |\psi^0_+(X)\rangle +
c^+_{2\sigma} |\psi^0_-(X)\rangle\right]\, .
\label{eq51}$$ $|\psi^0_{\pm}(X)\rangle$ denote oscillator states in real space when the electron is either situated on site 1 or site 2, similar to the oscillator wave functions illustrated in fig. \[Shore2\]. In particular, in the cross-over regime, these oscillator wave-functions reflect the bimodal probability distribution of the equilibrium positions of the oscillator where the electron sits. Let us next investigate the evolution in time of the charge transfer together with that of the inter-molecular deformation, given by the correlation functions:
The results of this are reproduced in fig. \[DynCor3\] for a fixed value of $\alpha$ (which in the present notation corresponds to $\alpha = \sqrt 2 \cdot 1.2$ = 1.70) and where we cover the cross-over from the strong coupling anti-adiabatic limit ($\gamma$ = 0.1) to the strong coupling adiabatic limit ($\gamma= 2.0$) upon increasing the adiabaticity ratio.
![The evolution in time (in units of $\omega^{-1}_0$) of the charge and intra-molecular distance transfer between two adjacent molecular units for a two-site one-electron system (after ref. [@deMelloPRB97]) in the anti-adiabatic ($\gamma$ = 0.1), the adiabatic ($\gamma$ = 2.0) and in the cross-over regime ($\gamma$ = 1.1) and for (in the present notation) a coupling strength $\alpha=1.2\sqrt{2}\simeq 1.7$.[]{data-label="DynCor3"}](EMJR11aNW.eps "fig:"){width="6.5cm"} \[DynCor1\]
![The evolution in time (in units of $\omega^{-1}_0$) of the charge and intra-molecular distance transfer between two adjacent molecular units for a two-site one-electron system (after ref. [@deMelloPRB97]) in the anti-adiabatic ($\gamma$ = 0.1), the adiabatic ($\gamma$ = 2.0) and in the cross-over regime ($\gamma$ = 1.1) and for (in the present notation) a coupling strength $\alpha=1.2\sqrt{2}\simeq 1.7$.[]{data-label="DynCor3"}](EMJR11dNW.eps "fig:"){width="6.5cm"} \[DynCor2\]
![The evolution in time (in units of $\omega^{-1}_0$) of the charge and intra-molecular distance transfer between two adjacent molecular units for a two-site one-electron system (after ref. [@deMelloPRB97]) in the anti-adiabatic ($\gamma$ = 0.1), the adiabatic ($\gamma$ = 2.0) and in the cross-over regime ($\gamma$ = 1.1) and for (in the present notation) a coupling strength $\alpha=1.2\sqrt{2}\simeq 1.7$.[]{data-label="DynCor3"}](EMJR11bNW.eps){width="6.5cm"}
The anti-adiabatic limit is characterized by a smooth and slowly in time varying transfer of charge from one molecule to the next. This charge transfer then [*slaves*]{} the inter-molecular deformation $X$ by subjecting it to a slowly in time varying driving force $-{\lambda \over \sqrt{2}} [n_{1\sigma}(\tau) - n_{2\sigma}(\tau)]X(\tau)$. This in turn leads to a slowly in time varying sinusoidal intermolecular deformation, onto which are superposed the local intrinsic oscillations of the individual molecules with a frequency of the order of $\omega_0$. This is the exact opposite of what happens in the extreme other limit, when we approach the adiabatic situation.
In this adiabatic regime, considering here an adiabaticity ratio $\gamma=2.0$, the slowly in time varying quantity is now the inter-molecular deformation (showing almost no fluctuations of the individual molecules with frequency $\omega_0$), which [*slaves*]{} the intermolecular charge transfer characterized by $[n_{1\sigma}(\tau) - n_{2\sigma}(\tau)]$ . Averaged over several periods of oscillations of frequency $\simeq 1/t$, this charge transfer adiabatically follows again the slowly in time varying potential $-{\lambda \over \sqrt{2}} [n_{1\sigma}(\tau)- n_{2\sigma}(\tau)] X(\tau)$.
These results show that in these two extreme limits of adiabaticity and anti-adiabaticity the slowly in time varying component of the charge transfer processes and of the inter-molecular deformation dynamics are completely locked together. This is no longer the case in the cross-over regime between these two extreme limits, here shown in fig. \[DynCor3\] for the adiabaticity ratio $1.1$. One can clearly distinguish phase-slips occurring between the correlated motion of the charge and the deformation transfer, which, in a large system where many body effects become important, could possibly lead to a dynamically induced localization of the charge carriers.
Some early experimental insights {#earlyexperinsights}
================================
The local character of small polaronic charge carriers requires specific experimental probes able to track the polaron induced lattice as well as charge excitations on a short spatial ($\simeq$ 5 Å) as well as time ($10^{-13}-10^{-15}$ sec) scale. Generally such probes are quite adequate for insulating polaronic systems with either charge ordered bipolarons (Ti$_4$O$_7$), bipolaronic Mott type insulators (WO$_{3-x}$, the manganites and nickelates) and low density polaron systems arising from photo-induced or chemical doping and situated on the verge of a quantum phase insulator-superconductor transition (high $T_c$ cuprates). Concerning the cuprates, manganites and nickelates we refer the reader to the lectures by T. Egami and N. L. Saini in this volume, dealing with pulsed inelastic neutron scattering techniques and EXAFS as well as XANES. Here we shall restrict ourselves to the discussion of more classical probes such as optical absorption which can select specific local polaron-sensitive lattice modes.
As we have seen in the discussion presented above, one of the major issue here is to track the disintegration and reconstruction of a polaron during its transfer from one site to the next. This process implies a gradual stripping off of the electron’s phonon cloud upon leaving a given site and a subsequent rebuilding of this phonon cloud on the new site where the electron eventually ends up. Another important issue in the polaron problem is connected to the physics of Many-Polaron system, caused by the polaron induced local attraction which can bind two polarons on a given effective site into a bipolaron. We shall now discuss specific experiments which can illustrate these two characteristic polaron features.
Shortly after Anderson’s suggestion [@AndersonPRL75] of localized bipolarons in amorphous chalcogenide glasses, exhibiting a natural diamagnetism resulting from covalent bonding in locally deformed lattice structures, a variety of systems were found where bipolarons existed in more or less dense situations. Examples for dense bipolaronic systems, exhibiting spatially symmetry broken states related to bipolaron ordering are: Ti$_{4-x}$V$_x$O$_7$ [@LakkisPRB76] and Na$_x$V$_2$O$_5$ [@ChakravertyPRB78]. An example for a dilute bipolaron systems is WO$_{3-x}$ [@SchirmerJPC80]. In those systems the bipolarons form on adjacent cations, such as Ti$^{3+}$-Ti$^{3+}$, V$^{4+}$-V$^{4+}$ and W$^{5+}$-W$^{5+}$ bonds in strongly deformed octahedral ligand environments. Those bipolaronic units are imbeded in a corresponding background of Ti$^{4+}$-Ti$^{4+}$, V$^{5+}$-V$^{5+}$ and W$^{6+}$-W$^{6+}$ molecular units, which together with the bipolaronic entities, constitute those crystalline materials.
Bipolaron dissociation and recombination in WO$_{3-x}$
------------------------------------------------------
Bipolarons can be dissociated with light which, in the case of WO$_{3-x}$, leads to the creation of isolated W$^{5+}$ cation sites in vibrationally excited states of the ligand environments and whose concentration can be tracked by electron spin resonance (ESR) signals coming from those W$^{5+}$ sites. Optical absorption measurements [@SchirmerJPC80] in non-illuminated samples show a peak in the spectrum centered around 0.7 eV coming from pre-existing isolated W$^{5+}$ cations sites. When the crystal is illuminated with a broad spectral band centered around a suitable frequency (1.1 eV in this case) the optical absorption coming from the W$^{5+}$ cations sites increases while, concomitantly, an intrinsic shoulder of the absorption band at around 1 eV—attributed to the absorption coming from the intrinsic bipolaronic W$^{5+}$-W$^{5+}$ bonds—decreases correspondingly. After the illumination is shut off, the single-polaron vibrationally exited W$^{5+}$ units relax to bipolaronic ones, as can be tracked by ESR measurements as a function of time (see fig. \[Salje2\]).
![The photoinduced infrared absorption coming from the laser-stimulated, respectively laser-suppressed vibrational lattice modes (low frequency part) and the photomodulation response for electrical ingap excitations (high frequency part) in YBa$_2$Cu$_3$O$_{6.2}$ (after ref. [@Taliani90]).[]{data-label="Taliani"}](figSalje2.eps){width="6.5cm"}
![The photoinduced infrared absorption coming from the laser-stimulated, respectively laser-suppressed vibrational lattice modes (low frequency part) and the photomodulation response for electrical ingap excitations (high frequency part) in YBa$_2$Cu$_3$O$_{6.2}$ (after ref. [@Taliani90]).[]{data-label="Taliani"}](figTaliani.eps){width="6.5cm"}
The time evolution of this relaxation follows a bimolecular recombination process controlled by a rate equation given by $${{\rm d}n_{\rm W}\over {\rm d}t} = -B(t) n^2_{\rm W}\, ,
\label{eq54}$$ where $n_W$ denotes the density of the W$^{5+}$ ions and $B(t)$ is a function describing diffusion limited reactions. This relaxation behavior is clearly distinct from that of mono-molecular process which describe relaxation processes from vibrationally excited single polaron units to their ground state. Such time resolved studies permit to investigate the rebuilding of the phonon cloud for a bipolaron after its separation into two separate polarons, much as would be expected for processes of bipolarons hopping between neighboring sites.
Photo-induced polarionic charge carriers in high $T_{\rm c}$ cuprates
---------------------------------------------------------------------
A somewhat related in spirit technique was used in the study of polaronic features in the high $T_{\rm c}$ cuprates [@Taliani90]. It was based on examining the variation in the optical absorption spectrum upon doping those materials in the semiconducting phase with electron-hole pairs using so-called photo-induced doping. The idea was to use laser illumination with photon energies bigger than the semiconductor gap in the pump laser beam. In the specific case of the high $T_c$ cuprates this meant a photon energy of typically 2 to 2.5 eV and which created an estimated density of charge carriers of about $10^{19}$ per cm$^3$. The injected photoinduced charge carriers modify locally the lattice symmetry and, by doing so, lead to the activation of corresponding local phonon modes (the 434 cm$^{-1}$ out of plane mode and the 500 cm$^{-1}$ axial \[Cu-O$_4$\] stretching mode). These modes can be made evident as the steady state response of the system to the photo illumination in the absorption spectrum tested by a probe infrared beam (see fig. \[Taliani\]). In such an experimental set up, the 590 cm$^{-1}$ mode with the tetragonal symmetry (being associated with the undistorted lattice which characterize the non-illuminated materials) loose in intensity (bleaching effect) in the absorption spectrum after the samples have been illuminated. Examining the response of the system at higher frequencies, covering the energy range of the semiconducting gap and higher, by photo-modulation techniques (which test the photo-induced charge carriers by their absorption of light in a narrow frequency window) visualizes the excitations with a life time which is inversely proportional to this frequency. The corresponding absorption spectra show an activation of charge excitations over a broad background, covering the energy region of the semiconducting gap, and a bleaching for energies above that frequency (see fig. \[Taliani\]). These results imply a shift of spectral weight upon illumination from the states above the gap in the non-illuminated samples into in-gap states. The intensity of the absorption of the activated phonon modes as well as of the activated electronic in-gap excitations turns out to scale like the square root of the intensity of the laser pulse illuminating the sample. This suggests that (in analogy with studies on WO$_{3-x}$ discussed above) those excitations relax via bimolecular recombination processes. Since the studies on photo-induced charge carriers have features similar to those in low doped systems, obtained by chemical substitution, it has been tempting to conclude that the charge carriers in the low doped superconducting samples have resonant bipolaronic features of a finite lifetime rather than corresponding to well defined stable bipolaronic entities. Such entities can nevertheless condense into a superconducting state and are controlled by phase rather than amplitude fluctuations as discussed in my lecture: “[*From Cooper pairs to resonating bipolarons*]{}” in this volume.
Bipolaronic charge ordering in Ti$_4$O$_7$
------------------------------------------
A particularly interesting and physically very rich example of a dense bipolaronic system is Ti$_4$O$_7$. It exhibits a low temperature phase ($T \leq 140$ K) where bipolarons are in a symmetry broken ordered state, consisting of diamagnetic $T^{3+}-T^{3+}$ bipolaronic diatomic pairs with sensibly reduced intra-molecular distances as compared to $T^{4+} -T^{4+}$ pairs with which they alternate in quasi 2D slab like structures. Upon increasing the temperature, there is a small interval, \[140 K $\leq T \leq 150$ K\], were these bipolaronic electron pairs are dynamically disordered and eventually break up into individual electrons and a metallic phase for $T \geq 150$ K. The experimental measurements of this material involved x-ray diffraction, resistivity, susceptibility and specific heat measurements as well as electron paramagnetic resonance (EPR) studies, summarized in ref. [@LakkisPRB76]. The two low temperature phases in the regimes $T \leq 140$ K and 140 K $\leq T \leq 150$ K are semiconducting with a conductivity characterized by comparable activation energies of the order of 0.16 eV. The phase transition between those two semiconducting phases can be attributed to an order-disorder phase transition where the bipolarons are essentially the same as in the low temperature ordered phase, as evident from the absence of any change in the magnetic susceptibility as well as intra-molecular distance of the bipolarons when going through this phase transition. The transition to the high temperature phase at $T$ = 150 K is characterized by a breaking up of the bipolarons into itinerant electrons leading to a metal with an enhanced Pauli susceptibility and a substantial decrease in the unit-cell volume (see fig. \[Lakkis\]).
![The electrical conductivity for Ti$_4$O$_7$ (after ref. [@LakkisPRB76]) showing with increasing $T$ a transition from a low temperature activated hopping regime to a different activated hopping regime at $T\simeq 140$ K followed by a second transition from that to a metallic phase at $T \simeq 150$ K. The magnetic susceptibility, shown in the inset, does not change at the low temperature transition[]{data-label="Lakkis"}](figLakkis.eps){width="6.5cm"}
Both transitions are first order and the entropy change in the high temperature transition is due, in roughly equal amounts, to electronic and lattice contributions. The most important feature of this system is however that the disordered bipolaronic phase is a dynamical rather than static disorder as evidenced from EPR experiments which show a vanishing of the EPR line upon entering this phase from the low temperature ordered phase. It was on the basis of these experimental findings that the possibility of a condensation of bipolarons was initially proposed [@AlexandrovPRB81]. This broke with a traditionally severely guarded doctrine and stimulated to look for superconducting materials which (i) were oxides, (ii) have reduced dimensionality, (iii) are close to insulating parent compounds and (iv) are generally poor rather than good metals in the normal phase.
On the basis of such a scenario and the Hostein model one could expect for such systems a superfluid state of bipolarons, albeit with a very small critical temperature, since being inversely proportional to the bipolaron mass which is typically several orders of magnitude bigger than the electron band mass. The real difficulty to observe this type of superfluidity in crystalline materials might however be related to the fact that the standard polaron models, generally based on harmonic lattice potentials, totally neglect any relaxation processes. High $T_c$ cuprates are clearly not candidates for this extreme case of Bipolaronic Superconductivity but are likely to contain localized bipolarons as resonant states inside the Fermi sea of itinerant electrons which could result in a superconducting state controlled by phase rather than amplitude correlations.
Summary
=======
In this introductory lecture I briefly reviewed various kinds of electron-lattice couplings, characterizing different classes of materials and which give rise to two distinct categories of polarons: large Fröhlich and small Holstein polarons. The importance of treating the phonons as quantum rather than classical variables became evident in connection with the question of itinerancy of the polaronic charge carriers. The dynamics of the polaron motion, exemplified in real time, shows a highly non-linear physics involving the dynamics of coupled charge and the lattice fluctuations which mutually drive each other. The present discussion was restricted to the single polaron respectively bipolaron problem and to limiting cases (such as strong coupling anti-adiabatic limit) where the Many Polaron problem can be decomposed into a band of single polaron states with different wave vectors. The fundamental questions of the polaron problem which pose themselves today, are evidently beyond the topics touched upon in this preliminary discussion. These are questions which concern the cross-over between the adiabatic and anti-adibatic regimes, the polaron induced residual interactions in a Many Polaron system and its dependence on the range of electron-lattice coupling as well as on the density of charge carriers, which can result in possible transitions between insulating and metallic behavior of polarons. At this stage, in order to tackle this kind of problems we have to resort to highly sophisticates numerical techniques, which will be presented in this school. The hope is that eventually this will give us some insight into this complex Many Body problem so that sooner or later we can formulate this polaron physics in a way where analytical approaches can capture its main qualitative features.
[99]{} L. D. Landau, Phys. Z. Sowjetunion, [**3**]{}, 644 (1933). H. Fröhlich, Adv. in Phys. [**3**]{}, 325 (1954). J. Bardeen and W. Shockley, Phys. Rev.[**80**]{},72 (1950). D. J. Scalapino in “Superconductivity” ed. R. D. Parks, Marcel Dekker, Inc, (New York 1969). H. Fröhlich, H. Pelzer and S. Zienau, Phil. Mag., [**41**]{}, 221 (1950). L. N. Cooper, Phys. Rev., [**104**]{}, 1189 (1956). H. Fröhlich, Proc. Phys. Soc., [**A63**]{} , 778 (1950). J. Bardeen, L. N. Cooper and J. R. Schrieffer, Phys. Rev., [**108**]{}, 1175 (1957). Holstein T., Ann. Phys. (N.Y.) [**8**]{} (1959) 325; ibid 343. “Polarons and Excitons” ed. C. G. Kuper and G. D. Whitefield (Plenum Press, New York, 1963). D. Emin and T. Holstein, Ann. Phys. (N.Y.) [**53**]{}, (1969) 439. I. G. Lang and Yu. A. Firsov, Sov. Phys. JETP [**16**]{}, 1301 (1963). J. Ranninger, Phys. Rev. B [**48**]{}, 13166 (1993). L. Asbrink, Chem. Phys. Lett.[**7**]{}, 549 (1970). H. Handschuh, G. Ganteför, B. Kessler, P. S. Bechthold and W. Eberhardt, Phys, Rev. Lett., [bf 74]{}, 1095 (1995). D. Louca and T. Egami, Phys. Rev.B [**59**]{}, 6193 (1999). X.-X Bi and P. Ecklund, Phys. Rev .Lett., [**70**]{}, 1625 (1993). Toyozawa Y., Prog. Theor. Phys.. [**26**]{}, 29 (1961). D. M. Eagles, Phys. Stat. Solidi B [**48**]{}, 407 (1971). H. B. Shore and L. M. Sander, Phys. Rev. B [**7**]{}, 4537 (1973). Cho K. and Toyozawa Y., J. Phys. Soc. Jap., [**33**]{}, 1555 (1971). H. de Raedt and Ad. Lagendijk, Phys. Rev. B [**27**]{}, 6097 (1983). J. Ranninger and U. Thibblin, Phys. Rev. B [**45**]{}, 7730 (1992). E. V. L. de Mello and J. Ranninger, Phys. Rev. B [**55**]{}, 14872 (1997). P. W. Anderson, Phys. Rev. Letts., [**34**]{}, 953 (1975). S. Lakkis, C. Schlenker, B. K. Chakraverty, R. Buder and M. Marezio, Phys. Rev. B [**14**]{}, 1429 (1976). B. K. Chakraverty, M. J. Sienko and J. Bonnerot, Phys. Rev. B [**17**]{} 3781 (1978). O. F. Schirmer and E. Salje, J. Phys. C [**13**]{} L1067 (1980). C. Taliani, A. J. Pal, G. Ruani, R. Zamboni, W. Wei and Z. V. Vardeny, in “Electronic properties of High $T_c$ Superconductors and related Compounds” Ed. H. Kuzmany, M. Mehring and J. Fink, (Springer Series of Solid State Science, Berlin 1990), vol [**99**]{} p 208. A. S. Alexandrov and J. Ranninger, Phys. Rev. B[**23**]{}, 1796 (1981).
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abstract: 'The intrinsic spin Hall effect in semiconductors has developed to a remarkably lively and rapidly growing branch of research in the field of semiconductor spintronics. In this article we give a pedagogical overview on both theoretical and experimental accomplishments and challenges. Emphasis is put on the the description of the intrinsic mechanisms of spin Hall transport in III-V zinc-blende semiconductors, and on the effects of dissipation.'
author:
- |
John Schliemann\
Institute for Theoretical Physics, University of Regensburg,\
D-93040 Regensburg, Germany
title: Spin Hall Effect
---
-3.5cm
Introduction
============
Starting from about the late 1990s, a pronounced, and partially also renewed, interest in effects of spin-orbit coupling in semiconductors has been emerging in the solid-state community. This development is mainly fueled by the field of spintronics. The latter keyword summarizes an entire plethora of theoretical and experimental efforts towards using the spin degree of freedom of electrons, instead, or in combination with, their charge for information processing, or, even more ambitious, for quantum information processes. Thus, controlling the electron spin in semiconductor structures is a key challenge, and the relativistic effect of spin-orbit coupling is an important, if not indispensable, ingredient for reaching this goal. Among all the rapidly progressing activities in this field, a major and very recent development was the theoretical prediction and subsequent experimental investigation of the spin Hall effect in semiconductor structures. This effect amounts in a spin current, as opposed to a charge current, driven by a perpendicular electric field. In this article we review recent studies on spin Hall transport in semiconductors induced by intrinsic mechanisms.
A brief overview on important aspects and perspectives of the field of spintronics is given in the article by Wolf [*et al.*]{} [@Wolf01]. Selected topics are reviewed in more detail in a volume edited by Awschalom, Loss, and Samarth [@Awschalom02]. A comprehensive review of many parts of spintronics was given by Zutic, Fabian, and Das Sarma [@Zutic04]. Research directions not covered here include the field of ferromagnetic semiconductors; for a review on this we refer to Refs. [@Dietl02; @Konig03; @Dietl03; @Timm03; @Dietl04; @MacDonald05].
From a historical perspective, the notion of the spin Hall effect in systems of itinerant spinful charge carriers was considered first by Dyakonov and Perel [@Dyakonov71] in the early seventies, and in a more recent paper by Hirsch [@Hirsch99]. In these studies the predicted spin Hall effect is due to spin-orbit effects influencing scattering processes upon static impurities. Following the usual terminology of semiconductor physics, this effect is referred to as the [*extrinsic*]{} spin Hall effect since it necessarily requires spin-dependent impurity scattering. This is in contrast to the [*intrinsic*]{} spin Hall effect which is entirely due to spin-orbit coupling terms in the single-particle carrier Hamiltonian and occurs even in the absence of any scattering process. The story of the intrinsic spin Hall effect starts in summer of 2003, when Murakami, Nagaosa, and Zhang [@Murakami03], and, almost simultaneously, J. Sinova [*et al.*]{} in a collaboration based in Austin (Texas)[@Sinova03] predicted this phenomenon. The work by Sinova [*et al.*]{} considers a two-dimensional electron gas being subject to spin-orbit coupling of the so-called Rashba type, whereas the former paper investigates valence-band holes in three-dimensional bulk systems. Shortly later on, Schliemann and Loss added a study on spin Hall transport of heavy holes confined to a quantum well[@Schliemann05a]. Only a few months later, J. Wunderlich [*et al.*]{} reported on an experimental study of the latter type of system, confirming the predicted spin Hall effect via optical techniques [@Wunderlich05]. The paper by Wunderlich [*et al.*]{} was only shortly preceded by another experimental report by Kato [*et al.*]{} who detected spin Hall transport in n-doped bulk systems, again using an optical method [@Kato04]. In both experiments, the existence of spin Hall transport is signaled by spin accumulation at the boundaries of the sample, which seems to be the easiest method so far to detect this effect.
The theoretical and experimental developments sketched above have generated a still rapidly growing amount of preprints and (subsequent) journal publications, so far mostly theoretical. In this article we give a pedagogical overview on important aspects of intrinsic spin-Hall transport in III-V zinc-blende semiconductors. Emphasis is put on the the description of the intrinsic mechanisms of spin Hall transport induced by spin-orbit coupling,and on the effects of dissipation. This article also addresses researchers who are not particularly specialized in spin phenomena in semiconductors but are interested in this rapidly developing field. A brief review on spin-Hall transport was already given in a conference paper by Murakami [@Murakami05a], and Sinova [*et al.*]{} recently provided a short summary of important issues [@Sinova05].
This article is organized as follows. In section \[spin-orbit\] we make a few general comments on spin-orbit coupling in semiconductors and describe its effective contributions to the band structure of both electron and hole doped systems. In section \[theory\] we first make some general remarks on the notion of spin currents before reviewing the particularly rich body of recent theoretical work on spin Hall transport. Our analysis here includes the two-dimensional electron gas as well as p-doped bulk systems and quantum wells. Experimental work and proposed experiments are discussed in section \[exp\]. We close with conclusions and an outlook in section \[concl\].
Spin-orbit coupling in III-V semiconductors {#spin-orbit}
===========================================
The coupling between the orbital and the spin degree of freedom of electrons is a relativistic effect described by the Dirac equation and its nonrelativistic expansion in powers of the inverse speed of light $c$. In second order one obtains, apart from two spin-independent contributions, the following well-known spin-orbit coupling term, $${\cal H}_{so}=\frac{1}{2m_{0}c^{2}}\vec s\cdot\left(\nabla V\times\frac{\vec p}{m_{0}}\right)\,,
\label{sogeneral}$$ where $m_{0}$ is the bare mass of the electron, $\vec s$, $\vec p$ its spin and momentum, respectively, and $V$ is some applied external potential. On the other hand, the free Dirac equation, $V=0$, has two dispersion branches with positive and negative energy, $$\varepsilon(\vec p)=\pm\sqrt{m_{0}^{2}c^{4}+c^{2}p^{2}}\,,$$ which are separated by an energy gap of $2m_{0}c^{2}\approx 1{\rm MeV}$. In particular, the nonrelativistic expansion of the Dirac equation quoted above can be seen as a method of systematically including the effects of the negative-energy solutions on the states of positive energy starting from their nonrelativistic limit. Moreover, the large energy gap $2m_{0}c^{2}$ appears in the denominator of the right hand side of Eq. (\[sogeneral\]), suppressing the effects of spin-orbit coupling for weakly bound electrons.
On the other hand, the band structure of zinc-blende III-V semiconductors shows many formal similarities to the situation of free relativistic electrons, while the relevant energy scales are grossly different [@Zawadzki70; @Darnhofer93; @Rashba04]. For not too large doping of such semiconductors, one can concentrate on the band structure around the $\Gamma$ point. Here one has a parabolic $s$-type conduction band and a $p$-type valence band consisting of the well-known dispersion branches for heavy and light holes, and the split-off band. However, the gap between conduction and valence band is of order $1{\rm eV}$ or smaller. This heuristic argument makes plausible that spin-orbit coupling is an important effect in III-V semiconductors which actually lies at the very heart of the field of semiconductor spintronics.
In the following we give an overview on effective model Hamiltonians for conduction-band electrons and valence-band holes in III-V zinc-blende semiconductors in several spatial dimensions. These effective expressions can be obtained via the so-called $\vec k\cdot\vec p$-theory and related methods, as general references we refer to Refs. [@Kane66; @Winkler03; @Ivchenko05]. Here we shall just state the results and discuss their main physical implications.
Conduction-band electrons
-------------------------
Let us first consider three-dimensional bulk systems. For electrons in the $s$-type conduction band, the contribution to spin-orbit coupling being of lowest order in the electron momentum $\vec p$ has been derived by Dresselhaus [@Dresselhaus55] and reads $${\cal H}_{D}^{bulk}=\frac{\gamma}{\hbar^{3}}\left(
\sigma^{x}p_{x}\left(p_{y}^{2}-p_{z}^{2}\right)
+\sigma^{y}p_{y}\left(p_{z}^{2}-p_{x}^{2}\right)
+\sigma^{z}p_{z}\left(p_{x}^{2}-p_{y}^{2}\right)\right)\,,
\label{bulkdressel}$$ where $\vec\sigma$ is the vector of Pauli matrices describing the electron spin, and $\gamma$ is an effective coupling parameter. This Hamiltonian is trilinear in the momentum $\vec p$ and invariant under all symmetry operations of the tetrahedral group $T_{d}$, the point symmetry group of the zinc-blende lattice. As a result, the parameter $\gamma$ is different from zero because the zinc-blende lattice does not possess an inversion center. Therefore, the Dresselhaus spin-orbit coupling is due to [*bulk-inversion asymmetry*]{}.
In a sufficiently narrow quantum well grown along the \[001\] direction, and at sufficiently low temperatures, one can approximate the operators $p_{z}$ and $p^{2}_{z}$ by their expectation values $\langle p_{z}\rangle\approx 0$, $\langle p^{2}_{z}\rangle=\hbar^{2}\langle k_{z}^{2}\rangle $. Then neglecting terms of order $p_{x}^{2}$, $p_{y}^{2}$ leads to a spin-orbit coupling term linear in the momentum [@Dyakonov86; @Bastard92], $${\cal H}_{D}=\frac{\beta}{\hbar}\left(p_{y}\sigma^{y}-p_{x}\sigma^{x}\right)
\label{dressel}$$ with $\beta=\gamma \langle k_{z}^{2}\rangle $. Here $k_{z}$ is the wave number in the lowest subband of the well. Another important contribution to spin-orbit coupling occurs in quantum wells whose confining potential is lacking inversion symmetry. This contribution due to [*structure-inversion asymmetry*]{} is known as the Rashba term [@Rashba60; @Bychkov84], $${\cal H}_{R}=\frac{\alpha}{\hbar}\left(p_{x}\sigma^{y}-p_{y}\sigma^{x}\right),
\label{rashba}$$ where the coupling parameter $\alpha$ is essentially proportional to the potential gradient across the quantum well. Hence, $\alpha$ is in particular tunable by an electric gate and can therefore be varied experimentally.
A both theoretically [@Lommer88] and experimentally [@Jusserand92; @Jusserand95] well established value for the Dresselhaus parameter in GaAs is $\gamma=25{\rm eV}\AA^{3}$. Depending on the width of the quantum well, this leads to values for $\beta$ being of up to $10^{-11}{\rm eVm}$. Regarding the Rashba coefficient $\alpha$, values of a few $10^{-11}{\rm eVm}$ can be reached in InAs [@Nitta97; @Engels97; @Heida98; @Hu99; @Grundler00; @Sato01; @Hu03], whereas in GaAs this quantity is typically an order of magnitude smaller [@Miller03]. Thus, the characteristic energy scales $$\varepsilon_{R}=\frac{m\alpha^{2}}{\hbar^{2}}\,,$$ $$\varepsilon_{D}=\frac{m\beta^{2}}{\hbar^{2}}$$ for Rashba and Dresselhaus coupling, respectively, can be of order $0.1\dots1.0{\rm meV}$, depending on the effective band mass $m$.
Let us finally briefly discuss the spectrum and eigenstates generated by the above spin-orbit coupling terms. We consider the single-particle Hamiltonian for a two-dimensional electron system $${\cal H}=\frac{\vec p^{2}}{2m}+{\cal H}_{R}+{\cal H}_{D}\,.$$ The eigenenergies are given by $$\varepsilon_{\pm}\left(\vec k\right)
=\frac{\hbar^{2}k^{2}}{2m}
\pm\sqrt{\left(\alpha k_{y}+\beta k_{x}\right)^{2}
+\left(\alpha k_{x}+\beta k_{y}\right)^{2}}
\label{dispersion1}$$ with eigenstates $$\langle\vec r|\vec k,\pm\rangle=
\frac{e^{i\vec k\cdot\vec r}}{\sqrt{A}}\frac{1}{\sqrt{2}}
\left(
\begin{array}{c}
1 \\
\pm e^{i\chi(\vec k)}
\end{array}
\right)
\label{eigenstate}$$ where $A$ is the area of the system and $$\chi(\vec k)=\arg(-\alpha k_{y}-\beta k_{x}+i(\alpha k_{x}+\beta k_{y}))\,.
\label{chi}$$ The above spin–orbit coupling terms can be viewed as a momentum-dependent Zeeman field acting on the electron spin. Consequently, the spin state of the electron depends on its momentum, as seen in Eq. (\[eigenstate\]). Note that for pure Rashba or Dresselhaus coupling, the dispersions form two parabolas being shifted horizontally. This is different form a normal Zeeman field which shifts the dispersion parabolas vertically, i.e. along the energy axis. The case $\alpha=\pm\beta$ is particular [@Schliemann03a; @Schliemann03b]. Here a new conserved quantity given by $\Sigma:=(\sigma^{x}\mp\sigma^{y})/\sqrt{2}$ arises, and the spin state of the electrons becomes independent of the wave vector. This result for $\alpha=\pm\beta$ is a very general one, it also holds in the presence of any arbitrary scalar potential, or if interactions between electrons are included.
Valence-band holes
------------------
The valence band of III-V zinc-blende semiconductors is of $p$-type, i.e. it is predominantly composed out of atomic wave functions with angular momentum $l=1$. Adding this angular momentum to the electron spin $s=1/2$, we find to multiplets with total angular momentum $j=3/2$ and $j=1/2$. The dublett $j=1/2$ forms essentially an energetically separated so-called split-off band and will not be considered any further. The multiplet $j=3/2$ consists essentially of the so-called heavy and light hole states which are, to a good degree of approximation, described by Luttinger’s Hamiltonian [@Luttinger56], $${\cal H}=\frac{1}{2m_{0}}\left(\left(\gamma_{1}+\frac{5}{2}\gamma_{2}\right)
\vec p^{2}-2\gamma_{2}\left(\vec p\cdot\vec S\right)^{2}\right)\,.
\label{Luttinger}$$ Here $m_{0}$ is again the bare electron mass, and $\vec S$ are spin-$3/2$-operators. The dimensionless Luttinger parameter $\gamma_{1}$ and $\gamma_{2}$ describe the valence band of the specific material with effects of spin-orbit coupling being included in $\gamma_{2}$. The eigenstates of the above Hamiltonian can be chosen to be eigenstates of the helicity operator $\lambda=(\vec k\cdot\vec S)/k$, where $\vec k=\vec p/ \hbar$ is the hole wave vector. The heavy holes correspond to $\lambda=\pm 3/2$, while the light holes have $\lambda=\pm 1/2$. From the Hamiltonian (\[Luttinger\]) one finds the effective band mass for the heavy holes as $$m_{hh}=\frac{m_{0}}{\gamma_{1}-2\gamma_{2}}$$ and for the light holes $$m_{lh}=\frac{m_{0}}{\gamma_{1}+2\gamma_{2}}\,.$$ Well established values for the Luttinger parameters, among other band structure parameters, can be found in the literature [@Vurgaftman01]. For example, for GaAs one has $\gamma_{1}\approx 7.0$ and $\gamma_{2}\approx 2.5$ giving $m_{hh}\approx 0.5m_{0}$ and $m_{lh}\approx 0.08m_{0}$.
In a bulk system, the heavy and light hole states are degenerate at the $\Gamma$-point $\vec k=0$. This degeneracy is lifted in a quantum well due to size quantization, and for sufficiently narrow wells and low enough temperatures one can concentrate on the lower-lying heavy holes. Moreover, if the quantized wave vector in the growth direction is large enough, i.e. if the well is not too wide, the spin of these heavy holes points predominantly along the growth direction with a projection of $\pm 3/2$. For asymmetric wells these hole states are subject to a spin-orbit contribution due to structure-inversion asymmetry analogous to the Rashba term for electrons in the conduction band. Choosing the growth direction to point along the $z$-axis, the resulting effective Hamiltonian has the form [@Winkler00; @Gerchikov92] $${\cal H}=\frac{\vec p^{2}}{2m}+i\frac{\tilde\alpha}{2\hbar^{3}}
\left(p_{-}^{3}\sigma_{+}-p_{+}^{3}\sigma_{-}\right)\,,
\label{defham}$$ using the notations $p_{\pm}=p_{x}\pm ip_{y}$, $\sigma_{\pm}=\sigma^{x}\pm i\sigma^{y}$. Here the Pauli matrices operate on the total angular momentum states with spin projection $\pm 3/2$ along the growth direction; in this sense they represent a pseudospin degree of freedom rather than a genuine spin 1/2. In the above equation, $m$ is the heavy-hole mass, and $\tilde\alpha$ is Rashba spin-orbit coupling coefficient due to structure inversion asymmetry. This Hamiltonian has two dispersion branches given by $$\varepsilon_{\pm}(k)=\frac{\hbar^{2}k^{2}}{2m}\pm\tilde \alpha k^{3}
\label{dispersion}$$ with eigenfunctions $$\langle\vec r|\vec k,\pm\rangle=
\frac{e^{i\vec k\vec r}}{\sqrt{A}}\frac{1}{\sqrt{2}}
\left(
\begin{array}{c}
1 \\
\mp i\left(k_{x}+ik_{y}\right)^{3}/k^{3}
\end{array}
\right)\,.$$ We note that the Rashba parameter $\alpha$ entering the Hamiltonian (\[rashba\]) for electrons in a quantum well has a different dimension than the parameter $\tilde\alpha$ for holes in Eq. (\[defham\]). In the latter case, the quantity $m\tilde\alpha/ \hbar^{2}$ has dimension of length and can reach a magnitude of several nanometers in GaAs samples [@Winkler02].
Spin Hall transport: Theory {#theory}
===========================
In this section we summarize important theoretical results on intrinsic spin Hall effect. We start with some general considerations on spin currents.
Spin currents: General remarks
------------------------------
A type of current most familiar to physicists is certainly the usual charge or particle current. The particle density in a many-body system is described by the operator $$\begin{aligned}
\rho(\vec r)=\sum_{n}\delta(\vec r-\vec r_{n})\end{aligned}$$ where the index $n$ labels the particles. This density operator is a function of time via the time-dependence of the positions $\vec r_{n}$ entering the argument of the delta-functions. Let this time evolution be generated by an Hamiltonian of the form $${\cal H}=\sum_{n}h(\vec r_{n},\vec p_{n},\vec \sigma_{n})+V_{int}\,.$$ Here the term $V_{int}$ describes interaction among the particles and depends only on their spatial coordinates, and the single-particle Hamiltonian $h$ reads $$h(\vec r,\vec p_,\vec \sigma)=\frac{\vec p^{2}}{2m}+\gamma_{ij}p_{i}\sigma^{j}+V(\vec r)\,.$$ Summation over repeated cartesian indices is understood, and the matrix $\gamma$ parameterizes spin-orbit coupling of the Rashba and Dresselhaus type for electrons in a quantum well. Finally, the static potential $V(\vec r)$ describes, e.g., static impurities. Now, starting from the Heisenberg equation of motion, $$\frac{d}{dt}\rho=\frac{i}{\hbar}\left[{\cal H},\rho\right]$$ and performing some elementary algebraic manipulations, one derives the well-known continuity equation for the particle current, $$\frac{d}{dt}\rho+\nabla\cdot\vec j=0$$ with the current density operator $$\vec j(\vec r)=\frac{1}{2}\sum_{n}\left\{\vec v(\vec p_{n},\vec\sigma_{n}),
\delta(\vec r-\vec r_{n})\right\}\,.$$ Here $\{A,B\}=AB+BA$ denotes the anticommutator of two operators, and the velocity operator $\vec v$ is given by $$\vec v(\vec p_{n},\vec\sigma_{n})=\frac{i}{\hbar}\left[{\cal H},\vec r_{n}\right]$$ for each particle $n$. Note that this operator is in general spin-dependent if spin-orbit coupling is present.
Let us now consider a general observable described by a hermitian single-particle operator $A$ which can be a function of position, momentum, and spin: $A=A(\vec r,\vec p,\vec\sigma)$. The density operator corresponding to this physical quantity $A$ is naturally defined as $$\rho_{A}(\vec r)
=\frac{1}{2}\sum_{n}\left\{A(\vec r_{n},\vec p_{n},\vec\sigma_{n}),\delta(\vec r-\vec r_{n})\right\}\,,$$ where the symmetrization ensures hermiticity. Now proceeding as above one finds $$\frac{d}{dt}\rho_{A}+\nabla\cdot\vec j_{A}=s_{A}\,,$$ where the current density operator $\vec j_{A}$ is given by $$\vec j_{A}(\vec r)=\frac{1}{4}\sum_{n}\left\{A(\vec r_{n},\vec p_{n},\vec\sigma_{n}),
\left\{\vec v(\vec p_{n},\vec\sigma_{n}),
\delta(\vec r-\vec r_{n})\right\}\right\}\,,$$ and the additional source term on the right-hand-side reads $$s_{A}(\vec r)
=\frac{1}{2}\sum_{n}\left\{
\frac{i}{\hbar}\left[{\cal H},A(\vec r_{n},\vec p_{n},\vec\sigma_{n})\right],
\delta(\vec r-\vec r_{n})\right\}\,.$$ Thus, we only obtain the usual from of the continuity equation if the observable $A$ commutes with the Hamiltonian ${\cal H}$, which is of course just a restatement of Noether’s theorem. For the case electron spin components as observables, the corresponding spin-current densities are given by $$\vec j_{i}(\vec r)=\frac{1}{4}\sum_{n}\left\{\frac{\hbar}{2}\sigma^{i}_{n},
\left\{\vec v(\vec p_{n},\vec\sigma_{n}),
\delta(\vec r-\vec r_{n})\right\}\right\}\,,
\label{defspincurr}$$ and the source terms are due to spin-orbit coupling being present in the single-particle Hamiltonian. These source terms reflect the fact that magnetization, i.e. the density of magnetic moments, can be altered by two ways: by spatially moving spinful particles, or by manipulating their spin state. The latter process is described by the source terms. For instance, in the case of noninteracting electrons in a quantum well with spin-orbit coupling of the Rashba and Dresselhaus type, the source terms can be expressed via the components of the spin-current densities itself [@Erlingsson05a; @Burkov04], $$\begin{aligned}
\frac{d}{dt} \rho_{x}
+\nabla\cdot\vec j_{x}
& = & \frac{2m\alpha}{\hbar^{2}}j^{x}_{z}+\frac{2m\beta}{\hbar^{2}}j^{y}_{z}\,,
\label{spincont1}\\
\frac{d}{dt} \rho_{y}
+\nabla\cdot\vec j_{y}
& = & \frac{2m\alpha}{\hbar^{2}}j^{y}_{z}+\frac{2m\beta}{\hbar^{2}}j^{x}_{z}\,,
\label{spincon21}\\
\frac{d}{dt} \rho_{z}
+\nabla\cdot\vec j_{z}
& = & -\frac{2m\alpha}{\hbar^{2}}\left(j^{x}_{x}+j_{y}^{y}\right)
-\frac{2m\beta}{\hbar^{2}}\left(j^{x}_{y}+j^{y}_{x}\right).
\label{spincont3}\end{aligned}$$ In summary, the definition of the spin current density as given in Eq. (\[defspincurr\]) is the straightforward generalization of the usual particle current and widely used in the literature. As seen above, this spin current density is, however, not conserved, i.e. it does not fulfill a simple continuity equation. This fact might or might not be seen as a shortcoming of the above definition. Another peculiarity of this type of current operator was pointed out by Rashba [@Rashba03] who found that, for the situation of an asymmetric quantum well, the current densities with in-plane spin components, $\vec j_{x}$, $\vec j_{y}$, can have nonzero expectation values even in the absence of an electric field, i.e. in thermal equilibrium. Using periodic boundary conditions and considering an infinite disorder-free system of non-interacting electrons at zero temperature and positive Fermi energy, the full result for the case of both Rashba and Dresselhaus coupling reads [@Erlingsson05a] $$\begin{aligned}
\langle j^{x}_{x}\rangle=-\langle j^{y}_{y}\rangle & = &
\frac{\beta}{6\pi}\left(\frac{m}{\hbar^{2}}\right)^{2}
\left(\alpha^{2}-\beta^{2}\right)\,,
\label{equi1}\\
\langle j^{x}_{y}\rangle=-\langle j^{y}_{x}\rangle & = &
\frac{\alpha}{6\pi}\left(\frac{m}{\hbar^{2}}\right)^{2}
\left(\alpha^{2}-\beta^{2}\right)\,.
\label{equi2}\end{aligned}$$ Note that these equilibrium spin currents vanish in the case $\alpha=\pm\beta$ due to the additional conserved spin operator arising at this point [@Schliemann03a]. The findings shown in Eqs. (\[equi1\]),(\[equi2\]), however, certainly depend on the boundary conditions used and are altered in a more realistic description of finite systems [@Kiselev05].
In the recent literature, there are several proposals and discussions on alternative forms of spin currents which possibly fulfill proper continuity equations [@Murakami04a; @Rashba04a; @Jin05; @Zhang05; @Li05; @Sugimoto05]. However, these issues do not seem to be settled yet. Therefore, in the following we shall concentrate on spin current densities as defined in Eq. (\[defspincurr\]).
Conduction-band electrons in two dimensions
-------------------------------------------
We now discuss spin Hall transport of conduction-band electrons in III-V semiconductor quantum wells. As a great simplification used in almost the entire theoretical work so far, we will consider non-interacting electrons. Thus, the system is described by the single-particle Hamiltonian $${\cal H}=\frac{\vec p^{2}}{2m}+\frac{\alpha}{\hbar}\left(p_{x}\sigma^{y}-p_{y}\sigma^{x}\right)
+\frac{\beta}{\hbar}\left(p_{y}\sigma^{y}-p_{x}\sigma^{x}\right)\,,
\label{singpartham}$$ and instead of the many-body spin current density operators (\[defspincurr\]) we can use the single-particle operator $$\begin{aligned}
\vec j_{z} & = & \frac{\hbar}{4}\left(\sigma^{z}\vec v+\vec v\sigma^{z}\right)\\
& = & \frac{\vec p}{m}\frac{\hbar}{2}\sigma^{z}\,,\end{aligned}$$ where we have concentrated on the spin component along the growth direction of the quantum well (chosen as z-axis) and used the anticommutativity of Pauli matrices. To account for effects of disorder and confining boundaries of the system, appropriate potentials should be added to the Hamiltonian (\[singpartham\]), as we will discuss in detail below.
The linear response of this spin current to an electric field applied in the plane of the two-dimensional electron gas can be evaluated via the usual Kubo formula [@Mahan00]. For the off-diagonal (or Hall) components of the response tensor one has [@Sinova03; @Schliemann04] $$\sigma^{S,z}_{xy}(\omega)=\frac{e}{A(\omega+i\eta)}
\int_{0}^{\infty}e^{i(\omega+i\eta)t}\sum_{\vec k,\mu}f(\varepsilon_{\mu}(\vec k))
\langle\vec k,\mu|[j^{x}_{z}(t),v_{y}(0)]|\vec k,\mu\rangle\,.
\label{generalKubo}$$ Here $A$ is the volume of the system, $e$ is the elementary charge, and $f(\varepsilon_{\mu}(\vec k))$ is the Fermi distribution function for energy $\varepsilon_{\mu}(\vec k)$ at wave vector $\vec k$ in the dispersion branch $\mu=\pm$ as given in Eq. (\[eigenstate\]). The above quantity describes the linear response in terms of a spin current to a perpendicular electric field of frequency $\omega$. In the commutator on the right-hand side the time-dependent spin current operator in the Heisenberg picture enters, $$\vec j_{z} (t)=e^{i{\cal H}t/\hbar}\vec j_{z}e^{-i{\cal H}t/\hbar}\,.$$ Moreover, the right-hand side of Eq. (\[generalKubo\]) has to be understood in the limit of vanishing imaginary part $\eta>0$ in the frequency argument. This imaginary part in the frequency reflects the fact that the external electric field is assumed to be switched on adiabatically starting from the infinite past of the system, and it also ensures causality properties of the retarded Green’s function occurring in Eq. (\[generalKubo\]). In general, and as we will discuss in more detail below, the limiting process $\eta\to 0$ does not commute with other limits, and, in particular, the dc-limit $\omega\to 0$ has to be taken with care [@Mahan00]. In the presence of random impurity scattering, the retarded two-body Green’s function in Eq. (\[generalKubo\]) will generically have a frequency argument with positive imaginary part [@Mahan00]. In this case the limit $\eta\to 0$ is unproblematic, and the imaginary part of the frequency argument is just due to impurity scattering and/or other (many-body) effects. Generically, the imaginary part $\eta>0$ corresponds to a finite carrier quasiparticle lifetime.
Let us now for simplicity consider the case of Rashba spin-orbit coupling only and assume the electron density to be large enough such that the Fermi energy is positive (which is usually the case in realistic samples). Neglecting all possible disorder effects and concentrating on the case of zero temperature, the Kubo formula (\[generalKubo\]) can be evaluated straightforwardly, giving a spin Hall conductivity at zero frequency of $$\sigma^{S,z}_{xy}(0)=-\sigma^{S,z}_{yx}(0)=\frac{e}{8\pi}\,.$$ This result was obtained first by Sinova [*et al.*]{} [@Sinova03]. It is remarkable in the sense that the value of the spin Hall conductivity does not depend on the Rashba parameter $\alpha$. In particular, even in the limit of vanishing spin-orbit coupling, the above result still predicts a finite spin Hall current. However, no effects of disorder in the system have been included so far, and the infinitesimal parameter $\eta$ has been put to zero right away. Let us now take into account disorder effects by replacing $\eta$ with a phenomenological relaxation rate $1/\tau$. Here we find [@Schliemann04] $$\sigma^{S,z}_{xy}(0)=-\sigma^{S,z}_{yx}(0)
=\frac{e}{8\pi}
-\frac{e}{32\pi}\frac{\hbar/\tau}{\varepsilon_{R}}\tan^{-1}\left(4\frac{\varepsilon_{R}}{\hbar/\tau}
\left(1+8\frac{\varepsilon_{R}\varepsilon_{f}}
{(\hbar/\tau)^{2}}\right)^{-1}\right)\,.
\label{SHtau}$$ The first term is still the universal expression found in Ref. [@Sinova03], whereas in the second contribution three energy scales enter: The Fermi energy $\varepsilon_{f}$, the Rashba energy $\varepsilon_{R}$, and the energy scale of the scattering by disorder potentials $\hbar/ \tau$. Clearly, if the latter quantity dominates over the Rashba coupling, $\hbar/ \tau\gg\varepsilon_{R}$, the second term in Eq. (\[SHtau\]) cancels the first one, and the spin Hall conductivity is indeed suppressed by disorder. Analogous results can be found if both the Rashba and the Dresselhaus coupling are included [@Sinitsyn04]. In this case, the above approach also yields nonvanishing [*longitudinal*]{} spin conductivities [@Sinitsyn04].
Thus, we arrive at an apparently physically satisfactory picture. However, it turns out to be qualitatively incorrect for the following reason: When replacing the infinitesimal parameter $\eta$ in Eq. (\[generalKubo\]) with the inverse of a phenomenological relaxation time $\tau$, one neglects certain contributions in a systematic perturbational expansion with respect to the disorder potentials. These contributions are known as vertex corrections. As is was shown first by Inoue, Bauer, and Molenkamp [@Inoue04], the full dissipative contribution to the spin-Hall conductivity including the vertex corrections [*exactly cancels the universal value, independently of the strength of the disorder potentials and the Rashba coupling*]{}. This result was obtained for an infinitely large system and in lowest perturbational order with respect to the disorder potentials which were modeled by delta-functions. Subsequently, this finding was reproduced an generalized by several other authors [@Mishchenko04; @Khaetskii04; @Raimondi05; @Chalaev05; @Dimitrova05; @Liu06; @Grimaldi06] using different theoretical methods. The conclusions from these investigations can be summarized as follows: The spin-Hall conductivity for spin polarization along the growth direction in an infinite two-dimensional system with spin-orbit coupling of the Rashba and Dresselhaus type strictly vanishes in the presence of any spin-independent mechanism with forces, via spin-orbit coupling, the electron spins to relax to a constant value. This result is independent of any perturbational expansion with respect to disorder terms and holds also both at finite temperature and in the presence of interactions among the electrons. However, it does in general not hold in the presence of magnetic fields or other spin-dependent contributions in the Hamiltonian.
The proof of this very general statement was worked out by Chalaev and Loss [@Chalaev05], and by Dimitrova [@Dimitrova05]; a preliminary version can also be found in Ref. [@Erlingsson05a]. For the case of both Rashba and Dresselhaus spin-orbit coupling, the argument is as follows: To analyze the electron spin dynamics in an infinite homogeneous system, it is sufficient to consider just the spin operator of a single electron. From the Heisenberg equation of motion we find $$\begin{aligned}
\frac{d}{dt}\sigma^{x} & = & \frac{4m\alpha}{\hbar^{3}}j_{z}^{x}+\frac{4m\beta}{\hbar^{3}}j_{z}^{y}\,,
\label{timderv1}\\
\frac{d}{dt}\sigma^{y} & = & \frac{4m\beta}{\hbar^{3}}j_{z}^{x}+\frac{4m\alpha}{\hbar^{3}}j_{z}^{y}\,.
\label{timderv2}\end{aligned}$$ These relations hold also in the presence of any arbitrary spin-independent potential or interaction term in the Hamiltonian. The key observation is that the time derivatives of the spin components can be expressed as linear combinations of the spin current operators itself. This result crucially relies on the fact that the spin-orbit coupling is linear in the electron momentum. Moreover, in the presence of spin-orbit coupling disorder effects can generally be expected to make the electron spins relax to constant values. Thus, in a stationary state, the expectation values of the left-hand sides of Eqs. (\[timderv1\]), (\[timderv2\]) should vanish. Now it follows immediately that the expectation values of the spin current components $j_{z}^{x}$, $j_{z}^{y}$ must also vanish provided $\alpha\neq\pm\beta$. In the case $\alpha=\pm\beta$, however, spin Hall transport is generally absent due to the additional conserved spin quantity which occurs here [@Sinitsyn04; @Shen04a; @Schliemann03a].
The central argument of the above proof is closely related to studies by Erlingsson, Schliemann and Loss [@Erlingsson05a], and by Dimitrova [@Dimitrova05] on the relationship between spin currents and magnetic susceptibilities. For further developments in this direction see also Ref. [@Bernevig05a].
Another interesting observation regarding spin Hall transport in n-doped quantum wells was made by Rashba [@Rashba04b] who considered Rashba spin-orbit coupling in the presence of a magnetic field coupling to the orbital degrees of freedom only, neglecting the Zeeman coupling to the electron spin. In this model, the spin Hall conductivity vanishes in the limit of vanishing magnetic field even in the absence of disorder, an effect closely related to the abovementioned vertex corrections [@Rashba04b]. Thus, the case of zero magnetic field (coupling to the orbital degrees of freedom only) is different from the limit of vanishing field. This result certainly relies on the assumption of an infinite system, since the orbital effects of a magnetic field should be small if the typical cyclotron radius is large compared to the system size.
The case of a magnetic field coupling both to the orbital and spin degree of freedom of electrons was investigated by Shen [*et al*]{} [@Shen04b; @Shen05]. Due to the coupling to the spin this situation is not covered by the above general argument. Shen [*et al*]{} find a resonant behavior of the spin Hall conductance as a function of the magnetic field when a degeneracy of Landau levels occurs at Fermi level. These studies, however, do not include disorder effects so far [@Shen04b; @Shen05].
Coming back the case zero external magnetic field, Adagideli and Bauer chose yet another approach to the problem by considering the electron acceleration in the presence of Rashba coupling [@Adagideli05], $$\frac{d^{2}}{dt^{2}}\vec r=\frac{4m\alpha^{2}}{\hbar^{4}}\vec j_{z}\times\vec e_{z}
-\frac{1}{m}\left(e\vec E+\nabla V \right)\,,$$ where $\vec e_{z}$ is the unit vector in the $z$-direction, $\vec E$ is the in-plane electric field, and $V$ is the disorder potential. Since the acceleration should relax to zero in a disordered system, the spin Hall current vanishes if the effects of the electric field and the disorder potential cancel on average. As the authors show, this is indeed the case in the bulk of the system, but not necessarily at its edges. This observation gives rise to the notion of [*spin Hall edges*]{} [@Adagideli05].
The above general result on the absence of spin Hall transport in infinite systems with spin-orbit coupling being linear in the electron momentum is the outcome of an intense theoretical discussion in the last about two years. It was also confirmed by numerical studies carried out by Nomura [*et al.*]{} [@Nomura05a]. These authors performed a careful numerical evaluation of the Kubo formula for the spin Hall conductivity in the presence of Rashba coupling and delta-function type impurity potentials.
We stress again that the above general conclusion rules out spin Hall transport only in the limit of an infinite system. In fact, several numerical investigations on finite systems have appeared recently [@Hankiewicz04; @ShengL05a; @LiJ05a; @Nikolic05a; @Nikolic05b; @Hankiewicz05; @ShengD05a; @Moca05; @Nikolic06]. In these studies, the underlying semiconductor structure are described by tight-binding hopping models of finite-size coupled to semi-infinite leads. These hopping Hamiltonians also include local disorder potentials and discrete versions of the spin-orbit contributions in an n-doped quantum well. Transport quantities are then evaluated using the well-established Landauer-Büttiker approach combined with a Green’s function treatment of the semi-infinite leads[@Datta95]. In summary, it still remains an interesting and unsettled question, whether intrinsic spin Hall transport can be experimentally observed in mesoscopic systems as studied in the above references.
Spin Hall transport of holes
----------------------------
We now analyze intrinsic spin Hall transport in p-doped III-V semiconductors and start with the case of a three-dimensional bulk system pioneered by Murakami, Nagaosa, and Zhang [@Murakami03].
### Three-dimensional bulk case
We consider valence-band heavy and light holes governed by the Hamiltonian (\[Luttinger\]) and the conventionally defined spin current operator as described before. In this case a nonzero spin conductivity occurs if the direction of the spin current, its spin polarization, and the driving electric field are mutually orthogonal. For definiteness, let us assume the spin polarization to point alone the $z$-axis with the electric field being in the $xy$-plane. From the Kubo formula (\[generalKubo\]), the zero-frequency spin Hall conductivity can be evaluated as [@Schliemann04; @Murakami04a; @Culcer04] $$\sigma_{xy}^{S,z} (0)=\frac{e}{4\pi^{2}}\frac{\gamma_{1}+2\gamma_{2}}{\gamma_{2}}\left(k_{f}^{h}-k_{f}^{l}
-\int_{k_{f}^{l}}^{k_{f}^{h}}dk
\frac{1}{1
+\left(\frac{2}{\hbar/\tau}\frac{\hbar^{2}}{m}\gamma_{2} k^{2}\right)^{2}}
\right)\,,
\label{holesbulkcond}$$ where we have again assumed an infinite system at zero temperature, and $$k_{f}^{h/l}=\sqrt{\frac{2m}{\hbar^{2}}\varepsilon_{f}
\frac{1}{\gamma_{1}\mp2\gamma_{2}}}$$ are the Fermi wave numbers for heavy and light holes, respectively. Similarly to the approach leading to Eq. (\[SHtau\]) for conduction-band electrons, disorder effects are taken into account via an effective relaxation time $\tau$. In the case here this approach is justified because vertex corrections can be shown to be absent for scattering potentials described by delta-functions[@Murakami04b]. Thus, for this type of disorder, the result (\[holesbulkcond\]) is exact in lowest order Born approximation [@Mahan00]. The absence of vertex corrections in this case crucially relies on the fact that the underlying Hamiltonian (\[Luttinger\]) is not linear but of second order in the components of the momentum [@Murakami04b; @Bernevig05b]. The case of impurity potentials of longer spatial range was investigated in Ref. [@Liu05a].
The remaining integral in Eq. (\[holesbulkcond\]) is elementary leading to a rather tedious expression which shall not be given here. However, we see that the value of the above integral is governed by the ratio of energy scale of the impurity scattering $\hbar/\tau$ and the “spin-orbit energy” $$\varepsilon_{so}:=\hbar^{2}\gamma_{2}(k_{f}^{0})^{2}/m=2\varepsilon_{f}\gamma_{2}/\gamma_{1},$$ since $$k_{F}^{0}=\sqrt{2m\varepsilon_{f}/\gamma_{1}\hbar^{2}}$$ is a typical wave number in the integration interval [@Schliemann04]. If $\hbar/\tau\gg \varepsilon_{so}$ the spin Hall conductivity vanishes as $$\begin{aligned}
\sigma_{xy}^{S,z} (0) & = &
\frac{e}{\pi^{2}}4k_{f}^{0}
\left(\frac{\varepsilon_{so}}{\hbar/\tau}\right)^{2}
\frac{\gamma_{2}}{\gamma_{1}}\nonumber\\
& & +{\cal O}\left(\left(\frac{\varepsilon_{so}}{\hbar/\tau}\right)^{4}
,\left(\frac{\varepsilon_{so}}{\hbar/\tau}\right)^{2}
\left(\frac{\gamma_{2}}{\gamma_{1}}\right)^{2}\right)
\label{expansion1}\end{aligned}$$ where we have also assumed that the ratio $\gamma_{2}/\gamma_{1}$ is small as it is usually the case [@Vurgaftman01]. In the opposite case $\hbar/\tau\ll \varepsilon_{so}$ one finds $$\begin{aligned}
\sigma^{S,z}_{xy}(0)
& = & \frac{e}{4\pi^{2}}\frac{\gamma_{1}+2\gamma_{2}}{\gamma_{2}}
\left[k_{f}^{h}-k_{f}^{l}+\frac{\left(k_{f}^{0}\right)^{4}}{12}
\left(\left(\frac{1}{k_{f}^{h}}\right)^{3}
-\left(\frac{1}{k_{f}^{l}}\right)^{3}\right)
\left(\frac{\hbar/\tau}{\varepsilon_{so}}\right)^{2}\right]\nonumber\\
& & +{\cal O}\left(\left(\frac{\hbar/\tau}{\varepsilon_{so}}\right)^{4}\right)
\label{expansion2}\end{aligned}$$ Here the the contribution in leading order is the result obtained in Refs. [@Murakami04a; @Culcer04] for a disorder-free system (up to some definitorial prefactor [@Schliemann04]). The expression given originally in Ref. [@Murakami03], however, differs somewhat from the above one due some approximation employed there [@Murakami03]. Ref. [@Bernevig04a] contains calculations of the spin Hall conductivity in band structure models more general than Eq. (\[Luttinger\]). Numerical results based on [*ab initio*]{} band structure calculations were presented in Ref. [@Guo05]. A further numerical study of spin Hall transport within the Hamiltonian (\[Luttinger\]) in the presence of disorder was performed in Ref. [@Chen05a].
In summary, spin Hall transport in p-doped bulk III-V semicondcutors is robust against disorder of not too large strength, but naturally breaks down if impurity effects are overwhelming the spin-orbit coupling [@Schliemann04].
### Heavy holes in a quantum well
Let us now turn to the case of spin Hall transport of heavy holes in p-doped quantum wells which was studied first by Schliemann and Loss [@Schliemann05a]. An experimental observation of spin Hall effect in such a system was recently reported by Wunderlich [*et al.*]{} [@Wunderlich05].
We consider the Hamiltonian (\[defham\]) and a spin current $$\vec j_{z}=\frac{\vec p}{m}\frac{3\hbar}{2}\sigma^{z}$$ for heavy holes with spin $\pm 3/2$ polarized along the growth direction of the well. Proceeding as above, one finds for the zero-frequency spin Hall conductivity [@Schliemann05a] $$\sigma^{S,z}_{xy}(0)=-\sigma^{S,z}_{yx}(0)
=\frac{e}{\pi}\frac{9}{4}\frac{\hbar^{2}\tilde\alpha}{m}\int_{k_{f}^{+}}^{k_{f}^{-}}dk
\frac{k^{4}}{\left(\hbar/\tau\right)^{2}+
\left(2\tilde\alpha k^{3}\right)^{2}}\,,
\label{fullspinHall}$$ where $\tau$ is again the momentum relaxation time. Similar to the three-dimensional bulk case, vertex corrections to the spin-Hall conductivity turn out to be zero for delta-function shaped scatterers [@Bernevig05b], justifying the above approach. This result is again due to the fact that the spin-orbit coupling is not linear but of higher order in the particle momentum.
The Fermi wave numbers $k_{f}^{\pm}$ entering Eq. (\[fullspinHall\]) refer to the two dispersion branches (\[dispersion\]) and can be expressed in terms of the particle density $$n=\frac{1}{4\pi}
\left(\left(k_{f}^{+}\right)^{2}+\left(k_{f}^{-}\right)^{2}\right)
\label{density}$$ as[@Schliemann05a] $$\begin{aligned}
k_{f}^{\pm} & = & \mp\frac{1}{2}\frac{\hbar^{2}}{2m\tilde\alpha}
\left(1-\sqrt{1-\left(\frac{2m\tilde\alpha}{\hbar^{2}}\right)^{2}4\pi n}\right)
\nonumber\\
& & +\sqrt{-\frac{1}{2}\left(\frac{\hbar^{2}}{2m\tilde\alpha}\right)^{2}
\left(1-\sqrt{1-\left(\frac{2m\tilde\alpha}{\hbar^{2}}\right)^{2}4\pi n}\right)
+3\pi n}\,.
\label{kfpm}\end{aligned}$$ Moreover, the [*longitudinal*]{} spin conductivities $\sigma^{S,z}_{xx}$, $\sigma^{S,z}_{yy}$ turn out out be identically zero.
The remaining integral in the above expression (\[fullspinHall\]) is elementary; however, it leads to a rather cumbersome expression which shall again not be given here. Analogously to the previous case, the energy scale of impurity scattering $\hbar/\tau$ has to be compared with the “Rashba energy” $\tilde\varepsilon_{R}=\tilde\alpha (k_{f}^{0})^{3}$, where $k_{f}^{0}=\sqrt{2m\varepsilon_{f}/\hbar^{2}}$ is the Fermi wave number for vanishing spin-orbit coupling, which is a typical value for $k$ in the integration in Eq. (\[fullspinHall\]). If the impurity scattering dominates over the Rashba coupling, $\hbar/\tau\gg\tilde\varepsilon_{R}$, the spin Hall conductivity vanishes with the leading order correction given by $$\sigma^{S,z}_{xy}(0)=\frac{e}{\pi}\frac{9}{20}\frac{\tilde\alpha}{m}\tau^{2}
\left((k_{f}^{-})^{5}-(k_{f}^{+})^{5}\right)
+{\cal O}\left(\left(\frac{\tilde\varepsilon_{R}}{\hbar/\tau}\right)^{4}\right)
\,,
\label{spinHall1}$$ where the Fermi wave numbers are given by Eq. (\[kfpm\]). In the opposite case $\tilde\varepsilon_{R}\gg\hbar/\tau$, the leading contribution to the spin Hall conductivity reads $$\sigma^{S,z}_{xy}(0)=\frac{e}{\pi}\frac{9}{16}
\frac{\hbar^{2}}{m\tilde\alpha}
\left(\frac{1}{k_{f}^{+}}-\frac{1}{k_{f}^{-}}\right)
+{\cal O}\left(\left(\frac{\hbar/\tau}{\tilde\varepsilon_{R}}\right)^{4}\right)\,.
\label{spinHall2}$$ Note that this result for the spin Hall conductivity depends only on the length scale $m\tilde\alpha/\hbar^{2}$ of the Rashba coupling and the total hole density $n$, but not separately on quantities like the Fermi energy and the effective heavy hole mass. If $m\tilde\alpha/\hbar^{2}$ is small against the inverse square root of the total hole density (but still fulfilling $\tilde\varepsilon_{R}\gg\hbar/\tau$), the spin Hall conductivity approaches a value of $$\sigma^{S,z}_{xy}=9\frac{e}{8\pi}\,.
\label{lowdis}$$ This is the case if $\hbar/\tau\ll\tilde\varepsilon_{R}\ll\varepsilon_{f}$. This above value should be compared with the universal value of $e/8\pi$ found originally in Ref.[@Sinova03] for electrons in a fully clean asymmetric quantum well. In this sense the hole spin Hall conductivity is enhanced by a factor of $9$ compared to the naive result for electrons, which is partially due to the larger angular momentum of the heavy holes.
Zarea and Ulloa studied the above system in the clean limit but with a perpendicular homogeneous magnetic field coupling to the orbital degrees of freedom of the holes but not to their spin [@Zarea06], an investigation analogous to the one by Rashba on n-doped quantum wells already mentioned [@Rashba04b]. Again it is found that, for an infinite system, the case of zero magnetic field and the limit of vanishing magnetic field do not coincide [@Zarea06]. The details of this effect, however, seem to be somewhat different from the observations made in Ref. [@Rashba04b] and need further study. In any case, influence of a magnetic field coupling to the orbital degrees of freedom only should only be appreciable if the field is strong enough to produce typical cyclotron radii being of order of the system size or smaller. Therefore, arbitrarily small fields cannot be expected to have an effect in real experiments. Another study of spin Hall transport in the presence of a perpendicular magnetic field was performed in Ref. [@Ma05] where a more general band structure Hamiltonian was used [@Bernevig05b]
As seen above, spin Hall transport of heavy holes in a quantum well is robust against disorder effects, differently from the situation for electrons in n-doped wells. This effect is due to the different functional form of the effective spin-orbit coupling and was also confirmed numerically by Nomura [*et al.*]{} who performed a careful comparison between these two systems [@Nomura05a]. In a subsequent study, the edge- spin accumulation caused by the spin current was investigated numerically [@Nomura05b]. Further investigations of disorder effects can be found in Ref. [@Liu05] where an approach based on nonequilibrium Green’s functions was used.
Moreover, several groups have studied numerically tight-binding models for two-dimensional hole systems coupled to semi-infinite leads. [@Hankiewicz05; @Wu05; @Chen05]. These investigations are analogous in spirit and technical approach to the numerical work on models for n-doped systems mentioned earlier [@Hankiewicz04; @ShengL05a; @LiJ05a; @Nikolic05a; @Nikolic05b; @ShengD05a; @Moca05; @Nikolic06]. In particular, the numerical results on heavy-hole systems [@Hankiewicz05; @Chen05] in the limit of low disorder confirm quantitatively the the enhanced spin conductivity (\[lowdis\]) obtained analytically in Ref. [@Schliemann05a]. Finally, many of the abovementioned mainly numerical studies on p-doped quantum wells were inspired by the experiments by Wunderlich [*et al.*]{} which we will discuss in section \[exp\].
Spin Hall effect in other systems
---------------------------------
Intrinsic mechanisms of spin Hall transport in n-doped bulk III-V semiconductors where investigated by Bernevig and Zhang [@Bernevig05c; @Bernevig05d]. Here the leading contribution to spin-orbit coupling is given by the bulk Dresselhaus term (\[bulkdressel\]) being of third order in the electron momentum. On the other hand, Engel, Halperin, and Rashba have studied extrinsic spin Hall effect in such systems [@Engel05]. We will discuss these issues in section \[exp\] in the context of the experimental results by Kato [*et al.*]{} [@Kato04].
Spin Hall effect in graphene, i.e. single planes of graphite, was investigated by Kane and Mele [@Kane05]. Using a theoretical picture similar to the edge-state theory of the charge quantum Hall effect, these authors propose a spin current at the edges of a graphene sheet. Work following these theoretical predictions include Refs. [@Sheng05; @Yang06]. Finally, Shchelushkin and Brataas considered spin Hall transport in normal metals due to extrinsic mechanisms [@Shchelushkin05].
Detection of spin Hall transport: Experiments and Proposals {#exp}
===========================================================
We now turn to experimental investigations of spin Hall effect in semiconductors. The studies already carried out and many of the experimental proposals found in the literature use the spin accumulation caused by the spin current for detecting spin Hall transport. Using a simple spin diffusion model, this spin accumulation is expected to decay towards the bulk of the sample on a length scale given by the spin diffusion length [@Zhang00]. The latter quantity is determined by the semiconductor material, but possibly also by further details of the sample and the experimental setup [@Tse05]. Further recent theoretical studies on spin accumulation caused by the spin Hall effect include [@Nomura05b; @Ma04; @Hu04; @Usaj05; @Malshukov05; @Reynoso05].
Kato [*et al.*]{} have studied spin Hall transport in n-doped bulk epilayers of GaAs and InGaAs with a thickness of $2\mu{\rm m}$ and $500{\rm nm}$, respectively [@Kato04]. The electron density in both samples was $n=3\cdot10^{16}{\rm cm^{-3}}$. The spin Hall effect was detected via the optical technique of Kerr rotation microscopy in the presence of an external magnetic field in a Hanle-type setup. The Kerr rotation signal as a function of the applied magnetic field could be fitted well by a Lorentzian where the spin lifetime $\tau_{s}$ entered as a fit parameter. By scanning over the sample, the authors obtained spatial profiles of the spin density along the direction perpendicular to the applied electric field. Fitting this data to solutions of the spin diffusion equation, the spin diffusion length was extracted. The values for this quantity lie between two and four micrometers and are, within the error bars, insensitive to the electric field varying between zero and $25{\rm mV}\mu{\rm m^{-1}}$. Combining this data with results for the spin lifetime, the authors inferred a spin conductivity of about $0.5\Omega^{-1}{\rm m^{-1}}$, where the latter quantity was converted to units of charge transport by multiplying with a factor of $e/ \hbar$.
The samples investigated by Kato [*et al.*]{} are n-doped and clearly in the bulk regime. Therefore, the intrinsic spin-orbit coupling to conduction-band electrons is dominated by the bulk Dresselhaus term (\[bulkdressel\]) which was studied by Bernevig and Zhang as a possible intrinsic mechanism for spin Hall transport. [@Bernevig05d]. The authors apply their results to the experiments by Kato [*et al.*]{} [@Kato04]. However, the agreement between theoretical predictions and the experimental findings is certainly not very convincing. Moreover, Kato [*et al.*]{} find only a very negligible dependence of the spin Hall transport on strain applied to the system. This observation also strongly disfavors an intrinsic mechanism. In fact, Engel, Halperin, and Rashba [@Engel05] have developed a theory of extrinsic spin Hall transport in GaAs based on impurity scattering and found reasonable agreement with the results by Kato [*et al.*]{} [@Kato04].
Further results on this type of experiments were reported by Sih [*et al.*]{} from the same research group for the case of n-doped GaAs quantum wells [@Sih05] (as opposed to the bulk epilayers discussed so far). The quantum well here has a width of $75\AA$ with a sheet density of $n=1.9\cdot10^{12}{\rm cm^{-2}}$ and a mobility of $\mu=940{\rm cm^{2}/Vs}$. From the analysis of their Kerr rotation data Sih [*et al.*]{} conclude that the signatures of spin Hall transport seen in their experiment are also most likely due to an extrinsic mechanism.
We now turn to spin Hall transport of holes. Wunderlich[*et al.*]{} have investigated spin Hall effect in a p-doped triangular quantum well which is part of a p-n junction light emitting diode [@Wunderlich05]. The quantum well has a sheet hole density of $n=2\cdot10^{12}{\rm cm^{-2}}$ which is for the still low enough, for the given sample geometry, such that only the first heavy hole subband is occupied. Thus, the intrinsic spin-orbit coupling to these heavy holes can be expected to be governed by the Hamiltonian (\[defham\]), leading to a spin Hall conductivity given by Eq. (\[fullspinHall\]). The spin accumulation at the edges of the well is detected by the circular polarization of the light emitted from the diode. In a subsequent publication, the authors presented further details on this technique [@Kaestner05]. In summary, Wunderlich [*at al.*]{} conclude that the spin Hall transport seen in their results is most likely of intrinsic nature, i.e. due to Rashba spin-orbit coupling as described by the Eq. (\[defham\]), although further work is needed to fully establish this conclusion [@Wunderlich05].
Let us now mention some proposals for further possible experimental investigations of spin Hall transport. Hankiewicz [*et al.*]{} have considered an H-shape two-dimensional electron system using the Landauer-Büttiker formalism [@Hankiewicz04]. The sample consists of two, say, horizontal bars connected in their centers by a shorter vertical bar. A voltage applied along one of the long bars should drive a spin current through the shorter connecting bar into the other horizontal bar. There this spin current should manifest itself by a voltage along the bar which can be calculated by inverting the spin conductivity tensor. Numerical calculations support the feasibility of this experimental approach [@Hankiewicz04].
A scheme to determine the spin Hall conductance purely via measurements of charge transport was put forward by Erlingsson and Loss [@Erlingsson05b]. The authors study a planar four-terminal setup. Using conventional scattering formalism, they express the spin Hall conductance in terms of voltages, charge conductances, and charge current noise quantities. This allows, in principle, to infer the spin Hall conductance just from electric measurements, avoiding the necessity of any magnetic or optical element [@Erlingsson05b].
Conclusions and outlook {#concl}
=======================
We have given an overview on recent theoretical and experimental developments concerning spin Hall transport in semiconductors. This phenomenon has certainly been over the last years one of the most intensively worked on topics in the solid-state community, and research efforts still continue to grow.
The theoretical situation regarding spin Hall effect in the two-dimensional electron gas with spin-orbit coupling linear in the momentum is by now well settled: There is no spin Hall effect in an infinite system if any kind of dissipation mechanism is present. This is, however, not a statement about finite mesoscopic systems, and further both theoretical and experimental work is to be expected here. On the other hand, p-doped quantum wells, i.e. the two-dimensional hole gas, appears to be a particularly attractive system.
A challenge for future theoretical work is certainly to extend many present results to the case of finite temperature, and, more importantly, include the Coulomb interaction between charge carriers. The last point is addressed by only very few papers so far [@Shekhter05; @Dimitrova04a]. A further recent theoretical development are studies on the [*zitterbewegung*]{} of electron and hole wave packets due to spin-orbit interaction in semiconductors [@Schliemann05b; @Schliemann05c]. These systems offer the possibility to experimentally detect the relativistic phenomenon of [*zitterbewegung*]{} which appears to be quite inaccessible in the case of free electrons [@Schliemann05b; @Schliemann05c].
Future challenges for experiments on spin Hall transport include the clarification and discrimination of extrinsic and intrinsic mechanisms. Moreover, from both a theoretical and an experimental point of view, it is undoubtedly desirable to develop setups and techniques which allow to detect spin Hall transport independently from spin accumulation.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank E. S. Bernardes, C. Bruder, D. Bulaev, G. Burkard, O. Chalaev, M. Duckheim, J. C. Egues, S. I. Erlingsson, M. Lee, D. Loss, D. Saraga, and R. M. Westervelt for fruitful collaboration and/or discussions on effects of spin-orbit coupling in semiconductors. This work was supported by the SFB 689 “Spin Phenomena in reduced Dimensions”.
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|
---
abstract: 'Since 1999, a radial velocity survey of 179 red giant stars is ongoing at Lick Observatory with a one month cadence. At present $\sim$20$-$100 measurements have been collected per star with an accuracy of 5 to 8 ms$^{-1}$. Of the stars monitored, 145 (80%) show radial velocity (RV) variations at a level $>$20 ms$^{-1}$, of which 43 exhibit significant periodicities. Here, we investigate the mechanism causing the observed radial velocity variations. Firstly, we search for a correlation between the radial velocity amplitude and an intrinsic parameter of the star, in this case surface gravity ($\log g$). Secondly, we investigate line profile variations and compare these with theoretical predictions.'
address:
- '$^1$ Leiden Observatory, Leiden University, P.O. Box 9513, 2300 RA Leiden, The Netherlands'
- '$^2$ Instituut voor Sterrenkunde, KU Leuven, Celestijnenlaan 200D, 3001 Leuven, Belgium'
- '$^3$ Department of Astrophysics, University of Nijmegen, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands'
- '$^4$ ZAH, Landessternwarte Heidelberg, Königstuhl 12, D-69117 Heidelberg, Germany'
- '$^5$ California Polytechnic State University, San Luis Obispo, CA 93407, USA'
author:
- 'S Hekker$^1$, I A G Snellen$^1$, C Aerts$^{2,3}$, A Quirrenbach$^4$, S Reffert$^4$ and D S Mitchell$^5$'
title: 'Radial velocities of giant stars: an investigation of line profile variations.'
---
Introduction
============
Since 1999, a radial velocity survey of 179 red giant stars is ongoing at Lick Observatory, using the 60 cm Coudé Auxiliary Telescope (CAT) in conjunction with the Hamilton echelle spectrograph (R $\approx$ 60000). These stars have been selected from the Hipparcos catalogue [@esa1997], based on the criteria described by [@frink2001]. The selected stars are all brighter than 6 mag, are presumably single and have photometric variations $< 0.06$ mag in V. The system with an iodine cell in the light path has been developed as described by [@marcy1992] and [@valenti1995]. With integration times of up to thirty minutes for the faintest stars ($m_{v}$ = 6 mag) we reach a signal to noise ratio of about $80-100$ at $\lambda = 5500$ Å, yielding a radial velocity precision of $5-8$ ms$^{-1}$.
The initial aim of the survey was to check whether red giants would be stable enough to serve as reference stars for astrometric observations with SIM/PlanetQuest [@frink2001]. In [@hekker2006] it is shown that a large fraction of the red giants in a specific part of the absolute magnitude vs. B-V colour diagram are stable to a level of 20 ms$^{-1}$ and could be effectively searched for long period companions, as is required for astrometric reference stars. For other stars in the sample the radial velocity variations are larger, and for 43 stars these show significant periodicities. So far, sub-stellar companions have been announced for two stars from the present sample ($\iota$ Dra [@frink2002] and $\beta$ Gem [@reffert2006]).
Here, we investigate which physical mechanism causes the observed radial velocity variations. In cases for which we do not find a significant periodicity in the observed radial velocity variations, an intrinsic mechanism such as spots or pulsations, possibly multi-periodic, seems most likely. On the other hand, the periodic radial velocity variations can be caused by sub-stellar companion, an intrinsic mechanism, or by both these mechanisms simultaneously. In Section 2 we search for a relation between the amplitude of the radial velocity variations and an intrinsic parameter, i.e. $\log g$. In Section 3 we investigate line shape variations and compare these with theoretical predictions. Our conclusions are presented in Section 4. A more extended paper on this subject is submitted [@hekker2007b].
Radial velocity amplitude vs. surface gravity relation
======================================================
![\[klogg\]Half of the peak-to-peak variation of the radial velocity as a function of surface gravity ($\log g$). The black dots indicate the stars with periodic radial velocity variations (stellar binaries are excluded), and the red squares indicate stars with random radial velocity variations. The solid line is the best fit through the random stars, the dotted line indicates the $1\sigma$ interval around the best fit and the dashed line indicates the $3\sigma$ interval. Six of the 8 stars with periodic radial velocity variations and $\log g < 1.6$ are classified bright giants or supergiants [@esa1997].](klogg_prop.eps){width="\linewidth"}
Hatzes and Cochran 1998 [@hatzes1998] already investigated the origin of the observed radial velocities in K giant stars. Although their sample contained only 9 stars, they suggested that the amplitude of the radial velocity increases with decreasing surface gravity. In lower surface gravity it takes longer to decrease the velocity of a moving parcel which results in larger amplitudes and the relation suggested by [@hatzes1998] would therefore be evidence for an intrinsic mechanism for these long period radial velocity variations.
For the present sample, $\log g$ values were determined spectroscopically by [@hekker2007], by imposing excitation and ionisation equilibrium of iron lines through stellar models. The equivalent width of about two dozen carefully selected iron lines were used for a spectroscopic LTE analysis based on the 2002 version of MOOG [@sneden1973] and Kurucz model atmospheres which include overshoot effects [@castelli1997]. These authors estimated the error on $\log g$ to be 0.22 dex from the scatter found in a comparison with literature values. A detailed description of the stellar parameters for individual stars and a comparison with literature values are available in [@hekker2007] and is therefore omitted here.
![\[kloggres\]Half of the peak-to-peak variation of the radial velocity as a function of surface gravity ($\log g$) as in Figure \[klogg\], but showing only those stars with periodic radial velocity variations (dots) and their residual (plus). The solid line indicates the linear fit through the stars with non-periodic radial velocity variations (from Figure \[klogg\]).](klogg_propres.eps){width="\linewidth"}
In Figure \[klogg\] the half peak-to-peak value of the radial velocity is plotted as a function of $\log g$, together with the best linear fit. There clearly exists a correlation between the observed radial velocity amplitude variation and the surface gravity. Also, most of the stars with periodic radial velocity variations and $\log g > 1.6$ dex are located above the best fit, which could indicate that both intrinsic and extrinsic mechanisms are contributing. To investigate this, we subtracted the periodic fit from the radial velocity variations and plotted half the peak-to-peak value of the residuals as a function of surface gravity, see Figure \[kloggres\]. For stars with $\log g > 1.6$ dex the residuals are now located around the best fit through the random stars. This is an indication that intrinsic and extrinsic mechanisms are indeed contributing simultaneously in these stars.
For 8 out of 9 stars with $\log g < 1.6$ dex, we find a significant period and therefore, the best fit in Figure \[klogg\] may not be very accurate in this region. Furthermore, the atmospheres of stars with these low surface gravities are so diluted that instabilities occur easily, either periodic or random. We therefore think it most likely that these variations are not due to companions. Moreover, the radial velocity variations of these stars are located around the fit for random variables, while the residuals are mostly below this relation.
Line shape analysis
===================
In the previous section we treated a sample of stars, but we would also like to know what mechanism causes the radial velocity variations in each star individually. For a sub-sample of stars, we therefore obtained high resolution spectra (R $\approx$ 164000) with the SARG spectrograph mounted on the Telescopio Nazionale Galileo, La Palma, Spain. We have between 3 and 8 observations per star, which is not enough to do a full line shape analysis. But we tried to identify whether significant line depth variation is present in a star, which would indicate an intrinsic mechanism. To do this we shifted the spectra of each star, taken at different epochs, to the laboratorium wavelength and computed a time averaged profile. Residuals at each epoch provides us with the variation in line depth which is indicative of line shape variations and thus the presence of an intrinsic mechanism.
![\[hip53261\] Left: radial velocity variations of HIP53261 as a function of Julian Date. A Keplerian orbit is fitted through the data and the residuals are shown in the bottom panel. Right: the residual of the Fe I line at 6252.57 Å, taken at different epochs (indicated with different line styles), with respect to a time averaged profile, as a function of wavelength. The vertical lines in this panel indicate the spectral line in wavelength. An error estimate is indicated with the thick error bar in the left upper corner. ](hip53261.eps){width="18pc"}
![\[hip53261\] Left: radial velocity variations of HIP53261 as a function of Julian Date. A Keplerian orbit is fitted through the data and the residuals are shown in the bottom panel. Right: the residual of the Fe I line at 6252.57 Å, taken at different epochs (indicated with different line styles), with respect to a time averaged profile, as a function of wavelength. The vertical lines in this panel indicate the spectral line in wavelength. An error estimate is indicated with the thick error bar in the left upper corner. ](hip53261Felinedif.eps){width="18pc"}
In Figure \[hip53261\] we show the radial velocity variation of HIP53261 as a function of phase and the residuals of the line depth. For this star we see significant variation in the line depth, which is indicative of an intrinsic mechanism. Because we lack data to perform a frequency analysis it is not yet possible to verify whether the period of the radial velocity and line depth variation are related, in which case the variations are due to an intrinsic mechanism. In case the periods are not related, we probably have a companion orbiting an intrinsically active star.
![\[ressim\] Half peak-to-peak values of the first moment as a function of residuals in line depth for stars with pulsations with $\ell=(0,1,2)$, positive $m$ values (different modes are indicated with different symbols). Top: for inclination angles of 30, 50 and 70 degrees, projected rotational velocity of 2.0, 3.5 and 5.0 kms$^{-1}$, equivalent width of 30, 40 and 50 kms$^{-1}$, intrinsic line width of 3.0, 4.0 and 5.0 kms$^{-1}$ and pulsation amplitudes of 0.1 and 1.0 kms$^{-1}$. Top: all computed points. Others from left to right and top to bottom: dependence on projected rotational velocity, equivalent width, intrinsic line width, inclination angle and pulsation amplitude.](plotgridBW.eps){width="18pc"}
![\[ressim\] Half peak-to-peak values of the first moment as a function of residuals in line depth for stars with pulsations with $\ell=(0,1,2)$, positive $m$ values (different modes are indicated with different symbols). Top: for inclination angles of 30, 50 and 70 degrees, projected rotational velocity of 2.0, 3.5 and 5.0 kms$^{-1}$, equivalent width of 30, 40 and 50 kms$^{-1}$, intrinsic line width of 3.0, 4.0 and 5.0 kms$^{-1}$ and pulsation amplitudes of 0.1 and 1.0 kms$^{-1}$. Top: all computed points. Others from left to right and top to bottom: dependence on projected rotational velocity, equivalent width, intrinsic line width, inclination angle and pulsation amplitude.](plotgridzoom.eps){width="18pc"}
![\[ressim\] Half peak-to-peak values of the first moment as a function of residuals in line depth for stars with pulsations with $\ell=(0,1,2)$, positive $m$ values (different modes are indicated with different symbols). Top: for inclination angles of 30, 50 and 70 degrees, projected rotational velocity of 2.0, 3.5 and 5.0 kms$^{-1}$, equivalent width of 30, 40 and 50 kms$^{-1}$, intrinsic line width of 3.0, 4.0 and 5.0 kms$^{-1}$ and pulsation amplitudes of 0.1 and 1.0 kms$^{-1}$. Top: all computed points. Others from left to right and top to bottom: dependence on projected rotational velocity, equivalent width, intrinsic line width, inclination angle and pulsation amplitude.](plotgridvrot.eps){width="18pc"}
![\[ressim\] Half peak-to-peak values of the first moment as a function of residuals in line depth for stars with pulsations with $\ell=(0,1,2)$, positive $m$ values (different modes are indicated with different symbols). Top: for inclination angles of 30, 50 and 70 degrees, projected rotational velocity of 2.0, 3.5 and 5.0 kms$^{-1}$, equivalent width of 30, 40 and 50 kms$^{-1}$, intrinsic line width of 3.0, 4.0 and 5.0 kms$^{-1}$ and pulsation amplitudes of 0.1 and 1.0 kms$^{-1}$. Top: all computed points. Others from left to right and top to bottom: dependence on projected rotational velocity, equivalent width, intrinsic line width, inclination angle and pulsation amplitude.](plotgridEW.eps){width="18pc"}
![\[ressim\] Half peak-to-peak values of the first moment as a function of residuals in line depth for stars with pulsations with $\ell=(0,1,2)$, positive $m$ values (different modes are indicated with different symbols). Top: for inclination angles of 30, 50 and 70 degrees, projected rotational velocity of 2.0, 3.5 and 5.0 kms$^{-1}$, equivalent width of 30, 40 and 50 kms$^{-1}$, intrinsic line width of 3.0, 4.0 and 5.0 kms$^{-1}$ and pulsation amplitudes of 0.1 and 1.0 kms$^{-1}$. Top: all computed points. Others from left to right and top to bottom: dependence on projected rotational velocity, equivalent width, intrinsic line width, inclination angle and pulsation amplitude.](plotgridIW.eps){width="18pc"}
![\[ressim\] Half peak-to-peak values of the first moment as a function of residuals in line depth for stars with pulsations with $\ell=(0,1,2)$, positive $m$ values (different modes are indicated with different symbols). Top: for inclination angles of 30, 50 and 70 degrees, projected rotational velocity of 2.0, 3.5 and 5.0 kms$^{-1}$, equivalent width of 30, 40 and 50 kms$^{-1}$, intrinsic line width of 3.0, 4.0 and 5.0 kms$^{-1}$ and pulsation amplitudes of 0.1 and 1.0 kms$^{-1}$. Top: all computed points. Others from left to right and top to bottom: dependence on projected rotational velocity, equivalent width, intrinsic line width, inclination angle and pulsation amplitude.](plotgridinc.eps){width="18pc"}
![\[ressim\] Half peak-to-peak values of the first moment as a function of residuals in line depth for stars with pulsations with $\ell=(0,1,2)$, positive $m$ values (different modes are indicated with different symbols). Top: for inclination angles of 30, 50 and 70 degrees, projected rotational velocity of 2.0, 3.5 and 5.0 kms$^{-1}$, equivalent width of 30, 40 and 50 kms$^{-1}$, intrinsic line width of 3.0, 4.0 and 5.0 kms$^{-1}$ and pulsation amplitudes of 0.1 and 1.0 kms$^{-1}$. Top: all computed points. Others from left to right and top to bottom: dependence on projected rotational velocity, equivalent width, intrinsic line width, inclination angle and pulsation amplitude.](plotgridampl.eps){width="18pc"}
![\[RVdif\]Half of the peak-to-peak variation of the radial velocity as a function of residuals in line depth, for all stars with at least 3 high resolution, high signal to noise observations with SARG.](dRVddif.eps){width="\linewidth"}
Apart from the relation in frequency between the radial velocity and line depth variations we are also interested in the possible existence of a correlation between the amplitudes of both variations. Because the stars show hardly any photometric variation, spots similar to sunspots are not the most likely mechanism and therefore we focus here on stellar pulsations. We computed the half peak-to-peak value of the first moment (another diagnostic for the radial velocity) and residuals in line depth for line profiles with pulsation modes $\ell=0, 1, 2$ and positive $m$ values, inclination angles of 30, 50 and 70 degrees, projected rotational velocities of 2.0, 3.5 and 5.0 kms$^{-1}$, intrinsic width of 3.0, 4.0 and 5.0 kms$^{-1}$, equivalent width of 30, 40 and 50 kms$^{-1}$ and pulsation velocities of 0.1 and 1.0 kms$^{-1}$, and plotted these in the top panels of Figure \[ressim\]. From these plots it is clear that there is no simple relation between the amplitude in the radial velocity and residual line depth variations. To investigate the influence of different parameters, we also plotted the first moment amplitude as a function of residual line depth for all modes and both pulsation amplitudes, and vary one parameter, while keeping three other parameters fixed (Figure \[ressim\]). In this way it becomes clear that the projected rotational velocity, intrinsic line width and equivalent width of a line introduces variations in the line residuals at approximately the same value for the first moment. On the other hand, varying the inclination angle or pulsation velocity changes the amplitude of the first moment (radial velocity variation) at approximately the same residual line width.
In Figure \[RVdif\] we show the observed half peak-to-peak values of the radial velocity variation as a function of the residual line depth variation, for all stars with at least 3 high resolution, high signal to noise observations with SARG. There is no correlation between these points, which is as expected from the theoretical analysis for stars with different parameters. Strikingly, most of the observed points fall in a region where no theoretical values are found. This difference between the observations and computations can be caused by the fact that in the theoretical models the temperature variation of the stellar surface due to oscillations is not taken into account. This temperature variation causes variations in the intrinsic and equivalent width of the spectral line and these parameters have a large influence on the amplitude of the residual line depth. This effect could cause the higher amplitudes in the observed line depth residuals than expected from the calculations. In case the input values chosen for the models are too far off the actual values of the pulsation parameters of the stars, we would also expect a discrepancy between the observed and computed relation between radial velocity variations and residual line depth.
Conclusions
===========
There exists a clear correlation between the half peak-to-peak values of the radial velocity variations and the surface gravity of the red giant stars. This is a strong indication that the observed radial velocity variations are caused by a mechanism intrinsic to the star. Companions and an intrinsic mechanism might be present in stars with periodic radial velocity variations and $\log g > 1.6$ dex, while for stars with $\log g<1.6$ dex solely an intrinsic mechanism seems most likely.
We have investigated whether there is a relation between the radial velocity amplitude and the amplitude of the line depth residuals. From theory we find that the rotational velocity, intrinsic and equivalent line width largely influence the line depth residual and no clear correlation could be identified.
In the theoretical models temperature variations due to the pulsations is ignored, but these temperature variations influence the intrinsic and equivalent line width of the spectral line, which influences the line depth residuals. This might be an explanation why the theoretical models do not overlap with the observations.
References {#references .unnumbered}
==========
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---
abstract: 'Nowadays, evidence is mounting that the race of living organisms for adaptation to the chemicals synthesized by their neighbours may drive community structures. Particularly, some bacterial infections and plant invasions disruptive of the native community rely on the release of allelochemicals that inhibit or kill sensitive strains or individuals from their own or other species. In this report, an eco-evolutionary model for community assembly through resource competition, allelophatic interactions, and evolutionary branching is presented and studied by numerical analysis. Our major findings are that stable communities with increasing biodiversity can emerge at weak allelopathic suppression, but stronger allelophaty is negatively correlated with community diversity. In the former regime the allelopathic interaction networks exhibit Gaussian degree distributions, while in the later one the network degrees are Weibull distributed.'
address:
- 'Departamento de Física, Universidade Federal de Viçosa, 36570-000, Viçosa, MG, Brazil'
- 'National Institute of Science and technology for Complex Systems, Centro Brasileiro de Pesquisas Físicas, Rua Xavier Sigaud 150, 22290-180, Rio de Janeiro, Brazil'
author:
- Sylvestre Aureliano Carvalho
- Marcelo Lobato Martins
title: 'Community structures in allelopathic interaction networks: an eco-evolutionary approach'
---
Complex networks ,Community structure ,Competition ,Allelopathy
Introduction
============
Conventional explanations of biodiversity postulate that it is passively shaped by niche differentiation, density-dependent predation pressure, habitat heterogeneity, or fluctuations in the resources required by the biological communities. Furthermore, stabilizing mechanisms relying on negative intraspecific interactions, stronger than interspecific interactions, are essential for species coexistence [@Chesson] since they cause species to limit themselves more than other organisms. Without stabilizing mechanisms, the inhibitory effects of competition on inferior competitors will ultimately lead to their extinction. Classically, such stabilizing interactions have been thought to result from resource partitioning: competing species can coexist provided they are most limited by different resources and consume the resources they are most limited by at a higher rate than do other species [@Tilman].
However, the astonishing high diversity observed within microorganism communities in seemingly uniform environments — the famous paradox of the plankton [@Hutchinson] — challenges the conventional resource competition framework. Indeed, even a highly structured habitat can hardly maintain such astronomical species numbers. Moreover, experiments performed with plants have neither shown intraspecific unequivocally exceeding interspecific competition [@Goldberg] nor competing plants coexisting through resource partitioning [@Miller]. Also, abiotic supply rates seems to be relatively high and stable over time, whereas the resident species do neither reduce resource densities or interfere greatly with resource access [@Davis].
In contrast, interference competitions mediated by the production of toxic chemical compounds — antibiotic, phytotoxins, lactate, etc. — are ubiquitous in biological communities, from microorganisms, such as bacteria [@Cordero], yeasts [@Starmer], and other fungi [@Berdy], to cancer cells [@Gatenby; @Braganhol] and plant invasions [@Bais]. So, additionaly to other nontrophic interactions (e. g., the raise of mycorrhizal networks in plant communities [@Heijden], mutualism at weak direct competition [@Bastolla], and facilitation [@Carrion]), the biochemical warfare between living organisms may drive species coexistence and community composition. The alternative view that biological communities can emerge from allelopathy, i.e., from competing interactions between their species mediated by toxins, faces a difficulty: multiple toxic environments are the least expected to sustain species diversity. Indeed, some exotic invasive plants may use allelopathic suppression to disrupt inherent, coevolved interactions among long-associated native species constituting the communities they invade [@Bais2; @Souza]. Therefore, community and invasion ecology are naturally interconnected because both the persistence of a species in a community or its invasion success abroad its native habitat primarily depends on its ability to increase from low density [@Morton; @Shea].
In this report, our goal is to discuss how community structures of populations enforced to adapt and survive to the direct allelochemical suppression of each other is affected by the evolutionary history of the interaction. Specifically, we extend previously proposed models for the allelopathic warfare between two species [@Fassoni; @Sylvestre] by integrating ecological and evolutionary processes. In the model, the genetic diversity is generated by mutations that induce changes in the allelochemical traits of the evolving species and selection is driven by ecological interactions, namely, intra- and interspecific resource competition and allelopathic suppression. These interactions determine how species evolve and enhance or diminish the diversity of communities. The paper is organized as follows. In Section \[model\], the mathematical model is introduced. The major results concerning community structure and biological diversity are reported in Section \[results\]. In Section \[discussion\] our major findings are discussed and some conclusions are drawn.
\[model\] In order to model the community dynamics, a set $S$ of $l\in\mathbb{N}$ biological species with populations given by $\mathbf{N}=(N_1,N_2,\ldots)$ is considered. The interactions among these species occurs only via intra- and interspecific resource competition and allelopathic suppression. Thus, every species in $S$ synthesizes and releases toxic secondary chemical compounds (microcins, fitotoxins etc.) that enhance the mortality of other species. The strengths of such interactions depends on the toxin concentration $\mathbf{B}=(B_1,B_2,\ldots)$ and vary in time because $\mathbf{B}$ depends on the abundance of the species. Furthermore, the community assembly proceeds from an initial subset $S_0 \subseteq S$ by randomly adding new species through mutations fixed in a fraction of resident species’ offspring.
Ecological dynamics
-------------------
The temporal evolution of the biological community in a homogeneous environment is described by the coupled ordinary differential equations
$$\begin{aligned}
\label{pop_dyn}
\frac{\mathrm{d}N_{i}}{\mathrm{d}t} &=& r_{i}\,\left( 1-\sum_{j= 1}^{l}\nu_{ij}N_{j}\right)\,N_{i} - \sum_{j\neq i}^{l}\mu_{ij}\Phi_{ij}^{(k)} (y_j) \,N_{i}\nonumber\\
& & \\
\frac{\mathrm{d}B_{i}}{\mathrm{d}t} &=& \beta_{i}\,N_{i} - \delta_{i}\,B_{i}- \sum_{j\neq i}^{l}\gamma_{ji}\,N_{j}\,B_{i}. \nonumber\end{aligned}$$
Here, $N_{i}$ stands for the population density of the species $i$ that produces the allelochemical concentration $B_{i}$, respectively. Also, $r_i$, $\beta_i$ and $\delta_i$, $i = 1,2,\ldots$, respectively, are the reproduction rates, toxin release and natural degradation rates associated to the competing species. A classical interspecific competition for the environmental resources is assumed. The parameters $\nu_{ij}$ are the competition coefficients that measure the extent to which each species presses upon the resources used by the others. The quantity $y_j= \gamma_{ji} N_i B_j$ represents the overall consumption of the toxin $j$ by the species $i$, $i \neq j$, with per capta absorption rate $\gamma_{ij}$. These quantities depend on the toxin’s levels in a linear way. So, the term $-\sum_{j \neq i} \mu_{ij} \Phi_{ij}^{(k)}(y_j)$ represent species decreases as they uptake the allelochemicals released by their allelopathic suppressors, in which $\mu_{ij}$ is the mortality rate of the species $i$ induced by the toxin released by its competitor $j$. Different Holling type *I*, *II*, and *III* functional responses were assumed:
$$\begin{aligned}
\label{res_func}
\Phi_{ij}^{(k)}=
\left\{\begin{matrix}
B_{j} & (k=1) \\ \vspace{0.1cm}
\gamma_{j,i}N_{i}B_{j} & (k=2) \\ \vspace{0.1cm}
\frac{B_{j}}{c_{i}+B_{j}} & (k=3)\\ \vspace{0.1cm}
\frac{\gamma_{j,i}N_{i}B_{j}}{c_{i}+\gamma_{j,i}N_{i}B_{j}} & (k=4)\\ \vspace{0.1cm}
\frac{B_{j}^{2}}{c_{i}+B_{j}^{2}} & (k=5)\\\vspace{0.1cm}
\frac{(\gamma_{j,i}N_{i}B_{j})^{2}}{c_{i}+(\gamma_{j,i}N_{i}B_{j})^{2}} & (k=6),
\end{matrix}\right.\end{aligned}$$
where the parameters $c_i$ control the toxin’s efficiencies in poison their competing species. All these response functions assume null thresholds for toxin effects, but those with $k \geq 3$ impose saturation to the allelopathic suppression. Also, the response functions indexed by odd $k$’s involve the total toxin concentration, in contrast to those indexed by even $k$’s for which only the absorbed toxin can induce responses.
Equations \[pop\_dyn\] and \[res\_func\] for two species were extensively investigated through analytical and numerical methods in references [@Fassoni; @Sylvestre]. In the present paper up to $l=100$ competing species were considered and the interacting parameters $\nu_{i,j}$, $\gamma_{j,i}$, and $\mu_{i,j}$ define networks in which the species are the nodes. These parameters can be expressed as $\nu_{i,j}=\nu_{i,j}\,\varepsilon_{i,j}$, $\gamma_{j,i}=\gamma_{i,j}\,\zeta_{j,i}$, and $\mu_{i,j}=\mu_{i,j}\,\zeta_{i,j}$, in which $\varepsilon_{i,j}=1$ ($\zeta_{i,j}=1$) if species $i$ competes with (poisons) species $j$, but $\varepsilon_{i,j}=0$ ($\zeta_{i,j}=0$) if $i$ does not compete (poisons) $j$. Every $\varepsilon_{i,j}, \zeta_{i,j}=1$ is a link connecting two species. The set of values $\varepsilon_{i,j}$ and $\zeta_{i,j}$ define two matrices $\varepsilon$ and $\zeta$ which characterize the competition and allelochemical interaction networks, respectively. These matrices are examples of the adjacency matrix, central in network theory [@Barabasi]. The diagonal elements of $\varepsilon$ are $\varepsilon_{i,i}=1$ and represent intraspecific competition, with all $\nu_{i,i}=1$ by definition. In turn, we set all $\zeta_{i,i}=0$ in order to avoid self-allelopathic suppression.
The ecological interactions (competition and allelopathy) drive the dynamics, equation \[pop\_dyn\], towards an stationary state ($\mathbf{N}^*$, $\mathbf{B}^*$) in a short time scale. This stationary state depends on the species initially present and their interaction networks. Eventually, even in the weak interspecific competition (coexistence) regime, some populations are led to extinction by allelopathic suppression and the community diversity (species richness) decreases.
Evolutionary dynamics
---------------------
The origin and maintenance of biological communities depends on the interplay between evolutionary processes and ecological interactions that allow species coexistence [@Edwards]. Ecological and evolutionary processes are integrated in our model by assuming that mutations in one of the competing species present at the current stationary state of the ecological dynamics generate a new species. This fresh species must survive and evolve in response to novel conditions, and the old species in the community must in turn evolve in response to the new species. Ultimately, the ecological dynamics is driven to another stationary state characterized by distinct populations and interaction networks. After that, additional genetic diversity is generated by adding different species to the current community, and so on. Two mechanisms for species introduction were tested.
### Sequential invasion events (SIE)
An alien species, the node $n+1$, is added to an stationary state currently containing $n$ species. It is assumed that the alien species competes for resources with all the $n$ pre-existing species. Thus $\varepsilon_{n+1,i}=\varepsilon_{i,n+1}=1$ for $i=1,\ldots,n$. Concerning allelochemical suppression, the alien species affects $k_{n+1}^{out}$ of the old ones and is affected by $k_{n+1}^{in}$ of them. So, $k_{n+1}^{out}$ elements $\zeta_{n+1,i}$ in the line $n+1$ of the enlarged adjacency matrix $\zeta$ are fixed in $1$ and the remaining in $0$. In order to do this, an integer $i$ is randomly chosen in the interval $[1,n]$, and we set $\zeta_{n+1,i}=1$, with a probability $p=1-n^{out}/n$, or $\zeta_{n+1,i}=0$, with a probability $1-p=n^{out}/n$. Then, a distinct $i$ is randomly selected and the protocol repeated until $k_{n+1}^{out}$ elements in the $(n+1)$-th line of $\zeta$ are set to $1$. The value $n^{out} \in [1,n]$ defines the probability $p$ and, again, is an integer random number chosen with equal change. In average, $n^{out}$ determines the fraction of species in the community which do not interact with the alien species. Analogously, $k_{n+1}^{in}$ elements $\zeta_{i,n+1}$ in the column $n+1$ of the enlarged adjacency matrix are fixed in $1$ and the remaining in $0$. The same protocol is used to determine the $k_{n+1}^{in}$ nodes $i$ that suppress the node $n+1$ (i. e., $\zeta_{i,n+1}=1$). But now the probability used is $p=1-n^{in}/n$. Finally, the initial toxin concentration of the alien species is $B_{n+1}=0$ and its population density is $N_{n+1}=0.01 N_i^*$, with $N_i^*$ corresponding to the stationary population density of one species chosen at random between the $n$ current members of the community. Regarding the initial community structure, the SIE evolutionary dynamics starts from a single species.
### Branching process (BP)
The new species $n+1$ introduced in the network descends from one of the $n$ species present at the community stationary state. The ancestor species $i$ is randomly chosen and only their allelochemical traits are mutated in its descendant species $n+1$. Specifically, all the $k_i^{in}$ input and $k_i^{out}$ output connections of the ancestor node $i$ are inherited by the new node $n+1$, except one of them. With equal chance, either a randomly chosen input $\zeta_{j,i}$ or output $\zeta_{i,j}$ of the node $i$ will be activated ($\zeta_{j,n+1}=1$) in node $n+1$ if inactive ($\zeta_{j,i}=0$) in $i$, or vice-versa. Since its is supposed here that the resource competition traits are not changed by mutations, $\varepsilon_{i,j}=\varepsilon_{n+1,j}$ and $\varepsilon_{j,i}=\varepsilon_{j,n+1}$ for $j=1,\ldots,n$. Again, the initial toxin concentration of the new species is $B_{n+1}=0$ and its population density is $N_{n+1}=0.01 N_i^*$. Finally, concerning the initial community structure, the BP evolutionary dynamics starts from a network with $n_0<l$ nodes. Different starting graphs for the BP dynamics are shown in figure \[start\_graphs\].
![Allelopathic networks used as starting structures for the BP dynamics. The species interactions are indicated by arrows. In numerical integrations, the population densities $N_{i}(0)=0.7$ and toxin concentrations $B_{i}(0)=0$ were fixed.[]{data-label="start_graphs"}](subgrafos.pdf){width="9.5cm"}
Community structures: species diversity and allelochemical network topologies {#results}
=============================================================================
The previously described eco-evolutionary process was investigated through numerical integration using the fourth-order Runge-Kutta method. Distinct distributions for the values of the competition and allelochemical parameters $\varepsilon_{ij}$ and $\zeta_{ij}$ were employed. Also, $200$ independent evolutionary histories were generated for the SIE and BP dynamics, in the latter case for each initial graph shown in figure \[start\_graphs\]. From the numerical integrations, the adjacency matrix at the successive stationary states for each evolutionary history were obtained. Then, the community structures (interaction network topology) and species richness were determined for both SIE and BP dynamics.
SIE dynamics
------------
Since our primary interest relied on how allelopathic suppression affects the community structure, $\nu_{i,j}=\nu=0.1$ was fixed in order to ensure equal competition coefficients for every species in a regime of interspecific coexistence.
The scenario of equal (or homogeneous) allelopathic traits was investigated. Thus, each species has fixed toxin sensibility, $c_i=c=0.1$, release, degradation, and uptaken rates, $\beta_i=\beta=0.2$, $\delta_i=\delta=0.2$, and $\gamma_{j,i}=\gamma=0.1$, respectively, $ \forall i,j$. In turn, two mortality rates induced by allelochemicals were considered, namely, weak ($\mu_i=\mu=0.1$) and strong ($\mu_i=\mu=0.5$) $\forall i$.
In figure \[abundance\_sie\] it is shown the average diversity as a function of the number $n_{SIE}$ of SIE. The diversity or species richness is defined as the fraction of species that survive at the community stationary state. As expected, weak allelopathic suppression allows the assembly of communities exhibiting large diversities. This is true for all response functions tested and, as expected, the diversity decreases as the response to toxins increases. For instance, in our simulations, $\Phi^{(1)}(x) < \Phi^{(5)}(x) < \Phi^{(3)}(x)$ except for small ($x < 0.11$) or large ($x > 0.89$) toxin concentrations. In contrast, community diversity is drastically reduced at strong allelopathy for all response functions. As an example, the number of surviving species decreases from $\sim 100$, at weak, to $\sim 10$ at strong allelopathic suppression and response function $\Phi^{1}$. In this strong regime, diversity seems to decrease slowly after reaches a maximum as the number of invasion events increases. Also, the effect of toxin’s uptaken is significant as revealed by the right column in figure \[abundance\_sie\]. In these graphs the response functions depend on the absorved fraction of toxins, not on their total concentration present in the homogeneous environment. So, even the regime of strong allelopathic suppression ($\mu=0.5$) at low toxins’ absorption can become effectively equivalent to the weak ($\mu=0.1$) regime.
![Average diversity for $200$ independent eco-evolutionary dynamics observed after successive invasion events. The initial community is always composed of a single species. The top and bottom plots refer, respectively, to weak and strong allelopathic effects.[]{data-label="abundance_sie"}](diversity_sie.pdf){width="9cm"}
In figure \[connect\_sie\], the average connectivity of allelochemical networks is illustrated as a function of the number $n$ of surviving species observed at the stationary state reached after a SIE. In a network of size $n$, the connectivity $C(n)$ is defined as the fraction of non-null elements in its $n \times n$ adjacency matrix ($\zeta_{i,j}$ in this case). In terms of the adjacency matrix $C(n)$ is given by
$$C(n)=\frac{\sum_{i,j} \zeta_{i,j}}{n(n-1)}.$$
Our results indicate that the average connectivity is significantly larger at weak ($\mu=0.1$) than strong ($\mu=0.5$) allelopathy. So, the interaction network is much more sparsely connected at strong allelochemical suppression. Furthermore, the connectivity initially increases up to $n \sim 10$ and saturates to a constant value and, for large response functions ($\Phi^{(3,5)}$), exhibits significant fluctuations at strong allelopathic regime. This behavior is very distinct from the power law scaling for large $n$ values observed in random networks, $C(n) \sim n^{-1}$ [@May], and a model for growing random networks based on global stability, $C(n) \sim n^{-1.2}$ [@Perotti]. Therefore, our results indicate that the communities generated by the SIE dynamics markedly differs from random networks involving positive and negative interactions.
![Average network connectivity $C(n)$ in communities containing $n$ species after a SEI. Again, the initial community is always composed of a single species. The results for weak and strong allelopathic suppression are shown in frames (a,c) and (b,d), respectively. The dashed lines corresponds to power laws with different exponents.[]{data-label="connect_sie"}](connectivity_sie.pdf){width="9cm"}
The degree distributions $P(k)$ for allelochemical interaction networks generated by the SIE dynamics are shown in figure \[degree\_pdf\_sie\]. The distribution $P(k)$ gives the probability that a randomly selected node in a network has $k$ links, i. e., it is connected to $k$ nodes. Normal (Gaussian) and Weibull distributions was observed for in-degree distributions $P(k^{in})$ depending on the mortality $\mu$ and the functional response to allelopathy. For strong allelopathic suppression and functional responses $\Phi^{(1,3,5)}$, $P(k^{in})$ is a Weibull distribution. In contrast, at weak allelopathic suppression and for the response functions $\Phi^{(4,6)}$ at the strong regime, $P(k^{in})$ is Gaussian distributed. The apparent anisotropies observed in the insets for $\Phi^{(1,2,4,6)}$ are very weak, as supported by skewness $S \sim 0$ and kurtosis $K \sim 3$ (see the appendix). However, the ratio $\kappa=\langle k^2 \rangle / \langle k \rangle \sim \langle k \rangle$ is always obtained, indicating that the SIE allelochemical networks are homogeneous [@Barabasi]. In turn, the degree distributions $P(k^{out})$ for all scenarios are normal (Gaussian) distributions (see the appendix).
![Degree distribution $P(k^{in})$ for SIE allelochemical interaction networks in which the competition and allelochemical traits are the same for all species. The top and bottom graphs correspond, respectively, to weak and strong allelopathic suppression.[]{data-label="degree_pdf_sie"}](degree_sie.pdf){width="9cm"}
Lastly, typical allelochemical networks or community structures generated by the SIE dynamics are illustrated in figure \[SIE\_nets\]. The nodes in these networks represent species present in the community and the directed edges between them represent allelopathic interactions. As the allelopathic strength increases, the number of node (surviving species) decreases, the network topology changes from random to hierarchical structures, and the corresponding connectivity distributions change from normal (or Gaussian) to Weibull distributions.
![Typical allelochemical networks generated after $l=100$ SIEs for (a) weak ($\mu=0.1$) and (b)-(c) strong allelopathic suppression ($\mu=0.4$ and $\mu=0.5$, respectively). The competition and allelochemical traits are homogeneous (constant and equal for all species) and the functional response $\Phi_{i,j}^{(1)}$ was used.[]{data-label="SIE_nets"}](graphs_sie.pdf){width="9cm"}
The community structure was also investigated for scenarios in which only the competition or competition and allelopathy are heterogeneous, so that the coefficients $\nu_{i,j}$ and $\mu_{i,j}$ are drawn from random distributions. The results are qualitatively the same as those obtained for the homogeneous cases reported earlier (data not shown), but the introduction of heterogeneity in competition coefficients has smaller effects than in allelochemical parameters, as shown in figure \[disorder\_sie\]. Indeed, competition coefficients uniformly distributed in $[0,1]$ do not move the system from the competition coexistence regime, whereas allelochemical parameters, particularly, $\mu_i$ and $c_i$, uniformly distributed in $[0,1]$ effectively correspond to strong allelopathy ($\langle \mu \rangle =0.5$).
![Average diversity for $200$ independent eco-evolutionary dynamics observed after successive invasion events at the strong allelopathic regime. The initial community is always composed of a single species. In (a) both competition and allelochemical parameters are heterogeneous, i. e., randomly chosen from uniform distributions in $(0,1]$. In (b) only the competition coefficients are heterogeneous and the allelopathic traits are the same for all species. The value $\mu=0.1$ was used (weak allelopathy).[]{data-label="disorder_sie"}](diversity_sie.pdf){width="9cm"}
BP dynamics
-----------
The BP dynamics was analysed for three distinct scenarios. In the first one, called homogeneous, all the original and introduced species have equal competition and allelopathic traits: $\nu_{i,j}=0.1$ and $\varepsilon_{i,j}=1, \, \forall \, i,j$, $\mu_{i,j}=0.1$ and $\gamma_{i,j}=0.1, \, \forall \, i \neq j$, $c_{i}=0.1$, $\beta_{i}=0.2$, and $\delta_{i}=0.2, \, \forall \, i$. In the second scenario, called heterogeneous competition, the allelochemical traits are equal, as before, but the competition coefficients $\nu_{i,j}$ are disordered, i. e., randomly drawn from a uniform distribution on the interval $(0,1]$. Thus, the species can have different competition, but the same allelochemical capabilities. Finally, in the third scenario, called completely heterogeneous, it is supposed that both competition and allelochemical traits are disordered and independently drawn from uniform distributions on the interval $(0,1]$. Only the toxins’ degradation and uptaken rates, $\delta_{i}=\delta=0.3$ e $\gamma_{j,i}=\gamma=0.1$, are assumed the same for all species.
The average diversity as a function of the number $n$ of “speciations” is shown in figure \[abundance\_bp\]. Again, the response functions involving the total toxin concentration (odd $k$’s) induce more extinctions and lead to less diversity. Figures \[abundance\_bp\]b (disordered competition) and \[abundance\_bp\]c (disordered competition and allelopathy) evidence that heterogeneous competition and allelochemical traits decrease community diversity in comparison to homogeneous traits (figure \[abundance\_bp\]a).
![Average diversity for $200$ independent BP eco-evolutionary histories as function of the number $n$ of speciation events. The initial communities are the graphs shown in figure \[start\_graphs\]. The top, middle, and bottom plots refers, respectively, to homogeneous, heterogeneous competition, and completely heterogeneous (both competition and allelopathy) scenarios.[]{data-label="abundance_bp"}](diversity_bp.pdf){width="9cm"}
In figure \[connect\_bp\], the average connectivity of allelochemical networks is illustrated as a function of the community size $n$.
![Average connectivity for communities grown from all graphs exhibited in the figure \[start\_graphs\] as a function of the network size $n$ for the homogeneous (top), heterogeneous competition (middle), and completely heterogeneous (bottom) scenarios. The dashed lines corresponds to the power law with different exponents.[]{data-label="connect_bp"}](connectivityBP.pdf){width="9cm"}
The in- and out-degree distributions, $P(k^{in})$ and $P(k^{out})$, for the allelochemical interaction networks generated by the BP dynamics are shown in figure \[degree\_pdf\_bp\]. Indeed, the complementary cumulative degree distributions defined as
$$\begin{aligned}
G(k^{in}) & =& 1-\sum_{k_i^{in} < k} P(k_i^{in}) \\ \nonumber
G(k^{out}) &= & 1-\sum_{k_i^{out} < k} P(k_i^{out}),\end{aligned}$$
respectively, are ploted in figure \[degree\_pdf\_bp\]. The cumulative distributions are used because they exhibit smaller statistical fluctuations than those observed for the degree distribution $P(k)$. In all the tested scenarios, these cumulative degree distributions are fitted by power-laws truncated by stretched exponentials $G(k) \sim k^{-\alpha} \exp(-\eta k^\lambda)$ (Weibull-like distributions). Again, $\langle k^2 \rangle / \langle k \rangle \sim \langle k \rangle$ is fulfilled, indicating the homogeneous nature of such networks. The values of this ratio and the exponents characterizing the degree distributions for SIE and BP dynamics are listed in the Appendix.
![Complementary cumulative degree distribution functions for in- and out-degrees in directed allelochemical networks. The top, middle, and bottom plots refer, respectively, to homogeneous, heterogeneous competition, and completely heterogeneous scenarios.[]{data-label="degree_pdf_bp"}](degree_bp.pdf){width="9cm"}
The BP dynamics generates largely diverse community structures as illustrated in figure \[BP\_nets\]. However, the hierarchical and modular character of such networks seems to be a universal trait. In order to further characterize these BP networks, the clustering coefficient, the average degree among the nearest neighbours of a node with degree $k$, the betweenness centrality and the network entropy were also determined.
![Examples of allelochemical networks generated after $l=100$ speciations for the homogeneous scenario (constant and equal competition and allelopathic parameters for all species) at weak allelopathy. The starting community structures are those shown in figure \[start\_graphs\]. The response functions $\Phi^{(3)}$ (left), $\Phi^{(4)}$ (top-right), and $\Phi^{(2)}$ (bottom-right) were used.[]{data-label="BP_nets"}](graphs_bp.pdf){width="9cm"}
The clustering coefficient $Cc$ measures the average probability that two nodes linked to another node are themselves linked to each other. In effect $Cc$ quantifies the density of triangles in a network [@Barabasi]. In the same way, we can define the local clustering coefficient $Cc_i$ for every node $i$, and a directed network has two such coefficients defined as
$$\begin{aligned}
Cc_i^{in} & = & \frac{Tr(A^T A^2)}{k_i^{in}(k_i^{in}-1)} \\
Cc_i^{out} & = & \frac{Tr(A^2 A^T)}{k_i^{out}(k_i^{out}-1)},\end{aligned}$$
where $A$ is the network adjacency matrix. Figure \[cluster\_coef\] shows the average local clustering as a function of the in- and out-degree. At weak allelopathic suppression (top and middle) and small responses, our results reveal that both $Cc^{in}(k)$ and $Cc^{out}(k)$ are constant for small degrees, but exhibit an exponential cut off for large degrees. However, despite the large fluctuations observed in $Cc_i^{in}$, the curves for the stronger response functions, $\Phi^{(3)}$ at the weak and $\Phi^{(1,3,5)}$ at the strong allelopathic suppression, suggest a power law scaling for large degree $k$. In turn, for $Cc^{out}$ this scaling is more neat. The exponents $b$ characterizing the power laws $Cc(k) \sim k^{-b}$ are close to one, a signature of modular structures with hierarchical organization [@Ravasz]. Since the behavior of Cc is associated to the dynamical mechanisms controlling which new attached node survives or extinguishes, this result indicates that allelochemical networks grow primarily by adding nodes with few links.
![Average local clustering coefficient as a function of in- and out-degree. The top, middle, and bottom plots correspond to the homogeneous, heterogeneous competition, and completely heterogeneous scenarios. The initial networks were all those graphs shown in figure (\[start\_graphs\]).[]{data-label="cluster_coef"}](clustering_bp.pdf){width="9cm"}
The average degree $K_{nn}$ among the nearest neighbours of a node with degree $k$ measures the mixing by degree properties of networks. $K_{nn}$ quantifies if there is a tendency of nodes with high degree to connect to others with high degree, and similarly for low degree. If this is the case, the network is assortative; if not, i. e., nodes with high degree tend to connect to others with low degree, the network is disassortative. In figure \[nearest\_neighbours\], we see that $K_{nn}$ decays exponentially for $k \gtrsim 25$ at weak and $k \gtrsim 10$ at strong allelopathic suppression, in a clear disassortative behavior. Such a result is consistent with the observation that assortative mixing by degree makes a network more unstable [@Brede].
![Average nearest neighbours degree $K_{nn}(k)$ of a node with total degree $k$ for the homogeneous (top), heterogeneous competition (middle), and completely heterogeneous (bottom) scenarios.[]{data-label="nearest_neighbours"}](nearest_bp.pdf){width="9cm"}
The betweenness centrality measures the extent to which a node lies on paths of minimal length connecting to other nodes [@Barabasi]. Nodes with high betweenness centrality often have significant influence on the network dynamics. Mathematically, the betweenness centrality $x_i$ of a node $i$ is defined as
$$x_i= \sum_{j,k} n^i_{jk},$$
where $n^i_{jk}=1$ if the node $i$ lies on the path of minimal length from node $j$ to node $k$ and $n^i_{jk}=0$ if $i$ does not or if there is no such path. In figure \[centralityBP\], the average $\langle x_i \rangle$ is plotted for every node $i$ present at the stationary allelochemical network after $l=100$ speciation events. It can be noticed that $\langle x_i \rangle$ decreases dramatically as the strength of allelopathic suppression increase. Indeed, even at weak suppression ($\mu=0.1$, figure\[centralityBP\](a)), strong responses to toxins ($\Phi^{(3,4)}$ lead to small or almost null average $\langle x_i \rangle$. Furthermore, the hierarchical and modular character of BP networks at weak allelopathy shown in figure \[BP\_nets\] is reflected on the peaks for small $k$ and the small fluctuations around a constant value of $\langle x_i \rangle$ for large $k$. In turn, a $\langle x_i \rangle =0$ for the heterogeneous competition and allelopathy is a consequence of very small and sparsely connected network structures.
![Average betweenness centrality for each node (surviving species) in communities generated from the initial graphs shown in figure \[start\_graphs\]. (a) homogeneous, (b) competition heterogeneous, and (c) completely heterogeneous scenarios.[]{data-label="centralityBP"}](centrality_bp.pdf){width="9cm"}
Finally, the allelochemical network entropy was determined. Specifically, the degree distribution entropy $H$ for networks with size $n$, defined as
$$H[P(k)]=-\sum_{k=k_{min}}^{k_{max}}P(k)\log_2(P(k)),$$
was calculated. This Shannon entropy [@Shannon] is larger more homogeneous is the degree distribution and communities with greater diversities tend to be more homogeneous. As illustrated in figure \[entropy\], the entropy decreases as the allelochemical suppression increases. Particularly, $H[P(k^{out})]$ is more affected. Also, peaks at the initial community structures $G_2$, $G_6$, and $G_{11}$ are neatly observed in $H[P(k^{in})]$ for almost all response functions in the heterogeneous regimes. Peaks in the entropy $H[P(k^{out})]$ are less evident and restricted to few response functions, e. g., in $G_9$ for $\Phi^{(4)}$ and $G_4$ for $\Phi^{(6)}$ at the homogeneous and heterogeneous competition scenarios, respectively.
![In- and out-degree distribution entropies for networks generated after $l=100$ speciation events starting from each initial community structures shown in figure \[start\_graphs\]. The upper, middle, and bottom plots correspond to the homogeneous, heterogeneous competition, and completely heterogeneous scenarios, respectively. Averages were taken over $200$ independent histories.[]{data-label="entropy"}](EntropyBP.pdf){width="9cm"}
Discussion
==========
We have proposed and studied, through numerical methods, an eco-evolutionary model for community assembly involving two coupled processes. The first is a fast ecological dynamics in which species compete for common resources and suppress each other allelopathically. The second are slow evolutionary events in which new species are added to the biological community at its ecological stationary state. Clearly, our study address a basic question: the relation between stability and complexity in the ecology of many interacting species.
All the results obtained here must be analysed bearing in mind the scenario for pure intra- and interspecific competition. In the coexistence regime (weak competition, $\nu_{ij} < 1 \, \forall \, i,j$), all the surviving species at every stationary state constitute fully connected competition networks as assumed in our models. Since some introduced and/or resident species are eventually extinct, the community diversity tends to be smaller than the number of invasion or speciation events. Yet, communities with high diversity are the rule. This scenario changes if allelopathic interactions exist.
In the SIE dynamics, ecological networks grow through a succession of species imigration. These alien species allelochemically suppress and are suppressed by resident species at random, eventually leading to the eradication of either the invader or some resident species. Our results, shown in figures \[abundance\_sie\] and \[connect\_sie\], reveal that communities exhibiting large diversities can be assembled at weak allelopathy, but diversities and average connectivities of stationary networks are drastically reduced at strong allelopathy for all response functions. Furthermore, in the strong suppression regime, species richness either saturates or decreases slowly after reaches a maximum. The maxima occurs after $\sim 10-30$ invasion events, depending on the response function to toxins (see figure \[disorder\_sie\] also). At the maxima, the average number of species in the communities never exceeds $16-18$. So, the system of interacting species becomes unstable and the networks stop to grow, consistent with the limit found by May [@May]. Beyond these upper bounds, the number of surviving species decreases continuously after each SIE until rest only one (a successful invasion) or very few species. Accordingly, network topologies evolve towards marked hierarchical structures, as seen in figure \[SIE\_nets\], and the corresponding connectivity distributions change from normal (or Gaussian) to Weibull distributions (figure\[degree\_pdf\_sie\]). Such networks, a subset of almost null measure in a random ensemble, can only be generated through a constrained growth process.
The fundamental distinction between the SIE and BP dynamics is that new species are attached to the community with random or correlated connective patterns.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was partially supported by the Brazilian Agencies CAPES (Carvalho graduate fellowship), CNPq (306024/2013-6 and 400412/2014-4), and FAPEMIG (APQ-04232-10 and APQ-02710-14).
F R $\frac{<k_{In}^{2}>}{<k_{In}>}$ $\frac{<k_{out}^{2}>}{<k_{out}>}$ $\frac{<k_{In}^{2}>}{<k_{In}>}$ $\frac{<k_{out}^{2}>}{<k_{out}>}$ $\frac{<k_{In}^{2}>}{<k_{In}>}$ $\frac{<k_{out}^{2}>}{<k_{out}>}$
-------------- --------------------------------- ----------------------------------- --------------------------------- ----------------------------------- --------------------------------- -----------------------------------
$\Phi^{(1)}$ 44.2099 45.099 4.4801 47.5051 4.5526 54.7007
$\Phi^{(2)}$ 45.9834 45.5727 4.4781 47.8501 4.5162 48.7940
$\Phi^{(3)}$ 26.7295 47.8898 4.6144 49.3415 4.4764 54.8799
$\Phi^{(4)}$ 31.7409 45.643 4.4547 47.7538 4.5486 50.6062
$\Phi^{(5)}$ 46.2172 45.9558 4.1965 47.5544 4.6187 54.5946
$\Phi^{(6)}$ 46.1179 45.5262 4.4403 47.1904 4.5349 49.1369
: My caption[]{data-label="my-label"}
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---
abstract: 'In this short paper we follow the entropic gravity approach and demonstrate how $f(R)$ theories of gravity can be emergent. This is done by introducing an effective gravitational constant which is naturally arising from the $f(R)$’s equations of motion.'
author:
- Ali Teimouri
title: 'Entropic $f(R)$ Gravity'
---
\[sec:level1\]Introduction
==========================
Gravitation is a universal force which interacts with all particles that carry an energy and thus there is a link between gravity and thermodynamics. The understanding of this relation is matured in the past decades by studying the black holes’ thermodynamics. For instance, Jacobson [@Jacobson:1995ab] derived the Einstein’s equations from the thermodynamics of the near horizon. Another important advancement in understanding the relation between thermodynamics and gravity was achieved by studying the black holes’ entropy, where Hawking and Bekenstein [@refref] have shown that this entropy is proportional to the area of the event horizon. More recently, and inspired by the area law, the holographic principle [@Susskind:1994vu] was realised in the context of $AdS/CFT$ correspondence [@Maldacena:1997re].
Inspired by these developments, there is a new conjecture that sees gravity not as a fundamental force but as an emerging phenomenon. Examples of this approach can be found in [@Verlinde:2010hp; @Padmanabhan:2009kr]. Particularly, in [@Verlinde:2010hp], the gravity is thought to be an entropic force, where the gravitational formulation can be derived by obtaining a relation between temperature and acceleration, and then using the holographic principle and an equipartition rule relating the energy to the temperature and the number of degrees of freedom.
Gravity, as we know it so far from the Einstein’s theory of general relativity, is fairly successful in predicting the natural phenomena. One of the most recent phenomenon was the prediction of the gravitational waves. However, the limits of general relativity brought the alternative theories of gravity to existence. Many of these alternative theories are essentially wide range of modifications to the original general relativity. From these modifications, one can mention the $f(R)$ gravity [@Sotiriou:2008rp], which generalises the Einstein’s theory of general relativity and was used most famously by Starobinsky [@Starobinsky:1980te], to describe the cosmic inflation. There are also many other types of modified theories of gravity [@Clifton:2011jh], each constructed to explain different phenomena when the general relativity is not able to provide an appropriate description. For instance, one can recall Lovelock, Gauss-Bonnet, higher derivative gravity [@Biswas:2011ar] and so on.
Looking at the gravity as an emergent phenomenon, and deriving the Einstein’s theory of general relativity from the basic thermodynamics, raises a question about the modified theories of gravity. Is it possible to derive the modified theories of gravity from the basic laws of thermodynamics? How can the modified theories of gravity be emerged from the basic laws and how do these theories reflect themselves in those basic laws? In this paper we wish to show how the $f(R)$ gravity can be obtained from the basic principles. We are going to do so by implementing Verlinde’s approach in [@Verlinde:2010hp].
We start by giving a brief review of the entropic gravity and we then move onto the emergence of the $f(R)$ gravity.
\[sec:level1\]Entropic Gravity
==============================
The existence of the area law in general relativity implies that the space-time is nothing but a perfect storage for information and that this information can be read off from the boundary which is defined by the area law. The information stored on the area is maximal and thus finite. This is to satisfy the Bekenstein’s entropy bound [@Bekenstein:1980jp]. In similar analogy, it can be assumed that the information is stored on the so called *screens*. Screens separate points and therefore one can locate the stored particles in discrete bits on the screen.
Verlinde, [@Verlinde:2010hp], argued that it is possible to use the second law of thermodynamics and derive the Einstein’s theory of general relativity from the first principle. This had been done by considering a holographic screen on closed surface of constant redshift. The assumption is that there is a associated mass configuration to the screen with total mass $M$. The bit density on the screen is then simply: $$\label{bitden}
dN=\frac{dA}{G\hbar},$$ where $N$ is a number of bits, $A=4\pi r^{2}$ is the surface area, $G$ is the gravitational constant and $\hbar$ is the Planck’s constant. Given that the total energy of the system is denoted by $E$, the temperature can be determined by the equipartition principle as, $$\label{energy1}
E=\frac{1}{2}Nk_B T,$$ where $k_B $ is the Boltzman’s constant. We can use the mass-energy equivalence and drop out the constants appropriately, and thus determine the mass that each bit carries by simply integrating the mass, which is: $$M=\frac{1}{2}\int_{\sigma}T dN=\frac{1}{4\pi}\int_{\sigma}\nabla\varphi dA.$$ The above equation is known as Gauss’ law for gravity and can be re-expressed in terms of the Komar mass which is sufficient to derive the Einstein’s equations.
$f(R)$ Gravity Emergence
========================
$f(R)$ gravity is the generalisation of the Einstein’s theory of general relativity. The action can be written in the form of [@Sotiriou:2008rp]: $$S=\frac{1}{16\pi G}\int dx^{4}\sqrt{-g}f(R)+S_{M}(g_{\mu\nu},\psi),$$ where $f(R)$ is the function of scalar curvature, $R$, $S_{M}$ denotes the matter term with $\psi$ being the matter fields. The variation of the action with respect to the metric gives: $$\label{eom}
f'(R)R_{\mu\nu}-\frac{1}{2}f(R)g_{\mu\nu}-[\nabla_{\mu}\nabla_{\nu}-g_{\mu\nu}\Box]f'(R)=8\pi
G T_{\mu\nu}.$$ We can re-write above as: $$G_{\mu\nu}=\frac{8\pi G}{f'(R)}\Big(T_{\mu\nu}+T^{(eff)}_{\mu\nu}\Big),$$ where we introduced the effective gravitational coupling strength as, $$G_{eff}\equiv\frac{G}{f'(R)}.$$ Introducing the effective gravitational constant $G_{eff}$ is equivalent to the requirement that the graviton is not a ghost. Moreover, $T^{(eff)}_{\mu\nu}$ is called the effective stress-energy tensor and can be easily read off from Eq. (\[eom\]).
Newtonian Limit
---------------
In this setup, let us consider the weak-field approximation. The metric describing the gravitational field of a static distribution of matter is given by, $$\label{met}
ds^2=-(1+2\varphi)dt^{2}+(1-2\varphi)d\bar x^{2},$$ where $d\bar x^{2}=dx^{2}+dy^{2}+dz^{2}$ and $\varphi(r)$ is the Newtonian potential and it is the function of distance $r$. In asymptotically flat space-time, the weak-field expression given above, can be used to approximate the metric in the asymptotic domain. At far distance from the static gravitating object we have [@frolov], $$\varphi(r)=-\frac{G_{eff}M}{r}.$$ Thus, the free-fall acceleration can be obtained by taking the gradient of the Newtonian potential: $$a^{i}=-\nabla\varphi(r)=-\frac{G_{eff}M}{r^{2}}n^{i},$$ where $n^{i}$ is a unit vector of the external normal to a 2D sphere $\sigma$ of radius $r$. The mass of the object can then be found via: $$\label{newtonianmass}
M=\frac{1}{4\pi G_{eff}}\int_{\sigma}a^{i}n_{i}d^{2}\sigma.$$ Again, this is the familiar Gauss law for gravity.\
Komar mass
----------
It is possible to write Eq. (\[newtonianmass\]) in terms of the Komar mass by identifying the Killing vector associated to the metric in Eq. (\[met\]), [@komar], $$M=\frac{1}{4\pi G_{eff}}\int_{\sigma}\nabla^{\nu}\xi^{\mu}d\sigma_{\mu\nu},
\quad\text{with:}\quad d\sigma_{\mu\nu}=n_{[\mu}u_{\nu]}d^{2}\sigma.$$ In the above definition of mass, $\xi^{\mu}$ is the Killing vector field of a static space-time. Moreover, $n_{\alpha}$ and $u_{\alpha}$ are the time-like and space-like normals to $\sigma$. By using the cyclic identity for Riemann tensor, and the fact that all Killing vectors must satisfy $$\Box\xi^{\alpha}=-R^{\alpha}_{\ \beta}\xi^{\beta},$$ and also by using the Stokes’ theorem, one has: $$M=\frac{1}{4\pi G_{eff}}\int_{\partial\Sigma}\nabla^{\nu}\xi^{\mu}d\sigma_{\mu\nu}=-\frac{1}{4\pi
G_{eff}}\int_{\Sigma}R^{\mu}_{\ \nu}\xi^{\nu}d\sigma_{\mu}.$$ Here, $\Sigma$ is a 3-dimensional volume bounded by holographic boundary $\partial\Sigma$. We shall note that $d\sigma_{\mu}$ is proportional to the normal to $\Sigma$. It can be clearly seen that upon expanding the effective gravitational constant and taking the example of Einstein-Hilbert action, which is $f(R)=R$, one recovers the results for general relativity.
Emergence of gravity
--------------------
As we saw from Eq. (\[bitden\]), in order to derive gravity from entropy one has to start with the bit density on the holographic screen. In the example of the $f(R)$ gravity this shall be modified to, $$dN=\frac{dA}{G_{eff}\hbar}.$$ Again, the boundary can be thought as a surface where the information is stored. Upon satisfying the holographic principle, the maximal storage space (*i.e.* the total number of bits) is proportional to the area, $A$. It is now possible to repeat the same procedure to essentially find the total energy related to the number of bits and the temperature as in Eq. (\[energy1\]) and then find the associated mass in terms of Komar integral and satisfy the equations of motion.
Summary
=======
In this paper, we have shown that one can obtain the $f(R)$ theories of gravity by using the entropic analogy of gravity. This requires introducing an effective gravitational constant in the Newtonian potential. This effective gravitational constant comes immediately from the equations of motion. This explains how the modification of the original theory of general relativity affected the Newtonian potential.
It is clear that the same approach can be employed to derive other theories of modified gravity. This is due to the fact that it is possible to generalise the Komar integral for higher order terms. However, the quest to recognise the most appropriate theory of gravity remains open to study.
Acknowledgement {#acknowledgement .unnumbered}
===============
The author would like to thank Spyridon Talaganis for fruitful discussions and comments.
[99]{} T. Jacobson, Phys. Rev. Lett. [**75**]{}, 1260 (1995) doi:10.1103/PhysRevLett.75.1260 \[gr-qc/9504004\]. J. M. Bardeen, B. Carter and S. W. Hawking, “The Four laws of black hole mechanics,” Commun. Math. Phys. 31, 161 (1973). J. D. Bekenstein, “Extraction of energy and charge from a black hole,” Phys. Rev. D 7, 949 (1973). J. D. Bekenstein, “Black holes and entropy,” Phys. Rev. D 7, 2333 (1973). S. W. Hawking, “Particle Creation By Black Holes,” Commun. Math. Phys. 43, 199 (1975) \[Erratum-ibid. 46, 206 (1976)\].
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---
author:
- |
Pengyu Wang$^1$ Phil Blunsom$^{1,2}$\
$^1$Department of Computer Science, University of Oxford\
$^2$Google DeepMind\
`{pengyu.wang,phil.blunsom}@cs.ox.ac.uk`
title: Stochastic Collapsed Variational Inference for Hidden Markov Models
---
Introduction
============
Hidden Markov models (HMMs) [@rabiner90hmm] are popular probabilistic models for modelling sequential data in a variety of fields including natural language processing, speech recognition, weather forecasting, financial prediction and bioinformatics. However, their traditional inference methods such as variational inference (VI) [@beal03] and Markov chain Monte Carlo (MCMC) [@scott02] are not readily scalable to large datasets. For example, one dataset in our experiment consists of $100$ million observations.
An important milestone for scaling VI was made by Hoffman et al. [@hoffman13], who proposed stochastic VI (SVI) that computes cheap gradients based on minibatches of data, updating the model parameters before a complete pass of the full dataset. A recent scalable and more accurate algorithm was proposed by Foulds et al. [@foulds13], who applied such stochastic optimization to the collapsed latent Dirichlet allocation (LDA) [@teh07collapsed], and their stochastic collapsed variational inference (SCVI) algorithm has been successful in large scale topic modelling.
However, while these recent advances have been studied extensively for topic models that assume a simple bag-of-words data setting [@hoffman13; @teh07collapsed; @hoffman10; @wang12; @bryant12], there has been little research on whether and how we can apply them in a time dependent data setting. Some research such as SVI for Bayesian time series models [@johnson14] and collapsed VI (CVI) for HMMs [@wangandblunsom13] consider the settings where datasets consist of many independent time series, naturally avoiding to break the sequential dependencies. Perhaps the only true exception is the SVI algorithm for HMMs proposed by Foti et al. [@foti14] in the setting of a single long time series, where the sequential dependencies must be broken.
In this paper, we follow the success of SCVI for LDA [@foulds13] and study the SCVI algorithm applied to a single long time series. In a collapsed HMM, we break a long chain into subchains, and we propose a novel sum-product algorithm to update the posteriors of subchains, taking into account their edge transitions due to the sequential dependencies. Our sum product algorithm can be understood as an alternative buffering method to the one in [@foti14]. Our experiments on two discrete datasets show that our SCVI algorithm for HMMs is scalable to very large datasets, memory efficient and significantly more accurate than the existing SVI algorithm.
Background
==========
A hidden Markov model (HMM) [@rabiner90hmm] consists of a hidden state sequence $\textbf{z} = \{z_t\}_{t=0}^T$ and a corresponding observation sequence $\textbf{x} = \{x_t\}_{t=1}^T$. Let there be $K$ hidden states. For convenience, we let the start state be $0$ and set $z_0 = 0$. Let $\boldsymbol\theta$ be the transition matrix where $\theta_{k,k'}=p(z_t=k'|z_{t-1}=k)$, and $\theta_0$ be the initial state distribution where $\theta_{0,k'}=p(z_1=k')$. For $k=0,...,K$, we specify the Dirichlet priors with symmetric hyperparameters $\alpha$ on $\theta_k$, $\theta_k|\alpha \sim \text{Dir}(\alpha)$ in a Bayesian setting.
A hidden sequence is generated by a Markov process, and each observation is generated conditioned on its hidden state. We have for $t=1,...,T$, $$\begin{aligned}
& z_t|z_{t-1}=k \sim \text{Mult}(\theta_{k}) & x_t|z_t=k' \sim p(\cdot|\phi_{k'}), &\end{aligned}$$ where $\phi_{k'}$ parametrizes the observation likelihood for the hidden state $k'$, with $\phi_{k',w}=p(x_t=w|z_t=k')$. Without loss of generality, we assume that the observation likelihoods and their conjugate prior take exponential forms. The exponential family is a broad class of probability distributions including multinomial, Gaussian, gamma, Poisson, Dirichlet, Wishart and many others; and there is a conjugate prior distribution for each member in this class. We have for $k'=1,...,K$, $$\begin{aligned}
p(w|\phi_{k'}) &= h_l(w) \exp \{ \phi_{k'}^T t(w) - a_l(\phi_{k'}) \} \\
p(\phi_{k'}|\lambda^\circ) &= h_g(\phi_{k'}) \exp \{ (\lambda_1^\circ)^T \phi_{k'} + (\lambda_2^\circ)^T (-a_l(\phi_{k'})) - a_g(\lambda^\circ) \} .\end{aligned}$$
The base measure $h$ and log normalizer $a$ are scalar functions; and the parameter $\phi_{k'}$ and sufficient statistics $t$ are vector functions. The subscripts $l$ and $g$ represent the local hidden variables and global model parameters, respectively. The dimensionality of the prior hyperparameter $\lambda^\circ = (\lambda_1^\circ, \lambda_2^\circ)$ is equal to $\text{dim}(\phi_{k'})+1$.
Stochastic Collapsed Variational Inference
==========================================
There is substantial empirical evidence [@foulds13; @wangandblunsom13; @asuncion09] that marginalizing the model parameters is helpful for both accurate and efficient inference. Thus we integrate out the model parameters $(\boldsymbol\theta,\boldsymbol\phi)$ and the marginal data likelihood of an HMM is: $$\begin{aligned}
p(\textbf{x},\textbf{z}) = \prod_{k=0}^K \textstyle \frac{\Gamma(K\alpha)}{\Gamma(K\alpha+C_{k\cdot})} \prod_{k'=1}^K \frac{\Gamma(\alpha + C_{kk'})}{\Gamma(\alpha)}
\displaystyle \prod_{t=1}^T \textstyle h_l(x_t) \displaystyle \prod_{k'=1}^K \textstyle \exp \{ a_g(\lambda^{k'}) - \{ a_g(\lambda^\circ)\}.
\label{collapsedhdphmm}\end{aligned}$$
The gamma functions and log normalizers result from the marginalization. $C_{kk'}$ denotes the transition count from the hidden state $k$ to $k'$, $C_{kk'} = \# \{t: z_{t-1}=k,z_t={k'}\}$. dot denotes the summed out column, e.g., $C_{\cdot k'}= \sum_{k} C_{kk'}$. $\lambda^{k'}$ denotes the posterior hyperparameter for the hidden state $k'$, $\lambda^{k'}_1 = \lambda^\circ_1 + \sum_{t=1}^T t(x_t)\delta(z_t=k')$ and $\lambda^{k'}_2 =\lambda^\circ_2 + C_{\cdot k'}$, where $\delta$ is the standard delta function.
Given an observed sequence $\textbf{x}$, the task of Bayesian inference in the collapsed space is to compute the posterior distributions over the hidden sequence, $p(\textbf{z}|\textbf{x})$. The posteriors over the model parameters can be estimated by taking a variational Bayesian maximization step with our estimated $q(\textbf{z})$ [@beal03]. As the exact computation is intractable, we introduce a variational distribution $q(\textbf{z})$ in a tractable family and we maximize the evidence lower bound (ELBO) denoted by $\mathcal{L}(q)$, $$\begin{aligned}
\log p(\textbf{x}) \geq \mathbb{E}[\log p(\textbf{x},\textbf{z})] - \mathbb{E}[\log q(\textbf{z}) ] \triangleq \mathcal{L}(q).\end{aligned}$$
We consider the tractable family under the generalized mean field assumption [@xing02] in the collapsed space: we break a single long hidden sequence into a set of subchains. We have $q(\textbf{z}) = \prod_{n=1}^N q(\textbf{z}^n)$. We do not make any further assumptions about the inner structure of each subchain, preserving the inner transition information. It might be worth emphasizing that the time series dependencies in an HMM model are not broken; only the variational posterior is factorized. Therefore, the information can still flow across different subchains via edge transitions.
For notational simplicity, we let each subchain be of the length $L$ and $N=\left \lfloor {T/L}\right \rfloor$ be the number of subchains given a long chain. For each hidden subchain $\textbf{z}^n=\{z^n_l\}_{l=1}^L$, we denote the corresponding observed subchain by $\textbf{x}^n=\{x^n_l\}_{l=1}^L$. Combining the work of SCVI for LDA [@foulds13] and CVI for HMM [@wangandblunsom13], we uniformly sample an observation subchain $\textbf{x}^n$, and we derive the posterior update for $q(\textbf{z}^n)$ with a zeroth order Taylor approximation [@teh07collapsed], $$\begin{aligned}
q(\textbf{z}^n) &\approx \propto \hat{\theta}_{\cdot,z^n_1} \bigg(\prod_{l=2}^L \hat{\theta}_{z^n_{l-1},z^n_l} \bigg) \hat{\theta}_{z^n_{L},\cdot} \bigg( \prod_{l=1}^L \hat{\phi}_{z^n_l,x^n_l} \bigg) \label{scvihmm1-1} \\
\hat{\theta}_{\cdot,z^n_1} &\propto \sum_{z^n_0} q(z^n_0) \bigg(\mathbb{E}[C_{z^n_0,z^n_1}]+ \frac{\alpha}{Kq(z^n_0)} \bigg) \label{scvihmm1-2} \\
\hat{\theta}_{z^n_{l-1},z^n_l} &\propto \mathbb{E}[C_{z^n_{l-1}z^n_l}] + \alpha \label{scvihmm1-3} \\
\hat{\theta}_{z^n_L,\cdot} &\propto \sum_{z^n_{L+1}} \bigg(\frac{\mathbb{E}[C_{z^n_L,z^n_{L+1}}]+\frac{\alpha}{Kq(z^n_{L+1})}}{\mathbb{E}[C_{z^n_L,\cdot}]+K\alpha}\bigg) q(z^n_{L+1}) \label{scvihmm1-4} \\
\hat{\phi}_{z^n_l,x^n_l} &\propto h(x^n_l) \exp \{ a_g( \lambda^\circ_1+t(x^n_l)+\mathbb{E}[t_{z^n_l}(\textbf{x},\textbf{z})], \lambda^\circ_2+1+\mathbb{E}[C_{\cdot z^n_l}]) \},
\label{scvihmm1-5}\end{aligned}$$ where $\mathbb{E}[C_{kk'}]=\sum_{t=1}^T q(z_{t-1},z_t=k,k')$ denotes the global expected transition count from state $k$ to $k'$, and $\mathbb{E}[t_{k'}(\textbf{x},\textbf{z})] = \sum_{t=1}^T q(z_t=k')t(x_t)$ denotes the global emission statistics at hidden state $k'$. Unlike CVI for HMM [@wangandblunsom13], we do not need to maintain local statistics and thus our algorithm is memory efficient. We show the algorithmic procedure to infer $q(\textbf{z}^n)$ in Section 3.1.
Given $q(\textbf{z}^n)$, we can collect the local transition counts $\mathbb{E}[C^n_{kk'}]$ and emission statistics $\mathbb{E}[t_{k'}(\textbf{x}^n,\textbf{z}^n)]$ and update the global statistics with an online average weighted by a step size $\rho_n$, $$\begin{aligned}
\mathbb{E}[C_{kk'}] &= (1-\rho_n) \mathbb{E}[C_{kk'}] + \rho_n T/(L-1) \mathbb{E}[C^n_{kk'}] \label{scvihmm2-1}\\
\mathbb{E}[t_{k'}(\textbf{x},\textbf{z})] &= (1-\rho_n) \mathbb{E}[t_{k'}(\textbf{x},\textbf{z})] + \rho_n N \mathbb{E}[t_{k'}(\textbf{x}^n,\textbf{z}^n)].
\label{scvihmm2-2}\end{aligned}$$
Modified Forward Backward Algorithm
-----------------------------------
Given a subchain $\textbf{z}^n=\{z^n_l\}_{l=1}^L$, we denote the hidden variable before it $z^n_0$ and the hidden variable after it $z^n_{L+1}$ to be the guarding variables; and we denote $\hat{\theta}_{\cdot,z^n_1}$ and $\hat{\theta}_{z^n_{L},\cdot}$ to be the edge transitions. In (\[scvihmm1-1\]), the edge transitions prevent us from applying the standard forward backward algorithm [@baum66] to the HMM parametrized by the surrogate parameters $\hat{\theta}$ and $\hat{\phi}$. Therefore, we propose a modified sum-product algorithm to buffer subchain edges with guarding variables. We start by defining a joint distribution of a subchain and its guarding variables using a factor graph shown in figure \[figure1\], $$\begin{aligned}
q(\textbf{z}^{(n)},z^n_0,z^n_{L+1}) &\propto f^n_0(z^n_0) \bigg(\prod_{l=1}^{L+1} f^n_l(z^n_{l-1},z^n_l)\bigg) f^n_{L+2}(z^n_{L+1}) \label{sumproduct-1} \\
f^n_0(z^n_0) &\triangleq q(z^n_0) \label{sumproduct-2} \\
f^n_1(z^n_0,z^n_1) &\triangleq \bigg(\mathbb{E}[C_{z^n_0,z^n_1}]+ \frac{\alpha}{Kq(z^n_0)}\bigg) \hat{\phi}_{z^n_1,x^n_1} \\
f^n_l(z^n_{l-1},z^n_l) &\triangleq \hat{\theta}_{z^n_{l-1},z^n_l} \hat{\phi}_{z^n_l,x^n_l} \quad\quad\quad \text{for } l = 2,...,L \\
f^n_{L+1}(z^n_L,z^n_{L+1}) &\triangleq \frac{\mathbb{E}[C_{z^n_L,z^n_{L+1}}]+\frac{\alpha}{Kq(z^n_{L+1})}}{\mathbb{E}[C_{z^n_L,\cdot}]+K\alpha} \\
f^n_{L+2}(z^n_{L+1}) &\triangleq q(z^n_{L+1}) \label{sumproduct-6}.\end{aligned}$$
The functions associated with each factor node $\{f^n_l\}_{l=0}^{L+2}$ are given in (\[sumproduct-2\]-\[sumproduct-6\]). It is easy to verify that summing over the guarding variables of the joint probability in (\[sumproduct-1\]) reduces to $q(\textbf{z}^n)$ in (\[scvihmm1-1\]). Now we can use the sum product algorithm [@Kschischang06] to compute the required marginals of $q(\textbf{z}^n)$. Specifically, we first pick $f_{L+2}$ as the root node and pass the messages from the leaf node $f_0$, and then we pass messages in a reverse direction[^1] . We have, $$\begin{aligned}
u_{f^n_l \rightarrow z^n_l}(z^n_l) &= \sum_{z^n_{l-1}} u_{f^n_{l-1} \rightarrow z^n_{l-1}}(z^n_{l-1}) f^n_l(z^n_{l-1},z^n_l) \quad \text{ for } l=1,...,L+1 \\
u_{f^n_{l+1}\rightarrow z^n_l}(z^n_l) &= \sum_{z^n_{l+1}} u_{f^n_{l+2}\rightarrow z^n_{l+1}}(z^n_{l+1}) f^n_{l+1}(z^n_l,z^n_{l+1}) \quad \text{ for } l=L,...,0,\end{aligned}$$ where the initial messages are simply the distributions of the two guarding variables, $$\begin{aligned}
& u_{f^n_0 \rightarrow z^n_0}(z^n_0) = f^n_0(z^n_0) & u_{f^n_{L+2} \rightarrow z^n_{L+1}}(z^n_{L+1}) = f^n_{L+2}(z^n_{L+1}). & \label{initial}\end{aligned}$$
After the messages have been passed in both directions, we compute the required variable marginals $q(z^n_l)$ and pairwise marginals $q(z^n_{l-1},z^n_{l})$ by, $$\begin{aligned}
q(z^n_l) &\propto u_{f^n_l \rightarrow z^n_l}(z^n_l) u_{f^n_{l+1} \rightarrow z^n_l}(z^n_l) \\
q(z^n_{l-1},z^n_l) &\propto f^n_l(z^n_{l-1},z^n_l) u_{f^n_{l-2} \rightarrow z^n_{l-1}}(z^n_{l-1}) u_{f^n_{l+1} \rightarrow z^n_l}(z^n_l)\end{aligned}$$
The normalization constant can be obtained by normalizing any of these marginals. This completes our algorithm to infer $q(\textbf{z}^n)$.
Our modified sum product algorithm is an alternative buffering method to the one proposed by Foti et al. [@foti14] in their SVI algorithm for a single long time series. A key difference is that we assume the independent subchains and we allow messages to be passed across the boarders via local beliefs of the guarding variables in (\[initial\]), whereas the subchains in the SVI algorithm are naturally correlated. However, the price for preserving the correlation is that they assume the hidden chain is irreducible and aperiodic so that each subchain starts with the initial distribution equal to the stationary distribution of the whole chain. A second superficial difference is that we buffer a subchain by only two guarding variables, whereas Foti et al. buffered a subchain with more observations.
Experiments
===========
We evaluated the utility of our buffering method and compared the performances of our SCVI algorithm against the SVI algorithm on two synthetic datasets created from the Wall Street Journal (WSJ) and New York Times (NYT). Both corpora are made of sentences, which in turn are sequences of words. For each sentence, the underlying sequence can be understood as a Markov chain of hidden part-of-speech (PoS) tags [@jurafsky00] and words are drawn conditioned on PoS tags, making HMMs natural models. We shuffled both datasets, added special symbols after each sentence to denote the ends and concatenated them. We used the first $1$ million words in the concatenated WSJ and $100$ million words in the concatenated NYT as our two long time series, respectively. As the evaluation metrics, we used predictive log likelihoods by holding out $5\%$ words of each time series as testing sets.
For both the SVI and our SCVI algorithms: we set the transition and emission priors to be $\text{Dir}(0.1)$; we initialized the global statistics using exponential distributions suggested by Hoffman et al. [@hoffman13]; we set $K=12$ assuming a universal PoS tag set [@petrov12]; when buffering was turned off, we set the initial distribution to start a subchain to be the whole chain’s stationary distribution. For SVI, when buffering was turned on, we buffered a subchain with $20$ words on both sides. We varied the subchain lengths, $L=2,3,10$ and used minibatches of subchains to reduce the sampling variance. Following Foti et al. [@foti14], we fixed the total length of all subchains in a minibatch $L \times M = 1000$, where $M$ is the minibatch size. Increasing $L$ means decreasing $M$ and vice-versa. Also, we varied the forgetting rates $\kappa=0.5,0.9$, which parametrize the step sizes $\rho_n = (1+n)^{-\kappa}$. Under each of the combined settings, we ran both algorithms for $5000$ iterations.
Figure \[single\_combined\] presents the predictive log likelihood results on the WSJ (left and middle) and NYT (right). We see that in most settings our SCVI algorithm outperformed the SVI algorithm by large margins, extending the success of SCVI for LDA [@foulds13] to time series data. The only exception is when $\kappa=0.9$, both algorithms performed comparably on the NYT. For SCVI, a smaller forgetting rate was preferred, which further promotes the scalability; whereas SVI was less sensitive. When $L$ is small, there are noticeable improvements using respective buffering methods in both algorithms. For SCVI, we attribute the improvement to the inter subchain communication through guarding variables.
Conclusion
==========
We have presented a stochastic collapsed variational inference algorithm for HMMs in the setting of a single long time series and an alternative buffering method that modifies the standard forward backward recursions. Our SCVI algorithm is significantly more accurate than the SVI algorithm on two large datasets, and our buffering method is robust against the poor choices of subchain lengths. For future work, we aim to derive the true nature gradients of the ELBO to prove the convergence of our algorithm [@ruiz2014], although we never saw a nonconverging case in our experiments.
[^1]: In both recursions, we have eliminated the messages of the ‘variable node to factor node’ type [@bishop06].
|
---
abstract: |
A mathematical formulation of an estimation problem of a cavity inside a three-dimensional thermoelastic body using time domain data is considered. The governing equation of the problem is given by a system of equations in the linear theory of thermoelasticity which is a coupled system of the elastic wave and heat equations. A new version of the enclosure method in the time domain which is originally developed for the classical wave equation is established. For a comparison, the results in the decoupled case are also given.
AMS: 74J25, 35Q79, 74F05, 35R30, 35L05, 35K05, 35B40
KEY WORDS: enclosure method, inverse obstacle problem, dynamic theory of thermoelasticity, displacement-temperature equation of motion, coupled heat equation, coupled system, cavity, non destructive testing.
author:
- 'Masaru IKEHATA[^1]'
title: On finding a cavity in a thermoelastic body using a single displacement measurement over a finite time interval on the surface of the body
---
R[[**R**]{}]{} \#1[**]{} \#1[(\#1)]{}
Introduction
============
The purpose of this paper is to pursue the possibility of the enclosure method [@I1; @E00] itself in inverse obstacle problems governed by several partial differential equations in [*time domain*]{}. In particular, the enclosure method using a [*single*]{} pair of the input and output over a [*finite time interval*]{} has been developed for several inverse obstacle problems whose governing equations are given by scalar wave equations, heat equations and the Maxwell system, see [@I4; @IW00; @IEO2; @IEO3; @ICA; @Iwall; @IK2; @IK5; @IMP; @IMax; @IMax2]. In [@IE4] the author added a new idea to this time domain enclosure method for finding the geometry of an unknown obstacle from a single point of the graph of the so-called [*response operator*]{} on the [*outer surface*]{} of the domain in which the obstacle is embedded. The paper is focused on the explanation of the idea and so the governing equation of the wave therein is a classical wave equation.
In this paper, we consider the new case when the governing equation is given by a coupled system of hyperbolic and parabolic equations. How does this new enclosure method work for the system?
We restrict ourself to considering a classical system in the linear theory of thermoelasticity since it is a typical coupled system having a physical meaning. It is a coupled system of the elastic wave and heat equations. We refer the reader to the article [@C] which gives us whole knowledge about the system together with several references.
Now let us formulate the problem concretely. We denote by $\Omega$ and $D$ the reference body and an unknown cavity embedded in $\Omega$, respectively. It is assumed that $\Omega\setminus\overline D$ is homogeneous and isotropic. In this paper, for simplicity and considering the results in [@D], we assume that $\Omega$ is given by a bounded domain with $C^{\infty}$-boundary and $D$ a nonempty bounded open subset of $\Omega$ with $C^{\infty}$-boundary such that $\Omega\setminus\overline D$ is connected. We use the same symbol $\mbox{\boldmath $\nu$}$ to denote both the outer unit normal vectors of $\partial D$ and $\partial\Omega$.
Let $0<T<\infty$. Given $f=f(x,t)$ and $\mbox{\boldmath $G$}=\mbox{\boldmath $G$}(x,t)$ with $(x,t)\in\partial\Omega\times\,]0,\,T[$ and let $\mbox{\boldmath $u$}=
\mbox{\boldmath $u$}_{f,\mbox{\boldmath $G$}}(x,t)$ and $\vartheta=\vartheta_{f, \mbox{\boldmath $G$}}(x,t)$ with $(x,t)\in\,(\Omega\setminus\overline D)\times\,]0,\,T[$ denote the solutions of the following initial boundary value problem $$\displaystyle
\left\{
\begin{array}{ll}
\displaystyle
\rho\partial_t^2\mbox{\boldmath $u$}-\mu\Delta\mbox{\boldmath $u$}
-(\lambda+\mu)\nabla(\nabla\cdot\mbox{\boldmath $u$})-m\nabla\vartheta=\mbox{\boldmath $0$}
& \text{in}\,(\Omega\setminus\overline D)\times\,]0,\,T[,\\
\\
\displaystyle
c\partial_t\vartheta-k\Delta\vartheta-m\theta_0\nabla\cdot\partial_t\mbox{\boldmath $u$}=0
& \text{in}\,(\Omega\setminus\overline D)\times\,]0,\,T[,\\
\\
\displaystyle
\mbox{\boldmath $u$}(x,0)=\mbox{\boldmath $0$} & \text{in}\,\Omega\setminus\overline D,
\\
\\
\displaystyle
\partial_t\mbox{\boldmath $u$}(x,0)=\mbox{\boldmath $0$} & \text{in}\,\Omega\setminus\overline D,\\
\\
\displaystyle
\vartheta(x,0)=0 & \text{in}\,\Omega\setminus\overline D,\\
\\
\displaystyle
s(\mbox{\boldmath $u$}, \vartheta)\mbox{\boldmath $\nu$}
=\mbox{\boldmath $0$} & \text{on}\,\partial D\times\,]0,\,T[,\\
\\
\displaystyle
k\nabla\vartheta\cdot\mbox{\boldmath $\nu$}=0
& \text{on}\,\partial D\times\,]0,\,T[,\\
\\
\displaystyle
s(\mbox{\boldmath $u$},\vartheta)
\mbox{\boldmath $\nu$}=\mbox{\boldmath $G$} & \text{on}\,\partial\Omega\times\,]0,\,T[,
\\
\\
\displaystyle
-k\nabla\vartheta\cdot\mbox{\boldmath $\nu$}=f & \text{on}\,\partial\Omega\times\,]0,\,T[,
\end{array}
\right.
\tag {1.1}$$ where $$\
\displaystyle
s(\mbox{\boldmath $u$},\vartheta)
=2\mu\,\text{Sym}\,\nabla\mbox{\boldmath $u$}+
\lambda(\nabla\cdot\mbox{\boldmath $u$})I_3+m\,\vartheta I_3.$$ The constants $\theta_0$ and $m$ are the reference temperature and stress-temperature modulus of the body $\Omega\setminus\overline D$, respectively; $k$ the conductivity; $\lambda$ and $\mu$ are Lamé modulus and shear modulus, respectively; $\rho$ and $c$ the density and specific heat. $\vartheta$ denotes the temperature difference of the absolute temperature from the reference temperature $\theta_0$; $\mbox{\boldmath $u$}$ and $s(\mbox{\boldmath $u$}, \vartheta)\mbox{\boldmath $\nu$}$ the displacement vector field and the surface traction, respectively.
It is assumed that $\rho$, $c$, $\theta_0$ and $k$ are known positive constants; $m$, $\lambda$ and $\mu$ are known constants and satisfy $m\not=0$, $\mu>0$ and $3\lambda+2\mu>0$.
Before describing the problem, we specify the class where the solution of (1.1) lives. We employ a general result on the unique solvability and regularity of the initial boundary value problem for the coupled system of the parabolic and hyperbolic system with [*inhomogeneous Neumann-type boundary condition*]{} established in [@D]. It is based on the Hille-Yoshida theorem. By applying Theorem 2.1 in [@D] to the present case, we see that the initial boundary value problem (1.1) has a unique solution such that $$\left\{
\begin{array}{l}
\displaystyle
\mbox{\boldmath $u$}\in C^2([0,\,T];L^2(\Omega\setminus\overline D)^3)
\cap C^1([0,\,T];H^1(\Omega\setminus\overline D)^3)\cap C^0([0,\,T];H^2(\Omega\setminus\overline D)^3),\\
\\
\displaystyle
\partial_t^{1+l}\partial_x^{\alpha}\mbox{\boldmath $u$}
\in L^2([0,\,T];H^{-1/2}(\partial(\Omega\setminus\overline D))),\,l+\vert\alpha\vert\le 1,
\\
\\
\displaystyle
\vartheta\in
C^1([0,\,T];L^2(\Omega\setminus\overline D))
\cap C^0([0,\,T];H^2(\Omega\setminus\overline D)),
\\
\\
\displaystyle
\partial_t\vartheta\in L^2([0,\,T];H^1(\Omega\setminus\overline D))
\end{array}
\right.$$ provided
$\bullet$ $f\in C^0([0,\,T];H^{1/2}(\partial\Omega))$ with $\partial_tf\in L^2([0,\,T];H^{1/2}(\partial\Omega))$ and $f(0)=0$;
$\bullet$ $\mbox{\boldmath $G$}\in C^0([0,\,T];H^{1/2}(\partial\Omega)^3)$ with $\partial_t\mbox{\boldmath $G$}\in L^2([0,\,T];H^{1/2}(\partial\Omega)^3)$ and $\mbox{\boldmath $G$}(0)=0$.
We say that the pair $(f,\mbox{\boldmath $G$})$ is [*admissible*]{} if the conditions listed above are satisfied.
Note that, this framework which is based on the Hille-Yoshida theorem corresponds to that of [@IE4] in which a combination of a standard lifting argument and the results in [@I] about the unique solvability of the classical wave equation with [*homogeneous*]{} Neumann boundary condition are employed. See also [@S] for the hyperbolic systems with [*inhomogeneous*]{} Neumann boundary condition. For the classical results about the direct problem with several [*homogeneous*]{} boundary conditions for the equations of theoremoelasticity which is based on a method due to Višik, see [@DA].
In this paper, we consider the following problem.
$\quad$
[**Problem.**]{} Fix a large $T$ (to be determined later) and a single set of the admissible pair $(f, \mbox{\boldmath $G$})$ (to be specified later). Assume that set $D$ is unknown. Extract information about the location and shape of $D$ from the displacement field $\mbox{\boldmath $u$}_{f,\mbox{\boldmath $G$}}(x,t)$ and temperature difference $\vartheta_{f,\mbox{\boldmath $G$}}(x,t)$ given at all $x\in\partial\Omega$ and $t\in\,]0,\,T[$.
$\quad$
For this problem we employ the idea in the most recent enclosure method developed in [@IE4]. The method introduces so-called indicator functions.
Let $B$ be an open ball centered at $p$ with radius $\eta$ satisfying $\overline B\cap\overline\Omega=\emptyset$. We think that radius $\eta$ of $B$ is very small. Let $\mbox{\boldmath $v$}$ and $\Theta$ satisfy $$\displaystyle
\left\{
\begin{array}{ll}
\displaystyle
\rho\partial_t^2\mbox{\boldmath $v$}-\mu\Delta\mbox{\boldmath $v$}
-(\lambda+\mu)\nabla(\nabla\cdot\mbox{\boldmath $v$})-m\nabla\Theta=\mbox{\boldmath $0$}
& \text{in $\Bbb R^3\times\,]0,\,T[$,}\\
\\
\displaystyle
c\partial_t\Theta-k\Delta\Theta-m\theta_0\nabla\cdot\partial_t\mbox{\boldmath $v$}=0
& \text{in $\Bbb R^3\times\,]0,\,T[$,}
\end{array}
\right.
\tag {1.2}$$ and $$\displaystyle
\left\{
\begin{array}{ll}
\displaystyle
\mbox{\boldmath $v$}(x,0)=\mbox{\boldmath $0$} & \text{in $\Bbb R^3$,}\\
\\
\displaystyle
\text{supp}\,\partial_t\mbox{\boldmath $v$}(\,\cdot\,,0)\cup
\text{supp}\,\Theta(\,\cdot\,,0)\subset\overline B. &
\end{array}
\right.
\tag {1.3}$$ Note that, at this stage we do not specify the form of $\partial_t\mbox{\boldmath $v$}(\,\cdot\,,0)$ and $\Theta(\,\cdot\,,0)$.
The simplified version of the enclosure method employs special $f$ and $\mbox{\boldmath $G$}$ in (1.1) as follows.
Set $$\begin{array}{ll}
\displaystyle
\mbox{\boldmath $G$}(\mbox{\boldmath $v$},\Theta)=
s(\mbox{\boldmath $v$}, \Theta)\mbox{\boldmath $\nu$}
& \text{on $\partial\Omega\times\,]0,\,T[$}
\end{array}
\tag {1.4}$$ and $$\begin{array}{ll}
\displaystyle
f(\mbox{\boldmath $v$},\Theta)=-k\nabla\Theta\cdot\mbox{\boldmath $\nu$} &
\text{on $\partial\Omega\times ]0,\,T[$.}
\end{array}
\tag {1.5}$$ Note that both $\mbox{\boldmath $G$}(\mbox{\boldmath $v$},\Theta)$ and $f(\mbox{\boldmath $v$},\Theta)$ do not contain any [*large*]{} parameter.
We assume that the pair $(f,\mbox{\boldmath $G$})$ given by (1.4) and (1.5) is admissible.
Then, we introduce two indicator functions which play the central role in this paper.
[**Definition 1.1.**]{} Let $\mbox{\boldmath $u$}$ and $\vartheta$ solve (1.1) with $(\mbox{\boldmath $G$}, f)=(\mbox{\boldmath $G$}(\mbox{\boldmath $v$},\Theta), f(\mbox{\boldmath $v$},\Theta))$ given by (1.4) and (1.5), respectively. Let $\tau>0$ and define $$\begin{array}{ll}
\displaystyle
I^1(\tau;\mbox{\boldmath $v$},\Theta)=\int_{\partial\Omega}
s(\mbox{\boldmath $w$}_0,\Xi_0)\mbox{\boldmath $\nu$}
\cdot (\mbox{\boldmath $w$}-\mbox{\boldmath $w$}_0)\,dS
\end{array}
\tag {1.6}$$ and $$\begin{array}{lll}
\displaystyle
I^2(\tau;\mbox{\boldmath $v$},\Theta)=\int_{\partial\Omega}
k\frac{\partial\Xi_0}{\partial\nu}
(\Xi-\Xi_0)\,dS,
\end{array}
\tag {1.7}$$ where $$\left\{
\begin{array}{ll}
\displaystyle
\mbox{\boldmath $w$}(x)=\mbox{\boldmath $w$}(x,\tau)=\int_0^Te^{-\tau t}\mbox{\boldmath $u$}(x,t)dt,
&
x\in\Omega\setminus\overline D,\\
\\
\displaystyle
\mbox{\boldmath $w$}_0(x)=\mbox{\boldmath $w$}_0(x,\tau)=\int_0^Te^{-\tau t}\mbox{\boldmath $v$}(x,t)dt,
&
x\in\Bbb R^3,
\end{array}
\right.
\tag {1.8}$$ and $$\left\{
\begin{array}{lll}
\displaystyle
\Xi(x)=\Xi(x,\tau)=\int_0^Te^{-\tau t}\vartheta(x,t)dt,
&
x\in\Omega\setminus\overline D,
\\
\\
\displaystyle
\Xi_0(x)=\Xi_0(x,\tau)=\int_0^Te^{-\tau t}\Theta(x,t)dt,
&
x\in\Bbb R^3.
\end{array}
\right.
\tag {1.9}$$ Some remarks are in order.
$\bullet$ The indicator functions together with functions $\mbox{\boldmath $w$}$, $\mbox{\boldmath $w$}_0$, $\Xi$ and $\Xi_0$ depend on $T$. However, for simplicity of description, we omit to show their dependence on $T$ explicitly.
$\bullet$ The function $\mbox{\boldmath $w$}$ in (1.6) is the trace $\mbox{\boldmath $w$}\vert_{\partial\Omega}$ of $\mbox{\boldmath $w$}$ given by (1.8) onto $\partial\Omega$ and we have $$\displaystyle
\mbox{\boldmath $w$}\vert_{\partial\Omega}
=\int_0^T e^{-\tau t}\mbox{\boldmath $u$}(\,\cdot\,,t)\vert_{\partial\Omega}\,dt.$$ The integral on this right-hand is the integration for the functions with the values in a Banach space ([@DL]). The same remark works also for $\Xi$ in (1.7). Therefore, these indicator functions can be computed from the response $\mbox{\boldmath $u$}$ and $\vartheta$ on $\partial\Omega$ over time interval $]0,\,T[$ which are the solutions of (1.1) with $\mbox{\boldmath $G$}=\mbox{\boldmath $G$}(\mbox{\boldmath $v$},\Theta)$ and $f=f(\mbox{\boldmath $v$},\Theta)$.
$\bullet$ Using the Lumer-Phillips theorem [@Y], one can show that, given the initial data $\mbox{\boldmath $v$}(x,0)=\mbox{\boldmath $v$}_0\in H^2(\Bbb R^3)^3$, $\partial_t\mbox{\boldmath $v$}(x,0)=
\mbox{\boldmath $v$}_1\in H^1(\Bbb R^3)^3$ and $\Theta(x,0)=\Theta_0\in H^2(\Bbb R^3)$, there exists a unique pair of $\mbox{\boldmath $v$}\in C^2([0,\infty[, L^2(\Bbb R^3)^3)
\cap C^1([0,\infty[, H^1(\Bbb R^3)^3)\cap C^0([0,\infty[, H^2(\Bbb R^3)^3)$ and $\Theta\in C^1([0,\,\infty[, L^2(\Bbb R^3))\cap C^0([0,\,\infty[, H^2(\Bbb R^3))$ satisfying (1.2). However, in this paper, we do not employ this general fact since the desired solutions $\mbox{\boldmath $v$}$ and $\Theta$ of (1.2) satisfying (1.3) have been constructed from those of decoupled equations.
Now we state the main results of this paper.
Coupled case
------------
In this subsection $m$ in (1.1) is an arbitrary real number, and needless to say, the special case $m=0$ is not excluded.
Let $\mbox{\boldmath $a$}$ be an arbitrary unit vector. Let $\mbox{\boldmath $\Phi$}$ solve $$\left\{\begin{array}{ll}
\displaystyle
\rho\partial_t^2\mbox{\boldmath $\Phi$}-\mu\Delta\mbox{\boldmath $\Phi$}=\mbox{\boldmath $0$}
&
\text{in $\Bbb R^3\times\,]0,\,T[$,}\\
\\
\displaystyle
\mbox{\boldmath $\Phi$}(x,0)=\mbox{\boldmath $0$}
&
\text{in $\Bbb R^3$,}\\
\\
\displaystyle
\partial_t\mbox{\boldmath $\Phi$}(x,0)=(\eta-\vert x-p\vert)^2\chi_B(x)\mbox{\boldmath $a$}
&
\text{in $\Bbb R^3$,}
\end{array}
\right.$$ where $\chi_B$ denotes the characteristic function of $B$. Since the function $\Bbb R^3\ni x\longmapsto (\eta-\vert x-p\vert)^2\chi_B(x)\in\Bbb R$ belongs to $H^2(\Bbb R^3)$, it is known that one can construct such $\mbox{\boldmath $\Phi$}$ in the class $$\displaystyle
C^2([0,\,T], H^1(\Bbb R^3)^3)
\cap C^1([0,\,T], H^2(\Bbb R^3)^3)
\cap C([0,\,T], H^3(\Bbb R^3)^3).$$ Set $$\displaystyle
\mbox{\boldmath $v$}_s=\nabla\times\mbox{\boldmath $\Phi$}
\in
C^2([0,\,T], L^2(\Bbb R^3)^3)
\cap C^1([0,\,T], H^1(\Bbb R^3)^3)
\cap C([0,\,T], H^2(\Bbb R^3)^3).
\tag {1.10}$$ We have $\nabla\cdot\mbox{\boldmath $v$}_s=0$. We see that (1.2) and (1.3) are satisfied with the pair $(\mbox{\boldmath $v$},\Theta)
=(\mbox{\boldmath $v$}_s,0)$; the pair $(f,\mbox{\boldmath $G$})=
(0,s(\mbox{\boldmath $v$}_s,0)\mbox{\boldmath $\nu$})$ given by (1.4) and (1.5) is admissible. Note that the pair $(\mbox{\boldmath $v$},\Theta)=(\mbox{\boldmath $v$}_s,0)$ is a special version of the Deresiewicz-Zorski solution of the system (1.2), see page 330 in [@C].
In this case we have $\Xi_0=0$ in $\Bbb R^3$ and (1.7) gives $I^2(\tau;\mbox{\boldmath $v$}_s,0)=0$ for all $\tau$. This means that one cannot obtain any information about $D$ from the indicator function $\tau\longmapsto I^2(\tau;\mbox{\boldmath $v$}_s,0)$. However, another indicator function $I^1(\tau;\mbox{\boldmath $v$}_s,0)$ has the following asymptotic behaviour.
\(i) Let $T$ satisfy $$\displaystyle
T>\sqrt{\frac{\rho}{\mu}}\left(2\text{dist}\,(D,B)-\text{dist}\,(\Omega,B)\right).
\tag {1.11}$$ Then, there exists a positive number $\tau_0$ such that, for all $\tau\ge\tau_0$ $I^1(\tau;\mbox{\boldmath $v$}_s,0)>0$ and we have $$\displaystyle
\lim_{\tau\longrightarrow\infty}\frac{1}{\tau}\log I^1(\tau;\mbox{\boldmath $v$}_s,0)
= -2\sqrt{\frac{\rho}{\mu}}\text{dist}\,(D,B).
\tag {1.12}$$
\(ii) We have $$\displaystyle
\lim_{\tau\longrightarrow\infty}e^{\tau T}I^1(\tau;\mbox{\boldmath $v$}_s,0)
=
\left\{\begin{array}{ll}
\displaystyle
\infty &
\text{if $\displaystyle T>2\sqrt{\frac{\rho}{\mu}}\text{dist}\,(D,B)$,}
\\
\\
\displaystyle
0 &
\text{if $\displaystyle T<2\sqrt{\frac{\rho}{\mu}}\text{dist}\,(D,B)$.}
\end{array}
\right.
\tag {1.13}$$
\(iii) If $\displaystyle T=2\sqrt{\frac{\rho}{\mu}}\text{dist}\,(D,B)$, then we have, as $\tau\longrightarrow\infty$ $$\displaystyle
e^{\tau T}I^1(\tau;\mbox{\boldmath $v$}_s,0)=O(\tau^4).
\tag {1.14}$$
Note that $\mbox{\boldmath $G$}$ depends on $\mbox{\boldmath $a$}$ and there is no restriction on the direction of $\mbox{\boldmath $a$}$ relative to the unit normal $\mbox{\boldmath $\nu$}$ at the points on $\partial D$ which are nearest to the center point of ball $B$.
Since $T$ in (i) has the constraint (1.11), formula (1.12) does not automatically imply the validity of (1.13) in the case when $T<2\sqrt{\rho/\mu}\,\text{dist}\,(D,B)$. Note also that constraint (1.11) is reasonable since as pointed out in [@IW00] (see also [@IE4]) we have $$\displaystyle
2\text{dist}\,(D,B)-\text{dist}\,(\Omega,B)
\ge
\inf\{\vert x-y\vert+\vert y-z\vert\,\vert\,x\in\partial B, y\in\partial D,z\in\partial\Omega\}.$$ The bound $O(\tau^4)$ on (1.14) is a rough estimation. The point is: it is just at most algebraic.
Theorem 1.1 gives a solution to Problem in the case when $f=0$ and $\mbox{\boldmath $G$}=\mbox{\boldmath $G$}(\mbox{\boldmath $v$}_s,0)$, where $\mbox{\boldmath $v$}_s$ is given by (1.10). Needless to say, the indicator function $I^1(\tau;\mbox{\boldmath $v$}_s,0)$ contains information about thermal effect. However, remarkably enough, the choice of the prescribed traction $\mbox{\boldmath $G$}=s(\mbox{\boldmath $v$}_s,0)\mbox{\boldmath $\nu$}$ and heat flux $f=0$ in (1.1) enables us to [*ignore*]{} thermal effects inside the body. More precisely, with the help of the identity $I^2(\tau;\mbox{\boldmath $v$}_s,0)=0$ one can control the asymptotic behaviour of the indicator function $I^1(\tau;\mbox{\boldmath $v$}_s,0)$ as $\tau\longrightarrow\infty$ in terms of $\mbox{\boldmath $v$}_s$ only over time interval $]0,\,T[$. See Proposition 3.1 in Section 3 and formula (3.3) in the proof.
It should be pointed out that Theorem 1.1 does not give any solution to the case when the surface traction on the outer boundary vanishes, that is, $\mbox{\boldmath $G$}=\mbox{\boldmath $0$}$. This case shall be more difficult since it seems that system (1.2) does not posses a solution $(\mbox{\boldmath $v$},\Theta)$ in such a way that $G(\mbox{\boldmath $v$},\Theta)=\mbox{\boldmath $0$}$ on $\partial\Omega\times\,]0,\,T[$.
Decoupled case
--------------
The enclosure method presented in this paper covers also the decoupled case $m=0$. In this subsection, for a comparison with the main result we present two results in that case.
Let $m=0$. Let $\phi$ be the solution of $$\left\{
\begin{array}{ll}
\rho\partial_t^2\phi-(\lambda+2\mu)\Delta\phi=0
&
\text{in $\Bbb R^3\times\,]0,\,T[$,}
\\
\\
\displaystyle
\phi(x,0)=0 &
\text{in $\Bbb R^3$,}
\\
\\
\displaystyle
\partial_t\phi(x,0)=(\eta-\vert x-p\vert)^2\chi_B(x)
&
\text{in $\Bbb R^3$.}
\end{array}
\right.$$ The class where $\phi$ belongs to is the same as all the components of $\mbox{\boldmath $\Phi$}$ in the preceding section. Set $$\displaystyle
\mbox{\boldmath $v$}_p=\nabla\phi.
\tag {1.15}$$ The same comment for the class where $\mbox{\boldmath $v$}_p$ works also and we have $\nabla\times\mbox{\boldmath $v$}_p=\mbox{\boldmath $0$}$. We see that the pair $(\mbox{\boldmath $v$}, \Theta)= (\mbox{\boldmath $v$}_p,0)$ satisfies (1.2) and (1.3) under the assumption $m=0$; the pair $(f,\mbox{\boldmath $G$})=(0, s(\mbox{\boldmath $v$}_p,0)\mbox{\boldmath $\nu$})$ given by (1.4) and (1.5) is admissible.
Let $m=0$. (i) Let $T$ satisfy $$\displaystyle
T>\sqrt{\frac{\rho}{\lambda+2\mu}}\left(2\text{dist}\,(D,B)-\text{dist}\,(\Omega,B)\right).$$ Then, there exists a positive number $\tau_0$ such that, for all $\tau\ge\tau_0$ $I^1(\tau;\mbox{\boldmath $v$}_p,0)>0$ and we have $$\displaystyle
\lim_{\tau\longrightarrow\infty}\frac{1}{\tau}\log I^1(\tau;\mbox{\boldmath $v$}_p,0)
= -2\sqrt{\frac{\rho}{\lambda+2\mu}}\text{dist}\,(D,B).$$
\(ii) We have $$\displaystyle
\lim_{\tau\longrightarrow\infty}e^{\tau T}I^1(\tau;\mbox{\boldmath $v$}_p,0)
=
\left\{\begin{array}{ll}
\displaystyle
\infty &
\text{if $\displaystyle T>2\sqrt{\frac{\rho}{\lambda+2\mu}}\text{dist}\,(D,B)$,}
\\
\\
\displaystyle
0 &
\text{if $\displaystyle T<2\sqrt{\frac{\rho}{\lambda+2\mu}}\text{dist}\,(D,B)$.}
\end{array}
\right.$$
\(iii) If $\displaystyle T=2\sqrt{\frac{\rho}{\lambda+2\mu}}
\text{dist}\,(D,B)$, then we have, as $\tau\longrightarrow\infty$ $$\displaystyle
e^{\tau T}I^1(\tau;\mbox{\boldmath $v$}_p,0)=O(\tau^4).$$
Note that Theorem 1.1 covers also the case when $m=0$. Thus, we have two results in that case.
Let $\Theta_0$ solve $$\left\{
\begin{array}{ll}
\displaystyle
c\partial_t\Theta-k\Delta\Theta=0
& \text{in $\Bbb R^3\times\,]0,\,T[$,}\\
\\
\displaystyle
\Theta(x,0)=(\eta-\vert x-p\vert)^2\chi_B(x) & \text{in $\Bbb R^3$.}
\end{array}
\right.
\tag {1.16}$$ We see that the pair $(\mbox{\boldmath $v$},\Theta)=(\mbox{\boldmath $0$},\Theta_0)$ satisfies (1.2) and (1.3) under the assumption $m=0$; the pair $(f,\mbox{\boldmath $G$})=(-k\nabla\Theta_0\cdot\mbox{\boldmath $\nu$},\mbox{\boldmath $0$})$ given by (1.4) and (1.5) is admissible.
Let $m=0$.
\(i) Let $T$ be an arbitrary fixed positive number. Then, there exists a positive number $\tau_0$ such that, for all $\tau\ge\tau_0$ $I^2(\tau;\mbox{\boldmath $0$},\Theta_0)>0$ and we have $$\displaystyle
\lim_{\tau\longrightarrow\infty}\frac{1}{\sqrt{\tau}}\log I^2(\tau;\mbox{\boldmath $0$},\Theta_0)
= -2\sqrt{\frac{c}{k}}\text{dist}\,(D,B).$$
\(ii) We have $$\displaystyle
\lim_{\tau\longrightarrow\infty}e^{\sqrt{\tau} \,T}I^2(\tau;\mbox{\boldmath $0$},\Theta_0)
=
\left\{\begin{array}{ll}
\displaystyle
\infty &
\text{if $\displaystyle T>2\sqrt{\frac{c}{k}}\text{dist}\,(D,B)$,}
\\
\\
\displaystyle
0 &
\text{if $\displaystyle T<2\sqrt{\frac{c}{k}}\text{dist}\,(D,B)$.}
\end{array}
\right.$$
\(iii) If $\displaystyle T=2\sqrt{\frac{c}{k}}\text{dist}\,(D,B)$, then we have, as $\tau\longrightarrow\infty$ $$\displaystyle
e^{\sqrt{\tau}\,T}I^2(\tau;\mbox{\boldmath $0$},\Theta_0)=O(\tau^2).$$
Note that, in (i) of Theorem 1.3 there is no restriction on the size of $T$ same as several results in [@I4; @IFR; @IK1; @IK2; @IK5; @II]. This suggests the [*infinite propagation*]{} speed of the signal governed by the heat equation indirectly. By virtue of arbitrariness of $T$ one can easily deduce (ii) from (i). However, (i) in Theorems 1.1 and 1.2 there are restrictions on the size of $T$. Thus to derive the later half of (ii) in Theorems 1.1 and 1.2 is independent of (i) in those theorems.
Since the proof of Theorems 1.2-1.3 is easier than that of Theorem 1.1 and can be done as Theorem 1.1 in [@IE4], we omit to describe their proof.
Preliminaries
=============
Decomposition formulae of the indicator functions
-------------------------------------------------
In this and next subsection section, the form of the pair $(\mbox{\boldmath $v$},\Theta)$ satisfying (1.2) and (1.3) is not specified.
Set $$\begin{array}{ll}
\displaystyle
\mbox{\boldmath $R$}=\mbox{\boldmath $w$}-\mbox{\boldmath $w$}_0,
&
\displaystyle
\Sigma=\Xi-\Xi_0,
\end{array}$$ where $\mbox{\boldmath $w$}$, $\Xi$, $\mbox{\boldmath $w$}_0$ and $\Xi_0$ are given by (1.8) and (1.9).
It follows from (1.1) that $\mbox{\boldmath $w$}$ and $\Xi$ satisfies $$\left\{
\begin{array}{ll}
\displaystyle
\mu\Delta\mbox{\boldmath $w$}+(\lambda+\mu)\nabla(\nabla\cdot\mbox{\boldmath $w$})
-\rho\tau^2\mbox{\boldmath $w$}
+m\nabla\Xi=\rho e^{-\tau T}\mbox{\boldmath $F$}& \text{in}\,\Omega\setminus\overline D,\\
\\
\displaystyle
k\Delta\Xi-c\tau\Xi+m\theta_0\tau\nabla\cdot\mbox{\boldmath $w$}
=e^{-\tau T}h & \text{in}\,\Omega\setminus\overline D,
\\
\\
\displaystyle
s(\mbox{\boldmath $w$},\Xi)\mbox{\boldmath $\nu$}
=s(\mbox{\boldmath $w$}_0,\Xi_0)\mbox{\boldmath $\nu$} & \text{on}\,\partial\Omega,\\
\\
\displaystyle
-k\nabla\Xi\cdot\mbox{\boldmath $\nu$}=
-k\nabla\Xi_0\cdot\mbox{\boldmath $\nu$}
& \text{on}\,\partial\Omega,
\\
\\
\displaystyle
s(\mbox{\boldmath $w$},\Xi)\mbox{\boldmath $\nu$}
=\mbox{\boldmath $0$} & \text{on}\,\partial D,
\\
\\
\displaystyle
-k\nabla\Xi\cdot\mbox{\boldmath $\nu$}=0 &
\text{on}\,\partial D,
\end{array}
\right.
\tag {2.1}$$ where $$\left\{
\begin{array}{ll}
\displaystyle
\mbox{\boldmath $F$}=\mbox{\boldmath $F$}(x,\tau)=\partial_t\mbox{\boldmath $u$}(x,T)
+\tau\mbox{\boldmath $u$}(x,T), & x\in\Omega\setminus\overline D,\\
\\
\displaystyle
h=c\vartheta(x,T)-m\theta_0\nabla\cdot\mbox{\boldmath $u$}(x,T),
&
x\in\Omega\setminus\overline D.
\end{array}
\right.
\tag {2.2}$$ It follows from (1.2) that the $\mbox{\boldmath $w$}_0$ and $\Theta_0$ satisfy $$\left\{
\begin{array}{ll}
\displaystyle
\mu\Delta\mbox{\boldmath $w$}_0+(\lambda+\mu)\nabla(\nabla\cdot\mbox{\boldmath $w$}_0)
-\rho\tau^2\mbox{\boldmath $w$}_0
+m\nabla\Xi_0+\rho\mbox{\boldmath $v$}_0
=\rho e^{-\tau T}\mbox{\boldmath $F$}_0& \text{in}\,\Bbb R^3,\\
\\
\displaystyle
k\Delta\Xi_0-c\tau\Xi_0+m\theta_0\tau\nabla\cdot\mbox{\boldmath $w$}_0+cf_0
=e^{-\tau T}h_0& \text{in}\,\Bbb R^3,
\end{array}
\right.
\tag {2.3}$$ where $$\left\{
\begin{array}{ll}
\displaystyle
\mbox{\boldmath $v$}_0(x)=\partial_t\mbox{\boldmath $v$}(x,0),
& x\in\Bbb R^3,\\
\\
\displaystyle
f_0(x)=\Theta(x,0),
& x\in\Bbb R^3,
\\
\\
\displaystyle
\mbox{\boldmath $F$}_0=\mbox{\boldmath $F$}_0(x,\tau)=\partial_t\mbox{\boldmath $v$}(x,T)
+\tau\mbox{\boldmath $v$}(x,T), & x\in\Bbb R^3,\\
\\
\displaystyle
h_0=c\Theta(x,T)-m\theta_0\nabla\cdot\mbox{\boldmath $v$}(x,T),
&
x\in\Bbb R^3.
\end{array}
\right.
\tag {2.4}$$ Recall that the supports of $\mbox{\boldmath $v$}_0$ and $f_0$ are contained in $\overline B$ and $B$ satisfies $\overline B\cap\overline\Omega=\emptyset$. See assumption (1.3).
Then, integration by parts together with the first equations on (2.1) and (2.3) yields $$\begin{array}{l}
\,\,\,\,\,\,
\displaystyle
\int_{\partial\Omega}
\left(s(\mbox{\boldmath $w$}_0,\Xi_0)\mbox{\boldmath $\nu$}\cdot
\mbox{\boldmath $w$}-
s(\mbox{\boldmath $w$},\Xi)\mbox{\boldmath $\nu$}\cdot
\mbox{\boldmath $w$}_0\right)\,dS\\
\\
\displaystyle
=\int_{\partial D}s(\mbox{\boldmath $w$}_0,\Xi_0)\mbox{\boldmath $\nu$}\cdot
\mbox{\boldmath $w$}\,dS
+m\int_{\Omega\setminus\overline D}(\Xi_0\nabla\cdot\mbox{\boldmath $w$}
-\Xi\nabla\cdot\mbox{\boldmath $w$}_0)\,dx\\
\\
\,\,\,
\displaystyle
+e^{-\tau T}\rho
\int_{\Omega\setminus\overline D}
\left
(\mbox{\boldmath $F$}_0\cdot\mbox{\boldmath $w$}-\mbox{\boldmath $F$}\cdot\mbox{\boldmath $w$}_0
\right)
dx.
\end{array}$$ Hence $$\begin{array}{ll}
\displaystyle
I^1(\tau;\mbox{\boldmath $v$},\Theta)
&
\displaystyle
=\int_{\partial D}s(\mbox{\boldmath $w$}_0,\Xi_0)\mbox{\boldmath $\nu$}\cdot
\mbox{\boldmath $w$}\,dS
+m\int_{\Omega\setminus\overline D}(
-\Sigma\nabla\cdot\mbox{\boldmath $w$}_0
+
\Xi_0\nabla\cdot\mbox{\boldmath $R$})\,dx
\\
\\
\displaystyle
&
\displaystyle
\,\,\,
+e^{-\tau T}\rho
\int_{\Omega\setminus\overline D}
\left
(\mbox{\boldmath $F$}_0\cdot\mbox{\boldmath $w$}-\mbox{\boldmath $F$}\cdot\mbox{\boldmath $w$}_0
\right)
dx.
\end{array}
\tag {2.5}$$ This is the first representation of indicator function $I^1(\tau;\mbox{\boldmath $v$},\Theta)$.
It follows from the second equations on (2.1) and (2.3) that $$\begin{array}{l}
\,\,\,\,\,\,
\displaystyle
\int_{\partial\Omega}
\left(k\frac{\partial\Xi_0}{\partial\nu}\Xi-k\frac{\partial\Xi}{\partial\nu}\Xi_0\right)\,dS\\
\\
\displaystyle
=
\int_{\partial D}k\frac{\partial\Xi_0}{\partial\nu}\Xi\,dS
+m\theta_0\tau
\int_{\Omega\setminus\overline D}
\left(\Xi_0\nabla\cdot\mbox{\boldmath $w$}-\Xi\nabla\cdot\mbox{\boldmath $w$}_0\right)\,dx\\
\\
\displaystyle
\,\,\,
+
e^{-\tau T}
\int_{\Omega\setminus\overline D}
(h_0\Xi-h\Xi_0)\,dx.
\end{array}$$ This yields $$\begin{array}{ll}
\displaystyle
I^2(\tau;\mbox{\boldmath $v$},\Theta)
&
\displaystyle
=\int_{\partial D}k\frac{\partial\Xi_0}{\partial\nu}\Xi\,dS
+m\theta_0\tau
\int_{\Omega\setminus\overline D}
\left(-\Sigma\nabla\cdot\mbox{\boldmath $w$}_0+\Xi_0\nabla\cdot\mbox{\boldmath $R$}\right)\,dx
\\
\\
\displaystyle
&
\displaystyle
\,\,\,
+
e^{-\tau T}
\int_{\Omega\setminus\overline D}
(h_0\Xi-h\Xi_0)\,dx.
\end{array}
\tag {2.6}$$ This is the first representation of indicator function $I^2(\tau;\mbox{\boldmath $v$},\Theta)$.
Next we decompose the first term on the right-hand side of (2.5). The result yields the following decomposition formula for $I^1(\tau;\mbox{\boldmath $v$},\Theta)$.
We have $$\begin{array}{ll}
\displaystyle
I^1(\tau;\mbox{\boldmath $v$},\Theta)
&
\displaystyle
=J(\tau)+m\int_D\Xi_0\nabla\cdot\mbox{\boldmath $w$}_0
+
E(\tau)
+m\int_{\Omega\setminus\overline D}\Sigma\nabla\cdot\mbox{\boldmath $R$}\,dx\\
\\
\displaystyle
&
\displaystyle
\,\,\,
+
m\int_{\Omega\setminus\overline D}(
-\Sigma\nabla\cdot\mbox{\boldmath $w$}_0
+
\Xi_0\nabla\cdot\mbox{\boldmath $R$})\,dx
+{\cal R}^1(\tau),
\end{array}
\tag {2.7}$$ where $$\displaystyle
J(\tau)=
\int_D\left(2\mu\left\vert\text{Sym}\,\nabla\mbox{\boldmath $w$}_0\right\vert^2
+\lambda\vert\nabla\cdot\mbox{\boldmath $w$}_0\vert^2+\rho\tau^2\vert\mbox{\boldmath $w$}_0\vert^2\right)\,dx,
\tag {2.8}$$ $$\displaystyle
E(\tau)=\int_{\Omega\setminus\overline D}\left(2\mu\left\vert\text{Sym}\,\nabla\mbox{\boldmath $R$}\right\vert^2
+\lambda\vert\nabla\cdot\mbox{\boldmath $R$}\vert^2+\rho\tau^2\vert\mbox{\boldmath $R$}\vert^2\right)\,dx
\tag {2.9}$$ and $$\displaystyle
{\cal R}^1(\tau)
=\rho e^{-\tau T}\left\{
\int_D\mbox{\boldmath $F$}_0\cdot\mbox{\boldmath $w$}_0\,dx
+\int_{\Omega\setminus\overline D}\mbox{\boldmath $F$}\cdot\mbox{\boldmath $R$}\,dx
+\int_{\Omega\setminus\overline D}
(\mbox{\boldmath $F$}_0-\mbox{\boldmath $F$})\cdot\mbox{\boldmath $w$}_0
dx\right\}.
\tag {2.10}$$
[*Proof.*]{} Since $\overline B\cap\overline\Omega=\emptyset$, from the first equations on (2.1) and (2.3) we see that the $\mbox{\boldmath $R$}$ satisfies $$\left\{
\begin{array}{ll}
\displaystyle
\mu\Delta\mbox{\boldmath $R$}+(\lambda+\mu)\nabla(\nabla\cdot\mbox{\boldmath $R$})
-\rho\tau^2\mbox{\boldmath $R$}
+m\nabla\Sigma=\rho e^{-\tau T}(\mbox{\boldmath $F$}-\mbox{\boldmath $F$}_0) & \text{in}\,\Omega\setminus\overline D,\\
\\
\displaystyle
s(\mbox{\boldmath $R$},\Sigma)\mbox{\boldmath $\nu$}
=\mbox{\boldmath $0$} & \text{on}\,\partial\Omega,
\\
\\
\displaystyle
\displaystyle
s(\mbox{\boldmath $R$},\Sigma)\mbox{\boldmath $\nu$}=
-s(\mbox{\boldmath $w$}_0,\Xi_0)\mbox{\boldmath $\nu$}
& \text{on}\,\partial D.
\end{array}
\right.
\tag {2.11}$$ Then, one can write $$\displaystyle
\int_{\partial D}s(\mbox{\boldmath $w$}_0,\Xi_0)\mbox{\boldmath $\nu$}\cdot
\mbox{\boldmath $w$}\,dS
=\int_{\partial D}s(\mbox{\boldmath $w$}_0,\Xi_0)\mbox{\boldmath $\nu$}\cdot
\mbox{\boldmath $w$}_0\,dS
-\int_{\partial D}s(\mbox{\boldmath $R$},\Sigma)\mbox{\boldmath $\nu$}\cdot
\mbox{\boldmath $R$}\,dS.$$ It follows from the first equation on (2.3) that $$\displaystyle
\int_{\partial D}s(\mbox{\boldmath $w$}_0,\Xi_0)\mbox{\boldmath $\nu$}\cdot
\mbox{\boldmath $w$}_0\,dS
=J(\tau)+m\int_D\Xi_0\nabla\cdot\mbox{\boldmath $w$}_0\,dx
+\rho e^{-\tau T}\int_D\mbox{\boldmath $F$}_0\cdot\mbox{\boldmath $w$}_0\,dx.$$ It follows from (2.11) that $$\begin{array}{ll}
\displaystyle
-\int_{\partial D}s(\mbox{\boldmath $R$},\Sigma)\mbox{\boldmath $\nu$}\cdot
\mbox{\boldmath $R$}\,dS
&
\displaystyle
=\int_{\partial(\Omega\setminus\overline D)}s(\mbox{\boldmath $R$},\Sigma)\mbox{\boldmath $\nu$}\cdot
\mbox{\boldmath $R$}\,dS\\
\\
\displaystyle
&
\displaystyle
=E(\tau)+m\int_{\Omega\setminus\overline D}\Sigma\nabla\cdot\mbox{\boldmath $R$}\,dx
+\rho e^{-\tau T}\int_{\Omega\setminus\overline D}(\mbox{\boldmath $F$}
-
\mbox{\boldmath $F$}_0)\cdot\mbox{\boldmath $R$}\,dx.
\end{array}$$ Thus we obtain $$\begin{array}{ll}
\displaystyle
\int_{\partial D}s(\mbox{\boldmath $w$}_0,\Xi_0)\mbox{\boldmath $\nu$}\cdot
\mbox{\boldmath $w$}\,dS
&
\displaystyle
=J(\tau)+m\int_D\Xi_0\nabla\cdot\mbox{\boldmath $w$}_0\,dx
+E(\tau)+m\int_{\Omega\setminus\overline D}\Sigma\nabla\cdot\mbox{\boldmath $R$}\,dx
\\
\\
\displaystyle
\,\,\,
&
\,\,\,
\displaystyle
+\rho e^{-\tau T}
\left\{
\int_D\mbox{\boldmath $F$}_0\cdot\mbox{\boldmath $w$}_0\,dx+
\int_{\Omega\setminus\overline D}(\mbox{\boldmath $F$}
-
\mbox{\boldmath $F$}_0)\cdot\mbox{\boldmath $R$}\,dx\right\}.
\end{array}$$ Then a combination of this and (2.5) gives (2.7).
$\Box$
[**Remark 2.1.**]{} We have, for all real $3\times 3$-matrix $A$ $$\displaystyle
2\mu\left\vert\text{Sym}\,A-\frac{\text{trace}\,A}{3}I_3\right\vert^2+
\frac{3\lambda+2\mu}{3}\vert\text{trace}\,A\vert^2
=2\mu\vert\text{Sym}\,A\vert^2+\lambda\vert\text{trace}\,A\vert^2.
\tag {2.12}$$ See page 85 in [@Gu]. Thus, both $J(\tau)$ and $E(\tau)$ given by (2.8) and (2.9), respectively are nonnegative under the assumption $\mu>0$ and $3\lambda+2\mu>0$.
We have
$$\begin{array}{ll}
\displaystyle
I^2(\tau;\mbox{\boldmath $v$},\Theta)
&
\displaystyle
=j(\tau)-m\theta_0\tau\int_D\Xi_0\nabla\cdot\mbox{\boldmath $w$}_0\,dx
+e(\tau)-m\theta_0\tau\int_{\Omega\setminus\overline D}\Sigma\nabla\cdot\mbox{\boldmath $R$}\,dx
\\
\\
\displaystyle
\,\,\,
&
\displaystyle
\,\,\,
+m\theta_0\tau
\int_{\Omega\setminus\overline D}
\left(-\Sigma\nabla\cdot\mbox{\boldmath $w$}_0+\Xi_0\nabla\cdot\mbox{\boldmath $R$}\right)\,dx
+{\cal R}^2(\tau),
\end{array}
\tag {2.13}$$ where $$\displaystyle
j(\tau)=
\int_D
\left(
k\vert\nabla\Xi_0\vert^2
+c\tau\vert\Xi_0\vert^2\right)\,dx,
\tag {2.14}$$ $$\displaystyle
e(\tau)=\int_{\Omega\setminus\overline D}\left(
k\vert\nabla\Sigma\vert^2
+c\tau\vert\Sigma\vert^2\right)\,dx
\tag {2.15}$$ and $$\displaystyle
{\cal R}^2(\tau)
=e^{-\tau T}\left\{
\int_Dh_0\Xi_0\,dx
+\int_{\Omega\setminus\overline D}h\Sigma\,dx
+\int_{\Omega\setminus\overline D}
(h_0-h)\Xi_0
dx\right\}$$
[*Proof.*]{} Since $\overline B\cap\overline\Omega=\emptyset$, it follows from some of equations on (2.1) and (2.3) that the $\Sigma$ satisfies $$\left\{
\begin{array}{ll}
\displaystyle
k\Delta\Sigma-c\tau\Sigma+m\theta_0\tau\nabla\cdot\mbox{\boldmath $R$}
=e^{-\tau T}(h-h_0) & \text{in}\,\Omega\setminus\overline D,\\
\\
\displaystyle
-k\nabla\Sigma\cdot\mbox{\boldmath $\nu$}=0 & \text{on}\,\partial\Omega,\\
\\
\displaystyle
-k\nabla\Sigma\cdot\mbox{\boldmath $\nu$}
=k\nabla\Xi_0\cdot\mbox{\boldmath $\nu$}
& \text{on}\,\partial D.
\end{array}
\right.
\tag {2.16}$$ Then, one can write $$\displaystyle
\int_{\partial D}k\frac{\partial\Xi_0}{\partial\mbox{\boldmath $\nu$}}\Xi\,dS
=\int_{\partial D}k\frac{\partial\Xi_0}{\partial\mbox{\boldmath $\nu$}}\Xi_0\,dS
-\int_{\partial D}k\frac{\partial\Sigma}{\partial\mbox{\boldmath $\nu$}}\Sigma\,dS.$$ It follows from the second equation on (2.3) that $$\displaystyle
\int_{\partial D}k\frac{\partial\Xi_0}{\partial\mbox{\boldmath $\nu$}}\Xi_0\,dS
=j(\tau)
-m\theta_0\tau\int_D\Xi_0\nabla\cdot\mbox{\boldmath $w$}_0\,dx
+e^{-\tau T}\int_Dh_0\Xi_0\,dx.$$ It follows from (2.16) that $$\begin{array}{ll}
\displaystyle
-\int_{\partial D}k\frac{\partial\Sigma}{\partial\mbox{\boldmath $\nu$}}\Sigma\,dS
&
\displaystyle
=
\int_{\partial\,(\Omega\setminus\overline D)}
k\frac{\partial\Sigma}{\partial\mbox{\boldmath $\nu$}}\Sigma\,dS\\
\\
\displaystyle
&
\displaystyle
=e(\tau)
-m\theta_0\tau\int_{\Omega\setminus\overline D}\Sigma\nabla\cdot\mbox{\boldmath $R$}\,dx
+e^{-\tau T}\int_{\Omega\setminus\overline D}(h-h_0)\Sigma\,dx.
\end{array}$$ Thus we obtain $$\begin{array}{ll}
\displaystyle
\int_{\partial D}k\frac{\partial\Xi_0}{\partial\mbox{\boldmath $\nu$}}\Xi\,dS
&
\displaystyle
=j(\tau)-m\theta_0\tau\int_D\Xi_0\nabla\cdot\mbox{\boldmath $w$}_0\,dx
+e(\tau)-m\theta_0\tau\int_{\Omega\setminus\overline D}\Sigma\nabla\cdot\mbox{\boldmath $R$}\,dx
\\
\\
\displaystyle
\,\,\,
&
\,\,\,
\displaystyle
+e^{-\tau T}\left\{\int_Dh_0\Xi_0dx+\int_{\Omega\setminus\overline D}(h-h_0)\mbox{\boldmath $\Sigma$}\,dx\right\}.
\end{array}$$ Then a combination of this and (2.6) gives (2.13).
$\Box$
As a direct consequence of Propositions 2.1 and 2.2 we obtain $$\begin{array}{l}
\displaystyle
\,\,\,\,\,\,
\theta_0\tau I^1(\tau;\mbox{\boldmath $v$},\Theta)
+I^2(\tau;\mbox{\boldmath $v$},\Theta)\\
\\
\displaystyle
=(j(\tau)+\theta_0\tau\,J(\tau))+(e(\tau)+\theta_0\tau\,E(\tau))
\\
\\
\displaystyle
\,\,\,
+m\theta_0\tau\,\int_{\Omega\setminus\overline D}
(-\Sigma\nabla\cdot\mbox{\boldmath $w$}_0+\Xi_0\nabla\cdot\mbox{\boldmath $R$})\,dx
+\theta_0\tau\,{\cal R}(\tau),
\end{array}
\tag {2.17}$$ where $$\displaystyle
{\cal R}(\tau)={\cal R}^1(\tau)+\frac{1}{\theta_0\tau}\,{\cal R}^2(\tau).
\tag {2.18}$$ Note that the first and second terms in the right-hand side on (2.17) are non negative and the third term contains $\nabla\cdot\mbox{\boldmath $w$}_0$ and $\Xi_0$ linearly. In short, we choose special $(\mbox{\boldmath $v$},\Theta)$ in Theorem 1.1 in such a way that this third term vanishes.
In the next subsection we give an upper bound on $e(\tau)+\theta_0\tau\,E(\tau)$ in terms of $j(\tau)$ and $J(\tau)$.
Basic estimate
--------------
We have, as $\tau\longrightarrow\infty$ $$\displaystyle
e(\tau)+\theta_0\tau E(\tau)
=O(\tau j(\tau)+\tau^3 J(\tau)+\tau^3 e^{-2\tau T}).
\tag {2.19}$$
[*Proof.*]{} It follows from (2.16) that $$\begin{array}{ll}
\displaystyle
\int_{\partial D}k\frac{\partial\Xi_0}{\partial\mbox{\boldmath $\nu$}}\Sigma\,dS
&
\displaystyle
=e(\tau)
-m\theta_0\tau\int_{\Omega\setminus\overline D}\Sigma\nabla\cdot\mbox{\boldmath $R$}\,dx
+e^{-\tau T}\int_{\Omega\setminus\overline D}(h-h_0)\Sigma\,dx.
\end{array}
\tag {2.20}$$ On the other hand, it follows from (2.11) that $$\begin{array}{ll}
\displaystyle
\theta_0\tau\int_{\partial D}s(\mbox{\boldmath $w$}_0,\Xi_0)\mbox{\boldmath $\nu$}\cdot
\mbox{\boldmath $R$}\,dS
&
\displaystyle
=\theta_0\tau E(\tau)+m\theta_0\tau\int_{\Omega\setminus\overline D}\Sigma\nabla\cdot\mbox{\boldmath $R$}\,dx
\\
\\
\displaystyle
&
\displaystyle
\,\,\,
+\rho \theta_0\tau e^{-\tau T}\int_{\Omega\setminus\overline D}(\mbox{\boldmath $F$}
-
\mbox{\boldmath $F$}_0)\cdot\mbox{\boldmath $R$}\,dx.
\end{array}
\tag {2.21}$$ Summing both sides of (2.20) and (2.21), we obtain $$\begin{array}{l}
\displaystyle
\,\,\,\,\,\,
\int_{\partial D}k\frac{\partial\Xi_0}{\partial\mbox{\boldmath $\nu$}}\Sigma\,dS
+\theta_0\tau\int_{\partial D}s(\mbox{\boldmath $w$}_0,\Xi_0)\mbox{\boldmath $\nu$}\cdot
\mbox{\boldmath $R$}\,dS
\\
\\
\displaystyle
=e(\tau)+\theta_0\tau E(\tau)\\
\\
\displaystyle
\,\,\,
+\rho e^{-\tau T}\int_{\Omega\setminus\overline D}(h-h_0)\Sigma\,dx
+\rho \theta_0\tau e^{-\tau T}\int_{\Omega\setminus\overline D}(\mbox{\boldmath $F$}
-
\mbox{\boldmath $F$}_0)\cdot\mbox{\boldmath $R$}\,dx.
\end{array}
\tag{2.22}$$ Rewrite this right-hand side as $$\begin{array}{l}
\displaystyle
\,\,\,\,\,\,
\int_{\Omega\setminus\overline D}
\left(k\vert\nabla\Sigma\vert^2
+c\tau\left\vert
\Sigma+\frac{\rho e^{-\tau T}}{2c\tau}(h-h_0)\right\vert^2\right)\,dx
\\
\\
\displaystyle
\,\,\,
+\theta_0\tau
\int_{\Omega\setminus\overline D}
\left(2\mu\left\vert\text{Sym}\,\nabla\mbox{\boldmath $R$}\right\vert^2+\lambda\vert\nabla\cdot\mbox{\boldmath $R$}\vert^2
+\rho\tau^2
\left\vert\mbox{\boldmath $R$}+\frac{e^{-\tau T}}{2\tau^2}(\mbox{\boldmath $F$}-\mbox{\boldmath $F$}_0)\right\vert^2\right)\,dx\\
\\
\displaystyle
-\frac{\rho^2 e^{-2\tau T}}{4c\tau}\Vert h-h_0\Vert^2_{L^2(\Omega\setminus\overline D)}
-\frac{\rho\theta_0 e^{-2\tau T}}{4\tau}\Vert\mbox{\boldmath $F$}-\mbox{\boldmath $F$}_0\Vert^2_{L^2(\Omega\setminus\overline D)}.
\end{array}$$ Since $\Vert h-h_0\Vert_{L^2(\Omega)}=O(1)$ and $\Vert\mbox{\boldmath $F$}-\mbox{\boldmath $F$}_0\Vert_{L^2(\Omega\setminus\overline D)}=O(\tau)$, it follows from this expression and (2.22) that $$\displaystyle
e(\tau)+\theta_0\tau E(\tau)
\le
\int_{\partial D}k\frac{\partial\Xi_0}{\partial\mbox{\boldmath $\nu$}}\Sigma\,dS
+\theta_0\tau\int_{\partial D}s(\mbox{\boldmath $w$}_0,\Xi_0)\mbox{\boldmath $\nu$}\cdot
\mbox{\boldmath $R$}\,dS+O(\tau e^{-2\tau T}).
\tag {2.23}$$ Take a lifting $\tilde{\Sigma}$ of $\Sigma\vert_{\partial D}$ into $D$ such that $\Vert\tilde{\Sigma}\Vert_{H^1(D)}
\le C\Vert\Sigma\Vert_{H^1(\Omega\setminus\overline D)}$, where $C$ is a positive constant independent of $\Sigma$. Then, the second equation on (2.3) gives $$\begin{array}{ll}
\displaystyle
\int_{\partial D}k\frac{\partial\Xi_0}{\partial\mbox{\boldmath $\nu$}}\Sigma\,dS
&
\displaystyle
=\int_{D}
(c\tau\Xi_0-m\theta_0\tau\nabla\cdot\mbox{\boldmath $w$}_0)\tilde{\Sigma}\,dx
+\int_D k\nabla\Xi_0\cdot\nabla\tilde{\Sigma}\,dx\\
\\
\displaystyle
\,\,\,
&
\displaystyle
\,\,\,
+e^{-\tau T}\int_D h_0\tilde{\Sigma}\,dx.
\end{array}
\tag {2.24}$$ Here from (2.8), (2.14), (2.15) we have $$\left\{
\begin{array}{l}
\Vert\Sigma\Vert _{H^1(\Omega\setminus\overline D)}=O(e(\tau)^{1/2}),\\
\\
\displaystyle
\Vert\Xi_0\Vert_{L^2(D)}=O(\tau^{-1/2}j(\tau)^{1/2}),
\\
\\
\displaystyle
\Vert\nabla\Xi_0\Vert_{L^2(D)}=O(j(\tau)^{1/2}),
\\
\\
\displaystyle
\Vert\nabla\cdot\mbox{\boldmath $w$}_0\Vert_{L^2(D)}
=O(J(\tau)^{1/2}).
\end{array}
\right.$$ Applying these to the right-hand side on (2.24), we obtain $$\begin{array}{l}
\,\,\,\,\,\,
\displaystyle
\int_{\partial D}k\frac{\partial\Xi_0}{\partial\mbox{\boldmath $\nu$}}\Sigma\,dS
\le C e(\tau)^{1/2}(\tau^{1/2}j(\tau)^{1/2}+\tau J(\tau)^{1/2}+e^{-\tau T}).
\end{array}
\tag {2.25}$$ A similar technique together with the first equation on (2.3) gives also $$\begin{array}{l}
\displaystyle
\,\,\,\,\,\,
\theta_0\tau\int_{\partial D}s(\mbox{\boldmath $w$}_0,\Xi_0)\mbox{\boldmath $\nu$}\cdot
\mbox{\boldmath $R$}\,dS\\
\\
\displaystyle
\le C\tau\Vert\mbox{\boldmath $R$}\Vert_{H^1(\Omega\overline D)}
\left(\tau^2\Vert\mbox{\boldmath $w$}_0\Vert_{L^2(D)}
+\Vert\nabla\mbox{\boldmath $w$}_0\Vert_{L^2(D)}+
\Vert\nabla\Xi_0\Vert_{L^2(D)}
+\tau e^{-\tau T}\right).
\end{array}$$ Korn’s second inequality [@DuL] tells us that $$\left\{
\begin{array}{l}
\displaystyle
\Vert\mbox{\boldmath $w$}_0\Vert_{H^1(D)}
\le C''
\left(\Vert\mbox{\boldmath $w$}_0\Vert_{L^2(D)}^2
+\Vert\text{Sym}\,\nabla\mbox{\boldmath $w$}_0\Vert_{L^2(D)}^2\right)^{1/2},
\\
\\
\displaystyle
\Vert\mbox{\boldmath $R$}\Vert_{H^1(\Omega\setminus\overline D)}
\le C''
\left(\Vert\mbox{\boldmath $R$}\Vert_{L^2(\Omega\setminus\overline D)}^2
+\Vert\text{Sym}\,\nabla\mbox{\boldmath $R$}\Vert_{L^2(\Omega\setminus\overline D)}^2\right)^{1/2},
\end{array}
\right.
\tag {2.26}$$ where $C''$ is a positive constant independent of $\mbox{\boldmath $w$}_0$ and $\mbox{\boldmath $R$}$. Here we note that, for all real $3\times 3$-matrix $A$ we have $$\begin{array}{l}
\displaystyle
\,\,\,\,\,\,
\vert\text{Sym}\,A\vert^2
\\
\\
\displaystyle
\le
2\left(\left\vert\text{Sym}\,A-\frac{\text{trace}\,A}{3}I_3\right\vert^2+\frac{\vert\text{trace}\,A\vert^2}{3}
\right)
\\
\\
\displaystyle
=\frac{2}{2\mu}\cdot 2\mu\left\vert\text{Sym}\,A-\frac{\text{trace}\,A}{3}I_3\right\vert^2+
\frac{2}{3\lambda+2\mu}\cdot\frac{3\lambda+2\mu}{3}\vert\text{trace}\,A\vert^2\\
\\
\displaystyle
\le 2\max\left\{\frac{1}{2\mu}, \frac{1}{3\lambda+2\mu}\right\}
\left(
2\mu\left\vert\text{Sym}\,A-\frac{\text{trace}\,A}{3}I_3\right\vert^2+
\frac{3\lambda+2\mu}{3}\vert\text{trace}\,A\vert^2
\right).
\end{array}$$ Applying identity (2.12) to this right-hand side, we obtain $$\displaystyle
\vert\text{Sym}\,A\vert^2
\le C
\left(2\mu\vert\text{Sym}\,A\vert^2+\lambda\vert\text{trace}\,A\vert^2\right)
\tag {2.27}$$ provided $\mu>0$ and $3\lambda+2\mu>0$ with a suitable positive constant $C=C(2\mu,3\lambda+2\mu)$.
Thus, substituting $A=\nabla\mbox{\boldmath $w$}_0, \nabla\mbox{\boldmath $R$}$ into (2.27) and using (2.26), repectively, for all $\tau$ with $\rho\tau^2\ge 1$ we have $$\left\{
\begin{array}{l}
\displaystyle
\Vert\mbox{\boldmath $w$}_0\Vert_{H^1(D)}
=O(J(\tau)^{1/2}),\\
\\
\displaystyle
\Vert\mbox{\boldmath $R$}\Vert_{H^1(\Omega\setminus\overline D)}
=O(E(\tau)^{1/2}).
\end{array}
\right.
\tag {2.28}$$ We have also the following trivial estimates: $$\left\{
\begin{array}{l}
\displaystyle
\Vert\mbox{\boldmath $w$}_0\Vert_{L^2(D)}=O(\tau^{-1}J(\tau)^{1/2}),\\
\\
\displaystyle
\Vert\nabla\Xi_0\Vert_{L^2(D)}=O(j(\tau)^{1/2}).
\end{array}
\right.$$ Hence we obtain $$\displaystyle
\theta_0\tau\int_{\partial D}s(\mbox{\boldmath $w$}_0,\Xi_0)\mbox{\boldmath $\nu$}\cdot
\mbox{\boldmath $R$}\,dS
\le C^{'''}\tau E(\tau)^{1/2}(\tau J(\tau)^{1/2}+j(\tau)^{1/2}+\tau e^{-\tau T}).
\tag {2.29}$$ Summing both sides on (2.25) and (2.29) and then from (2.23) we obtain $$\begin{array}{ll}
\displaystyle
\,\,\,\,\,\,
e(\tau)+\theta_0\tau E(\tau)
&
\displaystyle
\le C_1 e(\tau)^{1/2}(\tau^{1/2}j(\tau)^{1/2}+\tau J(\tau)^{1/2}+e^{-\tau T})\\
\\
\displaystyle
&
\,\,\,
+C_2\tau E(\tau)^{1/2}(\tau J(\tau)^{1/2}+j(\tau)^{1/2}+\tau e^{-\tau T})
+C_3\tau e^{-2\tau T}.
\end{array}$$ Now a standard technique gives (2.19).
$\Box$
Three special solutions and their bounds
----------------------------------------
In this subsection we introduce three functions and describe their upper and lower bounds.
Let $\mbox{\boldmath $w$}_{s0}\in H^1(\Bbb R^3)^3$ solve $$\begin{array}{ll}
\displaystyle
\mu\Delta\mbox{\boldmath $w$}_{s0}-\rho\tau^2\mbox{\boldmath $w$}_{s0}
+\rho\mbox{\boldmath $v$}_0(x)=\mbox{\boldmath $0$} & \text{in $\Bbb R^3$,}
\end{array}$$ where $\mbox{\boldmath $v$}_0(x)=\partial_t\mbox{\boldmath $v$}_s(x,0)$ and $\mbox{\boldmath $v$}_s$ is giveny by (1.10). Since we have $$\displaystyle
\mbox{\boldmath $v$}_0(x)=-2\chi_B(x)(x-p)\times\mbox{\boldmath $a$},$$ $\mbox{\boldmath $w$}_{s0}$ takes the form $$\begin{array}{ll}
\displaystyle
\mbox{\boldmath $w$}_{s0}(x)
&
\displaystyle
=-\frac{\rho}{2\pi}
\left(
\int_B\frac{e^{-\tau\sqrt{\rho/\mu}\,\vert x-y\vert}}{\vert x-y\vert}\,(y-p)dy\right)\times\mbox{\boldmath $a$}
\end{array}
\tag {2.30}$$ and we have $\nabla\cdot\mbox{\boldmath $w$}_{s0}=0$.
Second, let $\mbox{\boldmath $w$}_{p0}\in H^1(\Bbb R^3)^3$ solve $$\begin{array}{ll}
\displaystyle
(\lambda+2\mu)\Delta\mbox{\boldmath $w$}_{p0}-\rho\tau^2\mbox{\boldmath $w$}_{p0}
+\rho\mbox{\boldmath $v$}_0(x)=\mbox{\boldmath $0$} & \text{in $\Bbb R^3$,}
\end{array}$$ where $\mbox{\boldmath $v$}_0(x)=\partial_t\mbox{\boldmath $v$}_p(x,0)$ and $\mbox{\boldmath $v$}_p$ is given by (1.15). Since we have $$\displaystyle
\mbox{\boldmath $v$}_0(x)=-2\chi_B(x)(\eta-\vert x-p\vert)\frac{x-p}{\vert x-p\vert},$$ one gets the expression $$\begin{array}{ll}
\displaystyle
\mbox{\boldmath $w$}_{p0}(x)
&
\displaystyle
=-\frac{\rho}{2\pi}
\int_B\frac{e^{-\tau\sqrt{\rho/(\lambda+2\mu)}\,\vert x-y\vert}}{\vert x-y\vert}\,(\eta-\vert y-p\vert)\frac{y-p}{\vert y-p\vert}\,dy
\end{array}
\tag {2.31}$$ and we have $\nabla\times\mbox{\boldmath $w$}_{p0}=\mbox{\boldmath $0$}$.
Third, let $\Theta_{00}\in H^1(\Bbb R^3)$ solve $$\begin{array}{ll}
\displaystyle
k\Delta\Theta_{00}-c\tau\Theta_{00}+cf_0=0 & \text{in $\Bbb R^3$,}
\end{array}$$ where $f_0(x)=\Theta_0(x,0)$ and $\Theta_0$ is the solution of (1.16). Then, we have the expression $$\begin{array}{ll}
\displaystyle
\Theta_{00}(x)
&
\displaystyle
=\frac{c}{4\pi k}
\int_B\frac{e^{-\sqrt{\tau}\sqrt{c/k}\,\vert x-y\vert}}{\vert x-y\vert}\,(\eta-\vert y-p\vert)^2\,dy.
\end{array}
\tag {2.32}$$ It is easy to see that from the expression (2.30) to (2.32) one gets the following estimates.
Let $U$ be an arbitrary bounded open subset of $\Bbb R^3$ such that $\overline B\cap\overline U=\emptyset$. Then, we have, as $\tau\longrightarrow\infty$ $$\left\{
\begin{array}{l}
\displaystyle
\tau
\Vert
\mbox{\boldmath $w$}_{s0}
\Vert_{L^2(U)}
+
\Vert\nabla
\mbox{\boldmath $w$}_{s0}
\Vert_{L^2(U)}
=O(\tau e^{-\tau\sqrt{\rho/\mu}\,\text{dist}\,(U,B)}),\\
\\
\displaystyle
\tau
\Vert
\mbox{\boldmath $w$}_{p0}
\Vert_{L^2(U)}
+
\Vert\nabla
\mbox{\boldmath $w$}_{p0}
\Vert_{L^2(U)}
=O(\tau e^{-\tau\sqrt{\rho/(\lambda+2\mu)}\,\text{dist}\,(U,B)}),
\\
\\
\displaystyle
\sqrt{\tau}
\Vert
\Theta_{00}
\Vert_{L^2(U)}
+
\Vert\nabla
\Theta_{00}
\Vert_{L^2(U)}
=O(\sqrt{\tau}\,e^{-\sqrt{\tau}\,\sqrt{c/k}\,\text{dist}\,(U,B)}).
\end{array}
\right.
\tag {2.33}$$
The estimates in the next two lemmas play the key role in this paper.
Let $R>\eta$. Then, there exist positive numbers $C$ and $\tau_0$ such that, for all $\tau\ge\tau_0$ and $x\in\Bbb R^3\setminus\overline D$ with $\vert x-p\vert\le R$ $$\displaystyle
\vert\mbox{\boldmath $w$}_{so}(x)\vert
\ge C\tau^{-1}e^{-\tau\sqrt{\rho/\mu}\,(\vert x-p\vert-\eta)}\,
\left\vert\frac{x-p}{\vert x-p\vert}\times\mbox{\boldmath $a$}\right\vert
\tag {2.34}$$ and $$\displaystyle
\vert\mbox{\boldmath $w$}_{po}(x)\vert
\ge C\tau^{-1}e^{-\tau\sqrt{\rho/(\lambda+2\mu)}\,(\vert x-p\vert-\eta)}.
\tag {2.35}$$
From (2.30) and (2.31) one gets $$\begin{array}{ll}
\displaystyle
\mbox{\boldmath $w$}_{s0}(x)
&
\displaystyle
=-\frac{\rho}{2\pi}\mbox{\boldmath $I$}_1(x;\tau\sqrt{\rho/\mu})\times\mbox{\boldmath $a$}
\end{array}
\tag {2.36}$$ and $$\begin{array}{ll}
\displaystyle
\mbox{\boldmath $w$}_{p0}(x)
&
\displaystyle
=-\frac{\rho}{2\pi}
(\eta\mbox{\boldmath $I$}_0(x;\tau\sqrt{\rho/(\lambda+2\mu)})-\mbox{\boldmath $I$}_1(x;\tau\sqrt{\rho/(\lambda+2\mu)})),
\end{array}
\tag {2.37}$$ where $$\left\{
\begin{array}{l}
\displaystyle
\mbox{\boldmath $I$}_0(x;\tau)=\int_B\frac{e^{-\tau\,\vert x-y\vert}}{\vert x-y\vert}\frac{y-p}{\vert y-p\vert}\,dy,
\\
\\
\displaystyle
\mbox{\boldmath $I$}_1(x;\tau)=\int_B\frac{e^{-\tau\,\vert x-y\vert}}{\vert x-y\vert}(y-p)\,dy.
\end{array}
\right.
\tag {2.38}$$ It follows from the middle equation on (A.3) and (A.5) that $$\begin{array}{l}
\displaystyle
\mbox{\boldmath $I$}_1(x;\tau)
\sim
\frac{4\pi}{\tau^2}
\left(\vert x-p\vert^2\cdot\tau\eta\cdot\frac{e^{\tau\eta}}{2}
+\frac{1}{\tau^2}\cdot(\tau\eta)^3\cdot\frac{e^{\tau\eta}}{2}
\right)
\cdot\frac{e^{-\tau\vert x-p\vert}}{\vert x-p\vert^2}\cdot
\frac{x-p}{\vert x-p\vert}\\
\\
\displaystyle
=\frac{2\pi\eta}{\tau}(\vert x-p\vert^2+\eta^2)e^{\tau\eta}\frac{x-p}{\vert x-p\vert}.
\end{array}$$ A combination of this and (2.36) yields (2.34).
(2.35) is proved as follows. From the first equation on (A.3) and (A.4) we have $$\begin{array}{l}
\displaystyle
\eta\mbox{\boldmath $I$}_0(x;\tau)
\sim
\pi\left(\frac{2}{\tau}\cdot\vert x-p\vert^2\cdot\frac{e^{\tau\eta}}{2}
+\frac{2}{\tau^3}\cdot(\tau\eta)^2\cdot\frac{e^{\tau\eta}}{2}
\right)
\frac{e^{-\tau\vert x-p\vert}}{\vert x-p\vert^2}\cdot\frac{x-p}{\vert x-p\vert}
\\
\\
\displaystyle
=\frac{\pi}{\tau}(\vert x-p\vert^2+\eta^2)\cdot\frac{e^{-\tau\,(\vert x-p\vert-\eta)}}{\vert x-p\vert^2}\cdot\frac{x-p}{\vert x-p\vert}.
\end{array}$$ From the second equation on (A.3) and (A.5) we have $$\begin{array}{l}
\displaystyle
\mbox{\boldmath $I$}_1(x;\tau)
\sim
\frac{4\pi}{\tau^2}
\left(\vert x-p\vert^2\cdot\tau\eta\cdot\frac{e^{\tau\eta}}{2}
+\frac{1}{\tau^2}\cdot(\tau\eta)^3\cdot\frac{e^{\tau\eta}}{2}
\right)
\frac{e^{-\tau\vert x-p\vert}}{\vert x-p\vert^2}\cdot\frac{x-p}{\vert x-p\vert}
\\
\\
\displaystyle
=\frac{2\pi}{\tau}(\vert x-p\vert^2+\eta^2)\cdot\frac{e^{-\tau\,(\vert x-p\vert-\eta)}}{\vert x-p\vert^2}\cdot\frac{x-p}{\vert x-p\vert}.
\end{array}$$ Thus, we obtain $$\displaystyle
\eta\mbox{\boldmath $I$}_0(x;\tau)
-\mbox{\boldmath $I$}_1(x;\tau)
\sim
-\frac{\pi}{\tau}(\vert x-p\vert^2+\eta^2)\cdot\frac{e^{-\tau\,(\vert x-p\vert-\eta)}}{\vert x-p\vert^2}\cdot\frac{x-p}{\vert x-p\vert}.$$ Then (2.37) yields the desired estimate.
$\Box$
There exist positive constants $C$ and $\tau_0$ such that fo all $\tau\ge\tau_0$ and all $x\in\Bbb R^3\setminus\overline B$, $$\displaystyle
\Theta_{00}(x)\ge C\tau^{-3/2}\frac{e^{-\sqrt{\tau}\sqrt{c/k}\,(\vert x-p\vert-\eta)}}{\vert x-p\vert}.
\tag {2.39}$$ From (2.32) one has the expression $$\begin{array}{ll}
\displaystyle
\Theta_{00}(x)
&
\displaystyle
=\frac{c}{4\pi k}
(\eta^2 I_0(x;\sqrt{\tau}\sqrt{c/k})-2\eta I_1(x;\sqrt{\tau}\sqrt{c/k})+I_2(x;\sqrt{\tau}\sqrt{c/k})),
\end{array}
\tag {2.40}$$ where $$\displaystyle
I_j(x;\tau)
=\int_B\frac{e^{-\tau\,\vert x-y\vert}}{\vert x-y\vert}\,\vert y-p\vert^j\,dy.
\tag {2.41}$$ From (A.1), (A.2), the last equation on (A.3) and (A.6) we have: $$\displaystyle
\eta^2I_0(x;\tau)
\sim
\frac{4\pi\eta^2}{\tau^3}\cdot
(\tau\eta-1)\frac{e^{\tau\eta}}{2}
\cdot\frac{e^{-\tau\vert x-p\vert}}{\vert x-p\vert}
=\frac{2\pi\eta^2}{\tau^3}(\tau\eta-1)
\cdot
\frac{e^{-\tau(\vert x-p\vert-\eta)}}{\vert x-p\vert};$$ $$\displaystyle
2\eta I_1(x;\tau)
\sim
\frac{8\pi\eta}{\tau^4}
\cdot
\tau\eta(\tau\eta-2)
\frac{e^{\tau\eta}}{2}
\cdot\frac{e^{-\tau\vert x-p\vert}}{\vert x-p\vert}
=\frac{4\pi\eta^2}{\tau^3}(\tau\eta-2)\cdot\frac{e^{-\tau(\vert x-p\vert-\eta)}}{\vert x-p\vert};$$ $$\displaystyle
I_2(x;\tau)
\sim\frac{4\pi}{\tau^5}\cdot
(\tau\eta)^2(\tau\eta+6-3)
\frac{e^{\tau\eta}}{2}
\cdot
\frac{e^{-\tau\vert x-p\vert}}{\vert x-p\vert}
=\frac{2\pi\eta^2}{\tau^3}\cdot
(\tau\eta+3)
\frac{e^{-\tau(\vert x-p\vert-\eta)}}{\vert x-p\vert}.$$ Thus we have $$\displaystyle
\eta^2I_0(x;\tau)
-2\eta I_1(x;\tau)
+I_2(x;\tau)
\sim \frac{12\pi\eta^2}{\tau^3}\frac{e^{-\tau(\vert x-p\vert-\eta)}}{\vert x-p\vert}.$$ A combination of this and (2.40) gives the desired conclusion.
$\Box$
Bounds for $J(\tau)$ and $j(\tau)$ for special $\mbox{\boldmath $v$}$ and $\Theta$
----------------------------------------------------------------------------------
The integrals $J(\tau)$ and $j(\tau)$ given by (2.8) and (2.14), respectively, depends on $m$ and the pair $\mbox{\boldmath $v$}$ and $\Theta$ which is a solution of (1.2) with (1.3). To make it clear, in this subsection, we denote $J(\tau)$ and $j(\tau)$ by $J_m(\tau;\mbox{\boldmath $v$},\Theta)$ and $j_m(\tau;\mbox{\boldmath $v$},\Theta)$, repectively. The same remark works also for the pair of $\mbox{\boldmath $w$}_0$ and $\Xi_0$ which are given by the second quations on (1.8) and (1.9). We denote them by $\mbox{\boldmath $w$}_0^m(\,\cdot\,,\mbox{\boldmath $v$},\Theta)$ and $\Xi_0^m(\,\cdot\,,\mbox{\boldmath $v$},\Theta)$, respectively.
Let $U$ be a bounded open subset of $\Bbb R^3$ with $\overline B\cap\overline U=\emptyset$. Then, we have, as $\tau\longrightarrow\infty$ $$\displaystyle
\tau\Vert\mbox{\boldmath $w$}_0^m(\,\cdot\,,\mbox{\boldmath $v$}_s,0)\Vert_{L^2(U)}+
\Vert\nabla\mbox{\boldmath $w$}_0^m(\,\cdot\,,\mbox{\boldmath $v$}_s,0)\Vert_{L^2(U)}
=O(\tau e^{-\tau\sqrt{\rho/\mu}\,\text{dist}\,(U,B)}+e^{-\tau T}),
\tag {2.42}$$ $$\displaystyle
\tau\Vert\mbox{\boldmath $w$}_0^0(\,\cdot\,,\mbox{\boldmath $v$}_p,0\Vert_{L^2(U)}+
\Vert\nabla\mbox{\boldmath $w$}_0^0(\,\cdot\,,\mbox{\boldmath $v$}_p,0)\Vert_{L^2(U)}
=O(\tau e^{-\tau\sqrt{\rho/(\lambda+2\mu)}\,\text{dist}\,(U,B)}+e^{-\tau T}),
\tag {2.43}$$ $$\displaystyle
\sqrt{\tau}\Vert\Xi_0^0(\,\cdot\,,\mbox{\boldmath $0$},\Theta_0)\Vert_{L^2(U)}+
\Vert\nabla\Xi_0^0(\,\cdot\,,\mbox{\boldmath $0$},\Theta_0)\Vert_{L^2(U)}
=O(\sqrt{\tau}e^{-\sqrt{\tau}\sqrt{c/k}\,\text{dist}\,(U,B)}+e^{-\tau T}).
\tag {2.44}$$
[*Proof.*]{} First we give a proof of (2.42). Set $$\displaystyle
\mbox{\boldmath $\epsilon$}_s=e^{\tau T}(\mbox{\boldmath $w$}_0-\mbox{\boldmath $w$}_{s0}).$$ We have $$\displaystyle
\mbox{\boldmath $w$}_0=\mbox{\boldmath $w$}_{s0}+e^{-\tau T}
\mbox{\boldmath $\epsilon$}_s
\tag {2.45}$$ and $\nabla\cdot\mbox{\boldmath $w$}_0=0$ since $\nabla\cdot\mbox{\boldmath $v$}_s=0$. These together with $\nabla\cdot\mbox{\boldmath $w$}_{s0}=0$ yield $\nabla\cdot\mbox{\boldmath $\epsilon$}_s=0$. Moreover, we have $\Xi_0=0$. Then, from the first equation on (2.3) we have $$\begin{array}{ll}
\displaystyle
(\mu\Delta-\rho\tau^2)\mbox{\boldmath $\epsilon$}_s=\rho\mbox{\boldmath $F$}_0 & \text{in}\,\Bbb R^3.
\end{array}
\tag {2.46}$$ Then, using the third equation on (2.4), we can easily see that $$\displaystyle \tau\Vert\mbox{\boldmath $\epsilon$}_s\Vert_{L^2(\Bbb R^3)}+
\Vert\nabla\mbox{\boldmath $\epsilon$}_s\Vert_{L^2(\Bbb R^3)}=O(1).
\tag {2.47}$$ A combination of this and the first estimate on (2.33) yields (2.42).
Next set $$\displaystyle
\mbox{\boldmath $\epsilon$}_p=e^{\tau T}(\mbox{\boldmath $w$}_0-\mbox{\boldmath $w$}_{p0}).$$ We have $$\displaystyle
\mbox{\boldmath $w$}_0=\mbox{\boldmath $w$}_{p0}+e^{-\tau T}
\mbox{\boldmath $\epsilon$}_p$$ and $\nabla\times\mbox{\boldmath $w$}_0=\mbox{\boldmath $0$}$ since $\nabla\times\mbox{\boldmath $v$}_p=\mbox{\boldmath $0$}$. These together with $\nabla\times\mbox{\boldmath $w$}_{p0}=\mbox{\boldmath $0$}$ yield $\nabla\times\mbox{\boldmath $\epsilon$}_p=\mbox{\boldmath $0$}$. Then, from the first equation on (2.3) with $m=0$ and the equation $\nabla(\nabla\cdot\mbox{\boldmath $\epsilon$}_p)=\Delta\mbox{\boldmath $\epsilon$}_p$, we have $$\begin{array}{ll}
\displaystyle
\{(\lambda+2\mu)\Delta-\rho\tau^2\}\mbox{\boldmath $\epsilon$}_p=\rho\mbox{\boldmath $F$}_0 & \text{in}\,\Bbb R^3.
\end{array}$$ Then, we have $$\displaystyle \tau\Vert\mbox{\boldmath $\epsilon$}_p\Vert_{L^2(\Bbb R^3)}+
\Vert\nabla\mbox{\boldmath $\epsilon$}_p\Vert_{L^2(\Bbb R^3)}=O(1).$$ A combination of this and the middle estimate on (2.33) yields (2.43). (2.44) is also a consequence of the last estimate on (2.33) and the second equation on (2.3) with $m=0$.
$\Box$
\(i) We have $$\left\{
\begin{array}{ll}
\displaystyle
J_m(\tau;\mbox{\boldmath $v$}_s,0)
\displaystyle
=O(\tau^2 e^{-2\tau\sqrt{\rho/\mu}\,\text{dist}\,(D,B)}+e^{-2\tau T})
&
\text{as $\tau\longrightarrow\infty$,}
\\
\\
\displaystyle
j_m(\tau;\mbox{\boldmath $v$}_s,0)=0
&
\text{for all $\tau>0$.}
\end{array}
\right.
\tag {2.48}$$ (ii) We have, as $\tau\longrightarrow\infty$ $$\left\{\begin{array}{l}
\displaystyle
J_0(\tau;\mbox{\boldmath $v$}_p,0)
\displaystyle
=O(\tau^2 e^{-2\tau\sqrt{\rho/(\lambda+2\mu)}\,\text{dist}\,(D,B)}+e^{-2\tau T}),
\\
\\
\displaystyle
j_0(\tau;\mbox{\boldmath $0$},\Theta_0)=O(\tau e^{-2\sqrt{\tau}\,\sqrt{c/k}\,\text{dist}\,(D,B)}+e^{-2\tau T}).
\end{array}
\right.$$ (iii) Let $T$ satisfies $$\displaystyle
T>\sqrt{\frac{\rho}{\mu}}\,\text{dist}\,(D,B).
\tag {2.49}$$ Then, there exist positive constants $\tau_0$ and $C$ such that, for all $\tau\ge\tau_0$ $$\displaystyle
\tau^{3} e^{2\tau\sqrt{\rho/\mu}\,\text{dist}\,(D,B)}J_m(\tau;\mbox{\boldmath $v$}_s,0)\ge C.
\tag {2.50}$$ (iv) Let $T$ satisfies $$\displaystyle
T>\sqrt{\frac{\rho}{\lambda+2\mu}}\,\text{dist}\,(D,B).$$ Then, there exist positive constants $\tau_0$ and $C$ such that, for all $\tau\ge\tau_0$ $$\displaystyle
\tau^{2} e^{2\tau\sqrt{\rho/(\lambda+2\mu)}\,\text{dist}\,(D,B)}J_0(\tau;\mbox{\boldmath $v$}_p,0)\ge C.$$ (v) Let $T$ be an arbitrary positive number. Then, there exist positive constants $\tau_0$ and $C$ such that, for all $\tau\ge\tau_0$ $$\displaystyle
\tau^{4} e^{2\sqrt{\tau}\,\sqrt{c/k}\,\text{dist}\,(D,B)}j_0(\tau;\mbox{\boldmath $0$},\Theta_0)\ge C.$$
[*Proof.*]{} Applying Proposition 2.4 in the case when $U=D$ to the expression (2.8) and (2.14) for $J(\tau)$ and $j(\tau)$, we obtain the first estimate on (2.48) and (ii). From $\Xi_0^m(\,\cdot\,;\mbox{\boldmath $v$}_s,0)=0$, we obtain $j_m(\tau;\mbox{\boldmath $v$}_s,0)=0$.
Next we give the proof of (iii). It follows from (2.8) and (2.45) and (2.47) $$\displaystyle
J_m(\tau;\mbox{\boldmath $v$}_s,0)
\ge \frac{1}{2}J_0(\tau)+O(e^{-2\tau T}),
\tag {2.51}$$ where $$\displaystyle
J_0(\tau)=\int_D(2\mu\left\vert\text{Sym}\,\nabla\mbox{\boldmath $w$}_{s0}\right\vert^2+\rho\tau^2\vert\mbox{\boldmath $w$}_{s0}\vert^2)dx.$$ From (2.34) in Lemma 2.2 we have $$\displaystyle
J_0(\tau)
\ge
C^2\int_D
e^{-2\tau\sqrt{\rho/\mu}\,(\vert x-p\vert-\eta)}
\left\vert\frac{x-p}{\vert x-p\vert}\times\mbox{\boldmath $a$}\right\vert^2
\,dx.$$ Applying Lemma A.1, we conclude that here exist positive constants $\tau_0$, $C'$ and $\kappa$ such that, for all $\tau\ge\tau_0$ $$\displaystyle
\tau^{\kappa}e^{2\tau\sqrt{\rho/\mu}\,\text{dist}\,(D,B)}\,J_0(\tau)\ge C'.$$ Now from this and (2.51) we see that (2.50) is valid under condition (2.49). Note that $\kappa=2$ or $\kappa=3$. Thus we have chosen the worst $\kappa$.
\(iv) and (v) are easy consequences of (2.35) in Lemma 2.2, (2.39) in Lemma 2.3, Lemma A.2.
$\Box$
[**Remark 2.2.**]{} (ii), (iv) and (v) are for the proof of Theorems 1.2 and 1.3.
Proof of Theorem 1.1.
=====================
Once we have the following lower and upper bounds for indicator function $I^1(\tau;\mbox{\boldmath $v$}_s,0)$ as $\tau\longrightarrow\infty$, then the proof of Theorem 1.1 is a due cource as we have done in [@IE4]. Just apply the first estimate on (2.48) and (iii) in Proposition 2.5.
Let $T$ be an arbitraly positive number. We have, as $\tau\longrightarrow\infty$ $$\displaystyle
I^1(\tau;\mbox{\boldmath $v$}_s,0)
=O(\tau^2 J_m(\tau;\mbox{\boldmath $v$}_s,0)
+\tau^2e^{-\tau T}e^{-\tau\sqrt{\rho/\mu}\,\text{dist}\,(\Omega,B)}
+\tau^2 e^{-2\tau T})
\tag {3.1}$$ and $$\displaystyle
I^1(\tau;\mbox{\boldmath $v$}_s,0)
\ge J_m(\tau;\mbox{\boldmath $v$}_s,0)
+O(\tau^2e^{-\tau T}e^{-\tau\sqrt{\rho/\mu}\,\text{dist}\,(\Omega,B)}
+\tau^2 e^{-2\tau T}).
\tag {3.2}$$
[*Proof.*]{} As pointed out in Section 1.1, we have $\Xi_0=0$ and hence $$\displaystyle
I^2(\tau;\mbox{\boldmath $v$}_s,0)=0.$$ Note also that a combination of the second equation on (1.8) and (1.10) gives $$\displaystyle
\nabla\cdot\mbox{\boldmath $w$}_0=0.$$ Thus from (2.17) we obtain $$\displaystyle
I^1(\tau;\mbox{\boldmath $v$}_s,0)
=J_m(\tau;\mbox{\boldmath $v$}_s,0)+
\left(E(\tau)
+\frac{1}{\theta_0\tau}
e(\tau)\right)
+{\cal R}(\tau),
\tag {3.3}$$ where ${\cal R}(\tau)$ is given by (2.18).
A combination of (2.19) and the second equation on (2.48) gives $$\displaystyle
E(\tau)+\frac{1}{\theta_0\tau}e(\tau)
=O(\tau^2J_m(\tau;\mbox{\boldmath $v$}_s,0)+\tau^2 e^{-2\tau T}).
\tag {3.4}$$ Thus, for the proof of (3.1) and (3.2) it sfficies to give an estimate of ${\cal R}(\tau)$. We show that, as $\tau\longrightarrow\infty$ $$\displaystyle
{\cal R}(\tau)
=O(\tau^2e^{-\tau T}e^{-\tau\sqrt{\rho/\mu}\,\text{dist}\,(\Omega,B)}
+\tau e^{-2\tau T}).
\tag {3.5}$$ A combination of (3.4) and the first estimate on (2.48) gives $$\displaystyle
E(\tau)+\frac{1}{\theta_0\tau}e(\tau)
=O(\tau^4 e^{-2\tau\sqrt{\rho/\mu}\,\text{dist}\,(D,B)}+\tau^2 e^{-2\tau T}).
\tag {3.6}$$ This and (2.9) yields $$\displaystyle
\Vert\mbox{\boldmath $R$}\Vert_{L^2(\Omega\setminus\overline D)}
=O(\tau e^{-\tau\sqrt{\rho/\mu}\,\text{dist}\,(D,B)}+e^{-\tau T}).$$ This together with the first equation on (2.2) gives $$\displaystyle
\int_{\Omega\setminus\overline D}\mbox{\boldmath $F$}\cdot\mbox{\boldmath $R$}\,dx
=O(\tau^2 e^{-\tau\sqrt{\rho/\mu}\,\text{dist}\,(D,B)}+\tau e^{-\tau T}).
\tag {3.7}$$ And also it follows from (2.2), (2.4) and (2.42) with $U=D, \Omega\setminus\overline D$ we obtain $$\displaystyle
\int_D\mbox{\boldmath $F$}_0\cdot\mbox{\boldmath $w$}_0dx=O(\tau e^{-\tau\sqrt{\rho/\mu}\,\text{dist}\,(D,B)}+e^{-\tau T})
\tag {3.8}$$ and $$\displaystyle
\int_{\Omega\setminus\overline D}(\mbox{\boldmath $F$}_0-\mbox{\boldmath $F$})
\cdot\mbox{\boldmath $w$}_0\,dx
=O(\tau e^{-\tau\sqrt{\rho/\mu}\,\text{dist}\,(\Omega,B)}+e^{-\tau T}).
\tag {3.9}$$ Applying (3.7), (3.8) and (3.9) to the right-hand side on (2.10), we obtain $$\displaystyle
{\cal R}^1(\tau)
=O(e^{-\tau T}(\tau^2 e^{-\tau\sqrt{\rho/\mu}\,\text{dist}\,(D,B)}+\tau e^{-\tau\sqrt{\rho/\mu}\,\text{dist}\,(\Omega,B)}
+\tau e^{-\tau T})).
\tag {3.10}$$
Note that ${\cal R}^2(\tau)$ becomes $$\displaystyle
{\cal R}^2(\tau)=e^{-\tau T}\int_{\Omega\setminus\overline D}h\Sigma\,dx.
\tag {3.11}$$ From (3.6) we have $$\displaystyle
e(\tau)
=O(\tau^5 e^{-2\tau\sqrt{\rho/\mu}\,\text{dist}\,(D,B)}+\tau^3 e^{-2\tau T}).$$ This together with (2.15) yields $$\displaystyle
\Vert\Sigma\Vert_{L^2(\Omega\setminus\overline D)}
=O(\tau^2e^{-\tau\sqrt{\rho/\mu}\,\text{dist}\,(D,B)}+\tau e^{-\tau T}).$$ Then, noting $h_0=0$ by (2.4) and $\Xi_0=0$, from the second equation on (2.2) and (3.11), we obtain $$\displaystyle
\frac{1}{\theta_0\tau}{\cal R}^2(\tau)
=O(e^{-\tau T}(\tau e^{-\tau\sqrt{\rho/\mu}\,\text{dist}\,(D,B)}
+e^{-\tau T})).$$ Now a combination of this and (3.10) gives $$\displaystyle
{\cal R}(\tau)
=O(\tau^2e^{-\tau T}e^{-\tau\sqrt{\rho/\mu}\,\text{dist}\,(D,B)}+\tau e^{-\tau T}e^{-\tau\sqrt{\rho/\mu}\,\text{dist}\,(\Omega,B)}
+\tau e^{-2\tau T}).$$ Finally, applying the trivial estimate $\text{dist}\,(\Omega,B)\le\text{dist}\,(D,B)$ to the first term on this right-hand side, we obtain (3.5).
$\Box$
Some corollaries
================
Some remarks on corollaries of Theorem 1.1 should be mentioned.
We introduce another indicator function defined by the expression $$\begin{array}{ll}
\displaystyle
I^s(\tau;\mbox{\boldmath $v$}_s,0)
=
\int_{\partial\Omega}
s(\mbox{\boldmath $w$}_{s0},0)\mbox{\boldmath $\nu$}
\cdot
(\mbox{\boldmath $w$}-\mbox{\boldmath $w$}_{s0})\,dS,
&
\tau>0,
\end{array}$$ where $\mbox{\boldmath $w$}_{s0}\in H^1(\Bbb R^3)^3$ is given by (2.36) and the second equation on (A.3) explicitly.
Theorem 1.1 remains valid if $I^1(\tau;\mbox{\boldmath $v$}_s,0)$ is replaced with $I^s(\tau;\mbox{\boldmath $v$}_s,0)$.
[*Proof.*]{} Write $$\begin{array}{ll}
\displaystyle
s(\mbox{\boldmath $w$}_{0},0)\mbox{\boldmath $\nu$}
\cdot
(\mbox{\boldmath $w$}-\mbox{\boldmath $w$}_{0})
&
\displaystyle
=s(\mbox{\boldmath $w$}_{s0},0)\mbox{\boldmath $\nu$}
\cdot
(\mbox{\boldmath $w$}-\mbox{\boldmath $w$}_{0})
+
s(\mbox{\boldmath $w$}_{0}-\mbox{\boldmath $w$}_{s0},0)
\mbox{\boldmath $\nu$}
\cdot
(\mbox{\boldmath $w$}-\mbox{\boldmath $w$}_{0})\\
\\
\displaystyle
&
\displaystyle
=
s(\mbox{\boldmath $w$}_{s0},0)\mbox{\boldmath $\nu$}\cdot
(\mbox{\boldmath $w$}-\mbox{\boldmath $w$}_{s0})
+s(\mbox{\boldmath $w$}_{s0},0)\mbox{\boldmath $\nu$}
\cdot(\mbox{\boldmath $w$}_{s0}-\mbox{\boldmath $w$}_{0})
\\
\\
\displaystyle
&
\displaystyle
\,\,\,
+s(\mbox{\boldmath $w$}_0-\mbox{\boldmath $w$}_{s0},0)
\mbox{\boldmath $\nu$}
\cdot
(\mbox{\boldmath $w$}-\mbox{\boldmath $w$}_0)
\\
\\
\displaystyle
&
\displaystyle
=s(\mbox{\boldmath $w$}_{s0},0)\mbox{\boldmath $\nu$}\cdot
(\mbox{\boldmath $w$}-\mbox{\boldmath $w$}_{s0})
\\
\\
\displaystyle
&
\,\,\,
-e^{-\tau T}s(\mbox{\boldmath $w$}_{s0},0)\mbox{\boldmath $\nu$}
\cdot\mbox{\boldmath $\epsilon$}_{s}
+e^{-\tau T}s(\mbox{\boldmath $\epsilon$}_s,0)
\mbox{\boldmath $\nu$}
\cdot
(\mbox{\boldmath $w$}-\mbox{\boldmath $w$}_0),
\end{array}
\tag {4.1}$$ where $\mbox{\boldmath $\epsilon$}_s=e^{\tau T}(\mbox{\boldmath $w$}_0-\mbox{\boldmath $w$}_{s0})$. A combination of second estimate on (2.28) and (3.6), we have $$\displaystyle
\Vert\mbox{\boldmath $w$}-\mbox{\boldmath $w$}_0\Vert_{H^{1/2}(\partial\Omega)}=
O(\tau^2 e^{-\tau\sqrt{\rho/\mu}\,\text{dist}\,(D,B)}
+\tau e^{-\tau T}
).
\tag {4.2}$$ From (2.46) together with $\Vert\mbox{\boldmath $F$}_0\Vert_{L^2(\Bbb R^3)}=O(\tau)$ and (2.47) we have $\Vert\Delta\mbox{\boldmath $\epsilon$}_s\Vert_{L^2(\Bbb R^3)}=O(\tau)$ and hence $$\displaystyle
\Vert\mbox{\boldmath $\epsilon$}_s\Vert_{H^2(\Bbb R^3)}=O(\tau).
\tag {4.3}$$ This gives $$\displaystyle
\left\Vert
s(\mbox{\boldmath $\epsilon$}_s,0)\mbox{\boldmath $\nu$}\right\Vert_{H^{1/2}(\partial\Omega)}=O(\tau).$$ A combination of this and (4.2) gives $$\displaystyle
\int_{\partial\Omega}
s(\mbox{\boldmath $\epsilon$}_0,0)\mbox{\boldmath $\nu$}
\cdot(\mbox{\boldmath $w$}-\mbox{\boldmath $w$}_0)\,dS
=O(\tau^3e^{-\tau\sqrt{\rho/\mu}\,\text{dist}\,(D,B)}
+\tau^2e^{-\tau T}
).$$ And also from (2.30) and (4.3) we have $$\displaystyle
\int_{\partial\Omega}s(\mbox{\boldmath $w$}_{s0},0)\mbox{\boldmath $\nu$}
\cdot\mbox{\boldmath $\epsilon$}_s
\,dS
=O(\tau e^{-\tau\sqrt{\rho/\mu}\,\text{dist}\,(\Omega,B)}).$$ Using $\text{dist}\,(\Omega,B)<\text{dist}\,(D,B)$ and Integrating both sides on (4.1) and applying these, we obtain $$\displaystyle
I^1(\tau;\mbox{\boldmath $v$}_s,0)
=I^s(\tau;\mbox{\boldmath $v$}_s,0)
+O(\tau^3e^{-\tau(T+\sqrt{\rho/\mu}\,\text{dist}\,(\Omega,B))}+
\tau^2e^{-2\tau T}).$$ The bound for the last term on this right-hand side is essentially same as that of ${\cal R}(\tau)$ given in (3.5). Thus, we can easily see that all the statements of Theorem 1.1 are transplanted.
$\Box$
And also one can localize the place where the data are corrected. Given $M>0$ define the localized indicator function by the formula $$\begin{array}{ll}
\displaystyle
I^s(\tau;\mbox{\boldmath $v$}_s,0;M)
=
\int_{\partial\Omega(B,M)}s(\mbox{\boldmath $w$}_{s0},0)\mbox{\boldmath $\nu$}
\cdot
(\mbox{\boldmath $w$}-\mbox{\boldmath $w$}_{s0})\,dS,
&
\tau>0,
\end{array}$$ where $$\displaystyle
\partial\Omega(B,M)
=\{x\in\partial\Omega\,\vert\, d_B(x)<M\}$$ and $d_B(x)=\inf_{y\in B}\vert y-x\vert$.
The second corollary of Theorem 1.1 is the following.
Let $M$ satisfy $$\displaystyle
\text{dist}\,(D,B)<M.
\tag {4.4}$$ Let $T$ satisfy $$\displaystyle
T\ge \sqrt{\frac{\rho}{\mu}}\left(2M-\text{dist}\,(\Omega,B)\right).
\tag {4.5}$$ Then, statement (i) in Theorem 1.1 remains valid if $I^1(\tau;\mbox{\boldmath $v$}_s,0)$ is replaced with $I^s(\tau;\mbox{\boldmath $v$}_s,0;M)$.
[*Proof.*]{} From the expression (2.30), we have $$\displaystyle
\left\Vert
s(\mbox{\boldmath $w$}_{s0},0)\mbox{\boldmath $\nu$}
\right\Vert_{L^2(\partial\Omega\setminus\partial\Omega(M,B))}
=O(\tau e^{-\tau\sqrt{\rho/\mu}\,M}).$$ It follows from (4.2) and (4.3) that $$\displaystyle
\Vert\mbox{\boldmath $w$}-\mbox{\boldmath $w$}_{s0}\Vert_{L^2(\partial\Omega)}
=O(\tau^2 e^{-\tau\sqrt{\rho/\mu}\,\text{dist}\,(D,B)}+\tau e^{-\tau T}
).$$ A combination of these gives $$\displaystyle
I^s(\tau;\mbox{\boldmath $v$}_0,0)=I^s(\tau;\mbox{\boldmath $v$}_s,0;M)
+O(\tau^3e^{-\tau\sqrt{\rho/\mu}\,\text{dist}\,(D,B)}e^{-\tau\sqrt{\rho/\mu}\, M}
+\tau^2 e^{-\tau T}e^{-\tau\sqrt{\rho/\mu}\, M}
).$$ Then, we can check the validity of the statement in Corollary 4.2 by using Corollary 4.1 and the following facts.
$\bullet$ One can write $$\left\{\begin{array}{l}
\displaystyle
e^{2\tau\sqrt{\rho/\mu}\,\text{dist}\,(D,B)}\tau^3e^{-\tau\sqrt{\rho/\mu}\,\text{dist}\,(D,B)}e^{-\tau\sqrt{\rho/\mu}\,M}
=\tau^3 e^{-\tau\sqrt{\rho/\mu}\,(M-\text{dist}\,(D,B))},\\
\\
\displaystyle
e^{2\tau\sqrt{\rho/\mu}\,\text{dist}\,(D,B)}\tau^2 e^{-\tau T}e^{-\tau\sqrt{\rho/\mu}\,M}
=\tau^2 e^{-\tau\{T+\sqrt{\rho/\mu}\,(M-2\text{dist}\,(D,B))\}}.
\end{array}
\right.$$
$\bullet$ a combination of (4.4) and (4.5) implies $$\displaystyle
T+\sqrt{\frac{\rho}{\mu}}\,\left(M-2\text{dist}\,(D,B)\right)>0.$$
$\Box$
[**Remark 4.1.**]{} Corollaries 4.1 and 4.2 remain valid if the function $\mbox{\boldmath $w$}_{s0}$ in $\mbox{\boldmath $w$}-\mbox{\boldmath $w$}_{s0}$ of $I^s(\tau;\mbox{\boldmath $v$}_s,0)$ and $I^s(\tau;\mbox{\boldmath $v$}_s,0;M)$ is replaced with $\mbox{\boldmath $w$}_0^m(\,\cdot\,,\mbox{\boldmath $v$}_s,0)$ which depends on $T$ and given by the second equation on (1.8) with $(\mbox{\boldmath $v$},\Theta)=(\mbox{\boldmath $v$}_s,0)$. This is the original style stated in [@IE4] for the scalar wave equation.
Further problems
================
In Theorem 1.1 we made use of only the first indicator function $I^1(\tau;\mbox{\boldmath $v$}_s,0)$ which employs only a single displacement field observed on the surface of the body. It is based on the simple fact that the system (1.2) has special solutions $\mbox{\boldmath $v$}=\mbox{\boldmath $v$}_s$ with $\nabla\cdot\mbox{\boldmath $v$}=0$ and $\Theta=0$. This choice yields that the second indicatior function $I^2(\tau;\mbox{\boldmath $v$}_s,0)$ becomes identially $0$ and thus does not enable us to extract any information about the cavity $D$ from a single temperature field observed on the surface of the body. Needless to say there are other choices of special solutions of the system (1.2). For example, just solve (1.2) with the initial conditions $$\displaystyle
\left\{
\begin{array}{ll}
\displaystyle
\mbox{\boldmath $v$}(x,0)=\mbox{\boldmath $0$} & \text{in $\Bbb R^3$,}\\
\\
\displaystyle
\partial_t\mbox{\boldmath $v$}(x,0)=\mbox{\boldmath $0$} & \text{in $\Bbb R^3$,}\\
\\
\displaystyle
\Theta(x,0)=(\eta-\vert x-p\vert)^2\chi_B(x) & \text{in $\Bbb R^3$}
\end{array}
\right.$$ and take $f$ and $\mbox{\boldmath $G$}$ in (1.1) given by (1.4) and (1.5). Then it would be interested to clarify the information about the cavity contained in both the first and second indicator functions $I^1(\tau;\mbox{\boldmath $v$},\Theta)$ and $I^2(\tau;\mbox{\boldmath $v$},\Theta)$. This needs further analysis on the behaviour of the [*reflected solutions*]{} $\mbox{\boldmath $R$}$ and $\Sigma$ which are solutions of (2.11) and (2.16) and appear in the representation formula of the indicator functions (2.7) and (2.13). For the scalar wave equation case the analysis of the effect of the corresponding reflected solution on the asymptotic behaviour of the indicator function has been done in [@ICA; @IEO3; @IEE; @IMP] and see also [@IMax] for the Maxwell system. The governing equation of the present problem is a coupled system of the elastic wave and heat equations and it seems more difficult. We leave it for future research. Finally, it should be mentioned that the numerical implementation of the method of this paper belongs to our future plan.
$\quad$
[**Acknowledgments**]{}
The author was partially supported by Grant-in-Aid for Scientific Research (C)(No. 17K05331) of Japan Society for the Promotion of Science. The author thanks Hiromichi Itou for pointing out the reference [@D] and useful discussions. Some part of this work was initiated when the author stayed in ICUB (The Research Institute of the University of Bucharest), Bucharest, Romania for 1th Nov.-18th Nov. in 2016 with support by ICUB the visiting professors fellowship program.
Appendix
========
Explicit computation of integrals on (2.38) and (2.41)
------------------------------------------------------
The following formulae have been derived in [@IE4] as (A.5) and (A.6) therein. Note that the first formula on (A.1) below is also a result of the mean value theorem for the modified Helmholtz equation [@CH].
Let $x\in\Bbb R^3\setminus\overline B$. Then, we have $$\left\{
\begin{array}{l}
\displaystyle
I_0(x;\tau)=\frac{4\pi\varphi_0(\tau\eta)}{\tau^3}\frac{e^{-\tau\vert x-p\vert}}{\vert x-p\vert},\\
\\
\displaystyle
I_1(x;\tau)
=\frac{4\pi\varphi_1(\tau\eta)}{\tau^4}
\frac{e^{-\tau\vert x-p\vert}}{\vert x-p\vert},
\end{array}
\right.
\tag {A.1}$$ where $$\left\{\begin{array}{l}
\varphi_0(s)=s\cosh s-\sinh s,
\\
\\
\displaystyle
\varphi_1(s)=\left(s^2+2\right)\cosh s
-2s\sinh s
-2.
\end{array}
\right.
\tag {A.2}$$
In this appendix, we add the following formulae.
We have $$\left\{
\begin{array}{l}
\displaystyle
\mbox{\boldmath $I$}_0(x;\tau)
=
\pi \frac{e^{-\tau\,\vert x-p\vert}}{\vert x-p\vert^2}
K_{\tau}^0\,(\vert x-p\vert,\tau\eta)
\frac{x-p}{\vert x-p\vert},
\\
\\
\displaystyle
\mbox{\boldmath $I$}_1(x;\tau)
=\frac{4\pi}{\tau^2}
\frac{e^{-\tau\,\vert x-p\vert}}{\vert x-p\vert^2}
K_{\tau}^1(\vert x-p\vert,\tau\eta)\frac{x-p}{\vert x-p\vert},
\\
\\
\displaystyle
I_2(x;\tau)
=\frac{4\pi\varphi_2(\tau\eta)}{\tau^5}\frac{e^{-\tau\vert x-p\vert}}{\vert x-p\vert},
\end{array}
\right.
\tag {A.3}$$ where $$\begin{array}{ll}
\displaystyle
K_{\tau}^0(\xi,s)
&
\displaystyle
=\frac{2}{\tau}\left\{
\left(1-\frac{1}{\tau}\right)\xi^2-\frac{2}{\tau^2}\xi-\frac{2}{\tau^3}
\right\}(\cosh s-1)\\
\\
\displaystyle
&
\displaystyle
\,\,\,
+\frac{2}{\tau^3}\left(1-\frac{1}{\tau}\right)
\left\{(s^2+2)\cosh s-2s\sinh s-2\right\}\\
\\
\displaystyle
&
\displaystyle
\,\,\,
+\frac{4}{\tau^3}(s\sinh s-\cosh s),
\end{array}
\tag {A.4}$$ $$\begin{array}{ll}
\displaystyle
K_{\tau}^1(\xi,s)
&
\displaystyle
=\left\{\left(1-\frac{1}{\tau}\right)\xi^2-\frac{2}{\tau^2}
\left(\xi+\frac{1}{\tau}\right)\right\}(s\cosh s-\sinh s)\\
\\
\displaystyle
&
\displaystyle
\,\,\,
+\frac{1}{\tau^2}\left(1-\frac{1}{\tau}\right)
\left\{s^2(s+6)\cosh s-3(s^2+2)\sinh s\right\}\\
\\
\displaystyle
&
\displaystyle
\,\,\,
+\frac{2}{\tau^2}
\left(\xi+\frac{1}{\tau}\right)
\left\{(s^2+1)\sinh s-2s\cosh s\right\}.
\end{array}
\tag {A.5}$$ and $$\displaystyle
\varphi_2(s)
=s^2(s+6)\cosh s
-3(s^2+2)\sinh s.
\tag {A.6}$$
[*Proof.*]{} It suffices to consider the case when $p=0$. Let $\mbox{\boldmath $A$}_x$ be the orthogonal matrix such that $(\mbox{\boldmath $A$}_x)^Tx=\vert x\vert\mbox{\boldmath $e$}_3$. The change of variables $y=r\omega\,(0<r<\eta,\,\omega\in S^2)$ and a rotation give us $$\begin{array}{ll}
\displaystyle
\mbox{\boldmath $I$}_0(x;\tau) & \displaystyle =\int_0^{\eta}r^2 dr\int_{S^2}\frac{e^{-\tau\vert x-r\omega\vert}}{\vert x-r\omega\vert}
\,\omega d\omega
\\
\\
\displaystyle
& \displaystyle =\int_0^{\eta}r^2 dr
\int_{S^2}
\frac{\displaystyle e^{-\tau\vert \vert x\vert\mbox{\boldmath $e$}_3-r\omega\vert}}
{\displaystyle\vert\vert x\vert\mbox{\boldmath $e$}_3-r\omega\vert}\,\mbox{\boldmath $A$}_x\omega d\omega
\\
\\
\displaystyle
& \displaystyle =
\int_0^{\eta}r^2dr\int_0^{2\pi}d\theta
\int_0^{\pi}\sin\varphi d\varphi
\frac{\displaystyle e^{-\tau\sqrt{\vert x\vert^2-2r\vert x\vert\cos\varphi+r^2}}}
{\displaystyle\sqrt{\vert x\vert^2-2r\vert x\vert\cos\varphi+r^2}}
\mbox{\boldmath $A$}_x
\left(\begin{array}{c}
\sin\varphi\cos\theta\\
\\
\displaystyle
\sin\varphi\sin\theta\\
\\
\displaystyle
\cos\varphi
\end{array}
\right)
\\
\\
\displaystyle
&
\displaystyle
=2\pi
\int_0^{\eta}r^2dr
\int_0^{\pi}\sin\varphi d\varphi
\frac{\displaystyle e^{-\tau\sqrt{\vert x\vert^2-2r\vert x\vert\cos\varphi+r^2}}}
{\displaystyle\sqrt{\vert x\vert^2-2r\vert x\vert\cos\varphi+r^2}}
\cos\varphi\mbox{\boldmath $A$}_x\mbox{\boldmath $e$}_3
\\
\\
\displaystyle
& \displaystyle =2\pi\int_0^{\eta}U(\vert x\vert,r)r^2 dr\frac{x}{\vert x\vert},
\end{array}
\tag {A.7}$$ where $$\begin{array}{lll}
\displaystyle
U(\xi,r)
=\int_0^{\pi}
\frac{\displaystyle e^{-\tau\sqrt{\xi^2-2r\xi\cos\varphi+r^2}}}
{\displaystyle\sqrt{\xi^2-2r\xi\cos\varphi+r^2}}\sin\varphi \cos\varphi\,d\varphi, & \xi>\eta, & 0<r<\eta.
\end{array}$$ Fix $\xi\in]\eta,\,\infty [$ and $r\in]0,\,\eta[$. The change of variable $$\displaystyle
s=\sqrt{\xi^2-2r\xi\cos\varphi+r^2},\,\varphi\in]0,\,\pi[$$ gives $$\displaystyle
\sin\varphi\,\cos\varphi d\varphi
=\frac{\xi^2+r^2-s^2}{2r^2\xi^2}s\,ds.$$ Thus we have $$\begin{array}{ll}
\displaystyle
U(\xi,r)
& \displaystyle
=\frac{1}{2r^2\xi^2}\int_{\xi-r}^{\xi+r}e^{-\tau s}(\xi^2+r^2-s^2)ds
\\
\\
\displaystyle
& \displaystyle =
\frac{1}{2r^2\xi^2}
\left\{(\xi^2+r^2)(-e^{-\tau(\xi+r)}+e^{-\tau(\xi-r)})
-\int_{\xi-r}^{\xi+r}e^{-\tau s}s^2ds\right\}.
\end{array}$$ Using the formula $$\begin{array}{ll}
\displaystyle
\int_{\xi-r}^{\xi+r}e^{-\tau s}s^2ds
&
\displaystyle
=-\frac{1}{\tau}
\left\{(\xi+r)^2+\frac{2}{\tau}(\xi+r)+\frac{2}{\tau^2}\right\}e^{-\tau(\xi+r)}\\
\\
\displaystyle
&
\displaystyle
\,\,\,
+\frac{1}{\tau}
\left\{(\xi-r)^2+\frac{2}{\tau}(\xi-r)+\frac{2}{\tau^2}\right\}e^{-\tau(\xi-r)},
\end{array}$$ we obtain $$\displaystyle
U(\xi,r)
=\frac{e^{-\tau, \xi}}{2\xi^2 r^2}
\left(A(\xi,r)
e^{\tau r}
-
B(\xi,r)
e^{-\tau r}
\right)$$ and $$\left\{
\begin{array}{l}
\displaystyle
A(\xi,r)=\left(\xi^2+r^2\right)-\frac{1}{\tau}(\xi-r)^2-\frac{2}{\tau^2}(\xi-r)-\frac{2}{\tau^3},
\\
\\
\displaystyle
B(\xi,r)=\left(\xi^2+r^2\right)-\frac{1}{\tau}(\xi+r)^2-\frac{2}{\tau^2}(\xi+r)-\frac{2}{\tau^3}.
\end{array}
\right.$$
From this and (A.7) we obtain $$\displaystyle
\,\,\,\,\,\,
\mbox{\boldmath $I$}_0(x;\tau)
=
\pi e^{-\tau\xi}
\int_0^{\eta}
\left(
A(\xi,r)e^{\tau r}-
B(\xi,r)
e^{-\tau r}
\right) dr
\vert_{\xi=\vert x\vert}\frac{x}{\vert x\vert^3}.
\tag {A.8}$$ We have $$\left\{
\begin{array}{l}
\displaystyle
\int_0^{\eta}e^{\pm\tau r}rdr
=\pm\frac{1}{\tau}\left(\eta\mp\frac{1}{\tau}\right)e^{\pm\tau\eta}+\frac{1}{\tau^2},
\\
\\
\displaystyle
\int_0^{\eta}e^{\pm\tau r}r^2dr
=\pm\frac{1}{\tau}
\left(\eta^2\mp\frac{2\eta}{\tau}+\frac{2}{\tau^2}\right)e^{\pm\tau \eta}
\mp\frac{2}{\tau^3},
\\
\\
\displaystyle
\int_0^{\eta}e^{\pm\tau r}r^3dr
=
\pm\frac{1}{\tau}e^{\pm\tau \eta}\eta^3
-
\frac{3}{\tau^2}
\left(\eta^2\mp\frac{2\eta}{\tau}+\frac{2}{\tau^2}\right)e^{\pm\tau\eta}
+\frac{6}{\tau^4}
\end{array}
\right.$$ and thus $$\left\{
\begin{array}{l}
\displaystyle
\int_0^{\eta}(e^{\tau r}-e^{-\tau r})rdr
=\frac{2}{\tau^2}(\tau\eta\cosh\tau\eta-\sinh\tau\eta),
\\
\\
\displaystyle
\int_0^{\eta}(e^{\tau r}+e^{-\tau r})rdr
=\frac{2}{\tau^2}(\tau\eta\sinh\tau\eta-\cosh\tau\eta),
\\
\\
\displaystyle
\int_0^{\eta}(e^{\tau r}+e^{-\tau r})r^2dr
=
\frac{2}{\tau^3}
\left\{(\eta^2\tau^2+1)\sinh\tau\eta-2\eta\tau\cosh\tau\eta\right\},
\\
\\
\displaystyle
\int_0^{\eta}(e^{\tau r}-e^{-\tau r})r^2dr
=
\frac{2}{\tau^3}
\left\{(\eta^2\tau^2+2)\cosh\tau\eta-2\eta\tau\sinh\tau\eta-2\right\},
\\
\\
\displaystyle
\int_0^{\eta}(e^{\tau r}-e^{-\tau r})r^3dr
=\frac{2}{\tau^4}
\left\{\eta^2\tau^2(\eta\tau+6)\cosh\tau\eta
-3(\eta^2\tau^2+2)\sinh\tau\eta\right\}.
\end{array}
\right.
\tag {A.9}$$ Now writing $$\left\{\begin{array}{l}
\displaystyle
A(\xi,r)
=\left(1-\frac{1}{\tau}\right)\xi^2-\frac{2}{\tau^2}\xi-\frac{2}{\tau^3}
+
\left(1-\frac{1}{\tau}\right)r^2
+\frac{2}{\tau}
\left(\xi+\frac{1}{\tau}\right)r,
\\
\\
\displaystyle
B(\xi,r)
=\left(1-\frac{1}{\tau}\right)\xi^2-\frac{2}{\tau^2}\xi-\frac{2}{\tau^3}
+
\left(1-\frac{1}{\tau}\right)r^2
-\frac{2}{\tau}
\left(\xi+\frac{1}{\tau}\right)r,
\end{array}
\right.
\tag {A.10}$$ from this and (A.9) we obtain $$\begin{array}{l}
\displaystyle
\,\,\,\,\,\,
\int_0^{\eta}
(A(\xi,r)e^{\tau r}-B(\xi,r)e^{-\tau r}) dr\\
\\
\\
\displaystyle
=
\left\{
\left(1-\frac{1}{\tau}\right)\xi^2-\frac{2}{\tau^2}\xi-\frac{2}{\tau^3}
\right\}
\int_0^{\eta}(e^{\tau r}-e^{-\tau r})\,dr
\\
\\
\displaystyle
\,\,\,
+
\left(1-\frac{1}{\tau}\right)
\int_0^{\eta}(e^{\tau r}-e^{-\tau r})r^2\,dr
\\
\\
\displaystyle
\,\,\,+\frac{2}{\tau}
\int_0^{\eta}(e^{\tau r}+e^{-\tau r})r\,dr\\
\\
\displaystyle
=\frac{2}{\tau}\left\{
\left(1-\frac{1}{\tau}\right)\xi^2-\frac{2}{\tau^2}\xi-\frac{2}{\tau^3}
\right\}(\cosh\tau\eta-1)\\
\\
\displaystyle
+\frac{2}{\tau^3}\left(1-\frac{1}{\tau}\right)
\left\{(\eta^2\tau^2+2)\cosh\tau\eta-2\eta\tau\sinh\tau\eta-2\right\}\\
\\
\displaystyle
+\frac{4}{\tau^3}(\tau\eta\sinh\tau\eta-\cosh\tau\eta)
\end{array}$$ A combination of this and (A.8) gives the desired formula for $\mbox{\boldmath $I$}_0(x;\tau)$.
Using the same changes of variable as used in the computation of $\mbox{\boldmath $I$}_0(x;\tau)$, we have $$\displaystyle
\,\,\,\,\,\,
\mbox{\boldmath $I$}_1(x;\tau)
=
\pi e^{-\tau\xi}
\int_0^{\eta}
\left(
A(\xi,r)e^{\tau r}-
B(\xi,r)
e^{-\tau r}
\right)r dr
\vert_{\xi=\vert x\vert}\frac{x}{\vert x\vert^3}.
\tag {A.11}$$ Then, from (A.9) and (A.10) we obtain $$\begin{array}{l}
\displaystyle
\,\,\,\,\,\,
\int_0^{\eta}
(A(\xi,r)e^{\tau r}-B(\xi,r)e^{-\tau r})r dr\\
\\
\\
\displaystyle
=
\frac{2}{\tau^2}
\left\{
\left(1-\frac{1}{\tau}\right)\xi^2-\frac{2}{\tau^2}\xi-\frac{2}{\tau^3}
\right\}
(\tau\eta\cosh\tau\eta-\sinh\tau\eta)\\
\\
\displaystyle
\,\,\,
+\frac{2}{\tau^4}
\left(1-\frac{1}{\tau}\right)\left\{\eta^2\tau^2(\eta\tau+6)\cosh\tau\eta
-3(\eta^2\tau^2+2)\sinh\tau\eta\right\}\\
\\
\displaystyle
\,\,\,+\frac{4}{\tau^4}
\left(\xi+\frac{1}{\tau}\right)\left\{(\eta^2\tau^2+1)\sinh\tau\eta-2\eta\tau\cosh\tau\eta\right\}.
\end{array}$$ A combination of this and (A.11) gives the dersired formula for $\mbox{\boldmath $I$}_1(x;\tau)$.
Similarily doing as above, one has the expression $$\displaystyle
\,\,\,\,\,\,
I_2(x;\tau)
=
\frac{2\pi}{\tau}\frac{e^{-\tau\vert x\vert}}{\vert x\vert}
\int_0^{\eta}
(
e^{\tau r}-
e^{-\tau r})r^3 dr.$$ Now (A.9) gives the dersired formula for $I_2(x;\tau)$.
$\Box$
Lower estimates for volume integrals
------------------------------------
Recall that $B$ is an open ball centered at $p$ with radius $\eta$ and satisfies $\overline B\cap\overline D=\emptyset$: we have $\text{dist}\,(D,B)=d_{\partial D}(p)-\eta$. In the following lemma it is assumed that $\partial D$ is $C^2$.
There exist positive constants $C$, $\tau_0$ and $\kappa$ such that, for all $\tau\ge\tau_0$ $$\displaystyle
\tau^{\kappa}e^{2\tau\sqrt{\rho/\mu}\,\text{dist}\,(D,B)}
\int_D e^{-2\tau\sqrt{\rho/\mu}\,(\vert x-p\vert-\eta)}
\left\vert\frac{x-p}{\vert x-p\vert}\times\mbox{\boldmath $a$}\right\vert^2 dx
\ge C.
\tag {A.12}$$
[*Proof.*]{} Choose a point $q\in\partial D$ such that $\vert q-p\vert=d_{\partial D}(p)$. Since $\partial D$ is $C^2$, one can find an open ball $B'$ with radius $\delta$ and centered at $q-\delta\mbox{\boldmath $\nu$}_q$ such that $B'\subset D$ and $\partial B'\cap\partial D=\{q\}$. Then $\text{dist}\,(B',B)=\text{dist}\,(D,B)$. Thus, it suffices to prove (A.12) in the case when $D=B'$.
Set $d=d_{\partial D}(p)$ and $-\mbox{\boldmath $\nu$}_q=\mbox{\boldmath $\omega$}_0$. Let $B''$ be the open ball with radius $d+\delta$ centered at $p$. First we give a parametrization of the domain $B''\cap B'$. Given $s\in\,]0,\,\delta[$ we find the set of all unit vectors $\mbox{\boldmath $\omega$}$ satisfying $p+(d+s)\mbox{\boldmath $\omega$}\in B'$. Since the center of $B'$ has the expression $p+(d+\delta)\mbox{\boldmath $\omega$}_{0}$, this condition is equivalent to the equation $$\displaystyle
\vert (d+s)\mbox{\boldmath $\omega$}-(d+\delta)\mbox{\boldmath $\omega$}_0\vert<\delta.$$ This is equivalent to the condition $$\displaystyle
\mbox{\boldmath $\omega$}\cdot\mbox{\boldmath $\omega$}_0>\frac{(d+\delta)^2+(d+s)^2-\delta^2}{2(d+s)(d+\delta)}.$$ Thus, we have $$\displaystyle
B''\cap B'
=\cup_{0<s<\delta}
\left\{p+(d+s)\omega\,\vert\,\omega\in S(s)\right\},$$ where $$\displaystyle
S(s)=\left\{\mbox{\boldmath $\omega$}\in S^2\,\vert\,\mbox{\boldmath $\omega$}\cdot\mbox{\boldmath $\omega$}_0
>\frac{(d+\delta)^2+(d+s)^2-\delta^2}{2(d+s)(d+\delta)}\right\}.$$ Choose two linearly independent vectors $\mbox{\boldmath $b$}$ and $\mbox{\boldmath $c$}$ in such a way that $\mbox{\boldmath $b$}\cdot\mbox{\boldmath $c$}=0$, $\mbox{\boldmath $b$}\times\mbox{\boldmath $c$}=
\mbox{\boldmath $\omega$}_0$. We denote by $\theta(s)\in\,]0, \pi/2[$ the unique solution of $$\displaystyle
\cos\theta
=\frac{(d+\delta)^2+(d+s)^2-\delta^2}{2(d+s)(d+\delta)}.
\tag {A.13}$$ Given $s\in\,]0,\,\delta[$, $r\in\,]0,\,(d+s)\sin\theta(s)[$ and $\gamma\in\,[0,\,2\pi[$ set $$\displaystyle
\mbox{\boldmath $\Upsilon$}(s,r,\gamma)
=p+(d+s)\cos\theta(s)\,\mbox{\boldmath $\omega$}_0
+r(\cos\gamma\,\mbox{\boldmath $b$}+\sin\gamma\,\mbox{\boldmath $c$})
+h\,\mbox{\boldmath $\omega$}_0,$$ where $h$ is an unknown parameter to be determined by the equation $$\displaystyle
\vert\mbox{\boldmath $\Upsilon$}(s,r,\gamma)-p\vert=d+s.$$ By solving this, we obtain $$\displaystyle
h=-(d+s)\cos\theta(s)+\sqrt{(d+s)^2-r^2}.$$ Thus, we have $$\displaystyle
\mbox{\boldmath $\Upsilon$}(s,r,\gamma)
=p+\sqrt{(d+s)^2-r^2}\,\mbox{\boldmath $\omega$}_0
+r(\cos\gamma\,\mbox{\boldmath $b$}+\sin\gamma\,\mbox{\boldmath $c$}).$$ Note that $\vert\mbox{\boldmath $\Upsilon$}(s,r,\gamma)-p\vert=d+s$ and the unit vector $(\mbox{\boldmath $\Upsilon$}(s,r,\gamma)-p)/(d+s)$ belongs to $S(s)$ for each fixed $s$. It is easy to check also that the map $$\displaystyle
G\ni (s,r,\gamma)\longmapsto \mbox{\boldmath $\Upsilon$}(s,r,\gamma)\in B''\cap B'$$ is bijective, where $$\displaystyle
G=\left\{(s,r,\gamma)\,\vert\,(s,r)\in G', \gamma\in\,[0,\,2\pi[\,\right\}$$ and $$\displaystyle
G'=\left\{(s,r)\,\vert 0<s<\delta,\, 0<r<(d+s)\sin\theta(s)\right\}.$$ A simple computation gives $$\displaystyle
\text{det}\,\mbox{\boldmath $\Upsilon$}'(s,r,\gamma)
=\frac{r(d+s)}
{\sqrt{(d+s)^2-r^2}}.$$ Here we have $$\begin{array}{ll}
\displaystyle
(\mbox{\boldmath $\Upsilon$}(s,r,\gamma)-p)\times\mbox{\boldmath $a$}
&
\displaystyle
=\sqrt{(d+s)^2-r^2}\mbox{\boldmath $\omega$}_0\times
\mbox{\boldmath $a$}
+r(\cos\gamma\,\mbox{\boldmath $b$}\times\mbox{\boldmath $a$}
+\sin\gamma\,\mbox{\boldmath $c$}\times\mbox{\boldmath $a$})\\
\\
\displaystyle
&
\displaystyle
\equiv \mbox{\boldmath $A$}(s,r,\gamma)\mbox{\boldmath $a$}.
\end{array}
\tag {A.14}$$ Using the change of variables formula and $B''\cap B'\subset B'$, we obtain $$\begin{array}{l}
\displaystyle
\,\,\,\,\,\,
e^{2\tau\sqrt{\rho/\mu}\,\,(d-\eta)}\int_{B'} e^{-2\tau\sqrt{\rho/\mu}\,(\vert x-p\vert-\eta)}
\left\vert\frac{x-p}{\vert x-p\vert}\times\mbox{\boldmath $a$}\right\vert^2 dx
\\
\\
\displaystyle
\ge
\int_0^{\delta}ds\int_0^{(d+s)\sin\theta(s)}dr\int_0^{2\pi}d\gamma
\frac{e^{-2\tau\sqrt{\rho/\mu}\,s}}{d+s}
\left\vert\mbox{\boldmath $A$}(s,r,\gamma)\mbox{\boldmath $a$}\right\vert^2
\frac{r}
{\sqrt{(d+s)^2-r^2}}.
\end{array}
\tag {A.15}$$ It is easy to see that from (A.14) one gets $$\begin{array}{l}
\displaystyle
\,\,\,\,\,\,
\int_0^{2\pi}\left\vert\mbox{\boldmath $A$}(s,r,\gamma)\mbox{\boldmath $a$}\right\vert^2 d\gamma
\\
\\
\displaystyle
=2\pi\left\{(d+s)^2-r^2\right\}
\vert\mbox{\boldmath $\omega$}_0\times
\mbox{\boldmath $a$}\vert^2
+\pi r^2
(\vert\mbox{\boldmath $b$}\times\mbox{\boldmath $a$}\vert^2
+\vert\mbox{\boldmath $c$}\times\mbox{\boldmath $a$}\vert^2).
\end{array}$$ And also $$\left\{
\begin{array}{l}
\displaystyle
\int_0^{(d+s)\sin\theta(s)}r\sqrt{(d+s)^2-r^2}dr
=\frac{(d+s)^3}{3}(1-\cos^3\theta(s)),\\
\\
\displaystyle
\int_0^{(d+s)\sin\theta(s)}
\frac{r^3dr}
{\sqrt{(d+s)^2-r^2}}
=(d+s)^3
\left\{(1-\cos\theta(s))
+\frac{1-\cos^3\theta(s)}{3}
\right\}.
\end{array}
\right.$$ Thus $$\begin{array}{l}
\displaystyle
\,\,\,\,\,\,
\int_0^{(d+s)\sin\theta(s)}\frac{r dr}{\sqrt{(d+s)^2-r^2}}
\int_0^{2\pi}\left\vert\mbox{\boldmath $A$}(s,r,\gamma)\mbox{\boldmath $a$}\right\vert^2 d\gamma\\
\\
\displaystyle
=\frac{2\pi}{3}(d+s)^3(1-\cos^3\theta(s))\vert\mbox{\boldmath $\omega$}_0\times
\mbox{\boldmath $a$}\vert^2\\
\\
\displaystyle
\,\,\,
+\pi
(d+s)^3\left\{(1-\cos\theta(s))
-\frac{1-\cos^3\theta(s)}{3}
\right\}
(\vert\mbox{\boldmath $b$}\times\mbox{\boldmath $a$}\vert^2
+\vert\mbox{\boldmath $c$}\times\mbox{\boldmath $a$}\vert^2).
\end{array}$$ From (A.13) we have $$\begin{array}{ll}
\displaystyle
(d+s)(1-\cos\theta(s))
&
\displaystyle
=\frac{s(2\delta-s)}{2(d+\delta)}
\\
\\
\displaystyle
&
\displaystyle
\ge\frac{\delta s}{2(d+\delta)}.
\end{array}$$ This gives $$\begin{array}{ll}
\displaystyle
(d+s)^3(1-\cos^3\theta(s))
&
\displaystyle
\ge
\frac{\delta (d+s)^2 s}{2(d+\delta)}
(1+\cos\theta(s)+\cos^2\theta(s))
\\
\\
\displaystyle
\,\,\,
&
\displaystyle
\ge\frac{\delta d(d+s)s}{2(d+\delta)}.
\end{array}$$ And also $$\begin{array}{ll}
\displaystyle
(d+s)^3\left\{(1-\cos\theta(s))
-\frac{1-\cos^3\theta(s)}{3}
\right\}
&
\displaystyle
\ge
\frac{\delta (d+s)^2s}{2(d+\delta)}
\left(1-\frac{1+\cos\theta(s)+\cos^2\theta(s)}{3}\right)
\\
\\
\displaystyle
&
\displaystyle
=\frac{\delta (d+s)^2s}{6(d+\delta)}
(1-\cos\theta(s))(2+\cos\theta(s))
\\
\\
\displaystyle
&
\displaystyle
\ge
\frac{\delta^2(d+s)s^2}{12(d+\delta)^2}(2+\cos\theta(s))
\\
\\
\displaystyle
&
\displaystyle
\ge
\frac{\delta^2(d+s)s^2}{6(d+\delta)^2}.
\end{array}$$ Thus we have $$\begin{array}{l}
\displaystyle
\,\,\,\,\,\,
\int_0^sds\frac{e^{-2\tau\sqrt{\rho/\mu}\,s}}{d+s}
\int_0^{(d+s)\sin\theta(s)}\frac{r dr}{\sqrt{(d+s)^2-r^2}}
\int_0^{2\pi}\left\vert\mbox{\boldmath $A$}(s,r,\gamma)\mbox{\boldmath $a$}\right\vert^2 d\gamma\\
\\
\displaystyle
\ge
\frac{\pi}{3}\frac{\delta d}{d+\delta}
\int_0^{\delta}se^{-2\tau\sqrt{\rho/\mu}\,s}\,ds
\,\vert\mbox{\boldmath $\omega$}_0\times
\mbox{\boldmath $a$}\vert^2\\
\\
\displaystyle
\,\,\,
+\frac{\pi}{6}
\left(\frac{\delta}{d+\delta}\right)^2
\int_0^{\delta}s^2 e^{-2\tau\sqrt{\rho/\mu}\,s}\,ds
\,
(\vert\mbox{\boldmath $b$}\times\mbox{\boldmath $a$}\vert^2
+\vert\mbox{\boldmath $c$}\times\mbox{\boldmath $a$}\vert^2).
\end{array}
\tag {A.16}$$ Here we have, for $j=1,2$ $$\displaystyle
\int_0^{\delta}s^je^{-2\tau\sqrt{\rho/\mu}\,s}\,ds
=\frac{1}{(2\tau\sqrt{\rho/\mu})^{j+1}}+O(\tau^{-1}e^{-2\tau\sqrt{\rho/\mu}\,\delta}).$$ Therefore, from these and (A.15) and (A.16) one can conclude that: if $\mbox{\boldmath $\omega$}_0\times\mbox{\boldmath $a$}\not=\mbox{\boldmath $0$}$, then, one can choose $\kappa=2$ in (A.12); if $\mbox{\boldmath $\omega$}_0\times\mbox{\boldmath $a$}=\mbox{\boldmath $0$}$, then $\mbox{\boldmath $a$}=\pm\mbox{\boldmath $\omega$}_0$ and thus $\vert\mbox{\boldmath $b$}\times\mbox{\boldmath $a$}\vert^2
+\vert\mbox{\boldmath $c$}\times\mbox{\boldmath $a$}\vert^2>0$, and one can choose $\kappa=3$ in (A.12).
$\Box$
There exist positive constants $C$ and $\tau_0$ such that, for all $\tau\ge\tau_0$ $$\displaystyle
\tau^2e^{2\tau\,\text{dist}\,(D,B)}
\int_De^{-2\tau\,\,(\vert x-p\vert-\eta)}\,dx
\ge C.$$
This is proved as follows. From the proof of Lemma A.1 we have $$\begin{array}{l}
\displaystyle
\,\,\,\,\,\,
e^{2\tau\,(d-\eta)}\int_{D}e^{-2\tau\,(\vert x-p\vert-\eta)}\,dx\\
\\
\displaystyle
\ge
e^{2\tau\,(d-\eta)}\int_{B'}e^{-2\tau\,(\vert x-p\vert-\eta)}\,dx\\
\\
\displaystyle
\ge
2\pi\int_0^{\delta}\,ds\int_0^{(d+s)\sin\theta(s)}\,dr
e^{-2\tau s}\frac{r(d+s)}{\sqrt{(d+s)^2-r^2}}\\
\\
\displaystyle
=2\pi\int_0^{\delta}(d+s)e^{-2\tau s}\,ds
\int_0^{(d+s)\sin\theta(s)}
\frac{r}{\sqrt{(d+s)^2-r^2}}\,dr\\
\\
\displaystyle
=\pi\int_0^{\delta}(d+s)^2(1-\cos\theta(s))e^{-2\tau s}\,ds\\
\\
\displaystyle
\ge
\pi\int_0^{\delta}(d+s)\cdot\frac{\delta s}{2(d+\delta)}\cdot e^{-2\tau s}\,ds
\\
\\
\displaystyle
\ge
\frac{\pi\delta d}{2(d+\delta)}
\int_0^{\delta}se^{-2\tau s}\,ds.
\end{array}$$ Since $$\displaystyle
\int_0^{\delta}se^{-2\tau s}\,ds
=\frac{1}{4\tau^2}\int_0^{2\tau\delta}\xi e^{-\xi}\,d\xi,$$ we obtain the desired conclusion.
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e-mail address
Masaru Ikehata: [email protected]
[^1]: Laboratory of Mathematics, Graduate School of Engineering, Hiroshima University, Higashihiroshima 739-8527, Japan
|
---
author:
- |
Dmitri Antonov [^1]\
[*INFN-Sezione di Pisa, Universitá degli studi di Pisa,*]{}\
[*Dipartimento di Fisica, Via Buonarroti, 2 - Ed. B - I-56127 Pisa, Italy*]{}\
[*and*]{}\
[*Institute of Theoretical and Experimental Physics,*]{}\
[*B. Cheremushkinskaya 25, RU-117 218 Moscow, Russia*]{}
title: ' **Accounting for the finiteness of the Higgs-boson mass in the 3D Georgi-Glashow model**'
---
**[Abstract]{}**
(2+1)-dimensional Georgi-Glashow model is explored in the regime when the Higgs boson is not infinitely heavy, but its mass is rather of the same order of magnitude as the mass of the W boson. In the weak-coupling limit, the Debye mass of the dual photon and the expression for the monopole potential are found. The cumulant expansion applied to the average over the Higgs field is checked to be convergent for the known data on the monopole fugacity. These results are further generalized to the $SU(N)$-case. In particular, it is found that the requirement of convergence of the cumulant expansion establishes a certain upper bound on the number of colours. This bound, expressed in terms of the parameter of the weak-coupling approximation, allows the number of colours to be large enough. Finally, the string tension and the coupling constant of the so-called rigidity term of the confining string are found at arbitrary number of colours.
Introduction
============
Since the second half of the seventies [@1], (2+1)-dimensional Georgi-Glashow model is known as an example of the theory allowing for an analytic description of confinement. However, confinement in the Georgi-Glashow model is typically discussed in the limit of infinitely large Higgs-boson mass, when the model is reduced to compact QED. In ref. [@nd], possible influence of the Higgs field to the dynamics of the Georgi-Glashow model has been studied both at zero and nonzero temperatures. This has been done not only under the assumption that the Higgs-field mass is finite rather than infinite, which allows this field to propagate, but in the Bogomolny-Prasad-Sommerfield (BPS) limit [@bps]. This is the limit when the Higgs field is [*much*]{} lighter than the W boson (but is still much heavier than the dual photon). The first aim of the present paper is to generalize the zero-temperature results of ref. [@nd] to the case when the mass of the Higgs boson is of the same order of magnitude as the mass of the W boson. This situation is thus intermediate between the BPS limit and the limit of compact QED. In this way, we shall find the monopole potential and the Debye mass, and prove the convergence of the cumulant expansion associated to the average over the Higgs field. This will be done in the next Section.
Another aim of the present paper, which will be realized in Section 3, is to generalize this analysis to the $SU(N)$-case. The Debye mass and the parameter of the cumulant expansion will then be $N$-dependent quantities. The $N$-dependence of the latter will yield a certain upper bound on $N$ necessary to ensure the convergence of the cumulant expansion. This bound will turn out to be the exponent of the inverse parameter of the weak-coupling approximation, that will allow $N$ to vary in a wide enough range. We shall also find the values of the two leading coupling constants of the confining-string Lagrangian at arbitrary $N$.
The main results of the paper will be summarized in the Conclusions. In the Appendix, some technical details of the performed calculations will finally be outlined.
SU(2)-case
==========
The Euclidean action of the (2+1)D Georgi-Glashow model reads [@1]
$$\label{GG}
S=\int d^3x\left[\frac{1}{4g^2}\left(F_{\mu\nu}^a\right)^2+
\frac12\left(D_\mu\Phi^a\right)^2+\frac{\lambda}{4}\left(
\left(\Phi^a\right)^2-\eta^2\right)^2\right].$$
Here, the Higgs field $\Phi^a$ transforms by the adjoint representation and $D_\mu\Phi^a\equiv\partial_\mu\Phi^a+\varepsilon^{abc}A_\mu^b
\Phi^c$. Next, $\lambda$ is the Higgs coupling constant of dimensionality \[mass\], $\eta$ is the Higgs [*v.e.v.*]{} of dimensionality $[{\rm mass}]^{1/2}$, and $g$ is the electric coupling constant of the same dimensionality.
At the one-loop level, the partition function of the theory (\[GG\]) takes the following form [@dietz]:
$$\label{1}
S=\int d^3x\left[\frac12(\nabla\chi)^2+\frac12(\nabla\psi)^2
+\frac{m_H}{2}\psi^2-2\zeta{\rm e}^{g_m\psi}\cos(g_m\chi)\right].$$
Here, $\chi$ is the dual-photon field, and the field $\psi$ accounts for the Higgs field, when it is not infinitely heavy ([*i.e.*]{} one deviates from the compact-QED limit). Next, $g_m$ is the magnetic coupling constant related to the electric one as $g_mg=4\pi$. The Higgs-boson mass, $m_H$, reads $m_H=\eta\sqrt{2\lambda}$, and the monopole fugacity $\zeta$ has the form:
$$\label{Ze}
\zeta=\frac{m_W^{7/2}}{g}\delta\left(\frac{\lambda}{g^2}\right)
{\rm e}^{-(4\pi/g^2)m_W\epsilon}.$$
In this formula, $m_W=g\eta$ stands for the W-boson mass, and $\epsilon=\epsilon(\lambda/g^2)$ is a certain monotonic, slowly varying function, $\epsilon\ge 1$, $\epsilon(0)=1$ [@bps], $\epsilon(\infty)\simeq 1.787$ [@kirk]. As far as the function $\delta$ is concerned, it is determined by the loop corrections. It is known [@ks] that this function grows in the vicinity of the origin ([*i.e.*]{} in the BPS limit). However, the speed of this growth is so that it does not spoil the exponential smallness of $\zeta$ in the standard weak-coupling regime $g^2\ll m_W$ (or $g\ll\eta$) which we adapt in this paper.
Integrating further in eq. (\[1\]) over $\psi$ by virtue of the cumulant expansion, we get:
$$\label{2}
S\simeq\int d^3x\left[\frac12(\nabla\chi)^2-2\xi\cos(g_m\chi)\right]-
2\xi^2\int d^3xd^3y\cos(g_m\chi({\bf x})){\cal K}({\bf x}-{\bf y})
\cos(g_m\chi({\bf y})).$$
In this expression, we have disregarded all the cumulants of the orders higher than the second, and the limits of applicability of this so-called bilocal approximation will be discussed below. In eq. (\[2\]), ${\cal K}({\bf x})\equiv {\rm e}^{g_m^2D_{m_H}({\bf
x})}-1$ with $D_{m_H}({\bf x})\equiv{\rm e}^{-m_H|{\bf x}|}/(4\pi|{\bf
x}|)$ standing for the Higgs-field propagator, and
$$\label{fug}
\xi\equiv\zeta{\rm e}^{\frac{g_m^2}{2}D_{m_H}(0)}=\frac{m_W^{7/2}}{g}
\delta\left(\frac{\lambda}{g^2}\right){\rm e}^{\frac{2\pi m_W}{g^2}
\left(-2\epsilon+{\rm e}^{-c}\right)}$$
denotes the modified fugacity. In the derivation of eq. (\[fug\]), we have in the standard way set $m_W$ for the UV cutoff in the weak-coupling regime and denoted $c\equiv m_H/m_W$.
As it is clear from eq. (\[2\]), the compact-QED limit is achieved when $m_H$ formally tends to infinity, [*i.e.*]{} $c\to\infty$. In ref. [@nd], there has been explored the opposite, BPS, limit $c\ll 1$. Since $D_{m_H}({\bf x}-{\bf y})\sim m_H$, one can impose the inequality $g_m^2m_H\ll 1$, which together with the weak-coupling approximation yields $c\ll 1$, and obtain from eq. (\[2\]) the following action:
$$S\simeq\int d^3x\left[\frac12(\nabla\chi)^2-2\xi\cos(g_m\chi)\right]-
2(g_m\xi)^2\int d^3xd^3y\cos(g_m\chi({\bf x}))D_{m_H}({\bf x}-{\bf y})
\cos(g_m\chi({\bf y})).$$ Note that according to eq. (\[fug\]), the modified fugacity $\xi$ remains to be exponentially small in this limit. That is firstly because $\epsilon>\frac{{\rm e}^{-c}}{2}\simeq\frac12$ and secondly because, as it was discussed above, according to ref. [@ks], the function $\delta$ entering eq. (\[Ze\]) grows at $c\ll 1$ slower than exponentially. Next, the fact that $D_{m_H}({\bf x})$ rapidly vanishes at $|{\bf x}|\to\infty$ enables one to estimate the parameter of the cumulant expansion, which in this case reads $\xi g_m^2\int d^3xD_{m_H}({\bf x})=g_m^2\xi/m_H^2$. This quantity is exponentially small due to the exponential smallness of $\xi$, which proves the convergence of the cumulant expansion.
In what follows, we shall explore the action (\[2\]) in the regime intermediate between the BPS- and compact-QED limits, namely $c\sim 1$. First of all note that since $c^2=2\lambda/g^2$, $\xi$ will be exponentially small provided that $\epsilon(x)>
{\rm e}^{-\sqrt{2x}}/2$ at $x\sim 1/2$. One can see that this inequality is always satisfied, since its r.h.s. is not larger than 1/2, while $\epsilon\ge 1$. Next, analogously to the case $c\ll 1$, by noting that ${\cal K}({\bf x})$ rapidly vanishes at $|{\bf x}|\to\infty$, the parameter of the cumulant expansion can be estimated as $\xi I$, where $I\equiv\int d^3x{\cal K}({\bf x})$. This integral is evaluated in the Appendix. At $\frac{a}{c}{\rm e}^{-c}\gg 1$ with $a\equiv4\pi m_H/g^2$ (which is obviously true in the weak-coupling regime), it reads
$$\label{3}
I\simeq\frac{4\pi}{m_H^3}\left\{\sum\limits_{n=1}^{[1/c]}
\frac{a^n}{nn!}\left[n^{n-2}\Gamma(3-n,cn)-c^{2-n}{\rm e}^{-cn}\right]
+c^2\left[\exp\left(\frac{a}{c}{\rm e}^{-c}\right)\left(1-\frac{c}{a}{\rm e}^c\right)+
({\rm e}-1)\ln a\right]\right\}.$$
Here, $[1/c]$ stands for the largest integer, smaller or equal to $1/c$, and $\Gamma(b,x)=\int\limits_{x}^{\infty}
dt{\rm e}^{-t}t^{b-1}$ denotes the incomplete Gamma-function. In the case $c\sim 1$ under study, the sum entering eq. (\[3\]) contains a few terms, among whose the dominant one is of the order of $a$. These terms can thus be disregarded with respect to the term of the order of ${\rm e}^a$ standing in that equation, and we finally obtain: $I\simeq\frac{4\pi}{m_Hm_W^2}\exp\left(\frac{a}{c}{\rm e}^{-c}\right)$. Consequently, the parameter of the cumulant expansion, $\xi I$, will be exponentially small, provided that $\epsilon(x)>\frac32
{\rm e}^{-\sqrt{2x}}$ at $x\sim 1/2$. In particular, we should have $\epsilon(1/2)>3/(2{\rm e})\simeq 0.552$, which is clearly true, since $\epsilon\ge 1$. Thus, cumulant expansion is convergent in the case $c\sim 1$ under study.
One can further straightforwardly read off from eq. (\[2\]) the squared Debye mass of the dual photon. It has the form $m_D^2=2g_m^2\xi(1+2\xi I)$, where as it was just discussed, the second term in the brackets is exponentially small with respect to the first one, and therefore $m_D=g_m\sqrt{2\xi}(1+\xi I)$. Obviously, unity and $\xi I$ here are the contributions to $m_D$ brought about by the first and the second cumulants in eq. (\[2\]), respectively. Note also that this result for $m_D$ obviously reproduces the compact-QED one (see [*e.g.*]{} [@nd]), $g_m\sqrt{2\zeta}$. Indeed, at $m_H\to\infty$, $\xi\to\zeta$ and, as it follows directly from the definition of $I$, $I\to 0$, that proves our statement.
Similarly to how it was done for the case $c\ll 1$ in ref. [@nd], it is also possible in our case $c\sim 1$ to derive the representation of the action (\[2\]) in terms of dynamical monopole densities $\rho$’s. To this end, one should perform in the partition function the following substitution
$$\exp\left[-\frac12\int d^3x(\nabla\chi)^2\right]=\int {\cal D}\rho
\exp\left[-\frac{g_m^2}{2}\int d^3xd^3y\rho({\bf x})D_0({\bf x}-{\bf y})
\rho({\bf y})-ig_m\int d^3x\chi\rho\right],$$ where $D_0({\bf x})\equiv 1/(4\pi|{\bf x}|)$ is the Coulomb propagator. After that, it is necessary to solve the resulting saddle-point equation
$$\label{sp}
\sinh(\phi({\bf x}))\left[1+2\xi\int d^3y{\cal K}({\bf x}-{\bf y})
\cosh(\phi({\bf y}))\right]=\frac{\rho({\bf x})}{2\xi},$$
where $\phi\equiv ig_m\chi$. This equation can be solved iteratively by imposing the Ansatz $\phi=\phi_1+\phi_2$ with $|\phi_2|\ll|\phi_1|$. Introducing the notation $f\equiv\sqrt{1+\left(\frac{\rho}{2\xi}\right)^2}$, we then obtain:
$$\phi_1({\bf x})={\rm arcsinh}\left(\frac{\rho({\bf x})}{2\xi}\right),~~
\phi_2({\bf x})=-\frac{\rho({\bf x})}{f({\bf x})}\int d^3y{\cal K}({\bf x}-{\bf y})
f({\bf y}).$$ On the other hand, the average monopole density stemming from eq. (\[2\]) reads $\frac{\partial\ln\int {\cal D}\chi{\rm e}^{-S}}{{\cal V}\partial\ln\xi}
\simeq 2\xi(1+2\xi I)$, where ${\cal V}$ is the 3D-volume of observation. Therefore, at $|\rho|\le\xi$, we have $f\sim 1$, $|\phi_1|\sim|\rho|/\xi$, and $|\phi_2|\sim\xi I |\phi_1|\ll |\phi_1|$ thus justifying our Ansatz. The obtained solution to the saddle-point equation yields the representation of the theory (\[2\]) in terms of $\rho$’s in the form
$$\label{Act}
S=\frac{g_m^2}{2}\int d^3xd^3y\rho({\bf x})D_0({\bf x}-{\bf y})
\rho({\bf y})+V[\rho].$$
Here, the multivalued potential of monopole densities reads
$$V[\rho]=\int d^3x\left[\rho{\,}{\rm arcsinh}\left(\frac{\rho}{2\xi}\right)
-2\xi f\right]-2\xi^2\int d^3xd^3yf({\bf x}){\cal K}({\bf x}-{\bf y})f({\bf y}).$$ Note that the multivaluedness of this potential realizes the world-sheet independence of the Wilson loop in the theory (\[2\]). This is the essence of the string representation of the Georgi-Glashow model, discussed for the compact-QED limit, $c\to\infty$, in ref. [@cs] and for the BPS-limit, $c\ll 1$, in ref. [@nd].
Note also that at very low densities, $|\rho|\ll\xi$, up to an inessential constant addendum, $V[\rho]\simeq\frac{g_m^2}{2m_D^2}
\int d^3x\rho^2$, [*i.e.*]{} the action (\[Act\]) becomes quadratic. Therefore, in this limit, any (even) correlator of $\rho$’s can be evaluated explicitly. In particular, the bilocal one reads $\left<\rho({\bf x})\rho(0)\right>=-(m_D/g_m)^2
\nabla^2D_{m_D}({\bf x})\simeq 2\xi(1+2\xi I)\delta({\bf x})$, where in the derivation of the last equality we have used the exponential smallness of $m_D$. This yields the average squared density: $\overline{\rho^2}={\cal V}^{-1}
\int d^3x\left<\rho({\bf x})\rho(0)\right>\simeq 2\xi{\cal V}^{-1}(1+2\xi I)$. Next, at $|\rho|\le\xi$, the average distance between monopoles, $\bar r$, is not smaller than $\xi^{-1/3}$. The volume of observation, ${\cal V}$, should be much larger than $\bar r^3$ and therefore ${\cal V}$ is much larger than $\xi^{-1}$ as well. This yields the relation $\overline{\rho^2}\sim\xi{\cal V}^{-1}\ll\xi^2$, which justifies the initial approximation $|\rho|\ll\xi$.
SU(N)-case
==========
The $SU(N)$-generalization of the action (\[1\]), stemming from the $SU(N)$ Georgi-Glashow model, has the form
$$\label{s}
S=\int d^3x\left[\frac12(\nabla\vec\chi)^2+\frac12(\nabla\psi)^2+
\frac{m_H^2}{2}\psi^2-2\zeta{\rm e}^{g_m\psi}\sum\limits_{i=1}^{N(N-1)/2}
\cos\left(g_m\vec q_i\vec\chi\right)\right].$$
Here, $\vec q_i$’s are the positive root vectors of the group $SU(N)$. As well as the field $\vec\chi$, these vectors are $(N-1)$-dimensional. Note that the $SU(3)$-version of the action (\[s\]), which incorporates the effects of the Higgs field, has been discussed in ref. [@nd]. The compact-QED limit of the $SU(N)$-case has been studied in refs. [@wd], [@sn], and [@suN]. The string representation of the compact-QED limit has been studied for the $SU(3)$-case in ref. [@epl] both in 3D and 4D. Here, similarly to all the above-mentioned papers, we have assumed that W bosons corresponding to different root vectors have the same masses.
Straightforward integration over $\psi$ then yields the following analogue of eq. (\[2\]):
$$S\simeq\int d^3x\left[\frac12(\nabla\vec\chi)^2-2\xi\sum\limits_{i=1}^{N(N-1)/2}
\cos\left(g_m\vec q_i\vec\chi\right)\right]-$$
$$\label{N}
-2\xi^2\int d^3xd^3y\sum\limits_{i,j=1}^{N(N-1)/2}
\cos\left(g_m\vec q_i\vec\chi({\bf x})\right){\cal K}({\bf x}-{\bf y})
\cos\left(g_m\vec q_j\vec\chi({\bf y})\right).$$
The Debye mass of the field $\vec\chi$ can be derived from this expression by virtue of the formula [@group] $\sum\limits_{i=1}^{N(N-1)/2}q_i^\alpha q_i^\beta\propto
\delta^{\alpha\beta}$. The proportionality coefficient which should stand on the r.h.s. of this relation can easily be found from the requirement that all root vectors have the unit length. This coefficient is equal to $(N/2)$, and the square of the Debye mass turns out to be $m_D^2=g_m^2\xi N[1+\xi IN(N-1)]$. Note that this formula reproduces both the $SU(2)$-result of the previous Section and the $SU(3)$-result of the compact-QED limit [@epl], [@nd] $m_D^2=3g_m^2\zeta$.
The new parameter of the cumulant expansion, $\xi IN(N-1)$, will be exponentially small provided that at $x\sim 1/2$,
$$\epsilon(x)>\frac12\left[3{\rm e}^{-\sqrt{2x}}+\frac{g^2}{2\pi m_W}\ln(N(N-1))\right].$$ Setting in this inequality $x=1/2$ and recalling that [^2] $\epsilon(1/2)<\epsilon(\infty)\simeq 1.787$, we obtain the following upper bound on $N$, which guarantees the convergence of the cumulant expansion: $N(N-1)<{\rm e}^{15.522m_W/g^2}$. Clearly, in the weak-coupling regime under study, this bound is exponentially large, that allows $N$ to be large enough too.
Next, the representation of the theory with the action (\[N\]) in terms of the monopole densities can be derived similarly to the $SU(2)$-case by virtue of the formula
$$\exp\left[-\frac12\int d^3x(\nabla\vec \chi)^2\right]=$$
$$=\int\prod\limits_{i=1}^{N(N-1)/2} {\cal D}\rho_i
\exp\left[-\frac{g_m^2}{2}\int d^3xd^3y\rho_i({\bf x})D_0({\bf x}-{\bf y})
\rho_i({\bf y})-ig_m\sqrt{\frac{2}{N}}\int d^3x\vec q_i\vec\chi\rho_i\right].$$ The analogue of the saddle-point equation (\[sp\]) then reads
$$\sinh(\phi_i({\bf x}))\left[1+2\xi\sum\limits_{j=1}^{N(N-1)/2}\int d^3y{\cal K}({\bf x}-{\bf y})
\cosh(\phi_j({\bf y}))\right]=\frac{\rho_i({\bf x})}{\sqrt{2N}\xi},$$ where $\phi_j\equiv ig_m\vec q_j\vec\chi$. Solving this equation iteratively with the Ansatz $\phi_i=\phi_i^1+\phi_i^2$, where $|\phi_j^2|\ll |\phi_j^1|$, we obtain:
$$\phi_i^1({\bf x})={\rm arcsinh}\left(\frac{\rho_i({\bf x})}{\sqrt{2N}\xi}\right),~~
\phi_i^2({\bf x})=-\sqrt{\frac{2}{N}}\frac{\rho_i({\bf x})}{F_i({\bf x})}
\sum\limits_{j=1}^{N(N-1)/2}
\int d^3y{\cal K}({\bf x}-{\bf y})F_j({\bf y}),$$ where $F_i({\bf x})\equiv\sqrt{1+\frac{1}{2N}\left(\frac{\rho_i({\bf x})}{\xi}\right)^2}$. Next, the average density of monopoles of [*all*]{} kinds reads $\frac{\partial\ln\int {\cal D}\vec\chi{\rm e}^{-S}}{{\cal V}\partial\ln\xi}
\simeq\xi N(N-1)[1+\xi IN(N-1)]$, that, in particular, reproduces the $SU(2)$-result. Therefore, the average density of monopoles of only one kind is of the order of $\xi$. Thus, at $|\rho_i|\le\xi$, $F_i\sim 1$, $|\phi_i^1|\sim|\rho_i|/(\xi\sqrt{N})$, and $|\phi_i^2|\sim\xi IN(N-1)|\phi_i^1|$. The quantity $|\phi_i^2|/|\phi_i^1|$ is therefore of the order of the parameter of the cumulant expansion, that justifies the adapted Ansatz. The desired representation of the theory described by the action (\[N\]) in terms of the monopole densities then has the form
$${\cal Z}=\int\left(\prod\limits_{i=1}^{N(N-1)/2} {\cal D}\rho_i\right)\exp\left\{
-\frac{g_m^2}{2}\int d^3xd^3y\sum\limits_{i=1}^{N(N-1)/2}\rho_i({\bf x})D_0({\bf x}-{\bf y})
\rho_i({\bf y})-V_N\left[\{\rho_i\}_{i=1}^{N(N-1)/2}\right]\right\},$$ where the monopole potential is given by the following formula:
$$V_N\left[\{\rho_i\}_{i=1}^{N(N-1)/2}\right]=\int d^3x\sum\limits_{i=1}^{N(N-1)/2}
\left[\sqrt{\frac{2}{N}}
\rho_i{\,}{\rm arcsinh}\left(\frac{\rho_i}{\sqrt{2N}\xi}\right)
-2\xi F_i\right]-$$
$$-2\xi^2\int d^3xd^3y\sum\limits_{i,j=1}^{N(N-1)/2}
F_i({\bf x}){\cal K}({\bf x}-{\bf y})F_j({\bf y}).$$
One can further naively assume that the criterion of the low-density approximation has the form $|\rho_i|\ll\sqrt{N}\xi$ (although the average density of monopoles of one kind was discussed to be of the order of $\xi$). Indeed, similarly to the $SU(2)$-case, already under this inequality, the potential factorizes and becomes quadratic, so that (again up to an inessential constant addendum) $V_N\left[\{\rho_i\}_{i=1}^{N(N-1)/2}\right]\simeq
\frac{g_m^2}{2m_D^2}\int d^3x\sum\limits_{i=1}^{N(N-1)/2}\rho_i^2$. Consequently, the bilocal correlator of monopole densities reads
$$\left<\rho_i({\bf x})\rho_j(0)\right>=-(m_D/g_m)^2\delta_{ij}
\nabla^2D_{m_D}({\bf x})\simeq \xi N[1+\xi IN(N-1)]\delta_{ij}\delta({\bf x}),$$ and, in particular, the $SU(2)$-result obviously recovers itself. Therefore, the average squared density of monopoles of any kind has the form: $\overline{\rho_i^2}\simeq \xi N{\cal V}^{-1}
[1+\xi IN(N-1)]\sim \xi N{\cal V}^{-1}$. The inequality $\overline{\rho_i^2}\ll N\xi^2$, necessary for the justification of the initial approximation, will thus be satisfied provided that ${\cal V}\xi\gg 1$. For the densities $|\rho_i|\le\sqrt{N}\xi$, we however have ${\cal V}\gg\bar r_i^3\ge N^{-1/2}\xi^{-1}$ (where $\bar r_i$ is an average distance between the monopoles of the $i$-th kind), [*i.e.*]{} ${\cal V}\xi\gg N^{-1/2}$, rather than ${\cal V}\xi\gg 1$. The initial naive low-density approximation $|\rho_i|\ll\sqrt{N}\xi$, which ensures the factorization of the potential, is then fully justified for not too large $N$, [*i.e.*]{} it should be replaced by the right one, $|\rho_i|\ll\xi$. In another words, for too large $N$, the requirement $|\rho_i|\ll\sqrt{N}\xi$ becomes no more the low-density approximation, since it then allows $|\rho_i|$ to exceed significantly its average value, which is of the order of $\xi$.
Note finally that the obtained results lead to obvious modifications of the values of the confining-string coupling constants (string tension, coupling constant of the rigidity term, and so on). These modifications, which are due to the change of the Debye mass of the dual photon, can be accounted for by virtue of the formulae obtained in ref. [@cu]. One should also take into account that the charges of quarks are distributed over the lattice of weight vectors of the group $SU(N)$, whose squares are equal to $(N-1)/(2N)$. We finally obtain the following values of the string tension and the inverse coupling constant of the rigidity term ([*cf.*]{} refs. [@nd] and [@epl]):
$$\sigma=8\pi^2g\sqrt{\xi}\frac{N-1}{\sqrt{N}}\Biggl[1+\frac12\xi IN(N-1)\Biggr],~~
\alpha^{-1}=-\frac{1}{16}\frac{g^3}{\sqrt{\xi}}\frac{N-1}{N^{3/2}}\Biggl[1-\xi IN(N-1)\Biggr].$$
Conclusions
===========
In the present paper, we have explored the influence of the Higgs field to the dynamics of the (2+1)D Georgi-Glashow model and its $SU(N)$-generalization. To this end, the Higgs field was not supposed to be infinitely heavy, as it takes place in the compact-QED limit of the model. Owing to this fact, the Higgs field starts propagating, that leads to the additional interaction between monopoles and, consequently, to the modification of the conventional sine-Gordon theory of the dual-photon field. Contrary to the previous analysis, performed in ref. [@nd] in the BPS limit, in the present paper the Higgs-boson mass was considered to be of the order of the W-boson mass. In this regime, combined with the standard weak-coupling approximation, the Debye mass of the dual photon and the potential of monopole densities have been found. In the low-density limit, the latter enables one to evaluate correlators of densities to any order. There has also been demonstrated that the existing data on the monopole fugacity provide the convergence of the cumulant expansion, which is used for the average over the Higgs field. This justifies the bilocal approximation adapted for the performed analysis.
After that, the above-described investigation has been generalized to the case of the $SU(N)$ Georgi-Glashow model with $N\ge 2$. The results obtained in this way reproduce, in particular, the respective $(N=2)$-ones. There has also been found the upper bound for $N$, necessary to ensure the convergence of the above-mentioned cumulant expansion. This bound is a certain exponent of the ratio of the W-boson mass to the squared electric coupling constant. It is therefore an exponentially large quantity in the weak-coupling regime, that yields an enough broad range for the variation of $N$. Finally, we have found the values of the two leading coupling constants of the confining-string Lagrangian at arbitrary $N$.
Acknowledgments
===============
The author is greatful for useful discussions to Prof. A. Di Giacomo and Dr. N.O. Agasian. He is also greatful to Prof. A. Di Giacomo and to the whole staff of the Physics Department of the University of Pisa for cordial hospitality. This work has been supported by INFN and partially by the INTAS grant Open Call 2000, Project No. 110.
Appendix. Evaluation of the integral $\int d^3x{\cal K}({\bf x})$. {#appendix.-evaluation-of-the-integral-int-d3xcal-kbf-x. .unnumbered}
==================================================================
Setting, as everywhere else in this paper, $m_W$ for an UV cutoff and using the notations for $a$, $c$, $[1/c]$, and the incomplete Gamma-function introduced in the main text, we have for the desired integral:
$$I=\frac{4\pi}{m_H^3}\int\limits_{c}^{\infty}dxx^2
\left[\exp\left(\frac{a{\rm e}^{-x}}{x}\right)-1\right]=
\frac{4\pi}{m_H^3}\sum\limits_{n=1}^{\infty}\frac{a^n}{n!}
\int\limits_{c}^{\infty}dx{\rm e}^{-nx}x^{2-n}=$$
$$=\frac{4\pi}{m_H^3}\sum\limits_{n=1}^{\infty}\frac{a^n}{n!}
n^{n-3}\Gamma(3-n,cn)\simeq
\frac{4\pi}{m_H^3}\left[\sum\limits_{n=1}^{[1/c]}\frac{a^n}{n!}
n^{n-3}\Gamma(3-n,cn)+c^2\sum\limits_{[1/c]+1}^{\infty}
\left(\frac{a}{c}\right)^n\frac{{\rm e}^{-cn}}{nn!}\right].\eqno(A.1)$$ Clearly, in the derivation of the last equality, we have used the asymptotics of the incomplete Gamma-function at large values of its second argument: $\Gamma(3-n, cn)\simeq (cn)^{2-n}
{\rm e}^{-cn}$. The last sum in eq. (A.1) can further be rewritten as $\sum\limits_{n=1}^{\infty}-\sum\limits_{n=1}^{[1/c]}$, and we obtain:
$$I\simeq\frac{4\pi}{m_H^3}\left\{\sum\limits_{n=1}^{[1/c]}
\frac{a^n}{nn!}\left[n^{n-2}\Gamma(3-n,cn)-c^{2-n}{\rm e}^{-cn}\right]
+c^2\sum\limits_{n=1}^{\infty}\left(\frac{a}{c}\right)^n
\frac{{\rm e}^{-cn}}{nn!}\right\}.\eqno(A.2)$$
Note that the last sum here is equal to ${\rm Ei}\left(\frac{a}{c}{\rm e}^{-c}\right)-\gamma-\ln\left(\frac{a}{c}{\rm
e}^{-c}\right)$, where $\gamma\simeq 0.577$ is the Euler constant, and Ei denotes the integral exponential function. However, in the interesting to us case $c\sim 1$, $a\gg 1$, such a representation of that sum does not help when one tries to express it explicitly in terms of $a$ and $c$. Instead, it is useful to rewrite it as follows:
$$\int\limits_{0}^{\infty}dt\sum\limits_{n=1}^{\infty}
\left(\frac{a}{c}\right)^n\frac{{\rm e}^{-(c+t)n}}{n!}=
\int\limits_{0}^{\infty}dt\left\{\exp\left[\frac{a}{c}
{\rm e}^{-(c+t)}\right]-1\right\}=\int\limits_{0}^{\frac{a}{c}
{\rm e}^{-c}}\frac{dz}{z}\left({\rm e}^z-1\right),\eqno(A.3)$$ where $z\equiv\frac{a}{c}{\rm e}^{-(c+t)}$. Integrating by parts we have at $\frac{a}{c}{\rm e}^{-c}\gg 1$:
$$\left.\left.(A.3)\simeq\left[\exp\left(\frac{a}{c}{\rm e}^{-c}\right)-1\right]
\left(\ln\frac{a}{c}-c\right)-\Biggl(\left<{\rm e}^z\right>\right|_{0}^{1}
\int\limits_{0}^{1}dz\ln z+\left<\ln z\right>\right|_{1}^{\frac{a}{c}{\rm e}^{-c}}
\int\limits_{1}^{\frac{a}{c}{\rm e}^{-c}}dz{\rm e}^z\Biggr)\simeq$$
$$\simeq\exp\left(\frac{a}{c}{\rm e}^{-c}\right)\left(1-\frac{c}{a}{\rm e}^c\right)+
({\rm e}-1)\ln a,$$ where in the derivation of the last equality we have kept the terms leading in $a$ and $(a/c)$. Together with the first sum standing on the r.h.s. of eq. (A.2) this finally yields eq. (\[3\]) of the main text.
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[^1]: E-mail address: [[email protected]]{}
[^2]: Similarly to ref. [@suN], we assume here that the function $\epsilon$ is one and the same for any $N$.
|
---
abstract: 'When holes move in the background of strong antiferromagnetic correlation, two spatial scales emerge which lead to a much reduced hopping integral with an additional phase factor. By taking these two effects into consideration, we propose an effective Hamiltonian to investigate pseudogap in cuprates. We argue that pseudogap is the consequence of dressed hole moving in the antiferromagnetic background and has nothing to do with the superconductivity. In the normal state, The pseudogap opens near the antinodal region and vanishes around the nodal region. In the superconducting state, the d-wave superconducting gap dominates the nodal region, while the pseudogap dominates the antinodal region. A two-gap scenario is concluded to describe the relation between the two gaps.'
author:
- 'Y. Zhou$^1$'
- 'H. Q. Lin$^2$, and C. D. Gong$^3,^1$'
title: Origin of the pseudogap and its influence on superconducting state
---
Introduction
============
One of the most fascinating properties of the cuprates is the opening of a gap (pseudogap) above $T_{c}$ in the underdoped and optimally doped regime, where most abnormalities observed [@TTimusk]. The origin of the pseudogap and its relation to the superconducting (SC) gap are fundamental questions to realize the physics underline the high-temperature superconductors. Although much progress has been made, the issue remains open. There are two distinct scenarios on the relation between the pseudogap and the SC gap[@STS]. 1) The one-gap scenario: The pseudogap is viewed as the precursor of the SC gap, reflecting pair fluctuation above $T_{c}$, and would acquire phase coherence below $T_{c}$[@Emery]. The argument was based on angle-resolved photoemission spectroscopy (ARPES)[@HDing; @Loeser; @Valla], electron tunnelling [@Renner], and thermal transport measurements[@Hawthorn], etc; 2) The two-gap scenario: The pseudogap is not directly related to the SC gap, but emerge from some ordered states such as antiferromagnetic (AF)[@AF], staggered flux[@SF], stripe[@STRIPE], spin or charge density wave[@SCDW], and orbital circulating currents[@OCC], etc. and competes with SC gap. Models account for these ordered states capture some aspects of the pseudogap, but a theoretical framework which gives a full picture from the pseudogap to SC state is not yet established. Nevertheless, many experiments, including ARPES[@Kaminski], electronic Raman scattering[@LeTacon], and elastic neutron diffraction[@Fauque], etc. seem to favor the two-gap scenario. More recently, the improved ARPES data[@Tanaka; @WSLee; @TKondo; @KTerashima; @Kanigel] showed that the gap opens near the antinodal region, and vanishes near the nodal region in the pseudogap state, which leads to the arc structure of the Fermi surface. In the SC state, the pseudogap dominates the antinodal region, while the simple d-wave type gap dominates the nodal region.
In this Letter, an effective single particle Hamiltonian is proposed to study the pseudogap. The main idea came from the well known fact that moving holes in the antiferromagnetic background leads to two spatial scales: a long distance one and a short distance one. On the large spatial scale, the effective hopping integral is much reduced due to accumulated strong AF correlation, while on the small spatial scale, the moving holes obtain an additional phase factor due to the surrounding AF correlation. Thus, pseudogap came directly from the AF correlation and is irrelevant to the superconductivity. Based on the effective Hamiltonian, we find that the pseudogap opens near the antinodal regime and vanishes around the nodal region. Its doping and momentum dependence qualitatively agree with recent ARPES measurements. The arc structure of the Fermi surface is also obtained, with its length expanding from underdoping to overdoping. In the SC state, the simple d-wave SC gap dominates the node region, while the pseudogap dominates the antinodal region. The resultant gap also agrees well with experiments.
The neutron scattering[@Fauque] and quantum oscillations measurements[@Sebastian] show that in the underdoped region, antiferromagnetic correlation remains on the CuO$_{2}$ plane. Its existence has also been found numerically[@Leung]. For each spin, its four nearest neighbors compose a plaquette. As a hole is introduced into the center of this plaquette, it will sense an effective magnetic field originated from its four neighboring opposite spins. Correspondingly, there is a doping dependent gauge flux $\Phi$ passing through the plaquette. In its journey of wanderings, the hole is affected by the staggered flux filed in which $\Phi$ and $-\Phi$ appear alternatively. Therefore, in the small spatial scale, when the hole hops, it encounters a phase shift $\delta /4$ (or $-\delta /4$) via Aharonov-Bohm effect, where $\delta =2\pi \Phi /\Phi _{0}$ ($\Phi _{0}=hc/e$ is the flux quanta). On the other hand, in the large spatial scale, the effective hopping, represented by the hopping matrix $I_{n}=\langle c_{i}^{+}c_{j}+c_{i}^{+}c_{j}\rangle _{n}$ ($n=1$, $2$, and $3$ for the nearest, second nearest, and third nearest neighbor respectively), evaluated by all possible configurations, is substantially reduced by the strong AF correlation. Take the $t-J$ model as an example, we derive the following effective single particle Hamiltonian by considering the effects of two spatial scales as, $$\begin{aligned}
H=\sum_{k\sigma }\left( \gamma _{k}d_{k\sigma }^{+}e_{k\sigma
}+hc\right) +\sum_{k\sigma }\epsilon _{k}\left( d_{k\sigma
}^{+}d_{k\sigma }+e_{k\sigma }^{+}e_{k\sigma }\right) \text{,}
\label{e.1}\end{aligned}$$where $\gamma _{k}=-2(I_{1}+J'\chi_{0})(e^{i\delta /4}\cos k_{x}+e^{-i\delta
/4}\cos k_{y})$, and $\epsilon _{k}=-4I_{2}\cos k_{x}\cos k_{y}-2I_{3}\left(
\cos 2k_{x}+\cos 2k_{y}\right) -\mu$. $\chi_{0}=\sum_{\sigma}\langle
d_{i\sigma}^{+}d_{i \sigma}+e_{i\sigma}^{+}e_{i \sigma}\rangle$ is the uniform bond order, $J'=3J/8$. The summation is restricted in the AF Brillouin zone. The effective Hamiltonian is diagonalized with the quasiparticle dispersion $%
\epsilon _{k}^{\pm }=\epsilon _{k}\pm |\gamma _{k}|$.
To determine the phase factor $\delta/4$, we calculate the effective staggered antiferromagnetic field, $B_{eff}$, which can be estimated by the spin-spin correlation between a spin 1/2 sits on the site of the given hole and its surrounding spins. That is $B_{eff}=J\sum_{j\in NN}\langle S_{i}\cdot S_{j}\rangle/g\mu
_{B}S_{z}$ with $g$, and $\mu _{B}$ denoting Land$\acute{e}$ factor and Bohr magneton, respectively. Correspondingly the staggered flux $\Phi =B_{eff}a^{2}$. $a$ is the lattice constant, which is taken as $0.45nm$ throughout this paper. $\langle
S_{i}\cdot S_{j}\rangle$ is the nearest neighbor (NN) spin-spin correlation, which is calculated by the exact diagonalization calculations (ED) as shown in Table. \[T1\]. Since the effect of long-range spin-spin correlations and spatial variation of the effective magnetic field are neglected, the resultant flux and corresponding phase factor are overestimated to some degrees. The effective hopping matrix $I_{n}$ is also evaluated by the ED technique. For more hole cases, the hopping matrix should be divided by a factor approximately equal to hole number $n_{h}$. In the above calculations, we have assumed that the extend $t-J$ model can be used to describe the many body effect qualitatively. $t$ (about $0.4eV$ in the real system) is taken as energy unit, $t_{1}=-0.3$, $t_{2}=0.2$ and $J=0.4$. Comparing with the slave-boson treatment[@LJX], the effective hopping terms are all enhanced due to many body effect.
-----------------------------------------------------------------------
$n_{h}$ $\langle S_{i}\cdot $I_{1}$ $I_{2}$ $I_{3}$
S_{j}\rangle$
--------- --------------------- ---------- ----------- ---------- -- --
$0$ $-0.3454$ $ 0$ $0$ $0$
$1$ $-0.2745$ $0.0703$ $-0.0297$ $0.0125$
$2$ $-0.2179$ $0.1483$ $-0.0350$ $0.0355$
$3$ $-0.1670$ $0.2116$ $-0.0469$ $0.0510$
$4$ $-0.1399$ $0.2635$ $-0.0594$ $0.0732$
$5$ $-0.0855$ $0.3126$ $-0.0709$ $0.0960$
-----------------------------------------------------------------------
: The ED results of the spin-spin correlation function and hopping matrix in the 20-site cluster with different hole concentration.[]{data-label="T1"}
We further include the effect of the spin fluctuation under the random-phase approximation (RPA)[@YQS]. Since partial spin fluctuation has been taken into account for the contribution of additional phase factor in Eq. (1), we introduce an adjustable parameter to set the AF instability at experimental value $x\simeq 0.03$ to avoid the overestimating[@LJX]. The bare spin susceptibilities are expressed as
$$\begin{aligned}
\chi^{t}_{0}(q,\omega)=\frac{1}{2}
\sum_{k}[\kappa_{+}(F_{++}+F_{--})+\kappa_{-}(F_{+-}+F_{-+})]\nonumber\\
\chi^{t}_{0}(q,q+Q,\omega)=\frac{i}{2}
\sum_{k}\kappa_{3}(F_{++}+F_{--}-F_{+-}-F_{-+})\end{aligned}$$
with abbreviations $F_{\eta \eta ^{\prime }}=\frac{f(\epsilon _ {k}^{\eta
^{\prime }})-f(\epsilon _{k+q}^{\eta })}{\omega +\epsilon _{k+q}^{\eta
}-\epsilon _{k}^{\eta ^{\prime }}}$ $(\eta ,\eta ^{\prime }=+,-)$. $\kappa_{\pm}=1\pm (Re\gamma_{k}Re\gamma_{k+q}+ Im\gamma_{k}Im\gamma
_{k+q})/|\gamma_{k}\gamma_{k+q}|$ and $\kappa_{3}=(Re\gamma _{k}Im\gamma
_{k+q}-Im\gamma _{k}Re\gamma _{k+q})/|\gamma_{k}\gamma_{k+q}|$. Since $\pm $ band is not independent due to the AF correlation, the nondiagonal term $\chi
_{0}(q,q+Q,\omega) $ arises from the umklapp processes. Under this treatment, the Dyson’s equation can be expressed as $\hat{G}(k,i\omega_{n})^{-1}=
\hat{G}_{0}(k,i\omega_{n})^{-1}-\hat{\Sigma}(k,i\omega_{n})$ with $\hat{\Sigma}(k,i\omega_{n})=
\sum_{q,m}J_{q}^{2}\chi^{t}(q,i\omega_{m})\hat{G}(k-q,i\omega_{n}-i\omega_{m})$, where $J_{q}=J(cos q_{x}+cos q_{y})$, $\omega_{n}$ and $\omega_{m}$ are the fermionic and bosonic Matsubara frequency. Both the Green’s function and self-energy are $2\times 2$ matrices. The spectral function is obtained via $A(k,\omega)=-(1/\pi)Im[G(k,\omega+i\delta)]$.
The quasiparticle band shown in Fig. \[f.1\] is rather rigid against doping along the nodal line with much reduced bandwidth $0.6-0.7t (\sim
$240-280meV$)$, well consistent with the ARPES measurement[@Marshall]. The upper and lower band coincide at $(\pi /2,\pi /2)$. This means that no full gap opens, unlike the Mott insulator state. On the other hand, the lower band around $(\pi ,0)$ shows clear flatness below $E_{f}$. Correspondingly, a pseudogap (denoted by $\Delta _{PG} $) opens around the antinodal regime. The distance between the flatness part and Fermi level decreases with increasing doping. The results obtained here qualitatively coincide with the ARPES data[@Marshall]. It should be noted that as far as the QP dispersion is concerned, even the effective single particle Hamiltonian (Eq. (1)) has already succeeded to describe the main features mentioned above as shown in the insert of Fig. \[f.1\].
![Lower band QP dispersion for different hole concentration under RPA treatments. Insert is the results obtained by direct diagonalizing the Eq. (1).[]{data-label="f.1"}](QPdispersion.ps){width="3.0in"}
![Pseudogap as functions of angle $\protect\theta$ for different doping. Solid lines, and open squares correspond to results with and without RPA treatment, respectively. Insert shows the experimental data on Bi2212 obtained from Ref. [@WSLee], squares, circles, and triangles are for UD75K, UD92K, and OD86K, roughly corresponding to underdoped (x=0.1), optimal doped (x=0.15), and overdoped (x=0.2) case. The dotted line are guided for eyes.[]{data-label="f.2"}](PGgap.ps){width="3.0in"}
Fig. \[f.2\] shows the evolution of the pseudogap in the momentum space at different doping density. The magnitude of pseudogap is determined by evaluating the minimal distance of the lower band from Fermi energy in the given direction. $\Delta
_{PG}(\theta)$ decreases with increasing $\theta$, and disappears at a critical value of $\theta$. This can be easily realized from Fig. \[f.1\], where the gap opens near the antinodal regime due to the flatness of QP dispersion induced by the staggered magnetic flux, while the gap closes due to the crossing of the QP dispersion over $E_{f}$. Such behavior had been confirmed by various experiments [@WSLee; @TKondo; @KTerashima]. Both the pseudogap region and its magnitude decrease with doping as evidenced by ARPES data[@Valla]. The largest $\Delta_{PG}$ decreases from about $92meV$ at deep underdoped region to about $40meV$ at optimal doping with almost linear doping dependence. The numerical results consistent with ARPES measurement, but the range of angle seem to have about $10\%$ reduction[@WSLee]. Results obtained by direct diagonalizing the Eq. (1) show similar doping and momentum dependence. However, the magnitude and the variation range of angle are more reduced. Comparing with the experiments, the present pseudogap vanishes somewhat slower with increasing $\theta$. In fact, the value of angle $\theta_{0}
(\Delta_{PG}(\theta)=0)$ is intimately related to the AF correlation in momentum space, large $\theta_{0}$ means smaller AF correlation. This implies that the present treatment is more appropriate to describe situations with strong AF correlation. In the present treatments, we do not adopt any pairing potential, which is distinct from the previous slave-boson calculation[@LJX]. Furthermore, the behavior of pseudogap differs from the early ARPES discovery[@HDing; @Loeser], and is far from that of the simple d-wave gap. Therefore, the pseudogap is not related to the SC pairing. Our theoretical results imply that the pseudogap exists even in the overdoped range.
![Density plots for integration of spectral intensity times Fermi function over an energy interval $[-0.025, 0.025]t$ around the Fermi energy for different doping density.[]{data-label="f.3"}](FScounter.ps){width="3.0in"}
The evolution of the Fermi surface (FS) is depicted in Fig. \[f.3\]. The intensity is strong near the nodal region, and becomes weak gradually toward antinode due to the opening of the pseudogap in this region. Thus, an arc FS structure[@Norman] forms around the nodal region. Its length expands with increasing hole concentration. The arc extends to a large hole FS above $x=0.20$, where the pseudogap vanishes gradually. So, the arc structure is a direct consequence of the pseudogap. The evolution of the arc structure and its length also qualitatively agree with the recent ARPES experiment[@KMShen].
Now we discuss the existence and influence of pseudogap in the SC state. Based on experimental observations[@WSLee; @TKondo; @KTerashima] and above analysis, we start with the assumption that the total gap of the quasiparticle dispersion contains two components: the pseudogap, and the SC gap of the standard d-wave BCS form $\Delta_{k}=\Delta_{SC}(cosk_{x}-cosk_{y})$. After some manipulations, the corresponding Green’s function is $\hat
G_{SC}(k,i\omega)=[\hat G_{PG}(k,i\omega)^{-1}+\Delta_{k}^{2}\hat
G_{PG}(k,-i\omega)]^{-1}$, where $G_{PG}(k,i\omega)$ contains pseudogap $\Delta_{PG}(0)$, is the diagonal Green’s function in the pseudogap state under RPA treatment. We then compare our results with the experiments to see if $\Delta_{PG}(0)/\Delta_{SC}
<< 1$ (one-gap scenario) or it is substantial (two-gap scenario). We take the $x=0.15$ case, which is near the optimal doping, as an example. In the nodal region, the gap increases almost linearly with decreasing angle $\theta$ as shown in Fig. \[f.4\]. This is the typical d-wave behavior, showing that d-wave SC pairing dominates this region. In fact, the gap comes from the SC condensation entirely because the pseudogap vanishes along the arc. In the antinodal region, large deviation from the standard d-wave is rather obvious. The shape and the value of the gap in the SC state with small value of $\Delta_{SC}$ are quite similar to that of the pseudogap state. Even one increases $\Delta_{SC}$ up to $0.06$ such that $2\Delta_{SC}>\Delta_{PG}(0)$, the shape keeps unchanged and the value of the gap enhances little. Therefore, the pseudogap dominates the antinodal region. Between the two regions, the pseudogap and the SC gap coexist and compete with each other, so the deviation from the simple d-wave enhances gradually when approaching to the antinodal region. These features consist well with recent experiments on the optimal doped Bi2212[@WSLee], Bi2201[@TKondo], and LSCO[@KTerashima], as shown in the inset of Fig. \[f.4\].
![The total gap in the SC state for different magnitude of $\Delta_{SC}$. The doping density is set to be $x=0.15$, near the optimal doping. Insert shows experimental data at optimal doping, together with our theoretical results with $\Delta_{SC}=0.04$. Circles are for LSCO (Ref. [@KTerashima]), and triangles are for Bi2201 (Ref. [@TKondo]).[]{data-label="f.4"}](SCgap.ps){width="3.0in"}
The degree of the deviation depends on the ratio of $\Delta_{SC}/\Delta _{PG}(0)$. Smaller ratio means larger deviation. Such effect has been manifested in optimal doped Bi2201[@TKondo] and Bi2212[@JMesot]. The pseudogap in the two cuprates is almost the same at optimal doping, but the SC critical temperature in the former is three times of the latter. Correspondingly, the deviation from the simple d-wave is much enhanced in Bi2201. The enhancement of the deviation with decreasing doping can also be explained following this way. Such deviation had also been obtained in the previous one-gap scenario by applying the spin fluctuation theory[@LJX]. However, the correction was too small to account for the large deviation in underdoped Bi2201 and Bi2212. Based on our analysis, the two-gap scenario seems more appropriate to account for the observed gap behavior. The experiments support the one-gap scenario, mostly came from the earlier measurements[@HDing]. The data near the nodes were not easy to distinguish at that time. Additionally, some measurements, such as the thermal transport properties[@Hawthorn] may concern mainly on the nodal region, where the simple d-wave SC gap dominates. At last, we would like to point out that the enhancement of the SC gap with doping in the underdoped cuprates is an outcome of the decreasing pseudogap, which leads to the suppression of the density of state near the Fermi energy.
In conclusion, we have proposed an effective single particle Hamiltonian extracted from the exact diagonalization studies of the $t-J$ model to investigate pseudogap in the high-Tc cuprates. The main idea is to take the effects of two spatial scales, one leads to much reduced hopping and one adds to a phase factor, both reflect physics of doped holes moving in AF background, into account. Our results revealed that the pseudogap is the single particle property, and is not directly related to the SC state. It opens near the antinodal region and vanishes around nodal region. The doping and momentum dependence of the pseudogap are qualitatively consistent with recent ARPES measurements. The arc structure of the Fermi surface is a natural product of the pseudogap. In the SC state, the simple d-wave SC gap dominates the nodal region, while the pseudogap dominates the antinodal region. The two-gap scenario is therefore concluded.
This work was supported in part by NSFC Projects No. 10804047, 973 Projects No. 2009CB929504 and 2006CB601002. H. Q. Lin acknowledges RGC Grant of HKSAR.
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abstract: 'We study the effects of dust grain size on the spectral energy distribution (SED) of spherical circumstellar envelopes. Based on the self-similarity relations of dusty SEDs derived by @ive97, we expect an approximate invariance of the IR SED for models with different grain sizes. Approximate invariance follows from the fact that differently sized grains have similar optical properties at long wavelengths where the dust reprocesses the starlight. In this paper, we discuss what are the physical requirements on the model parameters to maintain the approximate invariance of the IR SED. Single grain size models are studied for a wide range of grain sizes in three optical depth regimes: optically thin models, moderate opacities, and very optically thick models. In this study, we find limits for the cases where the IR SED is and is not capable of conveying information about grain sizes, and to what extent it does so. We find that approximate invariance occurs for a much larger range of grain sizes than previously believed, and, when approximate invariance holds, the SED is controlled mainly by one parameter, the reprocessing optical depth, a quantity that measures the fraction of starlight that is absorbed by the dust grains. Models with a grain size distribution are studied as well. For these models, we find that, in many instances, the concept of approximate invariance may be extended from the IR SED to all wavelengths. This means that, for a wide range of optical depths, models with different grain size distributions will produce very similar SEDs and, hence, the reprocessing optical depth is the only quantity that can be unambiguously obtained from the SED. The observational consequences of this result are discussed in detail. Finally, in models with a size distribution, the different grain sizes each have different equilibrium temperatures. The consequences of this effect for the model SED are discussed as well.'
author:
- 'A. C. Carciofi'
- 'J. E. Bjorkman'
- 'A. M. Magalhães'
title: Effects of Grain Size on the Spectral Energy Distribution of Dusty Circumstellar Envelopes
---
Introduction \[introduction\]
=============================
Dust is associated with many astronomical objects, such as stars and galaxies. Dust grains typically absorb radiation at short wavelengths, and since the dust grains are usually cooler than the radiation sources, they reemit the absorbed radiation at longer wavelengths. Consequently, the presence of dust is usually detected by a flux excess at infrared wavelengths. The usual challenge is to use the spectral energy distribution (SED) of the object, to extract information about the nature of the underlying luminous source and the properties of the surrounding dust; i.e., its spatial distribution, optical depth and intrinsic properties. The intrinsic properties of the dust grains are their chemical composition, condensation temperature, shape and size.
Knowledge of the dust grain sizes present in circumstellar matter is desirable for many astrophysical situations. For example, it is believed that about 60% of the dust injected into the ISM originates from the winds of AGB stars [@geh89], so knowledge of the size of the dust grains present in the winds of such stars is fundamental for understanding the properties of interstellar dust. Similarly, grain size is important for understanding the details of grain formation and mass loss mechanisms for cool stars. If we assume that the source spectrum and the spatial distribution of the dust are known for these objects, the question becomes to what extent is it possible to obtain the intrinsic properties of the dust from knowledge of the SED alone. In this paper, we investigate how grain size affects the SED and to what extent the SED can be used to constrain the grain sizes.
It is well known that models of the SED are not unique. For this reason it is imperative to identify the fundamental parameters controlling the SED. Recently, @ive97, hereafter IE97, investigated the scaling properties of the radiation transfer problem in spherically symmetric dust-envelopes. They found that the radiation transfer problem in dusty media depends only on the following quantities:
i\) The geometry (shape) of the system (i.e., all distances are proportional to a scale factor);
ii\) The sublimation temperature of the material;
iii\) The spectral shape of the input radiation, $\lambda F_\lambda/F$;
iv\) The shape of the dust absorption and scattering opacities, $\kappa_\lambda/\kappa_{\lambda_0}$ and $\sigma_\lambda/\sigma_{\lambda_0}$, respectively, where $\lambda_0$ is the fiducial wavelenght;
v\) The dust scattering phase function (SPF)[^1];
vi\) The overall optical depth at the fiducial wavelength.
The above list can be regarded as a set of [*invariance requirements*]{}. Two models that have different physical parameters, but meet the above requirements exactly, are equivalent and have the same SED. For example, the physical dimensions of the dusty region can be freely changed (say by changing the stellar luminosity, which increases the dust condensation radius) with no effect on the SED, as long as the overall optical depth and shape factors do not change. If an invariance requirement is violated, however, then the models are no longer equivalent and the SEDs are expected to be different. For instance, if one were to change the grain size, the shape of the opacity and the SPF would change, violating requirements (iii) and (v).
For a given class of astronomical objects (e.g., AGB stars), most of the quantities above are likely to be similar (geometry and grain composition), so the optical depth becomes the single most important parameter that controls the SED.
It is reasonable to expect that if an invariance requirement is weakly violated, the SED will still be approximately the same. This was noted by IE97 who pointed out that if the grains are very small (about a tenth of wavelength of the peak of the source spectrum) then the shape of the opacities are very similar and the grain size is irrelevant for the problem. For a 3000 K source, for example, the upper limit for the grain size is about $0.05 \mu \rm m$. Grains larger than this upper limit will, according to IE97, significantly alter the results.
In the case of optically thin envelopes, this condition on the maximum grain size can be further relaxed, because the star completely dominates the optical SED while the grains produce the IR SED. For the IR SED to remain similar between models with different grain radii, it is evident that both the shape of the IR emissivity and the reprocessed luminosity must be the similar. For the shape of the IR emissivity to be the similar, it is necessary that the grains have similar absorption efficiencies [*only in the IR*]{}; this relaxed requirement increases the maximum grain size to at least $15 \mu \rm m$, assuming the grains have equilibrum temperatures of about 1000 K. The condition that the reprocessed luminosity be similar is equivalent to imposing a condition on the optical depth (see next section).
This example shows that, under certain circumstances, it is possible to [*relax the invariance requirements and still maintain the invariance of the SED*]{}, at least approximately. We call this concept [*approximate invariance*]{}, and in this paper we demonstrate how it reveals the similarities between models with different grain sizes. In particular, we study to what extent the grain sizes can be changed, while maintaining the approximate invariance of the SED under a wide range of model parameters. For each case presented, we identify the primary parameter (like optical depth) that controls the SED.
In the next section, we explore in more detail the physical requirements for approximate invariance when the dust grain size is varied. In section \[mc\], we briefly describe the Monte Carlo code used for the calculations. In section \[single\], we present the effects of grain size on the SED for single grain size models. Section \[dist\] extends the study to models with a grain size distribution, and a discussion and summary of the results is presented in section \[discussion\].
Approximate Invariance \[as\]
=============================
Consider an astrophysical system, which consists of a luminous source (e.g., a star) surrounded by a dusty envelope. The physical description of this system must include both the shape and spectrum of the source and the dust properties (chemical composition, grain size, spatial distribution and optical depth). One characteristic of such a model is the presence of an inner cavity. Typically, for a star that is losing mass (e.g., an AGB star), the location of the inner cavity will be controlled by dust sublimation/condensation. In contrast, a star that has stopped losing mass (e.g., a planetary nebulae) may have a much larger cavity, with a correspondingly cooler radiative equilibrium temperature at the inner edge of the cavity.
Now consider different models of the system, where we vary only three parameters: the dust grain size ($a$), the cavity inner radius ($r_{\rm i}$), and the optical depth ($\tau$). Such models violate some of the invariance requirements of section \[introduction\], thus they should have different SEDs. However, it follows from the idea of approximate invariance (AI) that their IR SED will be similar, provided that the three conditions bellow are satisfied:
1\. The shape of the absorption efficiency factor for the different grain sizes is similar in the spectral range where the grains emit most of their thermal flux;
2\. The integrated (bolometric) IR luminosity is the same; and
3\. The temperature of the grains at the inner edge of the envelope is the same for all grain sizes.
In the following, we investigate the requirements these conditions impose on the three model parameters (optical depth, grain size and cavity radius). Let us first establish the spectral region where we expect the absorption efficiencies $Q_{\rm abs}$ of differently sized grains to be approximately equal (condition 1). Figure \[effsil\] shows $Q_{\rm abs}$ for cosmic silicate of various grain sizes (optical data from @oss92). Mie theory defines three spectral regions where the grains have common properties:
i\) $\lambda \lesssim \pi a$. In this region, called the geometrical limit, the optical properties are roughly independent of $\lambda$, and $Q_{\rm abs} \approx \rm const.$;
ii\) $\lambda \gtrsim 5\pi a$. In this interval, the efficiencies are in the so-called Rayleigh limit, where the size of the particle is much smaller than the wavelength. In this limit, $Q_{\rm abs} \propto 1/\lambda$ for amorphous grains and $Q_{\rm abs} \propto 1/\lambda^2$ for crystalline grains;
iii\) $\pi a \lesssim \lambda \lesssim 5\pi a$. In this region, the efficiencies are calculated using Mie theory (spherical particles) and are dependent on grain size. We call this interval the [*Mie region*]{}.
These three regions are easily recognizable in Figure \[effsil\]. We see that all grains have approximately the same efficiencies for short wavelengths (region i) and that the shape of the efficiency factors are the same for $\lambda \gtrsim 10 \,\mu \rm m$ (region ii). In the visible and NIR region, the efficiencies are very different. From the above discussion, it is easy to see that condition 1 of AI is satisfied when most of the IR radiation is emitted in the Rayleigh limit. In practice, this sets a constraint on the maximum grain size that satisfies condition 1 of AI. For a given temperature of the inner cavity, $T_{\rm i}$, the largest grain size satisfying condition 1 of AI is given by $$a \lesssim 185/T_{\rm i}\,\mu \rm m. \label{eq1}$$ For example, for $T_{\rm i}=1000 \,\rm K$ (approximately the sublimation temperature for most grain species), the maximum grain size satisfying condition 1 of AI is about $0.2 \,\mu \rm m$.
Condition 2 of AI requires that the emergent envelope luminosity, $L_{\rm rep}$, remain fixed. The envelope is powered by absorbing some fraction of the stellar luminosity, so condition 2 is equivalent to requiring that this absorbed fraction remain fixed. To characterize this condition, we introduce the quantity $\tau_{\rm rep}$, the [*reprocessing optical depth*]{}. This quantity is defined so that envelopes with the same $\tau_{\rm rep}$ will reprocess (i.e., absorb) the same fraction of the input energy. The reprocessing optical depth is thus given by the condition $$1-e^{-\tau_{\rm rep}}\equiv L_{\rm rep}/L_\star \ . \label{trep}$$
To define the quantity $L_{\rm rep}$ it is useful conceptually to divide the emergent radiation into stellar photons and envelope photons. In a Monte Carlo simulation, the stellar luminosity is divided into $N$ equal energy photon packets (note that packets with different frequencies contain a different number of physical photons). The energy per packet is given by $$E_\gamma = L_\star \Delta t / N \ ,$$ where $L_\star$ is the stellar luminosity and $\Delta t$ is an arbitrary simulation time. Stellar photons are emitted from the star and they propagate through the envelope, where they may be scattered or absorbed. If a stellar photon is absorbed, radiative equilibrium requires that it be reemitted as an envelope photon. Note that scattering does not change the photon type (stellar vs. envelope). The fraction of the stellar luminosity reprocessed (absorbed) by the envelope may easily be determined by counting the number of stellar photon packets absorbed by the envelope, $N_\star^{\rm abs}$. This is equivalent to counting the number of stellar photons that emerge from the envelope without absorption, $N_\star^{\rm em}$. Thus the luminosity reprocessed by the envelope is $$L_{\rm rep} = N_\star^{\rm abs} E_\gamma/\Delta t = (N-N_\star^{\rm em}) E_\gamma/\Delta t = L_\star-L_\star^{\rm em} \ ,$$ where $L_\star^{\rm em}$ is the emergent stellar luminosity. This is the method we use to measure $L_{\rm rep}$ in this paper. Note that scattering affects $L_{\rm rep}$ because a photon packet that is scattered can be absorbed subsequently. Owing to these scattering effects, it is not straightforward to obtain $L_{\rm rep}$ from standard radiative transfer quantities. The reprocessed luminosity is given by the emergent luminosity of the envelope, so $$L_{\rm{rep}} = \int \left[
%\kappa_\lambda B_\lambda +
4\pi j_\lambda +
\int \frac{d \sigma}{d\Omega}\left(\hat{\rm{n}},\hat{\rm{n}}^\prime \right)
I_{\lambda}^{\rm{env}} \left( \hat{\rm{n}}^\prime \right) d\Omega^\prime
\right]
\rho~
e^{-\tau_\lambda(\hat{\rm{n}})}
d\Omega d\lambda dV
%= E_\gamma N^{\rm{esc}}_{\rm{env}}
\ , \label{lenv}$$ where $d \sigma/d\Omega$ is the differential scattering cross section, which is a function of both incoming and outgoing directions ($\hat{\rm{n}}^\prime$ and $\hat{\rm{n}}$, respectively), and $I_{\lambda}^{\rm{env}}$ is the envelope contribution (Monte Carlo envelope photons) to the specific intensity (this includes scattered envelope photons but does not include scattered stellar photons). Energy conservation requires that $L_{\rm rep}$ equals the energy absorbed in the envelope, so $L_{\rm rep}$ can also be written as $$L_{\rm rep} = 4 \pi \int \kappa_\lambda \rho J^\star_\lambda d\lambda dV \ , \label{lenv2}$$ where $J^\star_\lambda$ is the stellar contribution to the mean intensity, including scattered starlight.
For low optical depths in the point source approximation ($r_i \gg R_\star$), $J_\lambda \rightarrow H^\star_\lambda$, where $H^\star_\lambda$ is the source flux. Defining $H^\star \equiv \int_{0}^{\infty} H^\star_\lambda d\lambda$ and the [*flux mean opacity*]{} $$\kappa_{\rm F} = {\int_{0}^{\infty} \kappa_\lambda H^\star_\lambda d\lambda \over H^\star} \ , \label{sigp}$$ eq. (\[lenv2\]) can be rewritten, in the optically thin limit, as $$L_{\rm rep} = 4\pi \int \kappa_\lambda \rho H^\star_\lambda d\lambda ~ 4\pi r^2 dr =
16\pi^2 \int \kappa_{\rm F} \rho H^\star r^2 dr =
L_\star \int \kappa_{\rm F} \rho dr
\equiv L_\star \tau_{\rm F} \ ,
\label{tflux}$$ where $\tau_{\rm F}$ is the [*flux mean optical depth*]{}. From eqs. (\[trep\]) and (\[tflux\]) we see that [*in the optically thin limit, $\tau_{\rm rep}$ is equivalent to $\tau_{\rm F}$*]{}.
For larger opacities, $I_{\lambda}^{\rm{env}}$ of eq. \[lenv\] is not zero, and its determination must take into account the full radiative transfer problem, including multiple scattering effects. In general, multiple scattering will increase the effective opacity of the envelope, making $\tau_{\rm rep}$ larger than $\tau_{\rm F}$. From this it follows that the detailed shape of the scattering phase function (SPF) can be of importance for determining the amount of reprocessing in the wind. That is the main reason why the SPF is listed above (item v in section \[introduction\]) as one of the quantities necessary to completely specify the radiative transfer problem in dusty media. In section \[SPF\] we will expand on this issue with further details.
An important consequence of eq. (\[lenv\]) it that, because $L_{\rm rep}$ depends on the details of the radiative transfer, it is not straightforward to obtain a relation between $L_{\rm rep}$ (or $\tau_{\rm rep}$) and the optical depth at the fiducial wavelength (say, $\tau_{V}$). In this paper we demonstrate that, because of AI, $\tau_{\rm rep}$ is the most suitable parameter for SED classification; it follows, then, that the optical depth $\tau_{V}$ is [*not*]{} a suitable parameter, contrary to the usual approach in the literature.
Although $\tau_{\rm rep}$ (or $L_{\rm rep}$) is a well defined quantity, relating this quantity to the physical parameters of the model (such as mass loss rate) is a rather involved problem, requiring the full solution of the radiative transfer. On the other hand, in many instances $L_{\rm rep}$ is a well defined observational quantity. This is the case, for example, for optically thin envelopes surrounding a hot star, in which the attenuated stellar spectrum can be easily separated from the IR emission of the grains. In the following, where we discuss the detailed consequences of AI, we will show that this issue is at the heart of the fundamental uncertainties in determining mass loss rates using the SED.
Finally, condition 3 of AI determines the radius of the cavity. In particular, the radius will be the dust destruction radius when the presence of the cavity is controlled by dust sublimation/condensation. Note that by specifying the cavity radius in stellar radii we are not introducing another radial scale in the problem.
The Monte Carlo Code \[mc\]
===========================
For the calculations shown in this paper, we have used a Monte Carlo code developed to solve the general problem of the radiative transfer plus radiative equilibrium in dusty media [@car01]. The code uses a standard Monte Carlo simulation, which basically follows the path of a large number of [*photon packets*]{} as they are scattered, absorbed and reemited within a prescribed medium (for further details see, e.g., @cod95). The radiative equilibrium is solved using the method described in @bjo01, in which the grain temperature and emitted spectrum are corrected in the course of the simulation.
Although the radiative equilibrium calculation does not require any iteration, quantities like the dust condensation radius $r_{\rm i}$ and the reprocessing optical depth $\tau_{\rm rep}$ (see section \[as\]) do require iteration, since they depend on the results of the radiation transfer. Each iteration returns a temperature of the inner cavity and the amount of reprocessed energy. The values of $r_{\rm i}$ and the optical depth are modified accordingly, and a new simulation is run, until the correct values are obtained.
The code is fully 3-D, so it is capable of solving the radiative transfer problem for many circumstellar geometries and density distributions, including the presence of multiple sources. The transfer of polarized radiation is included, so the code is capable of providing the entire SED, the polarization as a function of wavelength, images at specific wavelengths as well as polarization maps.
The code treats an arbitrary mixture of grain species (amorphous carbon, silicate, etc.) and a different size distribution can be assigned to each grain species. The grain scattering and absorption properties are calculated using Mie theory, without approximations, to obtain the correct form of the scattering phase function for polarized incident radiation. The radiative equilibrium condition is imposed separately for each grain type and size, so the code calculates independent radiative equilibrium temperatures for each grain type and size. This procedure contrasts with the usual procedure in the literature, which consists of assigning a single temperature to all grain sizes, whose opacity and scattering properties are obtained by averaging over the size distribution. In section 5, we will show that, for hot stars mainly, individual grain sizes can have very different equilibrium temperatures, which has important effects for the model IR SED.
In summary, the only approximation used in our dust model is that the shape of dust grain is spherical.
Single Grain Size Models \[single\]
===================================
In this section, we explore the effects of grain size on the SED for single grain size models, and interpret the results in the framework of approximate invariance. Our goal is to determine to what extent the grain size can be varied while maintaining the approximate invariance of the SED. We begin by defining the basic model we use in this and in the next section. It consists of a central star of unit radius that emits a black body spectrum of temperature $T_{\rm eff}$, surrounded by a spherical dust shell with internal radius $r_{\rm i}$ and external radius $1000\, r_{\rm i}$. The internal radius is set by the temperature of the cavity wall, $T_{\rm i}$, often taken to be the dust condensation temperature. The dust density profile is proportional to $r^{-2}$, and the dust grains are spherical. This is a simple model, but useful with respect to many astrophysical situations, such as the nearly spherical circumstellar envelopes found around many RGB and AGB stars.
Three different chemical compositions were studied: cosmic silicate, with optical constants given by @oss92, amorphous carbon (optical constants by @zub96) and silicon carbide (optical constants by @peg88). The sublimation temperature for these materials was taken to be $1000\,\rm K$ for silicate, $800\,\rm K$ for amorphous carbon and $1500\,\rm K$ for silicon carbide. These numbers are somewhat arbitrary, but they reflect approximately the temperatures used in the literature (e.g., IE97, and @lor94).
The model parameters we vary are the dust properties, optical depth, source temperature, and temperature of the cavity wall. To comply with the three conditions of AI, stated in section \[as\], we compare models with the same chemical composition, the same temperature of the cavity wall, and the same reprocessing optical depth.
We shall distinguish between three different cases, corresponding to three optical depth regimes, which will be studied separately. In case A, the envelope is optically thin, both in the wavelengths emitted by the source (the extinction region) and in the IR (the reprocessing region). In case B, the envelope is optically thick in the extinction region, but is optically thin to reprocessed radiation, and in case C the envelope is optically thick in both regions (up to the wavelength for which the Rayleigh limit is achieved for all grain sizes considered).
Case A: Optically Thin Envelopes \[caseas\]
-------------------------------------------
In this case, we determine the V-band optical depth $\tau_{V}$ for each model using the condition $\tau_{\rm rep}=0.1$, which corresponds to reprocessing 9.5% of the input radiation. In the extinction region, about 90% of the flux is stellar in origin, so we expect SEDs from different models will be similar in the extinction region, at least at the 10% level. The IR SED will also be similar, as a result of the AI. Hence, we expect to observe a similarity between all models for all wavelengths.
We illustrate this situation in Figure \[sedas\]. Each panel shows the SED for four different models, each with a different grain size, for cosmic silicate grains with a given stellar temperature. In Figure \[tauas\], we show the extinction optical depth as a function of wavelength for the models with $T_{\rm eff}=2500\,\rm K$. The other model parameters, cavity radius, grain mass and V-band optical depth are listed in Table \[tab1\].
Figure \[sedas\] corroborates our above qualitative discussion. For all grain sizes and stellar temperatures, the short wavelength side of the curve is very similar, as a consequence of the low optical depth combined with the same total attenuation. Similarly, the overall amplitude of the IR emission scales approximately with $\tau_{\rm rep}$ and the shape is set by the grain absorption efficiency, which is the same for each grain size. As a result, the overall SED is very similar for all models. The mean difference between the models with $T_{\rm eff}=2500\, \rm K$ is about 7%, and for the models with $T_{\rm eff}=20000\,\rm K$, the mean difference is 9%. The similarity between the model SEDs, not only in the IR but also in the extinction region, allows us to extend the consequence of AI. We conclude that in the optically thin limit [*the entire SED is similar when the conditions of AI are met*]{}.
The conditions for approximate invariance are violated when we consider grains larger than about $a = 0.25 \,\mu \rm m$. These grain sizes fail to satisfy condition 1 of AI, so large differences in the IR SED of these models are observed. This is shown in Figure \[grande\], where we plot the SEDs for models with grain sizes $0.005, 0.50$ and $1.0\,\mu \rm m$. The large differences in the IR SED are apparent. However, as shown in eq. (\[eq1\]), the largest grain that satisfies condition 2 of AI depends on the temperature of the cavity wall. Consequently, if we lower the condensation temperature, these larger grains will also satisfy the AI conditions. This is illustrated in Figure \[sed300\], which shows the SED for envelope models with $T_{\rm i} = 300\,\rm K$. These plots show that a distinction can hardly be made between envelope models with grain sizes now ranging up to $1 \,\mu \rm m$. For $T_{\rm eff}=2500 \rm K$ the mean difference between the SEDs is at most 3%. This has the consequence that the SED is a poor constraint for grain sizes, at least for sizes that satisfy the conditions of AI.
We conclude that, for case A, given a grain composition, a temperature for the inner cavity, and a spatial distribution of the dust, the SED depends on a single parameter, $\tau_{\rm rep}$. Another important result is that $\tau_{\rm rep}$ is not uniquely related to the $V$-band optical depth (or any other wavelength), as shown in Figure \[tauas\]. All curves are for $\tau_{\rm rep}=0.1$, and yet show very large differences in the optical depth for most wavelengths. From this we can conclude that [*the optical depth is not a suitable parameter for SED classification*]{}; if we were to group the case A models according to $\tau_{V}$, for example, the models would have large differences in the IR SED, leading us to the wrong conclusion that the grain size does have an important effect on the SED, which we have demonstrated not to be the case.
The optical depth depends on the product of the grain opacity (which depends on the grain size) and the grain density. The fact that grain sizes are very difficult to determine from the SED implies a corresponding uncertainty in the dust mass of the envelope, as shown in table \[tab1\]. This uncertainty directly affects the ability to measure the mass loss rates from stars using the SED.
Case B: Optically Thick Extinction with Optically Thin IR Emission \[casebs\]
-----------------------------------------------------------------------------
In this case, the envelope is optically thick in the UV and visible, and optically thin in the IR. Based on AI, we expect the behavior of the IR SED to be similar to case A; i.e., the shape and level of the IR emission should be the same for all models with the same $\tau_{\rm rep}$. In contrast to case A, the larger optical depths result in a much larger attenuation of the stellar flux. Therefore, the shape of the SED in the extinction region is controlled by the shape of the extinction optical depth, which strongly depends on grain size.
In Figure \[sedbs\] we present the SED for the same envelope models as in Figure \[sedas\], but we maintain $\tau_{\rm rep}=1$. This implies 63.2% of the input radiation is reprocessed by the envelope. As before, other models parameters are listed in Table \[tab2\], and in Figure \[taubs\] we display the extinction optical depth as a function of wavelength.
We see in Figure \[sedbs\] that indeed the IR SED for the different models are similar, while the SEDs for short wavelengths are very different because they depend on the shape of the grain efficiencies. It follows that case B is different from case A, in the sense that the SED does depend strongly on the dust grain size. There are, however, two points to be considered. First, in order to obtain information about grain size, one has to observe the SED in the wavelengths where the effects of grain sizes are important (i.e., UV, visible or NIR, depending on the model). Second, single grain size models are very crude approximations for actual stellar envelopes, where a distribution of grain sizes is expected. In section \[dist\] we study this case and show how it affects our conclusions for case B.
At this point it is useful to compare our results with those of previous authors. For example, IE97 show SEDs in their Figure 10 for spherical envelopes of silicate and amorphous carbon dust grains, of varying sizes and optical depths. In their plots on the right, where the results for silicate grains are displayed, the upper two plots can be compared roughly with our Figures \[sedas\] and \[sedbs\]. In Figure 10 of IE97, large differences between the models with $a = 0.05$ and $0.1 \mu \rm m$ are evident. By looking at these plots, the reader is led to believe that the IR SED for grains with sizes $0.05$ and $0.1 \mu \rm m$ is intrinsically different. However, our results show that this is not the case. The IR SED of single grain size models whose sizes comply with the conditions of AI have very similar shapes and levels.
The differences between Figure 10 of IE97 and our Figures \[sedas\] and \[sedbs\] owe to the choice of the model optical depth. In IE97, all models have the same extinction $V$-band optical depth. Our models, on the contrary, have all the same $\tau_{\rm rep}$ but very different $\tau_{V}$, as seen in Figures \[tauas\] and \[taubs\]. Thus in case B, we also find that $\tau_{\rm rep}$ is the best parameter to reveal the essential similarities of models with different grain sizes (which must be lower than the limit set by condition 2 of AI).
Case C: Optically Thick Envelopes also in the IR \[casecs\]
-----------------------------------------------------------
In this case, the envelope is so optically thick that 100% of the input radiation is reprocessed by the dust grains. As a consequence, the quantity $\tau_{\rm rep}$ becomes ill-defined, so another parameter must be found to reveal the similarities resulting from AI.
By definition, all models, in this case, are optically thick to the reprocessed radiation. This means that most of the radiation will emerge at wavelengths larger than the Wien peak of the cavity wall (i.e., $\lambda > 3 \,\mu \rm m$ for $T_{\rm i}=1000 \rm K$). For a given wavelength the envelope can be divided into two different regions with different properties concerning the radiation field: an inner region with radius between $r_{\rm i}$ and $r_{\tau=1}$, and an outer region, between $r_{\tau=1}$ and $r_{\rm e}$, where $r_{\rm e}$ is the envelope outer radius, and $r_{\tau=1}$ is given by $$\int_{r_{\tau=1}}^{r_{\rm e}}\rho \, \kappa_{\lambda} dr = 1 \ , \label{tauc}$$ i.e., $r_{\tau=1}$ is the point where the radial optical depth of the envelope is unity (this quantity depends on the wavelength). By definition, radiation emitted inside the inner region is unlikely to leave the envelope because the optical depth is larger than 1; hence, a reasonable approximation for the emerging IR flux is given by the volume integral of the radiation emitted by the outer (optically thin) region $$F_\lambda = \int_{V} 4\pi j_\lambda dV = 16\pi^2 \int_{r_{\tau=1}}^{r_{\rm e}} r^2 \rho(r)\, \kappa_\lambda B_\lambda[T(r)] dr \ . \label{emis1}$$
In order to find an analytic expression for $F_\lambda$, we will suppose that both the temperature in the outer part of the envelope and the grain opacity can be approximated by power-laws, so we write $$T(r)=T_0 \left({r_{\rm i} \over r}\right)^s \ , \label{temp}$$ and $$\kappa_\lambda=\kappa_0 \left({\lambda_0 \over \lambda}\right)^p \ , \label{kappa}$$ where $\lambda_0$ is an arbitrary constant. The assumption for the opacity is very reasonable, because by definition most of the flux emerges at wavelengths larger than the Rayleigh limit (see Figure \[effsil\] and discussion in section \[as\]). The assumption for the temperature is justified by the fact that the outer part of the envelope is optically thin; it can be shown that, if the dust opacity is given by eq. (\[kappa\]), the radial dependence of the temperature of an optically thin envelope is $T(r) \propto r^{-2/(p+4)}$ (e.g. @mar78, eq. 7.3). Our detailed calculations, shown below, validate this assumption. Finally, assuming a dust density $\rho(r)$, also given by a power-law $$\rho(r)=\rho_0\left({r_{\rm i} \over r}\right)^n \ , \label{dens}$$ we calculate $r_{\tau=1}$ using eqs. (\[tauc\]), (\[kappa\]) and (\[dens\]), which gives $$\frac{r_{\tau=1}}{r_{\rm i}}=\left(\frac{\lambda_0}{\lambda}\right)^{\frac{p}{n-1}} \tau_0^{\frac{1}{n-1}} \ , \label{r1}$$ where $$\tau_0 \equiv \frac{\kappa_0\rho_0 r_{\rm i}}{n-1}$$ is the envelope optical depth at $\lambda = \lambda_0$.
Substituting eqs. (\[temp\]), (\[kappa\]), (\[dens\]), and [(\[r1\])]{} in eq. (\[emis1\]) and letting $r_{\rm e} \rightarrow \infty$, we obtain the spectral shape of the radiation emitted by the outer part of the envelope $${\lambda F_\lambda \over L_\star} = K \lambda^{\frac{3-n}{s}-p-5} \int_{u_1}^{\infty} {u^{2-n} \over e^{u^s}-1}du \ , \label{flux}$$ where $$u_1 = \Gamma \left( \frac{1}{\lambda} \right)^{\frac{n-1+sp}{s(n-1)}} \ ,$$ and $$\Gamma \equiv \frac{\left({k T_0 \over h c}\right)^{s}}{\left(\tau_0 \lambda_0^p \right)^{n-1}} \ . \label{gamma}$$ The normalization constant $K$ is defined by the condition $$\frac{\int_{0}^{\infty} \lambda F_\lambda d\lambda}{L_\star} = 1 \ .$$
In eq. (\[flux\]) the only parameter that depends on the grain size is $\Gamma$ in the lower integration limit, $u_1$. It follows that, if the values of $\tau_0$ and $T_0$ of models with different grain sizes are such that $\Gamma$ is the same, [*these models will produce the same SED*]{}.
We show in Figures \[sedcs2\] and \[tempc\] the SEDs and grain equilibrium temperatures for $20000\,\rm K$ models with different grain sizes. Figure \[tempc\] shows that the temperature power-law exponent has a value $s=0.4$ at large $r$. Other model parameters are listed in Table \[tab3\]. Note that the optical depths are very high (ranging from $\tau_0 = 3.4$ to 5.0 for $\lambda_0=100\,\mu\rm m$), according to the assumption for case C. These optical depths were chosen so that the parameter $\Gamma$ is the same among the different models (i.e., we measured $T_0$ and adjusted $\tau_0$ so that $T_0^s \tau_0^{1-n}=$ constant). The resulting SEDs are very similar for $\lambda\gtrsim 20 \mu\rm{m}$, which indicates that the parameter $\Gamma$ is an adequate scaling parameter for very high optical depths. For shorter wavelengths the SED invariance breaks down, because the opacities are no longer described by a power-law, and the SEDs are different.
Also shown in Figure \[sedcs2\] is the analytic expression for the emerging flux, eq. (\[flux\]). This expression reproduces well the shape of the SED for long wavelengths; for shorter wavelengths the agreement is not good for the reason stated above: our assumption for the grain opacity fails in the well-known silicate spectral features at 10 and 20 $\mu\rm m$, and as we go into the optical limit.
We conclude that, given a dust grain composition and spatial distribution, the shape of the IR SED is controlled by a single parameter, $\Gamma$. From the observational point of view, we have a situation similar to case B, where the IR emission bump does not convey any information about the dust grain size. However at shorter wavelengths, AI breaks down, so one can determine information about the dust grain size, if there is sufficient observable flux at these wavelengths.
Effects of the Scattering Phase Function on the SED \[SPF\]
-----------------------------------------------------------
In the previous sections we studied in detail the effects of grain size on the SED, and determined to what extent the grain size (i.e., opacity) can be changed while maintaining the approximate invariance of the SED. An additional effect is that, when the grain sizes are changed, both the opacity and the SPF change. Changing the SPF breaks invariance requirement (v) of section \[introduction\] and thus will alter the SED. The results shown in the previous sections were calculated using Mie theory which has an anisotropic SPF, but a common simplification is to use an isotropic SPF. In this section, we briefly discuss how an isotropic SPF affects our previous results.
In general, the primary change is the $V$-band optical depth required to reproduce $\tau_{\rm rep}$; the SED is largely unaffected (except at the few percent level). In Table \[tab4\], we compare the $V$-band optical depth between models with an anisotropic SPF ($\tau_{V}$) and an isotropic SPF ($\tau_{V}^{\rm iso}$) for both case A and B. One should compare column 1 with column 2, and column 4 with column 5. Because all models are subject to the condition $\tau_{\rm rep} = 0.1$ (case A) or 1 (case B), the difference between $\tau_{V}$ and $\tau_{V}^{\rm iso}$ for each case gauges the effects of the SPF on the amount of reprocessing for each model.
For case A models, $\tau_{V}$ and $\tau_{V}^{\rm iso}$ are roughly the same (at the 10% level). This is to be expected because, in the optically thin limit, $I_\lambda^{\rm env}$ in eq. (\[lenv\]) is small, so the SPF ($d\sigma/d\Omega$) has little effect on $L_{\rm rep}$.
For case B models, the differences between $\tau_{V}$ and $\tau_{V}^{\rm iso}$ are larger, but still not very large (at most 22%). We conclude that for optically thin models the SPF is largely irrelevant, while for optically thick models, the SPF affects the conversion from $\tau_{V}$ to $\tau_{\rm rep}$, but at most to a level at 20% to 30%.
Also shown in Table \[tab4\] is the flux mean optical depth, $\tau_{\rm F}$, defined in eq. \[tflux\]. One should compare these values with the corresponding value of $\tau_{\rm rep}$. The difference between $\tau_{\rm F}$ and $\tau_{\rm rep}$ arises from multiple scattering effects, and for this reason they are much smaller for optically thin models than for optically thick models. Many authors use $\tau_{\rm F}$ as a parameter for SED classification; the results in Table \[tab4\] show that this is not a good parameter, at least for optical depths in the range of case B models, for which multiple scattering effects are important.
Models with a Grain Size Distribution \[dist\]
==============================================
The single grain size assumption can be a poor approximation for describing the dust of an astrophysical system. For example, fitting the interstellar extinction curve requires a distribution of grain sizes. @mat77 [hereafter MRN] used a simple power law size distribution with index $q=-3.5$, and a lower and upper cutoff size, $a_{\rm min} \sim 0.05\,\mu\rm m$ and $a_{\rm max} \sim 0.25\,\mu\rm m$. This distribution has been revised many times in the literature. For example, @kim94 proposed a distribution in the form $$\frac{dn}{da}\propto a^{-q}e^{-a/a_0}, a \geq a_{\rm min} \ , \label{eq2}$$ where $a_0=0.14\,\mu\rm m$ for silicate grains and $a_0=0.28\,\mu\rm m$ for graphite grains. The upper cutoff was replaced by an exponential decay, to allow for the presence of large dust grains ($a \gtrsim 0.2 \,\mu \rm m$).
Although the fact that the dust has a distribution of grain sizes is well established, the exact form of this distribution is still an open issue. As we will demonstrate below, part of this uncertainty is due to the AI of the problem, which makes two models with similar size distributions virtually indistinguishable. Another source of uncertainty lies on our yet incomplete knowledge of the details of grain formation and growth. To complicate matters even further, the grain size distribution is likely to vary across the wind because different grain sizes have different drift velocities [@eli01].
In this section, we study the SED of models with a grain size distribution. For simplicity, we adopt a power-law (MRN) grain size distribution function; although the quantitative details of the results shown below do depend on the adopted size distribution, the fundamental concepts discussed do not. Previously, in section \[single\], we used the concept of AI to reveal important similarities in the IR SED (and in the extinction SED for optically thin models) for single grain size models. In the following, we study what controls the spectral shape of the SED for models with a size distribution in an attempt to identify the same sort of similarities.
Spectral Shape of the SED \[sec51\]
-----------------------------------
Let us consider an envelope model, consisting of spherical dust grains of the same chemical composition and sizes distributed according to a given size distribution function, $dn/da$. The envelope IR SED, ${F}_{\rm IR}$, will be given by the sum of the contributions from each grain size $${F}_{\rm IR}(\lambda)= \int_{a_{\rm min}}^{a_{\rm max}} f(a) \mathcal{F}_{\rm IR}(\lambda,a) da \ ,$$ where $f(a)$ is the wavelength integrated emission for each grain size, a measure of how the absorbed energy is split between the different grain sizes, and $\mathcal{F}_{\rm IR}(\lambda,a)$ is the spectral shape of the spectrum emitted by the grains with radius $a$.
The simplest situation is when the spectrum emitted by all grains has approximately the same spectral shape (i.e., $\mathcal{F}_{\rm IR}(\lambda,a)$ is independent of $a$). In this case, the spectral shape of the IR SED of the envelope will be similar to the spectral shape of the individual grains, regardless of how the absorbed energy is split between the different grains. A somewhat similar situation occurs when a particular range of grain sizes, with a similar emission spectrum, completely dominates the emission. Here, even if the spectral shape of the other grain sizes is different, it does not affect the envelope IR SED. An important difference from the first situation is that, here, the manner in which the absorbed energy is distributed between the grain sizes is important. For both these cases, we expect a close similarity to the corresponding single-sized grain model. An opposite situation occurs when different grain sizes emit a different spectral shape with similar integrated emissions. In this case, the envelope IR SED can no longer be described using a single grain size model.
What controls $\mathcal{F}_{\rm IR}(\lambda,a)$ is the spectral shape of the absorption efficiencies and the grain temperature. From the conditions of AI, it is evident that the grain sizes with similar emission spectra will be the ones that comply with conditions 1 and 3. The first condition requires all grain sizes of the distribution must be smaller than a maximum size, given by eq. (\[eq1\]). The third condition implies that all grain sizes must have similar equilibrium temperatures. However, owing to their different absorption efficiencies, different grain sizes can have very different equilibrium temperatures. This effect can be inferred from the second column of Table \[tab1\]: different grain sizes have different condensation radii, which indicates that, if these grains were to coexist at the same point in space, the grain sizes with larger condensation radii will be hotter than those with lower condensation radii. Recently, @wol03 studied the condensation temperature of the individual grain sizes in a mixture of different grains sizes, both in 1-D and 2-D dust shells, and found that the temperature difference spans a range of up to $\approx 250$ K, although this value is highly dependent of the choice of the dust properties.
It is easy to show that, in the Monte Carlo radiative equilibrium scheme, the temperature $T(a)$ of the grains with radius $a$ at a given point of the envelope will depend on the number of photons absorbed by these grains, $N_{\rm abs}(a)$, and on the Planck mean opacity, $\kappa_{\rm P}(T,a)$ (see @bjo01, eq. \[5\]), $$T(a)^4 \propto {N_{\rm abs}(a) \over \kappa_{\rm P}(T,a)}.$$ The number of absorbed photons is proportional to the absorption cross section averaged over the incident photons, so the equation above can be rewritten as $$T(a)^4 \propto {\langle \kappa(a) \rangle \over \kappa_{\rm P}(T,a)} \ , \label{eq4}$$ where $$\langle \kappa(a) \rangle \equiv \frac{\int \kappa_\lambda(a)J_\lambda d\lambda}{J} \ .$$
The ratio $\kappa_\lambda(a)/\kappa_P $ is shown in Figure \[sabs\] for silicate grains with sizes ranging from $a=0.005$ to $1 \,\mu \rm m$, where the Planck mean opacity was calculated for $T = 1000\,\rm K$, the condensation temperature of silicate grains. This plot indicates the relative equilibrium temperature for different grain sizes as a function of the wavelength of the illuminating radiation field. For example, if the grains are illuminated by a black body radiation of temperature $T_{\rm eff} = 2500\,\rm K$, which peaks at $\lambda = 1.2 \,\mu \rm m$, the grains with intermediate sizes ($a \approx 0.25$ to $0.50 \,\mu \rm m$) will be the hottest. Note, however, the range of temperatures will be relatively small in this case. On the other hand, if the grains are illuminated by a $20000 \rm K$ black body radiation field, the hottest grains will be the ones with $a \approx 0.05 \,\mu \rm m$ and all the other grain sizes will have much lower equilibrium temperatures, including the very small grains ($a = 0.005 \,\mu \rm m$). From eq. (\[eq4\]) and Figure \[sabs\], the ratio between the maximum and minimum equilibrium temperature can be estimated to be $T_{\rm max} / T_{\rm min} \approx 2.5$. Hence, if the hottest grains have a temperature of $1000\,\rm K$, the coolest ones will be as cold as $400\,\rm K$.
This temperature difference between the different grain sizes is to be expected only for optically thin models. For optically thicker models, the temperature differences will be smaller, and for very optically thick envelopes the temperatures will be the same, regardless of the grain size. This occurs because at very large optical depths, $J_\lambda = B_\lambda$, and the grains are in thermal equilibrium with the radiation field.
We now can see an important difference between models of cool and hot stars. For cool stars, the equilibrium temperatures of the different grain sizes are not very different, so we expect the shape of the IR SED to be similar to those of the individual grain sizes. For hot stars, however, the spectral shape for each grain size will be very different. From the previous discussion, we infer that the IR SEDs of hot stars will depend on how the flux is split among the different grain sizes.
For an optically thin model, it is possible to estimate the values of $f(a)$, the wavelength integrated emission for grains of size $a$. From Kirchhoff’s law, the emission must be the same as the wavelength integrated absorption, i.e., $$f(a) \propto \frac{dn}{da} \langle \kappa(a) \rangle \ . \label{fa}$$ This relative emission is plotted in Figure \[faf\] for a standard MRN size distribution function, with $q = -3.5$. This figure shows the relative $f(a)$ as a function of the illuminating wavelength for the same grain sizes shown in Figure \[sabs\]. Note that, with the exception of the smallest grain considered, the curves of Figure \[faf\] scale approximately with the curves of Figure \[sabs\], which means that, in general, the coolest grains will have the lowest integrated emission and vice-versa. As we will see later, this has an important consequence for the hot star case, for which the equilibrium temperature of the grains can be very different; however, because the hotter grains dominate the emission spectrum, the envelope IR SED will still be similar to the spectral shape of the individual grains.
The smallest grains are an exception for this rule. If we compare them with the grains with $a \approx 0.05\,\mu\rm m$, we see that they reprocess about the same energy, but have lower equilibrium temperatures. This creates a situation in which grains with two different equilibrium temperatures both contribute equally to the final emission spectrum.
The above discussion can be summarized as follows. For cool stars, we expect the IR SED of the distribution to be similar to that of the single grain models because all the grains have similar temperatures and, hence, similar emission spectra. For hot stars, if we exclude the smallest grains $(\sim 0.005\,\mu \rm m)$, the IR SED of the distribution will be similar to the SED of the hottest grain, which dominates the emission. It is important to note, however, that these conclusions are valid only for the size distribution function assumed above. A different value of $q$, for example, will lead to changes in the relative values for $f(a)$.
Approximate Invariance for Grain Size Distributions \[sec52\]
-------------------------------------------------------------
In this section, we follow the same approach as section \[single\] and compare model SEDs for different grain size distributions. More specifically, we employ a MRN size distribution function with $q=-3.5$, $a_{\rm min}=0.05 \,\mu \rm m$, and different values of $a_{\rm max}$. As before, we compare models with the same $\tau_{\rm rep}$, to ensure that condition 2 of AI is satisfied. As was seen in section \[single\], different grain sizes can have different condensation radii. However, large grains typically form from smaller grains, so, as an approximation, we set the cavity radius to be the condensation radius of the smallest grain. In section \[sec53\], we address the consequences and validity of this approach. Finally, we defer discussion of the smallest grains ($a \approx 0.005 \,\mu \rm m$) until later, for the reasons mentioned in the last section.
### Case A: Optically Thin Envelopes \[casead\]
The case A results for models with a grain size distribution are essentially the same as case A for single grain size models. If we compare model SEDs for models with $a_{\rm max} \lesssim 0.25 \,\mu \rm m$, the maximum grain size to satisfy condition 1 of AI, the curves are identical at the 3 to 5% level, depending on the stellar temperature, as shown in Figure \[sedad\]. This reinforces the conclusion for case A in section \[caseas\], where we saw that the SED is a poor indicator of grain size (as long as the grain sizes present in the distribution each satisfy the condition of AI).
It is interesting to note that, this time, the smallest differences are observed for the hot star models, in contrast with the single grain size case, where the smallest differences were observed for cool star models. The reason for this can be understood from Figure \[faf\], which shows that, for hot stars, a single grain size ($a \approx 0.05 \,\mu \rm m$) dominates the emission. In contrast, all grain sizes contribute to the cool star IR SED with slightly different emission spectra owing to small differences in the equilibrium temperatures. Another interesting point is that, if we include distributions with $a_{\rm max}$ larger than that satisfying condition 1 of AI, we still obtain very similar SEDs. For example, when $a_{\rm max} = 0.50 \,\mu \rm m$, the differences between the model SEDs is less than 10%. This, again, contrasts with the results for single grain size models.
### Case B: Optically Thick Extinction with Optically Thin IR Emission \[casebd\]
Figure \[sedbd\] shows the results for models with $a_{\rm max} = 0.15$ and $0.25 \,\mu \rm m$ and optical depth set so that $\tau_{\rm rep}= 1$. Note the striking similarity in the extinction region, in contrast with the previous results for single grain size models (see Figure \[sedbs\]). The mean difference between the two SEDs is now only 11% for $T_{\rm eff} = 2500\,\rm K$ and 10% for $T_{\rm eff} = 20000\,\rm K$. The much closer similarity in the extinction region can be understood in terms of the extinction optical depth, shown in Figure \[taubd\]. As a result of the averaging of the optical properties with respect to the distribution function, the shape of the extinction optical depth of the two size distribution models is very similar.
The overall similarity of the SEDs for case B has important theoretical and observational consequences. From the theoretical point of view, this allows us to extend the consequences of AI from the IR SED to the entire spectrum, as was done for case A, but this time for a much broader range of optical depths. From the observational point of view, this extends the conclusion that SED is a poor indicator of grain size, to include all optical depths.
### Case C: Optically Thick Envelopes also in the IR \[casecd\]
This case, in all aspects, is similar to case C for single grain size models. Since the optical depth is very high, the grains tend to thermalize, approaching local thermal equilibrium, where no temperature difference is expected for the different grain sizes. Hence, the set of different grain sizes will behave as a single-sized grain whose optical properties are given by the average of the properties with respect to the distribution function.
Breakdown of Approximate Invariance \[sec53\]
---------------------------------------------
The similarity of the SEDs for models with different size distributions for all optical depth regimes, illustrated in the last section, changes when we consider very small grains.
Figure \[sedbd2\] shows the case B SEDs for models with $a_{\rm min} = 0.005\,\mu \rm m$ and $a_{\rm max} = 0.05, 0.15$ and $0.25 \,\mu \rm m$. As before, the models with large $a_{\rm max}$ have very similar SEDs in all spectral regions (compare Figure \[sedbd2\] with Figure \[sedbd\]); however, a significant difference in the extinction region arises for the model with $a_{\rm max} = 0.05\, \mu \rm m$. The reason for this difference is easily understood from the different shape of the extinction optical depth, shown in Figure \[taubd2\]. This difference results from the fact that the smallest grains (in contrast with the larger ones) did not reach the geometrical limit throughout the extinction region.
Breakdown of Grain Condensation Radius \[sec54\]
------------------------------------------------
In Figure \[comp\] we compare the SED for models with the same $a_{\rm max}$ ($0.25\,\mu \rm m$) and different $a_{\rm min}$ (0.005 and $0.05\,\mu \rm m$) for a stellar temperature of $20000\,\rm K$. A large difference in the IR SED of the two models is observed. This difference arises from our choice of the cavity radius as the condensation radius of the smaller grains. For the model with $a = 0.05$ to $0.25\,\mu\rm m$ this choice is consistent, because the grains with $a \approx 0.05\,\mu\rm m$ are the hottest and they dominate the IR emission (see section \[sec51\]). For the model with $a = 0.005$ to $0.25\,\mu\rm m$, however, the smallest grain are cooler than the ones with $a \approx 0.05\,\mu\rm m$. From the choice of the inner radius, it follows that the equilibrium temperatures of the grains with $a \approx 0.05\,\mu\rm m$ is about $1250\,\rm K$, much higher than the grain condensation temperature, which is physically impossible. This causes the IR emission to shift to lower wavelengths, as seen in Figure \[comp\]. It is evident that in this case a more consistent model should be considered, in which the intermediate (and larger) grains are not allowed to condense until their radiative equilibrium temperature drops below the condensation temperature.
Strictly speaking, this more consistent treatment should be used for all situations studied above, but it would demand a much more complicated procedure and would involve additional model parameters. The discussion of section \[sec51\] helps us set useful limits to when the approximation of identical condensation radii is valid for all grain sizes. It will be approximately valid for cool stars, because the differences between the equilibrium temperatures of the grains are not large. It will be valid also for hot stars, when the smallest grain considered has the largest condensation temperature. Finally, the assumption fails for hot stars, when the smallest grain is cooler and has large integrated emission, comparable with hot grains, as seen above.
We can summarize this as follows: if all grain sizes meet the AI requirements, as is generally the case for cool stars, then the approximation of same condensation radius for all grain sizes is valid. If the grains do not meet the requirements, but a small range of grain sizes completely dominates the IR emission, then the approximate invariance of the SED is still maintained. The more consistent treatment is required only when the grain sizes do not meet the AI requirements and different grain sizes have similar integrated emissivities. However, it is likely when one uses the correct condensation radius for each grain size that even in this last case AI will be recovered.
Summary \[discussion\]
======================
We have studied the effects of grain size on the SED of circumstellar envelopes with dust. To do so we introduced the concept of [*approximate invariance*]{}, as a very useful tool for revealing the essential similarities of the problem and for systematically exploring a large grid of model parameters.
The concept of AI follows from the fact that the optical properties of differently sized grains is similar in certain spectral regions, and from the idea that if a requirement for SED invariance is weakly violated (in this case, the shape of the grain opacity), the SED should still be approximately the same. We studied separately single grain size models and models with a grain size distribution. For both situations, we studied three optical depths regimes: optically thin models ([*case A*]{}), optically thick models in the extinction region ([*case B*]{}) and very optically thick models ([*case C*]{}), for which most of the radiation emerges in the MIR to FIR.
Our results for case A show that, given a grain composition and dust density distribution, the SED of models that comply with the conditions of AI are similar not only in the IR, but also in the extinction region. The spectral shape of the SED is controlled by a single parameter, the [*reprocessing optical depth*]{}, $\tau_{\rm rep}$, defined in eq. (\[trep\]). This parameter is [*independent of grain size*]{}. Our results for case C are similar but, as shown in section \[casecs\], another parameter ($\Gamma$, defined in eq. (\[gamma\])) controls the shape of the SED.
The results for case B differ for single grain size models and models with a size distribution. For single grain size models, the SEDs are similar only in the IR, while large differences are observed in the extinction region, where the grain opacities depend very much on grain size. Models with a grain size distribution, on the other hand, display a striking similarity in the extinction region, provided the lower limit of the grain size distribution, $a_{\rm min}$, is in the geometrical limit in the extinction region. It follows that, if $a_{\rm min}$ is chosen accordingly, the case B SEDs for models with a grain size distribution is controlled solely by the parameter $\tau_{\rm rep}$ (the same result as case A).
The fact that $\tau_{rep}$ (or $\Gamma$ for case C) is the appropriate parameter for SED classification indicates that the usual approach in the literature - using $\tau_V$ or $\tau_F$ - will not reveal the general invariance of the SED discussed in this paper. We conclude therefore that the SED is generally a very poor constraint of grain size. In most situations, observations must resolve differences of the order of a few percent to extract information about the grain size. If we add to this the fact that other model parameters, such as dust spatial distribution, composition and optical properties are somewhat uncertain, it becomes apparent that the task of extracting information about grain sizes from the SED alone may prove very difficult or even impossible for spherical geometries.
If follows from AI that in most circumstances $\tau_{\rm rep}$ is the [*only parameter related to the grain opacity that can be unambiguously extracted from the observations*]{}. Using $\tau_{\rm rep}$ to determine dust column density or mass is directly subject to the uncertainty in the determination of the grain sizes, which directly affects our ability to measure mass loss rates using the SED.
An important physical effect discussed in section \[sec51\] is that for models with a grain size distribution, the different grain sizes have different radiative equilibrium temperatures. This implies that a consistent model should include the correct condensation radius for each grain size of the distribution. However, we show in section \[sec51\] that the approximation of using identical condensation radii is reasonable when either all grain sizes have similar condensation temperatures or when a particular range of grain sizes dominate the IR emission. The only situation where the consistent treatment (each grain size having its correct condensation radius) is required is when two or more ranges of grain sizes have different equilibrium temperatures and similar integrated emission. This situation was found only for models of hot stars with very small grains ($a \lesssim 0.01\,\mu \rm m$) present in the distribution. It is important to notice that the results presented here depend on the particular choice of the distribution function, because this will control the relative contribution of each grain size to the emission. We suggest that a study of a given system could start with an analysis of the chosen dust size distribution along the lines described in section \[sec51\], Figures \[sabs\] and \[faf\]. Such analysis can be very useful because one would know beforehand if the approximation of identical condensation radii for all grain sizes of the distribution is a valid one.
We have presented, here, results only for silicate grains. However, this study was carried out for amorphous carbon and silicon carbide grains as well. The results for these materials are not shown because, although the specific details of the results are different (see @car01), the fundamental concepts discussed here remain valid.
Bjorkman, J.E. & Wood, K. 2001, , 554, 615 Carciofi, A.C. 2001, PhD Thesis, Universidade de São Paulo, São Paulo, Brazil Code, A.D. & Whitney, B.A. 1995, , 441, 400 Elitzur, M. & Ivezić, Ž. 2001, , 327, 403 Gehrz, R.D. 1989, in IAU Symp. 135: “Interstellar Dust”, ed. L.J. Allamandola & A.G.G.M. Tielens (Dordrecht: Kluwer), 445. Ivezić, Ž. & Elitzur, M. 1995, , 445, 415 Ivezić, Ž. & Elitzur, M. 1997, , 287, 799 (IE97) Kim, S.H, Martin, P.G. & Hendry, P.D. 1994, , 422, 164 Lorentz-Martins, S. and Lefévre, J. 1994, , 291, 831 Martin, P.G. 1978, Cosmic Dust: Its Impact on Astronomy (Oxford: Clarendon) Mathis, J.S., Rumpl, W. & Nordsieck, K.H. 1977,, 217, 425 Ossenkopt, V., Henning, Th. & Mathis, J.S. 1992, , 267,567 Pegourié, B. 1988, , 194, 335 Wolf, S. 2003, , 582, 859 Zubko, V.G., Mennella, V., Colangeli, L. e Bussoletti, E., , 282, 1321
[cccc]{}
0.005 & 4.5 & 1.00 & 0.308\
0.05 & 4.64 & 0.97 & 0.640\
0.15 & 5.62 & 0.85 & 2.98\
0.25 & 6.26 & 0.74 & 1.54\
0.005 & $1.00 \cdot 10^3$ & 1.00 & $2.32 \cdot 10^{-2}$\
0.05 & $1.53 \cdot 10^3$ & 1.00 & $2.25 \cdot 10^{-2}$\
0.15 & $1.12 \cdot 10^3$ & 0.92 & 0.300\
0.25 & $8.12 \cdot 10^2$ & 0.78 & 0.358
[cccccc]{}
0.005 & 4.9 & 1.00 & 3.49\
0.05 & 5.3 & 1.09 & 7.42\
0.15 & 7.7 & 1.48 & 37.0\
0.25 & 9.2 & 1.30 & 17.0\
0.005 & $1.04 \cdot 10^3$ & 1.00 & 0.273\
0.05 & $1.70 \cdot 10^3$ & 0.96 & 0.224\
0.15 & $1.17 \cdot 10^3$ & 0.74 & 2.81\
0.25 & $8.61 \cdot 10^2$ & 0.65 & 3.41
[cccc]{}
0.005 & $1.75 \cdot 10^3$ & 1.00 & 5.2\
0.05 & $1.77 \cdot 10^3$ & 0.99 & 5.1\
0.15 & $1.85 \cdot 10^3$ & 1.03 & 4.8\
0.25 & $1.97 \cdot 10^3$ & 1.04 & 4.3\
0.50 & $2.4 \cdot 10^3$ & 1.37 & 3.8
[cccccc]{}
0.308 & 0.307 & $1.01\cdot 10^{-1}$ & 3.49 & 3.47 & 1.15\
0.640 & 0.647 & $9.96\cdot 10^{-2}$ & 7.42 & 7.44 & 1.16\
2.98 & 2.83 & $9.48\cdot 10^{-2}$ & 37.0 & 35.7 & 1.18\
1.54 & 1.39 & $9.45\cdot 10^{-2}$ & 17.0 & 15.7 & 1.04\
$2.32 \cdot 10^{-2}$ & $2.32 \cdot 10^{-2}$ & $1.02\cdot 10^{-1}$ & 0.273 & 0.272 & 1.20\
$2.25 \cdot 10^{-2}$ & $2.09 \cdot 10^{-2}$ & $9.83\cdot 10^{-2}$& 0.224 & 0.183 & $9.79\cdot 10^{-2}$\
0.300 & 0.284 & $1.01\cdot 10^{-1}$& 2.81 & 2.31 & $9.46\cdot 10^{-2}$\
0.358 & 0.346 & $9.96\cdot 10^{-2}$& 3.41 & 2.83 & $9.49\cdot 10^{-2}$
[^1]: This condition was not required by @ive97 since they only considered isotropic scattering. We consider dust with anisotropic phase function, therefore this additional requirement is necessary.
|
---
abstract: 'The effects of downfolding a Brillouin zone can open gaps and quench the kinetic energy by flattening bands. Quasiperiodic systems are extreme examples of this process, which leads to new phases and critical eigenstates. We analytically and numerically investigate these effects in a two-dimensional topological insulator with a quasiperiodic potential and discover a complex phase diagram. We study the nature of the resulting eigenstate quantum phase transitions; a quasiperiodic potential can make a trivial insulator topological and induce topological insulator-to-metal phase transitions through a unique universality class distinct from random systems. Additionally, at the transition between trivial and topological insulators, a Dirac semimetal phase is formed that can host a “magic-angle” phase transition due to the quasiperiodic potential. This wealth of critical behavior occurs concomitantly with the quenching of the kinetic energy, resulting in flat topological bands that could serve as a platform to realize the fractional quantum Hall effect without a magnetic field.'
author:
- Yixing Fu
- 'Justin H. Wilson'
- 'J. H. Pixley'
bibliography:
- 'QP\_BHZ.bib'
title: Flat Topological Bands and Eigenstate Criticality in a Quasiperiodic Insulator
---
The interplay of topology and strong correlations produces a wide range of fascinating phenomena, with the fractional quantum Hall effect [@stormer1999fractional] serving as the quintessential example. Conventionally, the magnetic field induces topology in the electronic many-body wavefunction; however, Berry curvature of the band structure is sufficient to induce topology in single-particle wavefunctions, which can survive in the presence of interactions (see Ref. for a review). Despite strong numerical evidence of fractional Chern and $\mathbb{Z}_2$ insulators [@RegnaultBernevig2011FracChernIns; @liu2012fractional; @harper2014perturbative; @bandres2016topological; @maciejko2010fractional; @swingle2011correlated], identifying a clear experimental route to the many-body analog of the fractional quantum Hall effect without a magnetic field has remained challenging. A natural direction is to find lattices that host flat topological bands that quench the kinetic energy and promote strong correlations [@bergholtz2013topological; @parameswaran2013fractional; @wang2012fractional; @yang2012topological; @heikkila2011flat].
Recent work on twisted graphene heterostructures has opened up new platforms to study strongly correlated physics, including correlated insulators [@cao2018correlated], superconductivity [@cao2018SC; @yankowitz2019SC], and Chern insulators [@spanton2018FciTbg; @knapp2019fractional; @sharpe2019emergent]. Various proposals for realizing flat topological bands in these systems have followed [@zhang2019flat; @chittari2019gatetune; @wu2019topological; @wolf2018substrate; @tong2017topological; @san2014electronic; @lian2019flat; @ledwith2019fractional; @song2019all]. It was also recently shown in Refs. that the incommensurate effect of the twist could be emulated by a quasiperiodic potential. Consequently, a class of models, dubbed magic-angle semimetals, show similar phenomena to twisted bilayer graphene (e.g., the formation of minibands and the vanishing Dirac cone velocity) at or near an eigenstate phase transition. Similarly, in order to understand the theory for fractional Chern and $\mathbb{Z}_2$ insulators in incommensurate systems and how eigenstate criticality can play a role, it is essential to build a simple model to study theoretically and realize experimentally. This notion of flat band engineering with incommensuration can find broad applicability outside twisted heterostructures, including ultra-cold atomic gases and metamaterials (e.g., photonic waveguides, microwave resonators, and topo-electric circuits).
![[**Phase Diagram at zero and non-zero energy**]{}. (a) Phase diagram of the Bernevig-Hughes-Zhang model at charge neutrality (i.e., zero energy) with topological mass $M$ and quasiperiodic potential strength $W$. There are five illustrated phases: topological (TI), normal (NI), and Anderson (AI) insulators, Dirac semimetal (SM), and critical metal (CM). The lines between TI and NI are SMs in addition to the $M=2$ vertical line. The black dashed lines are the perturbation theory prediction for the SM lines. The green and red data points use the density of states to locate the phase boundaries while the blue circles use transport to determine the boundary from CM to AI. Machine learning on localized and critical eigenstates roughly agrees with these phase boundaries, illustrated with the pink curve. The orange line with square symbols mark the location where the topological bands become flat. (b) A cut of the phase diagram in energy space represented by the yellow line in (a). Notice the multiple phase transitions, all driven by quasiperiodicity ($W$) in addition to the higher energy metallic nature. The pink curve represents the boundary to machine learned eigenstates that are localized. []{data-label="fig:phase"}](MainPhaseDiagram.png "fig:"){width="0.98\columnwidth"} ![[**Phase Diagram at zero and non-zero energy**]{}. (a) Phase diagram of the Bernevig-Hughes-Zhang model at charge neutrality (i.e., zero energy) with topological mass $M$ and quasiperiodic potential strength $W$. There are five illustrated phases: topological (TI), normal (NI), and Anderson (AI) insulators, Dirac semimetal (SM), and critical metal (CM). The lines between TI and NI are SMs in addition to the $M=2$ vertical line. The black dashed lines are the perturbation theory prediction for the SM lines. The green and red data points use the density of states to locate the phase boundaries while the blue circles use transport to determine the boundary from CM to AI. Machine learning on localized and critical eigenstates roughly agrees with these phase boundaries, illustrated with the pink curve. The orange line with square symbols mark the location where the topological bands become flat. (b) A cut of the phase diagram in energy space represented by the yellow line in (a). Notice the multiple phase transitions, all driven by quasiperiodicity ($W$) in addition to the higher energy metallic nature. The pink curve represents the boundary to machine learned eigenstates that are localized. []{data-label="fig:phase"}](M4p9_Color.pdf "fig:"){width="0.98\columnwidth"}
In this letter, we study a minimal model for a two-dimensional topological insulator with a quasiperiodic potential to find a controllable route to create flat topological bands and to induce quantum phase transitions beyond the Landau-Ginzburg paradigm. Using analytic and numeric techniques, we find an intricate phase diagram, as shown in Fig. \[fig:phase\]. In particular, quasiperiodicity creates practically flat topological bands near where finite-energy states exhibit criticality. At the transition between topological and trivial insulators, the system realizes a magic-angle semimetal with all of the features previously studied [@MASM]. We further characterize the critical properties of the various eigenstate transitions, understanding them as localization and delocalization transitions in momentum- or real-space bases. While random systems exhibit analogous features [@groth2009theoryTAI], these phase transitions represent unique universality classes that, to the best of our knowledge, have not been characterized to date.
*Model and approach:* To model a two-dimensional topological insulator, we use the Bernevig-Hughes-Zhang (BHZ) lattice model [@BHZ] with an additional 2D quasiperiodic potential. The square-lattice Hamiltonian (with sites ${{\bf r}}$) is block diagonal such that $$\mathcal H = \sum_{{{\bf r}},{{\bf r}}'} c^\dagger_{{{\bf r}}'} \begin{pmatrix} h_{{{\bf r}}'{{\bf r}}} & 0 \\ 0 & h_{{{\bf r}}'{{\bf r}}}^* \end{pmatrix}c_{{{\bf r}}}
+ \sum_{{\bf r}}c^\dagger_{{{\bf r}}} V({{\bf r}})c_{{{\bf r}}} ,$$ where $c_{{\bf r}}$ are four-component annihilation operators and $V({{\bf r}}) = W\sum_{\mu=x,y} \cos(Q r_\mu + \phi_\mu)$ is the quasiperiodic potential (QP) with amplitude $W$, wavevector $Q$, and random phase $\phi_\mu$; $h_{{{\bf r}}'{{\bf r}}}$ is a two-by-two matrix describing one block of the BHZ model ($h^*$ is its complex conjugate). The nonzero elements of $h$ are $h_{{{\bf r}}{{\bf r}}} = (M-2)\sigma_z$ and $h_{{{\bf r}},{{\bf r}}+\hat{\mu}} = h_{{{\bf r}},{{\bf r}}-\hat{\mu}}^\dagger = -\frac{i}2 t_\mu \sigma_\mu $ for $\mu=x,y$, topological mass $M$, and hoppings $t_\mu = 1$. Additionally, most analyses are done on this smaller two-by-two matrix since time-reversal symmetry relates each block, and $V({{\bf r}})$ does not couple blocks. To reduce finite-size effects, we average over twisted boundary conditions implemented with $t_\mu \rightarrow t_\mu e^{i\theta_\mu/L}$ for a twist $\theta_\mu$ in the $\mu$-direction randomly sampled from $[0,2\pi)$. The model is invariant under $M\rightarrow 4 - M$, so we focus on $M\geq 2$. For $2<M<4$, the band structure (i.e. $W=0)$ is topological with a quantized spin Hall effect $\mathcal{Q} = \sigma_{xy}^+ - \sigma_{xy}^-$ where $\sigma_{xy}^\pm$ are Hall conductivities for the blocks defined by $h$ and $h^*$ respectively. At $M = 2$ \[$M=4$\], the model is a Dirac semimetal with Dirac points at $\mathbf X = (\pi,0)$ and $\mathbf Y = (0,\pi)$ \[$\mathbf M = (\pi,\pi)$\] that have a velocity $v_0=t$.
Quasiperiodicity is encoded in $Q$, which in the thermodynamic limit we define as $Q/(2\pi) = (2/(\sqrt{5} + 1))^2$. For simulations, we take rational approximates such that $Q\approx Q_L = 2\pi F_{n-2}/F_n$, where $F_n$ is the $n$th Fibonacci number, and the system size is $L = F_n$. In the supplement, we consider other values of $Q$.
In the following, we investigate eigenstates, eigenvalues, and the transport properties of the system. To compute the transport and density of states (DOS), we use the kernel polynomial method (KPM) [@weisse2006KPM]. The KPM utilizes a Chebyshev expansion truncated at an integer $N_c$, which controls the energy resolution. To evaluate the conductivity tensor we use the Kubo formula [@garcia2015conductivity] $$\sigma_{\alpha\beta} = \frac{2 e^2 \hbar}{L^2}\! \int\! f(E)dE \operatorname{\mathrm{Im}}\operatorname{\mathrm{Tr}}\left\llbracket v_\alpha \frac{dG^-}{d\epsilon} v_\beta \delta(E - H)\right\rrbracket$$ where $f(E) = [e^{\beta(E-\mu)}+1]^{-1}$ is the Fermi function at inverse temperate $\beta$ and chemical potential $\mu$, $v_\alpha$ is the velocity operator, $G^{-}$ is the retarded Green function, and $\llbracket \cdots \rrbracket$ denotes an average over phases in the QP ($\phi_{\mu}$) and twists ($\theta_{\mu}$) in the boundary condition; for numerical data, we average over 200 samples. In contrast to disordered systems, band gaps are cleanly identifiable with the DOS $$\rho(E) = \frac{1}{2L^2} \bigg\llbracket\sum_i \delta(E-E_i)\bigg\rrbracket$$ where $E_i$ denotes the energy eigenvalues.
To probe wavefunctions, we compute the inverse participation ratios (IPRs) in real and momentum space. The IPR in a basis indexed by $\bm \alpha$ is $$\mathcal{I}_{\alpha}(E) = \sum_{\bm\alpha} \left\llbracket\lvert\braket{\bm\alpha|\psi_E}\rvert^4\right\rrbracket$$ where the wavefunctions are normalized to unity and in the momentum space ($\bm{\alpha} = {{\bf k}}$) or real space ($\bm{\alpha} = {{\bf r}}$) basis. We use exact diagonalization on small system sizes and Lanczos for larger system sizes to compute $\braket{\bm\alpha|\psi_E}$. For systems localized in basis $\alpha$, the IPR is $L$-independent; for delocalized systems, it goes like $\mathcal{I}_{\alpha} \sim 1/L^2$. At a localization transition [@Mirlin-2000; @evers2008anderson] $\mathcal{I}_{\alpha} \sim 1/L^{\gamma}$ where $\gamma$ is related to the fractal dimension ($d_2$) and $0<\gamma<2$.
Lastly, due to the great deal of structure in the phase diagram we use machine learning [@SuppMat] on the eigenstates to identify different phases in the model. A neural network model is trained using a subset of manually labelled data on the wavefunctions as extended, critical, or localized. The neural net model generalizes the identification to any combination of parameters ($W$, $M$, and $E$) to efficiently calculate the phase boundary and mobility edge (to Anderson localized phases) with a high resolution in parameter space. We validate this approach by comparing with the conductivity and IPR to determine a comprehensive phase diagram.
*Phase Diagram*: Using analytic and numeric techniques, we obtain the phase diagrams shown in Fig. \[fig:phase\]. There are principally five phases pictured: topological insulator (TI), normal insulator (NI), critical metal (CM), Anderson insulator (AI), and lines of Dirac semimetals (SM) between TI and NI phases. Both band-insulating and SM phases are stable to weak quasiperiodicity. Finite band gaps and quantized (zero) spin Hall conductivity describe the TI (NI) phase. Low-energy scaling of the DOS captures the SM phases ($\rho(E)\sim \tilde v^{-2}|E|$ for a 2D Dirac cone with renormalized velocity $\tilde v$). The AI phase has a finite DOS but zero conductivity and localized wave functions (real space IPR that is $L$-independent). We further use a machine learning algorithm that identifies delocalized, localized, and critical eigenstates to supplement other measures. The structure revealed is $Q$-dependent [@SuppMat] and reminiscent of other studies of insulating phases perturbed by quasiperiodicity [@roux2008quasiperiodic].
Upon increasing $W$, we usually traverse the phases TI/NI $\rightarrow$ CM $\rightarrow$ AI. However, more complicated cuts are possible as shown in Fig. \[fig:phase\](b); simply increasing $W$ leads to the phases NI $\rightarrow$ SM $\rightarrow$ TI $\rightarrow$ CM $\rightarrow$ TI $\rightarrow$ SM $\rightarrow$ NI $\rightarrow$ CM $\rightarrow$ AI along with bands and mobility edges shown. We see that quasiperiodicity can drive trivial phases topological (for $4<M \lesssim 5.0$) and into-and-out-of metallic and topological phases at zero-energy. Intriguingly, near the dashed orange line (with square symbols) in Fig. \[fig:phase\](a) higher-energy bands (some with nonzero topological index) flatten, the effective mass effectively diverges ($\sim10^5$ increase), and the eigenstates appear critical as measured by the IPRs.
The physics on the SM lines emanating from $M=2$ or $M=4$ at $W=0$ agrees with the universal features found in Ref. and reveals magic-angle transitions marked by red stars in Fig. \[fig:phase\](a). Concentrating on $M=2$, the semimetal is stable with a velocity (calculated from the DOS) that vanishes like $ \tilde v \sim (W_c(M=2)-W)^{\beta/2}$ where $W_c(M=2) = 1.42\pm0.02$ and $\beta=2\pm 0.3$, which is close to the universal value $\beta \approx 2$ obtained in other models and symmetry classes [@MASM; @MASM_chiral]. This is demonstrated in Fig. \[fig:magicangle\](a) where $\tilde v$ vanishes when $\rho(0)$ rises. Additionally, the wave functions are localized in momentum space when $W<W_c(M=2)$, and delocalized in momentum space when $W>W_c(M=2)$ (as indicated in Fig. \[fig:magicangle\](b) by $\mathcal I_k$ being $L$-independent and $\mathcal I_k\sim 1/L^2$, respectively). When the wave function is localizing (indicated by real space IPR) and the resistivity is increasing with $L$ and $N_c$, there is a localization transition $W_A(M=2) = 1.50\pm 0.03$, indicating a small but finite CM phase.
From the neural net model we have determined an additional measure of the Anderson localization transition, shown as the magenta line in Fig. \[fig:phase\](a). In the current Hamiltonian, the critical eigenstates can appear very close to being localized and therefore are not straightforward to diagnose with the IPR and conductivity alone. Thus, we use machine learning to provide a more conservative measure of the Anderson localized phase, which in certain regimes matches the IPR and conductivity, but in the more non-trivial regimes of the phase diagram extends to larger values of $W$.
![[**The magic-angle transition for the semimetal line $M=2$.**]{} (a) Renormalized velocity $v/v(0)$ and the resulting finite density of states $\rho(0)$ at the transition, calculated with system size $L=144$ and Chebyshev cutoff $N_c = 2^{15}$. (b) These plots indicate the appearance of a critical metallic phase $1.4 \lesssim W \lesssim 1.5$ inferred from both the resistivity $\rho_{xx}$ and the scaling of the momentum- and real-space IPRs. The $L$-dependence of the IPRs is fitted from $L=89$, $L=144$, and $L=233$ to a power law form $\mathcal{I}_{\alpha}\sim 1/L^{\gamma_{\alpha}}$, and $\gamma_{\alpha}$ is shown as the right vertical axis. []{data-label="fig:magicangle"}](M2p00_dos_v.pdf "fig:"){width=".4925\columnwidth"} ![[**The magic-angle transition for the semimetal line $M=2$.**]{} (a) Renormalized velocity $v/v(0)$ and the resulting finite density of states $\rho(0)$ at the transition, calculated with system size $L=144$ and Chebyshev cutoff $N_c = 2^{15}$. (b) These plots indicate the appearance of a critical metallic phase $1.4 \lesssim W \lesssim 1.5$ inferred from both the resistivity $\rho_{xx}$ and the scaling of the momentum- and real-space IPRs. The $L$-dependence of the IPRs is fitted from $L=89$, $L=144$, and $L=233$ to a power law form $\mathcal{I}_{\alpha}\sim 1/L^{\gamma_{\alpha}}$, and $\gamma_{\alpha}$ is shown as the right vertical axis. []{data-label="fig:magicangle"}](M2p00_IPR_rxx.pdf "fig:"){width=".4925\columnwidth"}
*Perturbation theory and NI-to-TI transition:* For smaller values of $W$, we use perturbation theory to map out the phase diagram and estimate the location of the NI-to-TI and SM-to-CM transitions.
To perform perturbation theory on the BHZ Hamiltonian $h_0(\mathbf{k})$, we use Dyson’s equation $G(\mathbf k, \omega)^{-1} = \omega - h_0(\mathbf k) - \Sigma(\mathbf k,\omega)$ to evaluate the self-energy $\Sigma(\mathbf k,\omega)$ by treating $V(\mathbf r)$ perturbatively [@SuppMat]. To illustrate, near $M=4$ we take a single two-by-two block of the full Hamiltonian and expand it in the low-energy limit at $W=0$ (near the Dirac cone at ${\bf k} = {\bf M}$) so that $h_0(\mathbf M + \mathbf q) = v \mathbf q \cdot \bm \sigma + (M-4)\sigma_z$. Putting the resulting Green function in the form $G(\mathbf M + \mathbf q,\omega) = Z[\omega - \tilde v \mathbf q \cdot \bm \sigma - \tilde M\sigma_z]^{-1}$ defines the quasiparticle residue $Z$, renormalized velocity $\tilde v$, and topological mass $\tilde M$. In this regime (for $M=4$), we obtain up to second order for $\tilde{M}$ $$\tilde M - 4 = \frac{\left[(M-4) +W^2\frac{(4-M)+(\cos Q-1)}{(4-M)^2+2(3-M)(\cos Q-1)}\right]}{1+W^2/( (4-M)^2 + 2(3-M)(\cos Q - 1))}.
\label{eqn:M}$$ where the denominator is $Z$. By solving for $\tilde M = 4$, we obtain the phase boundary between distinct insulating phases, as illustrated by the black dotted line in Fig. \[fig:phase\](a) (at fourth-order in $W$), which is in excellent agreement with the numerics. The curvature to this line demonstrates that quasiperiodicity can drive a topological phase transition NI-to-TI, which is the deterministic analog of the disordered topological Anderson insulator [@groth2009theoryTAI]. Note that for $M=2$ the line $\tilde M=0$ is completely vertical. Using numerics to access higher $M$ and $W$, we find for $M\gtrsim 5.4$ the NI transitions directly into the CM phase. The magic-angle transition (i.e. SM-to-CM) is obtained by solving for when $\tilde v \rightarrow 0$ on the line $\tilde M=0$.
To quantify band flatness, we find the dispersion relation from the pole of the Green function $E_{\mathrm{eff}}(\mathbf q) = \pm(\tilde M + q^2/2m^*)$ with effective mass $m^* = \tilde M/(2\tilde v^2)$. At fourth-order we find that $m^* \sim 10^5(1/t)$ at $W=3t$, our first indication that the QP is flattening the topological bands; our numerics show that this effect is even more drastic.
![[**Demonstration of the TI to CM transition.**]{} (a) By tracking the density of states, we see the gap closes when the longitudinal conductivity becomes finite and the gap vanishes as a power law $\Delta = (W_c(M) - W)^{\nu z}$ with $\nu z \approx 1$ at the TI-to-CM transition across each value of $M$. (b) Shows the conductivity as a function of quasiperiodic strength $W$ for $M=4.0$. The Hall conductivity $\sigma_{xy}$ saturates to a finite value in the TI phase, but for $2\lesssim W \lesssim 3$ the longitudinal conductivity becomes finite and the Hall part is suppressed. The system is localized when $W\gtrsim 3$. Note that the feature near $W=0$ is due to $M=4$ being a SM. []{data-label="fig:TItoM"}](gapsize.pdf "fig:"){width="0.4925\columnwidth"} ![[**Demonstration of the TI to CM transition.**]{} (a) By tracking the density of states, we see the gap closes when the longitudinal conductivity becomes finite and the gap vanishes as a power law $\Delta = (W_c(M) - W)^{\nu z}$ with $\nu z \approx 1$ at the TI-to-CM transition across each value of $M$. (b) Shows the conductivity as a function of quasiperiodic strength $W$ for $M=4.0$. The Hall conductivity $\sigma_{xy}$ saturates to a finite value in the TI phase, but for $2\lesssim W \lesssim 3$ the longitudinal conductivity becomes finite and the Hall part is suppressed. The system is localized when $W\gtrsim 3$. Note that the feature near $W=0$ is due to $M=4$ being a SM. []{data-label="fig:TItoM"}](cond_W_M4p00.pdf "fig:"){width="0.4925\columnwidth"}
*TI-to-CM transition*: We use numerics to capture the full, nonperturbative transition to the CM phase. Generically, we denote as $W_c(M)$ the phase transition into the CM phase. Near the transition, we find that the gap closes as $\Delta \sim |W - W_c(M)|^{\nu z}$ and $\nu z\approx 1$ for each $M$ value we have considered ($\nu$ is the correlation length exponent and $z$ is the dynamical exponent), see Fig. \[fig:TItoM\](a). These exponents indicate a unique universality class driven by quasiperiodicity distinct from random systems, where $\nu\sim 2.7$ has been estimated in the case of random disorder [@yamakage2013criticality] and $\nu\sim 5$ for random impurities [@chen2015tunable], and they both have $z=d$ [@evers2008anderson].
As the gap closes at $W_c(M)$, the conductivity at $E=0$ becomes finite, and the Hall conductivity is no longer quantized, indicating the onset of the CM phase. As seen in Fig. \[fig:TItoM\](b), the Hall conductivity drops, and $\sigma_{xx}$ peaks at the transition, remaining finite for the duration of the CM. For larger values of $W$, we find a transition into an Anderson insulating phase [@evers2008anderson; @abrahams1979scaling] with exponentially localized wavefunctions in real space and a vanishing $\sigma_{xx}$. Additionally, as Fig. \[fig:phase\](b) shows, we can have a more complicated structure for various sequences of transitions as well.
![[**Flat Chern bands and eigenstate criticality.**]{} (a) The effective mass of the lowest band as a function of $W$ and gap size. Notice that the gap begins to decrease when the effective mass calculated numerically exceeds its perturbative value by $\sim 10^3$. (b) Color plot of the momentum-space IPR scaling at these higher energies. Notice that around $W\sim 0.95$ many parts of the spectrum start becoming delocalized in momentum space. At low energies the states become completely delocalized (and even localize in real space), while at higher energies $\mathcal{I}_k \sim L^{-\gamma_k} $ for $0<\gamma_k<2$ indicating critical eigenstates and the value of $\gamma_k$ is given by the color. The lowest energy band (and flattest) is actually the Chern band. (c) A demonstration of the flat Chern band for $W=1$. From figure (b) we know that it is made up of critical eigenstates, and yet the Chern number (as indicated by $\sigma_{xy}$) jumps abruptly across the band. []{data-label="fig:flatChern"}](effmass_gap.pdf "fig:"){width="0.96\columnwidth"} ![[**Flat Chern bands and eigenstate criticality.**]{} (a) The effective mass of the lowest band as a function of $W$ and gap size. Notice that the gap begins to decrease when the effective mass calculated numerically exceeds its perturbative value by $\sim 10^3$. (b) Color plot of the momentum-space IPR scaling at these higher energies. Notice that around $W\sim 0.95$ many parts of the spectrum start becoming delocalized in momentum space. At low energies the states become completely delocalized (and even localize in real space), while at higher energies $\mathcal{I}_k \sim L^{-\gamma_k} $ for $0<\gamma_k<2$ indicating critical eigenstates and the value of $\gamma_k$ is given by the color. The lowest energy band (and flattest) is actually the Chern band. (c) A demonstration of the flat Chern band for $W=1$. From figure (b) we know that it is made up of critical eigenstates, and yet the Chern number (as indicated by $\sigma_{xy}$) jumps abruptly across the band. []{data-label="fig:flatChern"}](IPRk_W_E_color_jet_modified.pdf "fig:"){width="0.96\columnwidth"} ![[**Flat Chern bands and eigenstate criticality.**]{} (a) The effective mass of the lowest band as a function of $W$ and gap size. Notice that the gap begins to decrease when the effective mass calculated numerically exceeds its perturbative value by $\sim 10^3$. (b) Color plot of the momentum-space IPR scaling at these higher energies. Notice that around $W\sim 0.95$ many parts of the spectrum start becoming delocalized in momentum space. At low energies the states become completely delocalized (and even localize in real space), while at higher energies $\mathcal{I}_k \sim L^{-\gamma_k} $ for $0<\gamma_k<2$ indicating critical eigenstates and the value of $\gamma_k$ is given by the color. The lowest energy band (and flattest) is actually the Chern band. (c) A demonstration of the flat Chern band for $W=1$. From figure (b) we know that it is made up of critical eigenstates, and yet the Chern number (as indicated by $\sigma_{xy}$) jumps abruptly across the band. []{data-label="fig:flatChern"}](flat_chern_cond_M4p00W1p00.pdf "fig:"){width="0.96\columnwidth"}
*Criticality and flat topological bands*: At small $W$, the gap *increases* for some values of $M$, as seen in Fig. \[fig:flatChern\](a) (and as indicated by perturbation theory for $\tilde M$ [@SuppMat]). For larger $W$, the gap begins decreasing (indicated roughly by the orange dashed line with square symbols in Fig. \[fig:phase\](a)). When the gap begins decreasing, several phenomena occur, seen in Fig. \[fig:flatChern\] for the cut $M=4.0$. We track the effective mass $m^*$ and find that it increases almost over four orders of magnitude \[Fig. \[fig:flatChern\](a)\], indicating the onset of flat bands and the breakdown of perturbation theory. Additionally, the states in the bands become critical, as measured by the IPR in momentum and position space ($1/\mathcal{I}_{\alpha}\approx L^{\gamma_{\alpha}}$ for $0<\gamma_{\alpha}<2$ to delocalized in both bases ($\alpha=x,k$) \[Fig. \[fig:flatChern\](b)\]. Right after the states begin exhibiting critical behavior, we can isolate flat topological bands as in Fig. \[fig:flatChern\](c) by studying the change in $\sigma_{xy}$ across a band (we also see edge states[@SuppMat]). These “bands” can be thought of as a collection of states for which quasiperiodicity still causes level repulsion, but at smaller energy scales. Intuitively, as quasiperiodicity downfolds the Brillioun zone, some states get pushed up (the topological) and others down (trivial states); as they pass through each other, they hybridize and split becoming critical while simultaneously flattening.
At other points along the dashed line in Fig. \[fig:phase\], it is less clear how to separate the topological band from the zoo of nearby trivial bands, but other instances of flat topological bands are not hard to isolate. Remarkably, flat bands with eigenstate criticality occurring in tandem is very similar to magic-angle semimetals [@MASM].
*Conclusion–* In a simple 2D TI, we demonstrated that the inclusion of quasiperiodicity induces flat bands, eigenstate criticality, and a phase diagram full of structure. This was achieved by generalizing a quasiperiodic perturbative analysis to TIs and extensive numerics (using KPM, exact diagonalization, Lanczos methods, and machine learning). The eigenstates go through several Anderson-like transitions (delocalizing in momentum space before localizing in real space), which leads to critical eigenstates in a metallic phase. Meanwhile, we see the onset of flat topological bands within the TI phase concomitant with critical high energy eigenstates. Just as Dirac semimetals with quasiperiodicity are analogous to twisted bilayer graphene, TIs with quasiperiodicity achieve flat topological bands similar to twisted heterostructures. This identification allows for cold atom labs and metamaterial labs (both of which have already realized 2D TIs [@peterson2018quantized; @susstrunk2015observation; @lustig2019photonic; @kennedy2013spin]) to emulate similar physics.
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abstract: |
In this paper we study inhomogeneous Diophantine approximation with rational numbers of reduced form. The central object to study is the set $W(f,\theta)$ as follows, $$\begin{aligned}
\left\{x\in [0,1]:\left |x-\frac{m+\theta(n)}{n}\right|<\frac{f(n)}{n}\text{ for infinitely many coprime pairs of numbers } m,n\right\},\end{aligned}$$ where $\{f(n)\}_{n\in\mathbb{N}}$ and $\{\theta(n)\}_{n\in\mathbb{N}}$ are sequences of real numbers in $[0,1/2]$. We will completely determine the Hausdorff dimension of $W(f,\theta)$ in terms of $f$ and $\theta$. As a by-product, we also obtain a new sufficient condition for $W(f,\theta)$ to have full Lebesgue measure and this result is closely related to the study of [ *Duffin-Schaeffer conjecture* ]{}with extra conditions.
address: |
Han Yu\
School of Mathematics & Statistics\
University of St Andrews\
St Andrews\
KY16 9SS\
UK\
author:
- Han Yu
title: A Fourier analytic approach to inhomogeneous Diophantine approximation
---
[^1]
Introduction of the results
===========================
We are interested in Diophantine approximation with inhomogeneous shifts. Although it may look similar, the nature of inhomogeneous Diophantine approximation is considered to be rather different from its homogeneous counterpart [@L]. A nice introduction to the field can be found in [@BU]. There are also some recent results on inhomogeneous Diophantine approximation that come from different aspects of metric number theory and dynamical systems, see [@LN], [@RA] for more details. In [@SC], Chow proved a result which is closely related to the inhomogeneous Littlewood conjecture. The conjectures [@SC Conjecture 1.6, 1.7] also give some motivation for the content of this paper.
We now introduce the following sets of well approximable numbers. In the statement, we write ‘i.m.’ for ‘infinitely many’.
Given any sequences $$f:\mathbb{N}\to [0,1/2] \text{ and }\theta:\mathbb{N}\to [0,1/2],$$ we define the following two sets
$$W_0(f,\theta)=\left\{x\in [0,1]:\left |x-\frac{m+\theta(n)}{n}\right|<\frac{f(n)}{n} \text{ for i.m. numbers } m,n\right\}$$ and $$W(f,\theta)=\left\{x\in [0,1]:\left |x-\frac{m+\theta(n)}{n}\right|<\frac{f(n)}{n} \text{ for i.m. coprime pairs of numbers } m,n\right\}.$$
We call the sequence $f$ an approximation function and the sequence $\theta$ an inhomogeneous shift. The sets $W_0(.,.),W(.,.)$ are called sets of well approximable numbers with respect to $f,\theta$.
When $\theta$ is constantly equal to $0$ or equivalently $\theta=\mathbf{0}$, the study of well approximable numbers is referred to as homogeneous Diophantine approximation. In this case we have a good understanding about the size (in terms of the Lebesgue measure) of $W_0(.,.)$ and some partial information about $W(.,.)$. See [@BV3 Chapter 2] for some detailed discussions. However, when $\theta$ is not the zero function, we encounter inhomogeneous Diophantine approximation. So far we do not have a complete understanding about the size of $W(.,.)$ in terms of either Lebesgue or the Hausdorff dimension.
The Hausdorff dimension is invariant under inhomogeneous shifts
---------------------------------------------------------------
The Hausdorff dimension of $W_0(.,\mathbf{0})$ was studied extensively by Hinokuma and Shiga in [@HS]. In fact there is an explicit formula for computing $\dim_{H} W_0(f,\mathbf{0})$ in terms of the approximation function $f$. We will introduce this formula later. In this paper we are interested in the Hausdorff dimension of $W(.,.)$. Our main theorem in this paper is as follows. In below we use $\phi(.)$ for the Euler totient function and $d(.)$ for the divisor function, see Section \[Notations\] for more details.
\[ThmDimMain\] For any approximation function $f$ and inhomogeneous shift $\theta$, we have the following equality $$\dim_H W(f,\theta)=\dim_H W_0(f,\mathbf{0}).$$
In particular the above result implies that the Hausdorff dimension of $W(f,\theta)$ depends only on $f$ and not on $\theta$. Thus we have completely determined the Hausdorff dimension of sets of well approximable numbers with reduced fractions and arbitrary inhomogeneous shift. We shall see that Theorem \[ThmDimMain\] is a consequence of the following result.
\[ThmBy\] For any approximation function $f$ and inhomogeneous shift $\theta$, we have the following result $$\limsup_{N\to\infty} \frac{\sum_{n=1}^N \phi(n)f(n)/n}{\sqrt{\sum_{n=1}^N f(n)d^3(n)\log^2 n}}=\infty\implies
W(f,\theta) \text{ has full Lebesgue measure.}$$
The above theorem is closely related with [@SC Conjecture 1.7]. Ideally we want to get rid of the logarithmic and divisor function in the denominator. It is actually possible to prove the following result if we use all fractions instead of only the reduced ones.
For any approximation function $f$ and inhomogeneous shift $\theta$, we have the following result $$\limsup_{N\to\infty} \frac{\sum_{n=1}^N f(n)}{\sqrt{\sum_{n=1}^N f(n)d(n)}}=\infty\implies
W_0(f,\theta) \text{ has full Lebesgue measure.}$$
In Section \[Chow\] we shall discuss these resuts further. We note here that it is also possible to estimate the growth of the number of approximating fractions for a Lebesgue typical point. For more precise descriptions and discussions, see Theorem \[ThmNm\] below.
If we use the result about the maximal order of the divisor function $d(.)$ we can get the following corollary which is easier to work with.
\[Thm1\] For any approximation function $f$ and inhomogeneous shift $\theta$, if $$\limsup_{N\to\infty} \frac{\sum_{n=1}^N \phi(n)f(n)/n}{\log^2 N\log\log N\exp(3\log 2 \log N/\log\log N)}=\infty$$ then $$W(f,\theta) \text{ has full Lebesgue measure.}$$ To obtain an even more convenient result, we can replace the denominator with $N^{\epsilon}$ for any $\epsilon>0$.
The homogeneous version of the above result, in a slightly different form, appeared in [@HPV] and was improved later in [@BV2]. We will discuss these results later in this paper.
Some further results about inhomogeneous Diophantine approximation
------------------------------------------------------------------
Our method can help us deal with the Lebesgue measure of $W(.,.)$ in some cases. Our next result is related with the study of the [ *Duffin-Schaeffer conjecture* ]{}with extra conditions. This topic was studied in [@BV2] and [@HPV]. With Theorem \[ThmBy\] and Corollary \[Thm1\] above, we can revisit [@HPV Theorem 1] for inhomogeneous Diophantine approximation and it is interesting to see how much more we can obtain for Diophantine approximation with inhomogeneous shift. Our general result is as follows. In below $\mathcal{H}^h(.)$ denotes the Hausdorff measure with dimension function $h$, more details can be found in [@BV Section 2] and the references therein.
\[ThDim\] For any approximation function $f$ and inhomogeneous shift $\theta$, let $h:\mathbb{R}^+\to\mathbb{R}^+$ be such that $h(x)\to 0$ as $x\to 0$ and $h(x)/x$ is monotonic. If the following condition holds $$\limsup_{N\to\infty} \frac{\sum_{n=1}^N \phi(n)h(f(n)/n)}{\log^{2.5} N\left(\max_{n\in [1,N]} h(f(n)/n)^{1/2}n\right)}=\infty,$$ then $$\mathcal{H}^h \left(W(f,\theta)\right)=\mathcal{H}^h([0,1]).$$
In Section \[sefurther\] we will provide an example to show that the above theorem is not covered by known results in the homogeneous case. The above theorem is rather complicated to use in practice and we shall obtain the following corollary which is more convenient to work with.
\[Co1\] For any approximation function $f$ and inhomogeneous shift $\theta$, if there exists a number $A>3$ such that $$f(n)=O\left(\frac{\log ^A n}{n}\right)$$ and $$\limsup_{N\to\infty}\frac{\sum_{n=1}^{N}\frac{f(n)}{n}\phi(n)}{\log ^{\frac{A}{2}+2.5}N}=\infty,\label{Con2}\tag{1}$$ then $$W(f,\theta) \text{ has full lebesgue measure.}$$ The conclusion holds true if we replace the condition $(1)$ with $$\sum_{n=2}^{\infty}\frac{f(n)}{\log ^{A/2+2.5+\epsilon} n}=\infty,\tag{2}$$ for some $\epsilon>0.$ If $\theta=\mathbf{0}$, then condition $(1)$ can be weakened slightly because of a result of Gallagher [@GA] to the following $$\limsup_{N\to\infty}\frac{\sum_{n=1}^{N}\frac{f(n)}{n}\phi(n)}{\log ^{\frac{A}{2}+2.5}N}>0.\tag{1'}$$
We set $h(x)=x$ in Theorem \[ThDim\] and the first conclusion is easy to see. For the second conclusion, we assume condition $(2)$ and consider the iterated exponential intervals $$I_k=\left[2^{2^k},2^{2^{k+1}}\right].$$ There are infinitely many $k>0$ such that $$\sum_{n\in I_k}\frac{f(n)}{\log ^{A/2+2.5+\epsilon}n}>\frac{1}{k^2},$$ otherwise the following sum $$\sum_{n=2}^{\infty} \frac{f(n)}{\log ^{A/2+2.5+\epsilon}n}$$ will not diverge. Then, we see that $$\sum_{n\in I_k}\frac{f(n)}{n}\phi(n)=\sum_{n\in I_k}\frac{f(n)}{\log^{A/2+2.5+\epsilon}n}\frac{\phi(n)}{n}\log^{A/2+2.5+\epsilon} n$$ $$\begin{aligned}
&\geq& \min_{n\in I_k}\frac{\phi(n)}{n}\log^{A/2+2.5+\epsilon} n \sum_{n\in I_k}\frac{f(n)}{\log^{A/2+2.5+\epsilon}n}\\
&\geq& C\frac{1}{\log\log 2^{2^{k+1}}}\log^{A/2+2.5+\epsilon}2^{2^{k}} \frac{1}{k^2}\\
&\geq& C'\frac{1}{k^3} 2^{(A/2+2.5+\epsilon)k}\\
&\geq& C'' 2^{(A/2+2.5+0.5\epsilon)(k+1)},\\
\end{aligned}$$ where $C,C',C''$ are constants which only depend on $A,\epsilon$. The choice of constant $C$ comes from the following well-known result concerning the Euler gamma $\gamma$: $$\liminf_{n\to\infty}\frac{\phi(n)}{n}\log\log n=e^{-\gamma}.\tag{***}\label{FACT}$$ Then, for all $k>0$ we have $$\sum_{n=2}^{2^{2^{k+1}}}\frac{f(n)}{n}\phi(n)\geq \sum_{n\in I_k}\frac{f(n)}{n}\phi(n)\geq C'' 2^{(A/2+2.5+0.5\epsilon)(k+1)}.$$ This implies condition $(1)$ because $2^{0.5\epsilon k}\to\infty$ as $k\to\infty$.
Some earlier results and discussions {#Se4}
====================================
Before the proofs we shall briefly introduce some known results in metric Diophantine approximation and discuss how our results can be related to them. In Section \[sefurther\] we will also discuss some related questions.
Some general historical remarks
-------------------------------
One of the most famous result was first proved by Khintchine and generalized by Groshev, see for example [@BHH17 Theorem 1].
If $f$ is a non-increasing approximation function such that $$\sum_{n=1}^{\infty} f(n)=\infty,$$ then for any $a\in [0,1/2]$ $
W_0(f,\mathbf{a}) \text{ has full Lebesgue measure.}
$
For convenience, the bold letter $\mathbf{a}$ denotes the constant sequence $\theta$ such that $\theta(n)=a$ for all $n$. Later Duffin-Schaeffer [@Du] generalized Khintchine’s result in the homogeneous case.
For any approximation function $f$, if $$\limsup_{N\to\infty} \frac{\sum_{n=1}^{N}\frac{f(n)\phi(n)}{n}}{\sum_{n=1}^{N}f(n)}>0,\label{Con1}\tag{3}$$ then $$\sum_{n=1}^{\infty} \frac{f(n)}{n}\phi(n)=\infty\implies
W(f,\mathbf{0}) \text{ has full Lebesgue measure.}$$
Theorem \[ThmBy\] and Theorem \[ThDim\] are two inhomogeneous versions of the above result. Duffin and Schaeffer also asked whether the condition (\[Con1\]) can be dropped. They made the following famous conjecture.
For any approximation function $f$ we have the following result $$\sum_{n=1}^{\infty} \frac{f(n)}{n}\phi(n)=\infty\implies
W(f,\mathbf{0}) \text{ has full Lebesgue measure.}$$
[ *Duffin-Schaeffer conjecture* ]{}with extra conditions: known results before [@ALMTZ18]
-----------------------------------------------------------------------------------------
A lot of work has been done since the birth of the above conjecture. Various replacements of condition (\[Con1\]) have been found and we think that the mathoverflow webpage [@Online] gives a nice and brief overview. Notably, the first result of this topic with extra divergence appeared in [@HPV Corollary 1] as follows
For any approximation function $f$ that satisfies the following divergence condition with a positive $\epsilon>0$ $$\sum _{n=1}^\infty \frac{\phi (n) f (n)/n}{ n^{\epsilon}}= \infty,$$ the set $W(f,\mathbf{0})$ has full Lebesgue measure.
In fact the $n^\epsilon$ in the denominator can be replaced by $\exp(c \log n/\log\log n)$ with a suitable constant $c>0$. This was the content of [@HPV Theorem 1]. Note that Corollary \[Thm1\] is an inhomogeneous version of the above result. Later, in [@BV2], the above result was improved to the following.
For any approximation function $f$ that satisfies the following divergence condition $$\sum _{n=1}^\infty \frac{\phi (n) f (n)/n}{ \exp (c(\log \log n)(\log \log \log n))}= \infty,\label{BV}\tag{4}$$ the set $W(f,\mathbf{0})$ has full Lebesgue measure.
Our motivation for Theorem \[ThDim\] was to replace the above condition $(4)$ with the following $$\sum _{n=1}^\infty \frac{\phi (n) f (n)/n}{ \log^c n}= \infty.$$ We are not able to achieve this without the following extra upper bound [^2] $$f(n)=O(\log^c n/n).$$ We remark that in [@VA], it was shown that if $f(n)=O(1/n)$ then $$\sum_{n=1}^{\infty} \frac{f(n)}{n}\phi(n)=\infty\implies
W(f,\mathbf{0}) \text{ has full Lebesgue measure.}$$ We see that $W(f,\mathbf{0})$ has full Lebesgue measure if either we have a strong extra divergence condition like $(4)$, or we have a strong upper bound condition like $f(n)=O(1/n)$ together with a weak divergence condition. Theorem \[ThDim\] and Corollary \[Co1\] show that we can also balance the strength of the upper bound and divergence conditions. We note here that our result holds in the inhomogeneous case as well. For convenience, we introduce the following notations.
Given two non negative numbers $a,b$, we call the condition $C(a,b)$ to be the following two conditions on the approximation function $f$, $$f(n)=O\left(\frac{\log ^a n}{n}\right),$$ $$\limsup_{N\to\infty}\frac{\sum_{n=1}^{N}\frac{f(n)}{n}\phi(n)}{\log ^{b}N}=\infty.$$ We say that the condition $C(a,b)$ is sufficient if the set $W(f,\mathbf{0})$ has full Lebesgue measure under the condition $C(a,b)$.
Thus the result in [@VA] that was mentioned earlier says that the condition $C(0,0)$ is sufficient. Corollary \[Co1\] says that the condition $C(A,A/2+2.5+\epsilon)$ is sufficient for any $A>3,\epsilon>0$. However, it is easy to check that if the condition $C(a,b)$ is sufficient, then for any positive number $c$, the condition $C(a+c,b+c)$ is sufficient as well. Indeed, if we assume the condition $C(a+c,b+c)$ we can find the following new approximation function $$f'(n)=\frac{f(n)}{\log^c n}.$$ It is then easy to check that $f'$ satisfies the condition $C(a,b)$ and because it is sufficient we see that $
W(f',\mathbf{0})
$ has full Lebesgue measure. It is clear that $f'(n)\leq f(n)$ for all sufficiently large integer $n$ and therefore $W(f,\mathbf{0})$ has full Lebesgue measure. Similarly if the condition $C(a,b)$ is sufficient then so is the condition $C(a',b)$ and $C(a,b')$ for all $0\leq a'\leq b$ and $b'\geq b$. We plot the following graph to indicate the known and new sufficient conditions $C(a,b)$ represented as points in the Euclidean plane.
![This picture shows sufficient conditions for $W(f,\mathbf{0})$ to have full Lebesgue measure. We remark that in the blue area, $W(f,\theta)$ also has full Lebesgue measure.[]{data-label="arc2"}](pic1.png){width="5cm"}
We only considered the condition $C(a,b)$ without worrying about whether it can be satisfied at all. In fact, it is easy to check that the condition $C(a,b)$ can be satisfied if and only if $b<a+1$. This is the reason for requiring $A>3$ in Corollary \[Co1\].
The Hausdorff dimension results {#HDR}
-------------------------------
The Hausdorff dimensions of sets $W_0(.,.)$,$W(.,.)$ are much better known. For example we have the following result from [@HPV].
For any approximation function $f$ and any positive real number $s\in (0,1]$ we have the following result $$\sum_{n=1}^{\infty} \left(\frac{f(n)}{n}\right)^s\phi(n)=\infty\implies
\dim_{H}W(f,\mathbf{0})\geq s.$$
By Theorem \[ThmDimMain\], we see that under the same condition as above, $\dim_{H}W(f,\theta)\geq s$ for all inhomogeneous shifts $\theta$. We now introduce a result for computing $\dim_H W_0(f,\mathbf{0})$ from [@HS].
For any approximation function $f$ and real number $\alpha\in [1,\infty)$ we set $$C_\alpha(N)=\text{Cardinality of the set }\left\{n\leq N:f(n)/n\geq \frac{1}{n^{\alpha}}\right\},$$ and $$\delta(\alpha)=\sup\left\{\delta:\limsup_{N\to\infty} \frac{C_{\alpha}(N)}{N^{\delta}}>0 \right\}.$$ Then $$\dim_{H}W_0(f,\mathbf{0})=\min\{1,\sup_{\alpha\geq 1}\kappa(\alpha)\},$$ where $\kappa(\alpha)$ is the following number $$\kappa(\alpha)=\begin{cases}
\frac{1+\delta(\alpha)}{\alpha}& \lim_{N\to\infty} C_{\alpha}(N)=\infty \\
0 & \text{otherwise}
\end{cases}$$
We shall see that the above theorem of Hinokuma and Shiga plays an important role in proving Theorem \[ThmDimMain\]. As a special case, we assume now that $f$ is a non-increasing sequence and define the following lower order of $f$ $$\lambda(f)=\liminf_{n\to\infty} \frac{-\log f(n)}{\log n}.$$ Then for any $\alpha>\lambda(f)+1$, there exists infinitely many $n$ such that $$f(n)\geq \frac{1}{n^{\alpha-1}}.$$ It follows that $\kappa(\alpha)=(1+\delta(\alpha))/\alpha$. Now, because $f$ is non increasing, we see that for any $n$ such that $f(n)\geq 1/n^{\alpha-1}$ we have $$f(\lfloor n/2 \rfloor)\geq\dots\geq f(n)\geq \frac{1}{n^{\alpha-1}}.$$ As there are infinitely many such $n$, we see that for any $\alpha'>\alpha$ there are infinitely many $N$ such that $$C_{\alpha'}(N)\geq \frac{N}{2}.$$ Because numbers $\alpha',\alpha$ such that $\alpha'>\alpha>\lambda(f)+1$ can be chosen arbitrarily, we see that $$\sup_{\alpha\geq 1}\kappa(\alpha)\geq \frac{2}{\lambda(f)+1}.$$ On the other hand, for any $\alpha<\lambda(f)+1$, there are at most finitely many $n$ with $$f(n)\geq\frac{1}{n^{\alpha-1}},$$ and therefore we see that $$\kappa(\alpha)=0.$$ This means that $$\sup_{\alpha\geq 1}\kappa(\alpha)=\frac{2}{\lambda(f)+1},$$ and therefore by Theorem \[ThmDimMain\] we see that for any inhomogeneous shift $\theta$ $$\dim_H W(f,\theta)=\min\left\{1,\frac{2}{\lambda(f)+1}\right\}.$$ This particular result was obtained with $W(f,\theta)$ replaced by $W_0(f,\theta)$ in [@L]. We note here that in [@L] general higher dimensional results were obtained as well.
Notation {#Notations}
========
- In this paper we always use $f$ to denote approximation functions and $\theta$ to denote inhomogeneous shifts. Unless explicitly mentioned otherwise, we assume that $f$ and $\theta$ take values in $[0,1/2]$.
- For any number $a\in\mathbb{R}$ we use $\mathbf{a}$ to denote the constant sequence whose terms are equal to $a$.
- We use $\dim_H$ for the Hausdorff dimension and $\mathcal{H}^h$ for the $h$-Hausdorff measure with dimension function $h$. We will not directly deal with definitions of the Hausdorff measure/dimension. For more details see [@Fa Chapter 3] and [@Ma Chapter 4].
- In this paper we use $\log n$ for the natural logarithm function. There is a small issue we could encounter. For an expression like $\log \log n$, we know that it is not defined at $n=1.$ Since all the results and arguments we have here deal with only the situation for $n\to\infty$, there is no problem if we simply re-define $\log 0=\log 1=2.$
- We shall use the following arithmetic functions:
- : The Euler function: For $n\in\mathbb{N}$,
$\phi(n)=\text{ number of natural numbers smaller than and are coprime to $n$}$.
- : The greatest common divisor function: For $a,b\in\mathbb{N}$
$(a,b)=\text{ the greatest common divisor of $a,b$}$.
- : The divisor function: For $n\in\mathbb{N},\alpha\in\mathbb{R}$,
$d(n)=\text{ the number of divisors of $n$}$.
- : The Möbius function:
For $n\in\mathbb{N}$, the Möbius function is defined as follows,
$$\mu(n)=
\begin{cases}
1 & \text{$n$ is squarefree with even number of prime factors}\\
-1 & \text{$n$ is squarefree with odd number of prime factors}\\
0 & \text{$n$ is not squarefree}
\end{cases}$$
- : The Ramanujan sum:
For $n,k\in\mathbb{N}$: $c_n(k)=\sum_{1\leq a\leq n, (a,n)=1} e^{2\pi i \frac{ak}{n}}=\mu\left(\frac{n}{(k,n)}\right)\frac{\phi(n)}{\phi\left(\frac{n}{(k,n)}\right)}$
- We use $P$ for general probability measure on a probability space $\Omega$ and $\lambda$ for Lebesgue measure on $[0,1]$.
- For a sequence of sets $A_n\subset X$: $\limsup_{n\to\infty} A_n=\{x\in X : x\in A_n \text{ for infinitely many $n\in\mathbb{N}$}\}$.
Results that will be used without proof
=======================================
The central idea we shall use in this paper is a Fourier analytic method introduced by LeVeque in [@LeV]. To start with, given a function $f:[0,1]\to\mathbb{R}$ which is in $L^2$ and thus in $L^1$ as well, the Fourier series of $f$ is given by $$\forall k\in\mathbb{N}, \hat{f}(k)=\int_0^1 e^{2\pi i kx}f(x)dx.$$
We will need the following facts: $$\hat{f}(0)=\|f\|_{L^1} \text{ whenever $f$ is non negative,}$$ $$\widehat{fg}(0)=\sum_{k=-\infty}^{\infty} \hat{f}(k)\hat{g}(-k) \text{ whenever $f,g$ are $L^2$ functions}.$$ The above results can be found in most text books on harmonic analysis for example in [@Ka chapter 1, section 5.5].
We specify the version of Borel-Cantelli lemma (see [@BV3 lemma 2.2]) which will be used later.
\[T1\] Let $A_n$ be a sequence of events in a probability space $(\Omega,P)$ such that $$\sum_{n=1}^{\infty} P(A_n)=\infty,$$ then $$P(\limsup_{n\to\infty} A_n)\geq \limsup_{m\to\infty} \frac{(\sum_{n=1}^m P(A_n))^2}{\sum_{n_1,n_2=1}^{m}P(A_{n_1}\cap A_{n_2})}.$$
\[Re2\] For homogeneous metric Diophantine approximation, to conclude the full measure result we only need to show $$\limsup_{n\to\infty} \frac{(\sum_{n=1}^m P(A_n))^2}{\sum_{n_1,n_2=1}^{m}P(A_{n_1}\cap A_{n_2})}>0.$$ This follows from a result of Gallagher [@GA].
In order to prove the general result Theorem \[ThDim\], we will also use the following version of the mass transference principle in [@BV].
\[ThMT\] Let $\{B_i\}_{i\in\mathbb{N}}$ be a countable collection of balls in $\mathbb{R}$ with $r(B_i)\to 0$ as $i\to \infty$. Let $h$ be a dimension function such that $h(x)/x$ is monotonic and suppose that for any ball $B$ in $\mathbb{R}$ $$\lambda(B\cap\limsup_{i\to\infty} B^h_i)=\lambda(B).$$ Then, for any ball $B$ in $\mathbb{R}$ $$\mathcal{H}^h(B\cap\limsup_{i\to\infty} B_i)=\mathcal{H}^h(B).$$
Here $B^h$ denotes the dilated ball. To be precise, let $B$ be a ball centred at $x\in\mathbb{R}$ with radius $r>0$ then $B^h$ is the ball centred at $x$ with radius $h(r)$.
Some asymptotic results on arithmetic functions {#Se2}
===============================================
In what follows, we will use some results about the Ramanujan sum. The following result is standard and can be found in [@Ha] chapter 16. For integers $n,k$ we have $$c_n(k)=\sum_{(a,n)=1}e^{2\pi i \frac{a}{n}k}=\mu\left(\frac{n}{(n,k)}\right)\frac{\phi(n)}{\phi\left(\frac{n}{(n,k)}\right)}.$$ We will now state and prove some technical lemmas that will be used later.
\[L1\] There is a constant $C>0$ such that for any integers $k,m >0$ $$\frac{1}{d(k)\log m} \sum_{n=1}^m\frac{|c_n(k)|}{\phi(n)}<C.$$ Here $d(k)$ is the divisor function, that is, the number of divisors of the integer $k$.
By properties of the Ramanujan sum and the Euler totient function $$\begin{aligned}
\sum_{n=1}^m\frac{|c_n(k)|}{\phi(n)}&=& \sum_{n=1}^{m}\frac{\left|\mu\left(\frac{n}{(n,k)}\right)\right|}{\phi\left(\frac{n}{(n,k)}\right)}\\
&=& \sum_{n=1}^{m}\frac{\left|\mu\left(\frac{n}{(n,k)}\right)\right|}{\frac{n}{(n,k)}\prod_{r|\frac{n}{(n,k)}, r\text{ prime}}(1-\frac{1}{r})}\\
&=& \sum_{n=1}^{m}\frac{\left|\mu\left(\frac{n}{(n,k)}\right)\right|}{\prod_{r|\frac{n}{(n,k)}, r\text{ prime}}(r-1)}\\
&=& \sum_{l=1, l\text{ squarefree}}^{m}\prod_{r|l, r\text{ prime}}\frac{1}{r-1}\left|\left\{n\in [1,m]|l=\frac{n}{(n,k)}\right\}\right|.
\end{aligned}$$ The cardinality of the set can be bounded by $$\left|\left\{n\in [1,m]|l=\frac{n}{(n,k)}\right\}\right|\leq d(k),$$ because $(n,k)$ must be a divisor of $k$, and for every such divisor $s|k$, the value of $n$ (if exists) can be uniquely determined by $sl$. Then we see that $$\begin{aligned}
\sum_{n=1}^m\frac{|c_n(k)|}{\phi(n)}&\leq& d(k)\sum_{l=1, l\text{ squarefree}}^{m}\prod_{r|l, r\text{ prime}}\frac{1}{r-1}\\
&\leq& d(k)\prod_{r\leq m, r\text{ prime}}\left(1+\frac{1}{r-1}\right).
\end{aligned}$$ Then this lemma follows by Mertens’ third theorem.
We have $$\lim_{m\to\infty} \frac{1}{\log m}\prod_{r\leq m, r\text{ prime}}\left(1+\frac{1}{r-1}\right)=e^{\gamma},$$
where $\gamma$ is the Euler gamma $\gamma\approx 0.5772156$.
\[L2\] There exists a constant $C>0$ such that for all integers $n>1$, $$\sum_{k=1}^{n}\frac{d^2(k)}{k}<C\log ^3 n.$$
First, observe that $$d^2(k)=\sum_{l|k}d(l^2).$$ Indeed for any integer with prime factorization $k=p_1^{a_1}\dots p_k^{a_n}$ we have that $$d(k)=\prod_{i=1}^{i=n}(a_i+1).$$ It follows that: $$\begin{aligned}
\sum_{l|k}d(l^2)&=&\sum_{0\leq b_i\leq a_i,i\in\{1,2\dots n\}} \prod_{i=1}^{i=n}(2b_i+1)\\
&=& \prod_{i=1}^{n}\left(\sum_{b_i=0}^{b_i=a_i} (2b_i+1)\right)\\
&=& \prod_{i=1}^{n}(a_i+1)^2=d^2 (k).
\end{aligned}$$ Then we have the following estimate $$\begin{aligned}
\sum_{k=1}^{n}\frac{d^2(k)}{k}&=&\sum_{l=1}^{n}d(l^2)\sum_{k:l|k}^{k\leq n}\frac{1}{k}\\
&\leq&\sum_{l=1}^{n}\frac{d(l^2)}{l} (\log n+1)\\
&\leq&(\log n+1)\sum_{m=1}^{n^2}\sum_{l:m|l^2}^{l\leq n}\frac{1}{l}\\
&\leq&(\log n+1)\sum_{m=1}^{n^2}\sum_{l:m|l,l\in [1,n^2]}\frac{1}{l}\\
&\leq&(\log n+1)\sum_{m=1}^{n^2}\frac{1}{m}\left(\log n^2+1\right)\\
&\leq&(\log n+1)^2(2\log n+1)\leq C\log^3 n,
\end{aligned}$$ for a suitable constant $C>0$.
\[L3\] There exists a constant $C>0$ such that for any positive integer $m$, $$\sum_{1\leq n\leq m} d(n)d(m)(n,m)\leq C d^3(m) m\log m .$$
$$\begin{aligned}
\sum_{1\leq n\leq m} d(n)d(m)(n,m)&=& d(m)\sum_{r|m} \sum_{n\leq m, (n,m)=r} r d(n)\\
&=& d(m)\sum_{r|m} r \sum_{a\leq n/r, (a,m/r)=1} d(ar)\\
&\leq& d(m)\sum_{r|m} r\sum_{a\leq n/r, (a,m/r)=1}d(a)d(r)\\
&\leq& C d(m) \sum_{r|m} rd(r) \frac{m}{r}\log \frac{m}{r}\\
&\leq& C m d(m)\log m \sum_{r|m} d(r)\\
&\leq& C m d(m)\log m \sum_{r|m} d(r^2)\\
&=& C m d^3(m) \log m.
\end{aligned}$$
Here we used Dirichlet theorem for the divisor summatory function (for the constant $C$) and the first part of the proof of lemma \[L2\].
Fourier series and Diophantine approximation {#Se1}
============================================
From the Borel-Cantelli lemma (Theorem \[T1\]), we see that it is important to show some properties of the measure of intersections. Now we are going to set up the Fourier analysis method.
Let $f,\theta$ be as mentioned above, we denote $$\epsilon_n=\frac{f(n)}{n}.$$ Then we define the function $$g_n(x):[0,1]\to \{0,1\}$$ via the following relation $$g_n(x)=1\iff \left|x-\frac{m+\theta(n)}{n}\right|< \frac{f(n)}{n}, \text{ for an integer }m \text{ with }(m,n)=1.$$ It is clear that $g_n(x)$ is just the characteristic function on the set $A_n$, namely, $$A_n=\left\{x\in[0,1]| \exists 1\leq m\leq n, (m,n)=1, \left|x-\frac{m+\theta(n)}{n}\right|<\frac{f(n)}{n}\right\}.$$ In our case $f(n)\leq\frac{1}{2}$ and therefore $A_n$ is a union of $\phi(n)$ many equal length disjoint intervals. The Lebesgue measure of $A_n$ is $$\|g_n\|_{L^1}=2\epsilon_n \phi(n).$$ Now we see that $\lambda(A_n\cap A_m)=\|g_n g_m\|_{L^1}$. We need only to compute the case $n\neq m$ since otherwise the case is trivial. By using Fourier series we can write the $L^1$-norm as $$\|g_n g_m\|_{L^1}=\sum_{k=-\infty}^{\infty} \hat{g_n}(k)\hat{g_m}(-k).$$ The above equality holds whenever the series is absolutely convergent. This happens whenever $g_n,g_m$ are both $L^1$ functions. This is the case in our situation. Now we need to evaluate the Fourier series of $g_n$, it is easy to see that $g_n$ is just the characteristic function of $$[-\epsilon_n,\epsilon_n]=\left[-\frac{f(n)}{n},\frac{f(n)}{n}\right]$$ convolved with a sum of Dirac deltas $$\sum_{(a,n)=1}\delta\left(\frac{a+\theta(n)}{n}\right).$$ We can also compute the Fourier series directly for $k\neq 0$ $$\begin{aligned}
\int_0^1 e^{2\pi i kx} g_n(x)dx&=&\sum_{(a,n)=1} \int_{\frac{a+\theta(n)}{n}-\epsilon_n}^{\frac{a+\theta(n)}{n}+\epsilon_n}e^{2\pi i kx}dx\\
&=& \sum_{(a,n)=1}\frac{1}{\pi k} \sin(2\pi\epsilon_n k)e^{2\pi i\frac{a+\theta(n)}{n}k}\\
&=& \frac{1}{\pi k} \sin(2\pi\epsilon_n k)c_n(k)e^{2\pi i \frac{\theta(n)}{n} k},\end{aligned}$$ where $c_n(k)=\sum_{(a,n)=1}e^{2\pi i \frac{a}{n}k}$is the Ramanujan sum. For $k=0$, $\hat{g}_n(0)$ is simply $\|g_n\|_{L^1}$. Hence we can express $\lambda(A_n\cap A_m)$ with the following series $$\begin{aligned}
\lambda(A_n\cap A_m)&=&4\epsilon_n\epsilon_m\phi(n)\phi(m)\\
&+&\frac{2}{\pi^2}\sum_{k=1}^{\infty}\frac{\sin(2\pi\epsilon_n k)c_n(k)\sin(2\pi\epsilon_m k)c_m(k)\cos\left(2\pi \left(\frac{\theta(n)}{n}-\frac{\theta(m)}{m}\right) k\right)}{k^2},\end{aligned}$$ where we have used the fact that the values of the Ramanujan sum are real numbers and for all pairs of integers $n,k$ $$c_n(k)=c_n(-k).$$ We see that inhomogeneous shifts $\theta$ create just an extra $\cos(.)$ term whose modulus is bounded by $1$.
proof of Theorem \[ThmBy\] and Corollary \[Thm1\] {#sepp}
==================================================
It follows from the arguments in previous section that $$\lambda(A_n\cap A_m)\leq 4\epsilon_n\epsilon_m\phi(n)\phi(m)+\frac{2}{\pi^2}\sum_{k=1}^{\infty}\frac{|\sin(2\pi\epsilon_n k)c_n(k)\sin(2\pi\epsilon_m k)c_m(k)|}{k^2}.$$ The basic strategy is to split the sum over $k$ up to a number $M$ which will be determined later $$\sum_{k=1}^{\infty}=\sum_{k=1}^{M}+\sum_{k=M+1}^{\infty}.$$
For the first part, we use the fact that $|\sin(x)|\leq \min\{|x|,1\}$ for all $x\in\mathbb{R}$, $$\begin{aligned}
& &\sum_{k=1}^{M}\frac{|\sin(2\pi\epsilon_n k)c_n(k)\sin(2\pi\epsilon_m k)c_m(k)|}{k^2}\\
&\leq&\sum_{k=1}^{M}\frac{\min\{2\pi\epsilon_nk,1\}\min\{2\pi\epsilon_mk,1\}|c_n(k)c_m(k)|}{k^2}\\
&\leq& 2\pi\sum_{k=1}^{M} \frac{1}{k}\min\{\epsilon_n,\epsilon_m\} |c_n(k)||c_m(k)|.\end{aligned}$$
Recalling the formula for the Ramanujan sum $$c_n(k)=\mu\left(\frac{n}{(n,k)}\right)\frac{\phi(n)}{\phi\left(\frac{n}{(n,k)}\right)},$$ we see that there exists an absolute constant $C>0$ satisfying the following inequality $$\begin{aligned}
& &\sum_{k=1}^{M}\frac{|\sin(2\pi\epsilon_n k)c_n(k)\sin(2\pi\epsilon_m k)c_m(k)|}{k^2}\\
&\leq& 2\pi\sum_{k=1}^{M} \frac{1}{k}\min\{\epsilon_n,\epsilon_m\} |c_n(k)||c_m(k)|\\
&=& 2\pi\sum_{k=1}^{M} \frac{1}{k}\min\{\epsilon_n,\epsilon_m\} \left|\mu\left(\frac{n}{(n,k)}\right)\frac{\phi(n)}{\phi\left(\frac{n}{(n,k)}\right)}\mu\left(\frac{m}{(m,k)}\right)\frac{\phi(m)}{\phi\left(\frac{m}{(m,k)}\right)}\right|\\
&\leq& 2\pi\sum_{k=1}^{M}\frac{1}{k} \min\{\epsilon_n,\epsilon_m\} (n,k)(m,k)\left|\mu\left(\frac{n}{(n,k)}\right)\mu\left(\frac{m}{(m,k)}\right)\right|\\
&\leq& 2\pi\sum_{k=1}^M \frac{1}{k}\min\{\epsilon_n,\epsilon_m\} (n,k)(m,k)\\
&\leq& C \log M d(n)d(m)(n,m)\min\{\epsilon_n,\epsilon_m\}.\end{aligned}$$ Here we used the fact that $\phi(n)=n\prod_{r|n,r \text{ prime}}\frac{r-1}{r}$. For the last step we see that for any divisor $s_n$ of $n$ and $r_m$ of $m$, we can sum those $k$ such that $$(n,k)=s_n, (m,k)=r_m.$$ Such $k$ must be a multiple of $[s_n,r_m]$ and therefore we obtain the following result $$\sum_{k:(n,k)=s_n, (m,k)=r_m}\frac{1}{k} (n,k)(m,k)=\sum_{l: l\leq M/[s_n,r_m]}\frac{s_nr_m}{l[s_n,r_m]}\leq C\log M (s_n,r_m).$$ The previous estimate follows from summing over all divisors of $n,m$ and using the fact that $(s_n,r_m)\leq (n,m)$.
For the second part $\sum_{k=M+1}^{\infty}$, we use the fact that $|\sin(x)|\leq 1$ and obtain an absolute constant $C'>0$ with the following property $$\sum_{k=M}^{\infty}\frac{|\sin(2\pi\epsilon_n k)c_n(k)\sin(2\pi\epsilon_m k)c_m(k)|}{k^2}\leq \frac{C'}{M}d(n)d(m)(n,m).$$ We can now set $M= d(n)d(m)(n,m)n^4m^4$. We assume that $\epsilon_n\epsilon_m\neq 0$ otherwise $\lambda(A_n\cap A_m)=0$ and there is nothing to show. Then we see that $\log M\leq 10\log n+10\log m$. In particular, if $n,m\leq N$ then $\log M\leq 20\log N$. We also see that $$\frac{d(n)d(m)(n,m)}{M}=\frac{1}{n^4m^4}.$$ Then, there exists an absolute constant $C''>0$ such that the following holds: $$\begin{aligned}
\lambda(A_n\cap A_m)\leq 4\epsilon_n\epsilon_m\phi(n)\phi(m)&+&C'' \min\{\epsilon_n,\epsilon_m\}d(n)d(m)(n,m)(10\log n\\
&+&10\log m)+C'\frac{1}{n^4m^4}.\end{aligned}$$ We can now use theorem \[T1\] and lemma \[L3\] to conclude the proof. First, observe that by Lemma \[L3\] there exists a constant $C'''>0$ such that $$\begin{aligned}
& &\sum_{n=1}^N\sum_{m\leq n} \min\{\epsilon_n,\epsilon_m\}d(n)d(m)(n,m)(10\log n+10\log m)\\
&\leq& \sum_{n=1}^N\sum_{m\leq n} 20\epsilon_n d(n)d(m)(n,m)\log n\\
&\leq& C'''\sum_{n=1}^N \epsilon_n n d^3(n)\log^2 n.\end{aligned}$$ Similarly, the result holds for the sum $\sum_{m=1}^N\sum_{n<m}$ as well, therefore for a constant $C''''>0$ we have the following inequality $$\begin{aligned}
(*)
\sum_{n,m=1}^N\lambda(A_n\cap A_m)&=&(\sum_{n=1}^{N}\sum_{m\leq n}+\sum_{m=1}^{N}\sum_{n< m})\lambda(A_n\cap A_m)\\&\leq& (\sum_{n=1}^N 2\epsilon_n\phi(n))^2+C''''\sum_{n=1}^{N} \epsilon_n n d^3(n)\log^2 n+100C'\zeta^2(4).\end{aligned}$$ From here Theorem \[ThmBy\] follows. In fact, by the Borel-Cantelli lemma (theorem \[T1\]), we see that $$\begin{aligned}
\lambda(\limsup_{n\to\infty} A_n)&\geq& \limsup_{N\to\infty} \frac{(\sum_{n=1}^N \lambda(A_n))^2}{\sum_{n,m=1}^{N}\lambda(A_n\cap A_m)}\\
&\geq& \limsup_{N\to\infty}\frac{ 1}{1+\frac{C''''\sum_{n=1}^{N} \epsilon_n n d^3(n)\log^2 n+100C'\zeta^2(4)}{(\sum_{n=1}^N 2\epsilon_n\phi(n))^2}}.\end{aligned}$$ The rightmost side of the above inequality is equal to $1$ under the condition of theorem \[ThmBy\]. Next, it is easy to see the following result for an absolute constant $C''''$ and for all integers $n$: $$\frac{n}{\phi(n)}d^3(n)\log^2 n\leq C'''' \log^2 n\exp(3\log2 \log n/\log\log n)\log\log n.$$ We have used here the following result relating to the divisor function: $$\limsup_{n\to\infty} \frac{\log d(n)}{\log n/\log\log n}=\log 2.$$ From here the proof of Corollary \[Thm1\] concludes.
Expected number of solutions {#Se3}
============================
Here we refine the result of the previous section. The content of this section will be used in the final proof of Theorem \[ThmDimMain\]. Previously, we have required that $f(n)\in [0,1/2]$ for all integers $n$. In this section we shall allow $f(n)$ to take any value in $[0,n/2)$. Care is needed regarding the interpretation when $f(n)>1/2$. The first thing to observe is that the following intervals for different $m$ such that $ (m,n)=1$ may overlap $$\left\{x: \left|x-\frac{m+\theta(n)}{n}\right|<\frac{f(n)}{n}\right\}.$$ The second thing to observe is that it is now possible that $$\left\{x: \left|x-\frac{m+\theta(n)}{n}\right|<\frac{f(n)}{n}\right\}\cap (1,\infty)\neq\emptyset.$$ To overcome these problems we need to consider $[0,1)$ as $\mathbb{R}/\mathbb{Z}$. For $x\in\mathbb{R}$ we use $\|x\|$ to be the following quantity $$\inf_{n\in\mathbb{Z}}|x+n|.$$ Given an approximation function $f$ such that for each integer $n\geq 2$, $f(n)\in [0,n/2)$ and inhomogeneous shift $\theta$ taking values in $[0,1/2)$. We want to study the following quantity for Lebesgue typical $x\in\mathbb{R}$, $$S(f,\theta,x,N)=\#\left|\left\{n,m\leq N, (n,m)=1:\left \|x-\frac{m+\theta(n)}{n}\right\|<\frac{f(n)}{n}\right\}\right|.$$ We will prove here the following result.
\[ThmNm\] For any $f:\mathbb{N}\to [0,\infty)$, $\theta:\mathbb{N}\to [0,1/2]$ and a positive number $\rho\in (0.5,1]$. If $$\limsup_{N\to\infty} \frac{\sum_{n=1}^N f(n)n^{-1} \phi(n)}{\exp(\log N\log\log\log N/\log\log N)}= \infty,$$ then for Lebesgue almost all $x\in [0,1]$, there exist infinitely many integers $N_i(x)$ such that $$\left|S(f,\theta,x,N_i(x))-\sum_{n=1}^{N_i(x)} 2\frac{f(n)}{n}\phi(n)\right|\leq \left(\sum_{n=1}^{N_i(x)} 2\frac{f(n)}{n}\phi(n)\right)^\rho.$$
As in section \[Se1\] we construct the function $$g_n(x)=\sum_{1\leq a\leq n:(a,n)=1,\left \|x-\frac{a+\theta(n)}{n}\right\|<\frac{f(n)}{n} }1$$ and see that $$S(f,\theta,x,N)=\sum_{n=1}^{N}g_n(x).$$ We note here that $g_n(x)$ can take integer values other than $0$ and $1$. It is easy to see that $$\int_{0}^1 S(f,\theta,x,N) dx=\sum_{n=1}^N 2\epsilon_n\phi(n)=E_N.$$ Now, we estimate the variance $$\int_0^1 |S(f,\theta,x,N)-E_N|^2 dx=\int_0^1 \sum_{n,m=1}^N g_n(x)g_m(x)dx-(E_N)^2.$$ We need to consider the following integral $$\int_0^1 \sum_{n,m=1}^N g_n(x)g_m(x)dx=\sum_{n,m=1}^N \|g_ng_m\|_{L^1}.$$ Although the functions $g_n$ are more complicated than the ones in Section \[Se1\], the computations of their Fourier coefficients are the same and results are unchanged. We omit the details here. Now we can use Fourier series to obtain the following equality as in the previous section $$\begin{aligned}
\|g_ng_m\|_{L^1}&=&4\epsilon_n\epsilon_m\phi(n)\phi(m)\\
&+&\frac{2}{\pi^2}\sum_{k=1}^{\infty}\frac{\sin(2\pi\epsilon_n k)c_n(k)\sin(2\pi\epsilon_m k)c_m(k)\cos\left(2\pi i \left(\frac{\theta(n)}{n}-\frac{\theta(m)}{m}\right) k\right)}{k^2}.\end{aligned}$$ The argument in the proof of theorem \[Thm1\] allows us to see that for some constant $C'''>0$ we have $$\int_0^1 |S(f,\theta,x,N)-E_N|^2 dx\leq C'''\sum_{n=1}^{N} \epsilon_n n d^3(n)\log^2 n.$$ By the Markov inequality we see that given any sequence of positive numbers $\{\beta(n)\}_{n\in\mathbb{N}}$ $$K(f,\theta,N,\beta):=\lambda(x:|S(f,\theta,x,N)-E_N|\geq \beta_N)\leq C'''\frac{1}{\beta^2_N}\sum_{n=1}^{N} \epsilon_n n d^3(n)\log^2 n.$$ If $\lim_{N\to\infty} K(f,\theta,N,\beta)=0,$ then there exist a subsequence $N_i$ such that $$\sum_i K(f,\theta,N_i,\beta)<\infty.$$ For Lebesgue almost every $x$, there are only finitely many $N_i$ such that $$|S(f,\theta,x,N_i)-E_{N_i}|\geq\beta_{N_i}.$$ Now we see from the discussions in previous section that $$K(f,\theta,N,\beta)\leq C''' \frac{1}{\beta^2_N} E_N \log^2 N \log\log N \exp(3\log 2 \log N/\log\log N).$$ Let us denote $$A_N=\frac{E_N}{ \exp(\log N\log\log\log N/\log\log N)},$$ and suppose that $\limsup_{N\to\infty} A_N=\infty$, then we see that for $\beta_N=E^\rho_N$ $$K(f,\theta,N,\beta)\to 0 \text{ as $N\to\infty$}.$$ This is because of the following inequality and the fact that $2\rho-1>0$ $$\begin{aligned}
K(f,\theta,N,\beta)&\leq& C''' \frac{1}{E^{2\rho-1}_N} \log^2 N \log\log N \exp(3\log 2 \log N/\log\log N)\\
&=& C''' \frac{1}{A^{2\rho-1}_N} \frac{\log^2 N \log\log N \exp(3\log 2 \log N/\log\log N)}{\exp((2\rho-1)\log N\log\log\log N/\log\log N)}.\end{aligned}$$ Hence for Lebesgue almost all $x\in \mathbb{R}$ there are infinitely many integers $N>0$ such that $$E_N+E^{\rho}_N \geq S(f,\theta,x,N)\geq E_N-E^{\rho}_N.$$ In particular if $\rho<1$, then for such $x$ we see that for infinitely many coprime pairs $n,m$ the following inequality holds $$\left \|x-\frac{m+\theta(n)}{n}\right\|<\frac{f(n)}{n}.$$
proof of theorem \[ThmDimMain\]
===============================
Recall Theorem (HS) in Section \[HDR\]. We now show that $
\dim_H W(f,\theta)\geq \dim_H W_0(f,\mathbf{0}).
$ The other direction can be proved by the same argument provided in [@HS], see also [@L Lemma 1]. Only for the lower bound are there some difficulties in estimating the size of the intersections $A_n\cap A_m$ by using direct number theoretic methods.
Let $f$ be any approximation function and $\theta$ be any inhomogeneous shift. As in the above theorem, for any $\alpha$, we find sets with cardinality $C_\alpha(N)$ and find the exponent $\delta(\alpha)$. Assume that $\kappa(\alpha)>0,$ otherwise there is nothing to show.
First, we consider the case when $\delta(\alpha)>0$ and we shall show that $$\dim_H W(f,\theta)\geq \frac{1+\delta(\alpha)}{\alpha}.$$ Now, for an arbitrarily small number $\sigma>0$ such that $\sigma<\delta(\alpha)$ we use the dimension function $h(x)=x^{\frac{1-\sigma+\delta(\alpha)}{\alpha}}$ in the mass transference principle (Theorem \[ThMT\]). We see that $\epsilon_n=f(n)/n\geq 1/n^\alpha$ for a subset $C_\alpha$ of $\mathbb{N}$ such that $$\limsup_{N\to\infty} \frac{\#|C_\alpha\cap [1,N]|}{N^{\delta(\alpha)-0.5\sigma}}=\infty.$$ We see that $h(\epsilon_n)\geq \frac{1}{n^{1-\sigma+\delta(\alpha)}}$ and $$\begin{aligned}
& &\limsup_{N\to\infty} \frac{\sum_{n=1}^N \phi(n) h(\epsilon_n)}{\log^2 N\log\log N\exp(3\log 2 \log N/\log\log N)}\\
&\geq& \limsup_{N\to\infty}\frac{\#|C_\alpha\cap [1,N]| \frac{1}{\log\log N} \frac{1}{N^{-\sigma+\delta(\alpha)}}}{\log^2 N\log\log N\exp(3\log 2 \log N/\log\log N)}\\
&\geq& \limsup_{N\to\infty} \frac{N^{0.5\sigma}}{\log^2 N\log\log^2 N\exp(3\log 2 \log N/\log\log N)}=\infty.\end{aligned}$$ By Theorem \[ThMT\], we have $$\mathcal{H}^{\frac{1-\sigma+\delta(\alpha)}{\alpha}}(W(f,\theta))=\infty.$$ This implies that for all $\sigma>0$ $$\dim_H W(f,\theta)\geq \frac{1-\sigma+\delta(\alpha)}{\alpha}.$$ This implies further that $$\dim_H W(f,\theta)\geq \frac{1+\delta(\alpha)}{\alpha}.$$
Now we consider the case when $\delta(\alpha)=0$ and $C_\alpha(N)\to\infty$. For a positive number $\rho<1$ which can be chosen close to $1$, we consider the dimension function $h(x)=x^{\rho/\alpha}$. Assume that $$f(n)\neq 0\iff \epsilon_n=f(n)/n\geq \frac{1}{n^{\alpha}},$$ and by shrinking some values of $f$ if necessary $$f(n)\neq 0\iff \epsilon_n=f(n)/n= \frac{1}{n^{\alpha}}.$$ Therefore we see that $$h(\epsilon_n)\neq 0\iff h(\epsilon_n) = \frac{1}{n^{\rho}}.$$ Because $1/n^{\rho}>1/n$ we are in the situation discussed in section \[Se3\]. Now if $\epsilon_n\neq 0$ we see that when $n$ is also large enough (see (\*\*\*) in proof of Corollary \[Co1\]) $$\phi(n)h(\epsilon_n)\geq 0.0001\frac{n}{\log\log n} \frac{1}{n^{\rho}}\geq n^{0.5-0.5\rho}.$$ This implies that $$\sum_{n=1}^N \phi(n)h(\epsilon_n)\geq N^{0.5-0.5\rho}$$ for infinitely many $N$. By theorem \[ThmNm\] (with $f(n)=n^{1-\rho}$ in the statement), we see that for Lebesgue almost all $x\in \mathbb{R}$ there are infinitely many coprime pairs $n,m$ such that $f(n)\neq 0$ and $$\left \|x-\frac{m+\theta(n)}{n}\right\|<\frac{1}{n^{\rho}}.$$ This is almost what we need, we want to find $x\in [0,1]$ such that there are infinitely many coprime pairs $n,m$ such that $f(n)\neq 0$ and $$\left |x-\frac{m+\theta(n)}{n}\right|<\frac{1}{n^{\rho}}.$$ Let $M$ be a large integer. Consider now $x\in [M^{-1},1-M^{-1}]$. Suppose that there is a coprime pair $n,m$ such that $$\left \|x-\frac{m+\theta(n)}{n}\right\|<\frac{1}{n^{\rho}}.$$ Then if $n$ is also large enough ($1/n^\rho<1/M^2$) we see that $$\left |x-\frac{m+\theta(n)}{n}\right|<\frac{1}{n^{\rho}}.$$ This observation implies that for Lebesgue almost all $x\in [M^{-1},1-M^{-1}]$ there are infinitely many coprime pairs $n,m$ such that $f(n)\neq 0$ and $$\left |x-\frac{m+\theta(n)}{n}\right|<\frac{1}{n^{\rho}}.$$ By letting $M\to\infty$ and using Theorem \[ThMT\] we see that $$\mathcal{H}^{\rho/\alpha}(W(f,\theta))=\infty.$$ This implies that $$\dim_H W(f,\theta)\geq \frac{\rho}{\alpha}.$$ Now we can choose $\rho$ arbitrarily close to $1$ and observe $$\dim_H W(f,\theta)\geq \frac{1}{\alpha}.$$ Then, combining this with the theorem by Hinokuma-Shiga we see that $$\dim_H W(f,\theta)\geq \dim_H W_0(f,\mathbf{0}).$$
proof of theorem \[ThDim\]
==========================
We now try to directly estimate the following sum $$\sum_{n,m=1}^{N}\lambda(A_n\cap A_m).$$
\[Thmain\] Let $f, \theta, \epsilon_n$ be as mentioned before. Then there is a constant $C>0$ such that for all integer $N>0$ $$\sum_{n,m=1}^{N}\lambda(A_n\cap A_m)\leq C\left(\max_{n\in [1,N]}\epsilon_n^{0.5}\phi(n)\right)^2\log^5 N+(\sum_{n=1}^{N}2\epsilon_n\phi(n))^2$$
By the arguments in Section \[Se1\] we see that $$\sum_{n,m=1}^{N}\lambda(A_n\cap A_m)\leq (\sum_{n=1}^{N}2\epsilon_n\phi(n))^2+\frac{2}{\pi^2}\sum_{k=1}^{\infty}\frac{1}{k^2}(\sum_{n=1}^{N}|\sin (2\pi\epsilon_n k)c_n(k)|)^2.$$ Since $|\sin(x)|\leq 1$, for any $\alpha\in [0,1]$ we have $$|\sin(x)|\leq |\sin(x)|^\alpha\leq |x|^\alpha.$$ The basic strategy is again to split the sum with respect to $k$, say, $$\sum_{k=1}^{\infty}=\sum_{k=1}^{M}+\sum_{k=M+1}^{\infty},$$ for a later determined integer $M>0$. For convenience, we make the following notation: $$I=\sum_{k=1}^{M},$$ $$II=\sum_{k=M+1}^{\infty}.$$ Then for part $I$ we use the estimate $|\sin(x)|\leq |\sin(x)|^{0.5}\leq |x|^{0.5}$ $$\begin{aligned}
I&=&\frac{2}{\pi^2}\sum_{k=1}^{M}\frac{1}{k^2}\left(\sum_{n=1}^{N}|\sin (2\pi\epsilon_n k)c_n(k)|\right)^2\\
&\leq& \frac{4}{\pi}\sum_{k=1}^{M}\frac{1}{k^2}\left(\sum_{n=1}^{N}\epsilon_n^{0.5} k^{0.5} |c_n(k)|\right)^2\\
&\leq& \frac{4}{\pi}\sum_{k=1}^{M}\frac{1}{k}\left(\sum_{n=1}^{N}\epsilon_n^{0.5}\phi(n) \frac{|c_n(k)|}{\phi(n)}\right)^2\\
&\leq& \frac{4}{\pi}\sum_{k=1}^{M}\frac{\left(\max_{n\in [1,N]}\epsilon_n^{0.5}\phi(n)\right)^2}{k}\left(\sum_{n=1}^{N} \frac{|c_n(k)|}{\phi(n)}\right)^2.\end{aligned}$$ By lemma \[L1\],\[L2\], we see that for a constant $C_1>0$ $$I\leq C_1 \left(\max_{n\in [1,N]}\epsilon_n^{0.5}\phi(n)\right)^2 \log^2 N \log^3 M,$$ where $\log^2 N$ comes from lemma \[L1\] and $\log^3 M$ comes from lemma \[L2\]. For $II$ we use the trivial bound $|\sin(x)|\leq 1$ and see that $$\begin{aligned}
II&=&\frac{2}{\pi^2}\sum_{k=M+1}^{\infty}\frac{1}{k^2}\left(\sum_{n=1}^{N}|\sin (2\pi\epsilon_n k)c_n(k)|\right)^2\\
&\leq& \frac{2}{\pi^2}\sum_{k=M+1}^{\infty}\frac{1}{k^2}\left(\sum_{n=1}^{N}|c_n(k)|\right)^2\\
&\leq& \frac{2}{\pi^2}\sum_{k=M+1}^{\infty}\frac{1}{k^2} N^4\leq C_2 \frac{N^4}{M}\end{aligned}$$ for another constant $C_2>0$. Note that in above inequalities we used the fact $$|c_n(k)|\leq \phi(n)\leq n.$$ With some careful analysis we can replace the $N^4$ with $N^3$, but there is no essential difference as we shall see. Now we choose $M=N^5$. The following estimate holds for a suitable constant $C>0$ $$\begin{aligned}
I+II&\leq& 125C_1 \left(\max_{n\in [1,N]}\epsilon_n^{0.5}\phi(n)\right)^2 \log^2 N \log^3 N+C_2\frac{1}{N}\\
&\leq& C\left(\max_{n\in [1,N]}\epsilon_n^{0.5}\phi(n)\right)^2 \log^2 N \log^3 N.\end{aligned}$$ From here the result of this theorem follows.
We can now prove theorem \[ThDim\]:
By theorem \[Thmain\] we see that for a constant $C>0$ such that $$\begin{aligned}
\sum_{n,m=1}^{N}\lambda(A_n\cap A_m)&\leq& C\left(\max_{n\in [1,N]}\epsilon_n^{0.5}\phi(n)\right)^2\log^5 N+(\sum_{n=1}^{N}2\epsilon_n\phi(n))^2,
\end{aligned}$$ we have $$\begin{aligned}
\frac{(\sum_{n=1}^{N}2\epsilon_n\phi(n))^2}{\sum_{n,m=1}^{N}\lambda(A_n\cap A_m)}&\geq& \frac{(\sum_{n=1}^{N}2\epsilon_n\phi(n))^2}{C\left(\max_{n\in [1,N]}\epsilon_n^{0.5}\phi(n)\right)^2\log^5 N+(\sum_{n=1}^{N}2\epsilon_n\phi(n))^2}\\
&\geq& \frac{1}{C \left(\max_{n\in [1,N]}\epsilon_n^{0.5}\phi(n)\right)^2\frac{\log ^{5} N}{(\sum_{n=1}^{N}2\epsilon_n\phi(n))^2}+1}.
\end{aligned}$$ We can then apply the following condition for $h(x)=x$ $$\limsup_{N\to\infty} \frac{\sum_{n=1}^N \phi(n)h(\epsilon_n)}{\log^{2.5} N\left(\max_{n\in [1,N]} h(\epsilon_n)^{1/2}n\right)}=\infty,$$ and obtain $$\begin{aligned}
\limsup_{N\to\infty}\frac{(\sum_{n=1}^{N}2\epsilon_n\phi(n))^2}{\sum_{n,m=1}^{N}\lambda(A_n\cap A_m)}&\geq&1. \end{aligned}$$ The conclusion of this theorem holds for the special dimension function $h(x)=x$. For general dimension functions, we can combine the special case and the mass transference principle(Theorem \[ThMT\]) to concludes the proof.
Further discussions {#sefurther}
===================
Rigidity of the Hausdorff dimension
-----------------------------------
Our result Theorem \[ThmDimMain\] shows that the Hausdorff dimensions of sets of well approximation numbers stay unchanged under inhomogeneous shifts and dropping non-reduced fractions. We guess that this phenomena should hold in general. In order to formulate the problem we consider the following general Diophantine approximation system.
Given any integer $n$, let $B_n$ be a subset of $\{0,1,\dots, n-1\}$. For any approximation function $f$ and inhomogeneous shift $\theta$, define $$W_B(f,\theta)=\left\{x\in [0,1]: \left|x-\frac{m+\theta(n)}{n}\right|\leq \frac{f(n)}{n} \text{ for i.m. pairs } n,m \text{ such that } m\in B_n \right\}.$$
Thus $W_0(.,.)$ is equal to $W_B(.,.)$ with $B_n=\{0,\dots,n-1\}$ for all integers $n$. We formulate the following two conjectures.
For any approximation function $f$ and inhomogeneous shift $\theta$, we have the following equality $$\dim_H W_B(f,\theta)=\dim_H W_B(f,\mathbf{0}).$$
For any approximation function $f$ and inhomogeneous shift $\theta$, we have the following chain of inequalities $$\liminf_{n\to\infty}\frac{\log |B_n|}{\log n}\dim_H W_0(f,\mathbf{0})\leq \dim_H W_B(f,\theta)\leq \limsup_{n\to\infty}\frac{\log |B_n|}{\log n}\dim_H W_0(f,\mathbf{0}).$$ In particular, if $$\lim_{n\to\infty}\frac{\log |B_n|}{\log n}=1,$$ then $$\dim_H W_0(f,\mathbf{0})= \dim_H W_B(f,\theta).$$
Inhomogeneous Duffin-Schaeffer problems
---------------------------------------
The main motivation of this paper is to show that inhomogeneous metric Diophantine approximation is not too different than the homogeneous case. In fact it is a folklore conjecture that in order to prove the Duffin-Schaeffer conjecture, the homogeneous case is perhaps the hardest case. For example, in [@RA] it was asked whether for any inhomogeneous shift $\theta$ the following statement holds $$W(f,\textbf{0}) \text{ has full Lebesgue measure}\implies W(f,\theta) \text{ has full Lebesgue measure }.$$
There are several developments of Duffin-Schaeffer theorem in the homogeneous case. We are curious to see whether all known results about homogeneous Duffin-Schaeffer problem hold for the inhomogeneous situation as well. In particular we list below two such questions.
[(See also [@SC Conjecture 1.7])]{}\[q1\] For any approximation function $f$ and inhomogeneous shift $\theta$, if the following additional condition is satisfied $$\limsup_{N\to\infty} \frac{\sum_{n=1}^{N}\frac{f(n)\phi(n)}{n}}{\sum_{n=1}^{N}f(n)}>0,$$ is the following statement true $$\sum_{n=1}^{\infty} \frac{f(n)}{n}\phi(n)=\infty\implies
W(f,\theta) \text{ has positive Lebesgue measure?}$$
For any approximation function $f$ that satisfies the following divergence condition $$\sum _{n=1}^\infty \frac{\phi (n) f (n)/n}{ \exp (c(\log \log n)(\log \log \log n))}= \infty,$$ does the set $W(f,\theta)$ has positive Lebesgue measure for all inhomogeneous shift $\theta$?
Cancellation of trigonometric functions {#Chow}
---------------------------------------
So far we have completely ignored the effect of inhomogeneous shift. In fact in [@RA] some dynamical shift was considered. This shed some lights on another important feature of Fourier analysis, the cancellation. Although rather technical, carefully analysis of the cancellation of trigonometric sums often provides nice results. We are curious to see whether in this case we can perform any cancellation in the main formula: $$\sum_{n,m=1}^N\frac{2}{\pi^2}\sum_{k=1}^{\infty}\frac{\sin(2\pi\epsilon_n k)c_n(k)\sin(2\pi\epsilon_m k)c_m(k)\cos\left(2\pi i \left(\frac{\theta(n)}{n}-\frac{\theta(m)}{m}\right) k\right)}{k^2},$$ if in the above expression we replace the Ramanujan sums $c_n(k)$ with the full trigonometric sum $$\Delta_n(k)=\sum_{a\in\{0,\dots,n-1\}}e^{2\pi i ka/n}=n1_{n|k}.$$ The last notation indicates the function equal to $n$ when $k$ is a multiple of $n$ and $0$ otherwise. In this case, in [@LeV page 217, inequality (5)], LeVeque showed by using Fourier series method $$\sum_{k=1}^{\infty}\frac{\sin(2\pi\epsilon_n k)\Delta_n(k)\sin(2\pi\epsilon_m k)\Delta_m(k)\cos\left(2\pi i \left(\frac{\theta(n)}{n}-\frac{\theta(m)}{m}\right) k\right)}{k^2}\leq
2(n,m)\min\{\epsilon_n,\epsilon_m\}.$$ Compare with our method in Section \[sepp\], the most significant point is that the above bound does not have any logarithmic factor. In fact, by the above inequality and the fact that for all integer $n$, $$\sum_{1\leq m\leq n} (n,m)=\sum_{r:r|n} r\sum_{m:m\in [1,n], (n,m)=r} 1=\sum_{r:r|n} r\phi(n/r)=n\sum_{r|n}\frac{\phi(r)}{r}\leq nd(n).$$ With the same method as in Section \[sepp\] we can show the following result.
\[Thmpo2\] For any approximation function $f$ and inhomogeneous shift $\theta$, we have the following result $$\limsup_{N\to\infty} \frac{\sum_{n=1}^N f(n)}{\sqrt{\sum_{n=1}^N f(n)d(n)}}=\infty\implies
W_0(f,\theta) \text{ has full Lebesgue measure.}$$
Because our method in Section \[sepp\] completely ignored the cancellation of trigonometric sums we think that by carefully performing the cancellation one can actually get rid of the logarithmic factor, $$\begin{aligned}
& &\sum_{n,m=1}^N\frac{2}{\pi^2}\sum_{k=1}^{\infty}\frac{\sin(2\pi\epsilon_n k)c_n(k)\sin(2\pi\epsilon_m k)c_m(k)\cos\left(2\pi \left(\frac{\theta(n)}{n}-\frac{\theta(m)}{m}\right) k\right)}{k^2}\\
&\leq^?& C(n,m)\min\{\epsilon_n,\epsilon_m\}.\end{aligned}$$ Where $C>0$ is a constant and $\leq^?$ indicates our uncertainty. If the above would be true then one could obtain the following result which is a better version of Theorem \[ThmBy\] and a weaker version of the content of Question \[q1\].
For any approximation function $f$ and inhomogeneous shift $\theta$, we have the following result $$\limsup_{N\to\infty} \frac{\sum_{n=1}^N \phi(n)f(n)/n}{\sqrt{\sum_{n=1}^N f(n)d(n)}}=\infty\implies
W(f,\theta) \text{ has full Lebesgue measure.}$$
The above argument can help us derive some new results as well. In fact, the main task is to find a good estimate for $\lambda(A_n\cap A_m)$ (see Section \[Se1\]). We are now going to show a much weaker version of the above conjecture.
\[Thmpo\] For any approximation function $f$ and inhomogeneous shift $\theta$, we have the following result $$\begin{aligned}
& &\sum_{n}{f(n)}=\infty \text{ and }
\limsup_{N\to\infty} \frac{\sum_{n=1}^N \phi(n)f(n)/n}{\sum_{n=1}^N f(n)d(n)}>0\\
&\implies&
W(f,\theta) \text{ has positive Lebesgue measure.}
\end{aligned}$$
We introduce the sets $\tilde{A}_t$ for integers $t$, $$\tilde{A}_t=\left\{x\in[0,1]| \exists 1\leq m\leq t, \left|x-\frac{m+\theta(t)}{t}\right|<\frac{f(t)}{t}\right\}.$$ It is easy to see that $A_t\subset\tilde{A}_t$ for all integers $t$ and therefore we have that $$\lambda(A_n\cap A_m)\leq \lambda(\tilde{A}_n\cap \tilde{A}_m).$$ With the help of [@LeV page 217, inequality (5)] we see that $$\lambda(\tilde{A}_n\cap \tilde{A}_m)\leq 4f(m)f(n)+2(n,m)\min\{\epsilon_n,\epsilon_m\}.$$ Then, for all integer $N>0$ and a constant $C>0$, we have $$\sum_{n,m=1}^N \lambda(A_n\cap A_m)\leq (2\sum_{n=1}^N f(n))^2+C\sum_{n=1}^N f(n) \tilde{d}(n),$$ where $\tilde{d}(n)$ is defined by $$\tilde{d}(n)=\sum_{s|n} \frac{\phi(s)}{s}.$$ It is easy to see that $\tilde{d}(n)\leq d(n)$ and this is enough to prove this theorem. We see that $$\begin{aligned}
\lambda(\limsup_{n\to\infty} A_n)&\geq& \limsup_{N\to\infty} \frac{(\sum_{n=1}^N \lambda(A_n))^2}{\sum_{n,m=1}^{N}\lambda(A_n\cap A_m)}\\
&\geq& \limsup_{N\to\infty} \frac{1}{\left(\frac{\sum_{n=1}^N f(n)}{\sum_{n=1}^N f(n)\phi(n)/n}\right)^2+C\frac{\sum_{n=1}^N f(n)d(n)}{(\sum_{n=1}^N f(n)\phi(n)/n)^2}}.\end{aligned}$$ In order to obtain a positive measure of the $\limsup$ set it is enough to find infinitely many integers $N_i$ and a positive number $c>0$ such that $$\frac{\sum_{n=1}^{N_i} f(n)\phi(n)/n}{\sum_{n=1}^{N_i} f(n)}>c,\tag{*}$$ and $$\frac{(\sum_{n=1}^{N_i} f(n)\phi(n)/n)^2}{\sum_{n=1}^{N_i} f(n)d(n)}>c.\tag{**}$$ This is almost the Duffin-Schaeffer theorem in [@Du] which does not require condition $(**)$. Notice that under the condition of this theorem, $(*)$ is trivially satisfied because $f(n)\leq f(n)d(n)$ for all integers $n$. However we see that $(**)$ is satisfied even for $c=\infty$. This concludes the proof.
Approximation functions with nice support
-----------------------------------------
By Theorem \[ThmBy\] and the Hardy-Ramanujan-Turán-Kubilius theorem on the normal order of the logarithm of the divisor function, we see that if the approximation function $f$ is supported on a large subset of $\mathbb{N}$ on which $d(n)\leq \log^{1+\epsilon} n$, then we can provide an inhomogeneous Duffin-Schaeffer type result. For a positive number $\epsilon>0$, let $A\subset\mathbb{N}$ is such that: $$a\in A\iff d(a)\leq\log^{1+\epsilon} a.$$ Note that $A$ is of natural density $1$. Then, for an approximation function supported on $A$ and any inhomogeneous shift $\theta$: $$f(n)\neq 0\implies n\in A,$$ and $$\limsup_{n\to\infty} \frac{\sum_{n=1}^N \phi(n)f(n)/n}{\log ^{3+3.5\epsilon}N}=\infty\implies
W(f,\theta) \text{ has full Lebesgue measure.}$$ Or in a more convenient form: $$\sum_{n=1}^{\infty}\frac{f(n)}{\log ^{3+4\epsilon} n}=\infty.\implies
W(f,\theta) \text{ has full Lebesgue measure.}$$ Note that the power $3+4\epsilon$ here is probably not optimal.
An example
----------
We shall now discuss more about Theorem \[ThDim\] and Corollary \[Co1\]. A result due to Vaaler [@VA] says that if $f(n)=O(1/n)$, then $$\sum_{n=1}^{\infty} \frac{f(n)}{n}\phi(n)=\infty\implies
W(f,\mathbf{0}) \text{ has full Lebesgue measure.}$$ We can provide an approximation function $f$ that does not satisfy the Duffin-Schaeffer condition $(3)$ and the extra divergence condition $(4)$ in Section \[Se4\] nor Vaaler’s condition $f(n)=O(1/n)$. To begin with, we decompose the integer set into dyadic intervals $$D_k=[2^k,2^{k+1}),k=0,1,\dots$$ For each $k$, we choose an integer $m(k)$ such that $$\liminf_{k\to\infty} m(k)\to\infty, \sum_{k=0}^{\infty}\frac{k^{2.4}}{m(k)!}=\infty.$$ Then in each $n\in D_k$ we assign the value $f(n)=\log^{10} n/n$ if $n$ is a multiple of $m(k)!$. Otherwise, set $f(n)=0$. It is easy to see that for large enough $n$ $$\frac{\phi (n) f (n)}{n \exp (c(\log \log n)(\log \log \log n))}\leq \frac{1}{n\log^2 n}.$$ Therefore, condition (\[BV\]) is not satisfied. Next, $f(n)$ is only non zero if $n$ is a multiple of $N_n!$ for a suitable integer $N_n$ and as $n\to\infty$, $N_n\to\infty$. Then, we see that $$\frac{\phi(n)}{n}\leq \prod_{r \text{ prime}, r\leq N_n}\left(1-\frac{1}{n}\right)\to 0, \text{ as } n\to\infty.$$ Hence the Duffin-Schaeffer condition (\[Con1\]) is not satisfied. For large enough $k$ there are more than $0.5 |D_k|/m(k)!$ numbers in $D_k$ which are multiples of $m(k)!$, so we see that $$\sum_{n\in D_k, m(k)| n} \frac{\log^{10} n}{n \log^{7.6} n}\geq 0.5 \frac{2^k}{m(k)!} \frac{k^{10}}{2^{k+1} (k+1)^{7.6}}\geq \frac{1}{2^{10}}\frac{k^{2.4}}{m(k)!}.$$ Here we used the fact that $k+1\leq 2k$ for all $k>1$. The conditions in Corollary \[Co1\]. Therefore we see that $W(f,\theta)$ has full Lebesgue measure for any inhomogeneous shift $\theta.$ In particular, this holds for $\theta=\mathbf{0}.$ As we have mentioned before, this homogeneous result can be also derived from [@ALMTZ18 Theorem 1].
Acknowledgement
===============
HY was financially supported by the University of St Andrews. We want to thank S. Chow for providing us a draft of [@SC] as well as an anonymous referee for acknowledging us the research article [@ALMTZ18]. We also want to thank the anonymous referee(s), S. Burrell and J. Fraser for carefully proofreading an early version of this paper.
[00]{} C. Aistleitner, T. Lachmann, M. Munsch, N. Technau and A. Zafeiropoulos, *The Duffin-Schaeffer conjecture with extra divergence*, preprint, arxiv:1803.05703, (2018).
D. Badziahin, S. Harrap and M. Hussain, *An inhomogeneous Jarník type theorem for planar curves.* Math. Proc. Cambridge Philos. Soc. **163** (2017), no. 1, 47–70.
V. Beresnevich, G. Harman, A. Haynes, and S. Velani, *The Duffin–Schaeffer conjecture with extra divergence II*, Math. Z. **275** (2013), no. 1, 127–133.
V. Beresnevich, F. Ramírez, and S. Velani, *Metric Diophantine approximation: Aspects of recent work*, London Mathematical Society Lecture Note Series, Cambridge University Press, (2016),1-–95.
V. Beresnevich and S. Velani, *A mass transference principle and the Duffin-Schaeffer conjecture for Hausdorff measures*, Ann. of Math. (2) **164** (2006), no. 3, 971–992.
Y. Bugeaud, *A note on inhomogeneous Diophantine approximation*, Glasg. Math. J. **45** (2003), no. 1, 105–110.
S. Chow, *Bohr sets and multiplicative Diophantine approximation*, to appear in Duke Math. J., arXiv:1703.07016, (2017)
R. Duffin and A. Schaeffer, *Khintchine’s problem in metric Diophantine approximation*, Duke Math. J. **8** (1941), no. 2, 243–255.
K. Falconer, *Fractal geometry: Mathematical foundations and applications*, 3rd Edition,Wiley, 2014.
P. Gallagher, *Approximation by reduced fractions*, J. Math. Soc. Japan **13** (1961), no. 4, 342–345.
A. Haynes, A. Pollington, and S. Velani *The Duffin–Schaeffer conjecture with extra divergence*, Math. Ann. **353** (2012), no. 2, 259–273.
T. Hinokuma and H. Shiga, *Hausdorff dimension of sets arising in Diophantine approximation*, Kodai Math. J. **19** (1996), no. 3, 365–377.
G. Hardy, E. Wright, R. Heath-Brown, and J. Silverman, *An introduction to the theory of numbers*, Oxford mathematics, OUP Oxford, (2008).
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*http://mathoverflow.net/questions/63514/weakening-the-hypotheses-in-the-duffin-schaeffer-conjecture*.
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F. Ramírez, *Counterexamples, covering systems, and zero-one laws for inhomogeneous approximation*, Int. J. Number Theory **13** (2017), 633–654.
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[^1]:
[^2]: A few months after the first public version of this paper, it was proven [@ALMTZ18] that this upper bound condition can be dropped for homogeneous cases. For inhomogeneous cases, it is not known whether one can get rid of this upper bound condition.
|
---
abstract: 'We provide a companion to the recent B[é]{}nyi-[Ć]{}urgus generalization of the well-known theorems of Ceva and Menelaus, so as to characterize both the collinearity of points and the concurrence of lines determined by six points on the edges of a triangle. A companion for the generalized area formula of Routh appears, as well.'
author:
- 'B.D.S. “Blue” McConnell\'
title: 'A Six-Point Ceva-Menelaus Theorem'
---
The venerable theorems of (Giovanni) Ceva and Menelaus (of Alexandria) concern points on the edge-lines of a triangle. Each point defines —and is defined by— the ratio of lengths[^1] of collinear segments joining it to two of the triangle’s vertices, and the theorems use a trio of such ratios to neatly characterize the special configurations in Figure 1.
[0.5]{} {width=".8\textwidth"}
[0.5]{} {width=".8\textwidth"}
Specifically, with $D$, $E$, $F$ on edge-lines opposite respective vertices $A$, $B$, $C$, we write $$d := \frac{|BD|}{|DC|} \qquad e := \frac{|CE|}{|EA|} \qquad f := \frac{|AF|}{|FB|}$$ and express the theorems as follows:
Lines $\overleftrightarrow{AD}$, $\overleftrightarrow{BE}$, $\overleftrightarrow{CF}$ pass through a common point if and only if $$\label{eqn_ceva}
d e f = \phantom{-}1$$
Points $D$, $E$, $F$ lie on a common line if and only if $$\label{eqn_menelaus}
d e f = -1$$
B[é]{}nyi and [Ć]{}urgus [@BC2012], and this author, independently (and nearly-simultaneously) considered separate aspects of the same approach to generalizing the above —namely, doubling the number of points on the triangle’s edges— arriving at equations whose terms, in the grand Ceva-Menelaus tradition, differ only in sign.
Interestingly, the B[' e]{}nyi-[' C]{}urgus result concerns [*Ceva*]{}-like elements (lines through vertices) and a [*Menelaus*]{}-like phenomenon (collinearity of points). This author’s contribution, on the other hand, concerns [*Menelaus*]{}-like elements (points on edges) and a [*Ceva*]{}-like phenomenon (concurrence of lines).
[0.5]{} {width=".8\textwidth"}
[0.5]{} {width=".8\textwidth"}
Place points $A^{+}$ and $A^{-}$ on the edge-line opposite vertex $A$; likewise, $B^{+}$ and $B^{-}$ opposite $B$, and $C^{+}$ and $C^{-}$ opposite $C$. Define these ratios:[^2]
$$\begin{array}{c}
\displaystyle a^{+} := \frac{|BA^{+}|}{|A^{+}C|} \qquad b^{+} := \frac{|CB^{+}|}{|B^{+}A|} \qquad c^{+} := \frac{|AC^{+}|}{|C^{+}B|} \\[10pt]
\displaystyle a^{-} := \frac{|CA^{-}|}{|A^{-}B|} \qquad b^{-} := \frac{|AB^{-}|}{|B^{-}C|} \qquad c^{-} := \frac{|BC^{-}|}{|C^{-}A|}
\end{array}$$
$\phantom{xyzzy}$
1. Lines ${\overleftrightarrow}{B^{+}C^{-}}$, ${\overleftrightarrow}{C^{+}A^{-}}$, ${\overleftrightarrow}{A^{+}B^{-}}$ pass through a common point if and only if $$\label{eqn_6ptconcurrence}
a^{+} b^{+} c^{+} \;+\; a^{-} b^{-} c^{-} \;=\; \phantom{-}1 \;-\; a^{+} a^{-} \;-\; b^{+} b^{-} \;-\; c^{+} c^{-}$$
2. (B[é]{}nyi-[Ć]{}urgus) Points[^3] ${\widehat}{B^{-}C^{+}}$, ${\widehat}{C^{-}A^{+}}$, ${\widehat}{A^{-}B^{+}}$ lie on a common line if and only if $$\label{eqn_6ptcollinearity}
a^{+} b^{+} c^{+} \;+\; a^{-}b^{-}c^{-} \;=\; -1 \;+\; a^{+} a^{-} \;+\; b^{+} b^{-} \;+\; c^{+} c^{-}$$
Note: Identifying $A^{-}$, $B^{-}$, $C^{-}$ with $C$, $B$, $A$ yields $a^{-} = b^{-} = c^{-} = 0$, so that (\[eqn\_6ptconcurrence\]) and (\[eqn\_6ptcollinearity\]) reduce to (\[eqn\_ceva\]) and (\[eqn\_menelaus\]). The Six-Point Theorem generalizes the traditional results.
For proof, one can invoke vector techniques, as indicated with Theorem \[thm\_sixpointrouth\] below.
Routh, too. {#routh-too. .unnumbered}
-----------
When Ceva’s lines fail to concur, and when Menelaus’ points fail to “colline”, they determine triangles. One might well ask how the area[^4] of each resulting triangle compares to that of the original figure. (Edward John) Routh provided answers. (See [@BC2012].)
[0.5]{} {width=".8\textwidth"}
[0.5]{} {width=".8\textwidth"}
1. (“Routh’s Theorem”). The triangle with (non-parallel) edge-lines ${\overleftrightarrow}{AD}$, ${\overleftrightarrow}{BE}$, ${\overleftrightarrow}{CF}$ has area $$\label{eqn_routh2}
|\triangle ABC| \cdot \frac{\left(\; d e f - 1 \;\right)^2}{
\left(\; 1 + d + d e \;\right)\left(\; 1 + e + e f \;\right) \left(\; 1 + f + f d \;\right)}$$
2. The triangle with (finite) vertices $D$, $E$, $F$ has area\
$$\begin{aligned}
\label{eqn_routh1}
|\triangle ABC| \cdot \frac{\; d e f + 1 \;}{
\left(\; 1 + d \;\right) \left(\; 1 + e \;\right) \left(\; 1 + f \;\right)}\end{aligned}$$
Observe that the numerator in each of these formulas —and, thus, the area of the triangle in question— vanishes, as it should, when (and only when) the conditions for Ceva’s or Menelaus’ theorems indicate that the triangle degenerates into a point or a line. The reader may verify that a denominator vanishes when (and only when) the triangle becomes unbounded, having non-finite vertices and parallel edges.
B[é]{}nyi and [Ć]{}urgus [@BC2012] specifically address the six-point generalization of Theorem 4b. At the suggestion of Mr. [Ć]{}urgus, this author derived the counterpart generalization of 4a.
[0.5]{} {width=".8\textwidth"}
[0.5]{} {width=".8\textwidth"}
$\phantom{xyzzy}$ \[thm\_sixpointrouth\]
1. The triangle with (non-parallel) edge-lines ${\overleftrightarrow}{B^{+}C^{-}}$, ${\overleftrightarrow}{C^{+}A^{-}}$, ${\overleftrightarrow}{A^{+}B^{-}}$ has area\
$$\begin{aligned}
\label{eqn_6ptrouth1}
|\triangle ABC| \cdot \frac{\left(\; a^{+} b^{+} c^{+} + a^{-} b^{-} c^{-} + a^{+} a^{-} + b^{+} b^{-} +
c^{+} c^{-} - 1 \;\right)^2}{
\begin{array}{c}
\phantom{\cdot}\left(\; 1 - a^{+} a^{-} + b^{-} ( 1 + a^{-} ) + c^{+} ( 1 + a^{+} ) \;\right) \\[3pt]
\cdot\left(\; 1 - b^{+} b^{-} + c^{-} ( 1 + b^{-} ) + a^{+} ( 1 + b^{+} ) \;\right) \\[3pt]
\cdot\left(\; 1 - c^{+} c^{-} + a^{-} ( 1 + c^{-} ) + b^{+} ( 1 + c^{+} ) \;\right)
\end{array}}\end{aligned}$$
2. (B[é]{}nyi-[Ć]{}urgus). The triangle with (finite) vertices ${\widehat}{B^{-}C^{+}}$, ${\widehat}{C^{-}A^{+}}$, ${\widehat}{A^{-}B^{+}}$ has area $$\label{eqn_6ptrouth2}
|\triangle ABC| \cdot \frac{\; a^{+} b^{+} c^{+} + a^{-} b^{-} c^{-} - a^{+} a^{-} - b^{+} b^{-} -
c^{+} c^{-} + 1 \;}{
\left(\; 1 + b^{-} + c^{+} \;\right)\left(\; 1 + c^{-} + a^{+} \;\right) \left(\; 1 + a^{-} + b^{+} \;\right)}$$
Treating points as vectors, we can write $$A^{+} = \frac{B (1+a^{+}) + Ca^{+}}{1+a^{+}} \qquad A^{-} = \frac{Ba^{-} + C(1+a^{-})}{1+a^{-}} \qquad \text{, etc.}$$ to find, after a bit of tedious algebra, that the vertices of the triangles in parts (a) and (b) of the theorem have the respective forms $$\frac{A ( 1 - a^{-} a^{+}) + B ( c^{+} + a^{-} b^{-} ) + C ( b^{-} + a^{+} c^{+} )}
{1 - a^{-} a^{+} + b^{-} ( 1 + a^{-} ) + c^{+} ( 1 + a^{+} )}
\qquad\text{and}\qquad
\frac{A + B c^{+} + Cb^{-}}{1 + c^{+} + b^{-}}$$ The area formulas follow from a bit more —and more-tedious— algebra. Of course, since ratios of lengths of collinear segments, and of areas of coplanar triangles, are preserved under affine transformation, one could simplify this analysis somewhat by assuming, say, $A = (0,0)$, $B=(1,0)$, $C=(0,1)$; even in generality, however, verification of these formulas amounts to just a few seconds’ effort from a computer algebra system.
Remarks {#remarks .unnumbered}
-------
We can accentuate the duality of the traditional results of Ceva and Menelaus by reciting them thusly: “[*Points determined by pairs of lines through the (vertex-)points of a triangle coincide if and only if ...*]{}” versus “[*Lines determined by pairs of points on the (edge-)lines of a triangle coincide if and only if ...*]{}”.
Taking a deep breath, we can do likewise for the parts of the Six-Point Theorem: “[*Points determined by pairs of lines determined by pairs of points on the (edge-)lines of a triangle coincide if and only if ...*]{}” versus “[*Lines determined by pairs of points determined by pairs of lines through the (vertex-)points of a triangle coincide if and only if ...*]{}”.
What formulas characterize the coincidence of lines and/or points at the next order of complexity? For that matter, what strategy for pairing lines and/or points best [*constitutes*]{} the next order of complexity?
[9]{}
B[' e]{}nyi, [Á]{}rp[á]{}d, and [Ć]{}urgus, Branko. “A generalization of Routh’s triangle theorem”. [*The American Mathematical Monthly*]{}, Vol. 120, No. 9 (Nov., 2013), pp. 841-846. (Preprint available at arXiv:[math.MG/1112.4813v2]{})
[^1]: Throughout, we consider segment lengths to be [*signed*]{}, with each of $\overrightarrow{AB}$, $\overrightarrow{BC}$, $\overrightarrow{CA}$ —for [*distinct*]{} $A$, $B$, $C$— indicating the direction of a positively-signed segment on the corresponding (extended) side of the triangle. Moreover, we adopt these conventions regarding ratios of these lengths: $$\frac{|PP|}{|PQ|} = 0 \qquad\qquad \frac{|PQ|}{|QQ|} = \infty \qquad\qquad \frac{|PX|}{|XQ|} = -1, \;\; \text{for $X$ the point at infinity on ${\overleftrightarrow}{PQ}$}$$
[^2]: Observe that the superscripts emphasize an opposing directionality in the definitions of the ratios. For instance, the points in ratio $a^{+}$ trace the path $B$-$A^{+}$-$C$, with endpoints oriented in the [*positive*]{} direction; in $a^{-}$, the path $C$-$A^{-}$-$B$ has endpoints oriented in the [*negative*]{} direction. Were we to define all six ratios in “matching” orientations —as was done in [@BC2012]— the resulting formulas would lose some clarity and symmetry.
[^3]: To amplify the duality with (a), we write “${\widehat}{P^{-}Q^{+}}$” for the point of intersection of lines ${\overleftrightarrow}{PP^{-}}$ and ${\overleftrightarrow}{QQ^{+}}$.
[^4]: As with length, we consider triangle area to be [*signed*]{}. Areas $|\triangle ABC|$ and $|\triangle DEF|$ agree in sign when vertex paths $A$-$B$-$C$-$A$ and $D$-$E$-$F$-$D$ trace their respective figures in the same direction; similarly for triangles defined by their edge-lines.
|
---
abstract: 'We discuss the form of the wave-function of a state subjected to a scalar linear potential, paying special attention to quantum tunneling. We analyze the phases acquired by the evolved state and show that some of them have a pure quantum mechanical origin. In order to measure one of these phases, we propose a simple experimental scenario. We finally apply the evolution equations to re-analyze the Stern&Gerlach experiment and to show how to manipulate spin by employing constant electric fields.'
address:
- '$^1$ Departamento de Física, Universidade Federal de Minas Gerais, 30123-970 Belo Horizonte, Brazil'
- '$^2$ Department of Physics, University of Oulu, Fin-90014 Oulu, Finland'
- '$^3$ Institut Néel-CNRS, BP 166, 25 rue des Martyrs, 38042 Grenoble Cedex 9, France '
author:
- 'F Fratini$^{1,2,3}$ and L Safari$^{2}$'
title: 'Quantum mechanical evolution operator in the presence of a scalar linear potential: discussion on the evolved state'
---
Introduction
============
Scalar linear potentials are widely used in physics, as they can be generated by homogeneous irrotational fields, like electrostatic or gravitational fields, which are very typical in physical problems. Moreover, linear potentials are of general use since they can approximate more sophisticated potentials for sufficiently small distances. Scalar linear potentials are also considered for quantum tunneling (e.g. the Sauter potential [@Cal1999]). Being able to rigorously evolve a quantum mechanical state subjected to a linear potential is therefore of fundamental and pedagogical importance.
The exact evolution of quantum systems subjected to linear and quadratic potentials has recently attracted some interest. The quantum propagator in the presence of a linear potential has been studied in several works [@Hol1997; @Arr1996; @Bro1994; @Rob1996]. The evolution operator related to the most generic time-dependent quadratic potential (with linear terms included) has been also analyzed in the literature by using quantum invariants [@Liu2004; @Har2011]. Here, we re-derive the evolution operator in the presence of a scalar time-independent linear potential by using the Zassenhaus formula [@Que2004]. Our simple approach allows for several physical considerations that are laid out in the article.
In Sec. \[sec:Ev\], we show that the wave-function of a state evolved in the presence of a linear potential is given, up to a phase, by the free evolved wave-function (i.e., evolved with no potential) whose argument is shifted by a certain quantity which depends on the potential. In Sec. \[sec:EvGauss\], we consider a gaussian wave-packet subjected to a linear potential. We pay some attention to the problem of quantum tunneling, or quantum diffusion, which is the phenomenon where a microscopic object (typically a particle or an atom) can penetrate a potential barrier whose height is larger than the object’s kinetic energy [@Mohsen; @Lau2000]. Since such phenomenon is forbidden by classical laws of mechanics, it is often referred to as a peculiar characteristic of quantum mechanics. We then move to discuss, in Sec. \[sec:PSG\], the form of the evolved state and the phases that it acquires. Some of these phases are shown to stem from the non-commutativity of momentum and position operators in quantum mechanics. A simple experimental scenario aimed at measuring one of those phases is proposed. In Sec. \[sec:SG\], as a pedagogic application of our evolution equations, we rigorously re-analyze the example of the Stern&Gerlach (SG) experiment. In Sec. \[sec:SpinE\], we show how to manipulate spin of charged particles by using constant electric fields, instead of the more commonly used magnetic fields. Finally, a summary is given in Sec. \[sec:SumC\].
Evolution Operator and evolved wave-functions {#sec:Ev}
=============================================
We consider a potential of the form $V_0 x$, where $V_0$ is an arbitrary constant or any operator which commutes with momentum and position operators. Without restriction of generality and for simplicity, we consider only one dimension, which is the $x$ direction. The hamiltonian may be thus written as $$\label{eq:Htot}
{\hat{\textrm H}}=\frac{{\hat{\textrm p}}^2}{2m}+V_0{\hat{\textrm x}} ~,$$ where ${\hat{\textrm p}}$ is the linear momentum operator along the $x$ direction and $m$ is the particle mass. Since the hamiltonian is time-independent, we may straightforwardly write down the correspondent evolution operator from an initial time $t_i$ to $t$ [@Sak1994]: $$\label{eq:UtotPrim}
{\hat{\textrm U}}(t,t_i)=e^{-\frac{i}{\hbar}\left(\frac{{\hat{\textrm p}}^2}{2m}+V_0{\hat{\textrm x}}\right)(t-t_i)} ~,$$ where $\hbar$ denotes the reduced Planck constant. However, since the argument of the exponential in the equation above is made of non-commutative operators, its application to ket states is non-trivial. In order to rewrite in a more manageable way, we make use of the Zassenhaus formula [@Que2004]: $$\label{eq:Zass}
e^{{\hat{\textrm A}}+{\hat{\textrm B}}}=e^{{\hat{\textrm A}}}\,e^{{\hat{\textrm B}}}\,\prod_{i=2}^{\infty}e^{{\hat{\textrm C}}_i}~,$$ where $$\begin{array}{lcl}
{\hat{\textrm C}}_2&=&{\displaystyle}\frac{1}{2}\left[{\hat{\textrm B}},{\hat{\textrm A}}\right]~,\\[0.4cm]
{\hat{\textrm C}}_3&=&{\displaystyle}\frac{1}{3}{\left[{\left[{\hat{\textrm B}},{\hat{\textrm A}}\right]},{\hat{\textrm B}}\right]}\,+\,\frac{1}{6}{\left[{\left[{\hat{\textrm B}},{\hat{\textrm A}}\right]},{\hat{\textrm A}}\right]}~,\\[0.4cm]
{\hat{\textrm C}}_4&=&{\displaystyle}\frac{1}{8}\left(
{\left[{\left[{\left[{\hat{\textrm B}},{\hat{\textrm A}}\right]},{\hat{\textrm B}}\right]},{\hat{\textrm B}}\right]}
\,+\,
{\left[{\left[{\left[B,A\right]},A\right]},B\right]}
\right)\,+\,
\frac{1}{24}{\left[{\left[{\left[{\hat{\textrm B}},{\hat{\textrm A}}\right]},{\hat{\textrm A}}\right]},{\hat{\textrm A}}\right]}
~,\\
...&&...~
\end{array}$$ By choosing ${\hat{\textrm A}}\equiv-\frac{i}{\hbar}V_0{\hat{\textrm x}}\,(t-t_i)$, ${\hat{\textrm B}}\equiv-\frac{i}{\hbar}\frac{{\hat{\textrm p}}^2}{2m}\,(t-t_i)$ and by using ${\left[{\hat{\textrm p}}^2,x\right]}={\hat{\textrm p}}{\left[{\hat{\textrm p}},x\right]}+{\left[{\hat{\textrm p}},x\right]}{\hat{\textrm p}}=-2i\hbar{\hat{\textrm p}}$, we readily get ${\hat{\textrm C}}_2=+\frac{i}{\hbar}\frac{V_0{\hat{\textrm p}}}{2m}(t-t_i)^2$, ${\hat{\textrm C}}_3=-\frac{i}{\hbar}\frac{V_0^2}{6m}(t-t_i)^3$, ${\hat{\textrm C}}_{i\ge4}=0$. Finally, by using the fact that $e^{{\hat{\textrm B}}}$ commutes with $e^{{\hat{\textrm C}}_2}$ and that $e^{{\hat{\textrm C}}_3}$ commutes with anything, we may write the (exact) evolution operator as $$\label{eq:Utot}
\begin{array}{l}
{\hat{\textrm U}}(t,t_i)=
e^{-\frac{i}{\hbar}\frac{V_0^2}{6m}(t-t_i)^3}
e^{-\frac{i}{\hbar}V_0{\hat{\textrm x}}(t-t_i)}\, e^{+\frac{i}{\hbar}\frac{V_0{\hat{\textrm p}}}{2m}(t-t_i)^2}
{\hat{\textrm U}}_0(t,t_i)~,
\end{array}$$ where $$\label{eq:Ufree}
{\hat{\textrm U}}_0(t,t_i)=e^{-\frac{i}{\hbar}\frac{{\hat{\textrm p}}^2}{2m}(t-t_i)}$$ is the evolution operator in the free case, i.e. if there were no potential. Alternatively, by choosing ${\hat{\textrm B}}\equiv-\frac{i}{\hbar}V_0{\hat{\textrm x}}\,(t-t_i)$, ${\hat{\textrm A}}\equiv-\frac{i}{\hbar}\frac{{\hat{\textrm p}}^2}{2m}\,(t-t_i)$, one may analogously derive $$\label{eq:Utot2}
\begin{array}{l}
{\hat{\textrm U}}(t,t_i)={\hat{\textrm U}}_0(t,t_i)e^{+\frac{i}{\hbar}\frac{V_0^2}{3m}(t-t_i)^3}\,
e^{-\frac{i}{\hbar}V_0{\hat{\textrm x}}(t-t_i)}e^{-\frac{i}{\hbar}\frac{V_0{\hat{\textrm p}}}{2m}(t-t_i)^2}~.
\end{array}$$ Equation coincides with Eq. (2.6) in Ref. [@Arr1996].
We notice that, for times much shorter than the characteristic time of interaction between potential and particle, we could neglect the terms $\sim(t-t_i)^2$ and $\sim (t-t_i)^3$ in the exponentials of Eqs. and , in favor of the linear terms $\sim(t-t_i)$. The form thus obtained for the evolution operator ${\hat{\textrm U}}(t,t_i)$ would be equal to the one obtainable from Eq. by considering as if the kinetic energy operator ($\frac{{\hat{\textrm p}}^2}{2m}$) and the potential energy operator ($V_0{\hat{\textrm x}}$) commuted. This means that, for short interaction times, the kinetic and the potential energy operators may be considered to approximately commute. This result is not unexpected, as it is indeed the basic step for the path integral formulation of quantum mechanics [@Alt2001].
Next, we apply to an arbitrary initial state ${\mathinner{|{\alpha(t_i)}\rangle}}$ defined at time $t_i$, so as to obtain the ket state at time $t$. By then multiplying by ${\mathinner{\langle{x}|}}$ from the left, we obtain the wavefunction of the state at time $t$ in the spatial representation: $$\begin{array}{lcl}
{\displaystyle}\Psi(x,t)&=& {\langle x|\alpha(t)\rangle}={\displaystyle}{\mathinner{\langle{x}|}}{\hat{\textrm U}}(t,t_i){\mathinner{|{\alpha(t_i)}\rangle}}\\[0.4cm]
&=&{\displaystyle}e^{-\frac{i}{\hbar}\frac{V_0^2}{6m}(t-t_i)^3}e^{-\frac{i}{\hbar}V_0 x(t-t_i)}\,
{\mathinner{\langle{x}|}}e^{+\frac{i}{\hbar}\frac{V_0{\hat{\textrm p}}}{2m}(t-t_i)^2}{\mathinner{|{\alpha(t)}\rangle}}_0
\end{array}$$ where we defined ${\mathinner{|{\alpha(t)}\rangle}}_0={\hat{\textrm U}}_0(t,t_i){\mathinner{|{\alpha(t_i)}\rangle}}$, which is just the state evolved by the free evolution operator . By using $\exp{\Big(-i{\hat{\textrm p}} \Delta x/\hbar\Big)}{\mathinner{|{x}\rangle}}={\mathinner{|{x+\Delta x}\rangle}}$, which follows from the definition of momentum operator as generator of spatial translations [@Sak1994], we finally obtain $$\label{eq:WFr}
\begin{array}{l}
\Psi(x,t)={\displaystyle}e^{-\frac{i}{\hbar}\frac{V_0^2}{6m}(t-t_i)^3}
e^{-\frac{i}{\hbar}V_0 x(t-t_i)} \,{\displaystyle}\Psi_0\left(x+\frac{V_0}{2m}(t-t_i)^2,t\right)~.
\end{array}$$ It is important to notice that, in the equation above, the free evolved wave-function $\Psi_0$ can be of any form. In the special case $\Psi_0$ is a plane-wave, then $\Psi$ will be a solution of the time-dependent Schrödinger equation related to the Hamiltonian [@Werner]. The modulus squared of the wave-function (i.e., the probability density) simply satisfies $$\label{eq:SWFr}
\begin{array}{l}
{\displaystyle}\left|\Psi(x,t)\right|^2= \left|\Psi_0\left(x+\frac{V_0}{2m}(t-t_i)^2,t\right)\right|^2~.
\end{array}$$ We may obtain analogous relations in the momentum representation: $$\begin{aligned}
\nonumber
{\displaystyle}\tilde\Psi(p,t)&=& {\langle p|\alpha(t)\rangle}={\displaystyle}{\mathinner{\langle{p}|}}{\hat{\textrm U}}(t,t_i){\mathinner{|{\alpha(t_i)}\rangle}}\\[0.4cm]
\label{eq:WFp}
&=&{\displaystyle}e^{+\frac{i}{\hbar}\frac{V_0^2}{3m}(t-t_i)^3}e^{+\frac{i}{\hbar}\frac{V_0p}{2m}(t-t_i)^2}
\,{\displaystyle}\tilde\Psi_0\Big(p+V_0(t-t_i),t\Big)~,\\[0.4cm]
{\displaystyle}\left|\tilde\Psi(p,t)\right|^2&=&{\displaystyle}\left|\tilde\Psi_0\Big(p+V_0(t-t_i),t\Big)\right|^2~,\end{aligned}$$ where the relation $\exp{\Big(i{\hat{\textrm x}} \Delta p/\hbar\Big)}{\mathinner{|{p}\rangle}}={\mathinner{|{p+\Delta p}\rangle}}$ has been used.
Equations , relate the wave-functions (in the spatial and linear momentum representations) of a general state evolved in the presence of a linear potential with the wave-functions of the same state evolved without the linear potential. The evolved wave-function is given, up to a phase, by the free evolved wave-function with its argument evolved following the classical equation of motion in the presence of the opposite potential. For example, in the case of spatial representation, the argument is evolved following $x\to x+\frac{V_0}{2m}(t-t_i)^2$, while the classical evolution of the position given by the potential $V_0$ would be $x\to x-\frac{V_0}{2m}(t-t_i)^2$. Although this might seem counterintuitive at a first sight, in the next section we shall see that it is not.
Evolution of a Gaussian wave-packet {#sec:EvGauss}
===================================
Let us consider at initial time $t_i$ a Gaussian wave-packet with momentum mean value $p_0$, spatial mean value $x_0$ and standard deviation (or width) $\sigma$, which represents a realistic state in standard experiments: $$\label{eq:Gauss}
\Psi^{G}(x,t_i)=\frac{1}{\pi^{1/4}\sigma^{1/2}}e^{\frac{i}{\hbar}p_0(x-x_0)-\frac{(x-x_0)^2}{2\sigma^2}}\,\equiv\,
{\langle x|G\rangle}~.$$ We shall denote with $\Psi^{G}_0(x,t)$ the free evolved Gaussian wave-packet at time $t$, i.e. $\Psi^{G}_0(x,t)={\mathinner{\langle{x}|}}{\hat{\textrm U}}_0(t,t_i){\mathinner{|{G}\rangle}}$. Using Eq. with the Gaussian state (i.e., replacing $\Psi_0(x,t)\to\Psi^{G}_0(x,t)$), the modulus squared of the wave-packet at time $t$ is of the form: $$\label{eq:Gstate}
\begin{array}{lcl}
\Big|\Psi^G(x, t)\Big|^2&=&
\Big| \Psi_0^G(x+\frac{V_0}{2m}(t-t_i)^2, t) \Big|^2\propto
e^{-\frac{\left(x+V_0\Delta t^2/2m-(x_0+p_0\Delta t/m)\right)^2}{\left|\sigma(\Delta t)\right|^2}}\\[0.4cm]
&\propto& e^{-\frac{\left(x-(x_0+p_0\Delta t/m-V_0\Delta t^2/2m)\right)^2}{\left|\sigma(\Delta t)\right|^2}}
\end{array}$$ where $\Delta t=t-t_i$, and $\sigma(\Delta t)$ denotes the spatial width of the Gaussian wave-packet free evolved for a time $\Delta t$. The term $p_0\Delta t/m$ comes from the free evolution of the Gaussian wave-packet, as it could be expected. The detailed expression of $\sigma(\Delta t)$ and $\Psi^{G}_0(x,t)$ can be found in standard textbooks [@Merz]. Evidently, equation describes the probability density of a Gaussian wave-packet whose spatial mean value obeys non-relativistic classical kinematics. In other words, The spatial mean value has been subjected to a constant acceleration equal to $-\frac{V_0}{m}$ for a time interval $\Delta t$. This result is also in line with Ehrenfest’s theorem for the mean values [@Sak1994]. In conclusion, the fact that the wave-packet argument follows the classical equation of motion in the presence of the opposite potential ensures that the wave-packet mean value follows the classical equation of motion with the correct potential. The same analysis can be conducted in the momentum representation, with the same classical results. We thus conclude that the evolution equations and are physically plausible, when analyzed from the point of view of classical kinematics. We furthermore notice that the presence of the linear potential does not affect the spatial width $\sigma$, which is rather fully determined by the free evolution in time.
![A gaussian wave-packet (blue curve) hits a potential barrier (black curve) whose initial part (D) can be approximated to linear. The potential barrier is in units of the initial average value for the kinetic energy of the wave-packet ($<E_{kin}^{in}>=p_0^2/(2m)$). The red solid vertical bars denote the classical turning points (a, b). The length D$'$ is chosen to be much larger than half wave-packet spatial width as given at the time $t_a$ (D$'$$\,\gg\frac{\sigma(t_a-t_i)}{2}$). Here $t_a=p_0/V_0$ is the time when the wave-packet mean value is at the turning point $a$. For these settings, as a consequence of the evolution given by Eq. , the wave-packet will be wholly pushed backwards and no tunneling will be permitted. []{data-label="fig:fig1"}](fig1.eps)
Equation may be also read in the following way: When the whole wave-packet is subjected to a linear potential, the (modulus squared of the) wave-packet follows the classical kinematics evolution. Now, let us consider a gaussian wave-packet that hits a potential barrier whose initial part can be approximated to linear. This initial (approximately) linear part of the potential shall be called D. The maximum value of the potential in D is supposed to be higher than the initial average kinetic energy of the wave-packet, where this latter is $<E_{kin}^{in}>=p_0^2/(2m)$. For convenience, this situation is depicted in Fig. \[fig:fig1\] and also showed in the animation available on line as supplementary material (in the animation, also the evolution of the wave-packet in the free case is displayed for comparison). Let us further suppose that the difference between the furthermost point of D and the first classical turning point (a) is much bigger than half wave-packet spatial width as given at the time $t_a$ (D$'$$\,\gg\frac{\sigma(t_a-t_i)}{2}$). Here $t_a=p_0/V_0$ is the time when the wave-packet mean value is at the turning point $a$. In other words, we suppose that the DeBroglie wave-length of the wave-packet is small compared to the characteristic distance over which the first derivative of the potential varies appreciably [^1]. In these chosen settings, the whole wave-packet will be subjected to the same linear potential up to the turning point a. The evolution given by Eq. then dictates that the wave-packet will be wholly pushed backwards, as one would expect from the classical point of view. Consequently, [*no tunneling will be permitted to the wave-packet*]{}. Therefore, in order to have any chance for quantum tunneling, the wave-packet spatial width must be somewhat larger or comparable to D$'$ ($\sigma\gtrsim$ D$'$), so that the evolution given by may not be applicable.
In the animation available on line as supplementary material, we chose the mean position, the standard deviation and the velocity to be $0$ m, $0.2$ cm and $1$ m/sec, respectively, at time $t=0$. The strength of the potential is chosen to allow the wave-packet to travel for a length of $0.3$ meters before reaching the classical turning point. At the classical turning point, the spatial width will have grown of only 10% (see animation). For the best comparison, we adopted the same length unit of the animation for the abscissa in Fig. \[fig:fig1\]. Other settings for smaller length scale could be analogously applied.
Based on the above considerations, we argue that quantum tunneling reflection and transmission coefficients should directly depend on the spatial width of the wave-packet. However, although studies on tunneling with Gaussian wave-packets have been made in literature (see Ref. [@Stamp1996] and references therein), to the best of our knowledge no direct relation between tunneling coefficients and spatial width of the wave-packet has been suggested. In fact, transmission coefficients in quantum tunneling are not normally given as dependent on the wave-packet spatial width but rather as solely dependent on the energy of the particle (E) and the thickness of the barrier. For instance, within WKB approximation, which is the most widely used approach for solving tunneling problems, the transmission coefficient is given by ${\displaystyle}T(E)=e^{-2\sigma_R}/\left(1+e^{-2\sigma_R}/4\right)^2$, where $\sigma_R=\int_a^b\sqrt{\frac{2m}{\hbar^2}\left(V(x)-E\right)}\,dx \,>\,0$, and $a$, $b$ are the classical turning points (i.e., $L=b-a$ is the classically forbidden region) [@Mohsen]. An experimental assessment of the dependence of tunneling coefficients on the spatial width would thus be desirable.
The direct dependence of tunneling coefficients on the spatial width of the wave-packet could have application in many areas of science: Quantum tunneling could be enhanced or suppressed by controlling the spatial width of the state, instead of controlling the energy of the state or the environment surrounding it [@Gri1998; @Cal1981].
Unfortunately, we cannot find here an explicit expression for transmission and reflection coefficients for a realistic potential barrier with the present quantum mechanical formalism. This is because the Zassenhaus formula does not converge for potentials of order higher than linear, and because a realistic potential barrier cannot be represented by a linear function. Nonetheless, any potential barrier can be approximated to linear for short distances. Based on this, our claim that the tunneling coefficients should depend on the spatial width of the wave-packet holds. In view of the fact that the wave-packet spatial mean value ($x_0$) follows the classical equation of motion, our conjecture is that the fraction of the wave-packet beyond the potential barrier at the classical turning point plays leading role in determining transmission coefficient in quantum tunneling: Given wave-packets with the same linear momentum mean value, those wave-packets with larger spatial width will tunnel more efficiently. This can be simply checked by preparing Gaussian wave-packets and delaying the arrival of some of them to the potential barrier. The spatial width $\sigma(\Delta t)$ of the wave-packets increases during the free evolution. Thus the delayed wave-packets will have larger spatial width with respect to the non-delayed wave-packets.
Discussion on the phases: Phase Shift Generator {#sec:PSG}
===============================================
We here discuss more extensively the expression for the evolution operator in Eq. .
The first $\left(-\frac{i}{\hbar}\frac{V_0^2}{6m}(t-t_i)^3\right)$ and third $\left(+\frac{i}{\hbar}\frac{V_0{\hat{\textrm p}}}{2m}(t-t_i)^2\right)$ phases which multiply the free evolution operator from the left in stem directly from the non-commutativity between momentum and position operators, and are thus purely quantum mechanical corrections. On the other hand, the second phase $\left(-\frac{i}{\hbar}V_0{\hat{\textrm x}}(t-t_i)\right)$ and the free evolution operator itself (${\hat{\textrm U}}_0(t,t_i)$) somehow represent the evolution given by the potential and kinetic energy gained by the traveling particle, respectively. If position and momentum operators commuted, then only these two latter terms would be present. It is somewhat interesting that the phase $\frac{V_0{\hat{\textrm p}}}{2m\hbar}(t-t_i)^2$ is directly responsible in equation for the shift in the argument of the free evolved wave-function in the spatial representation, such shift being $V_0/(2m)\,(t-t_i)^2$. In fact, that shift, together with the shift $p_0\Delta t/m$ given by the free evolution, gives rise to the classical motion of the spatial mean value of the state (see previous section). Therefore, the non-commutativity of momentum and position operators turns out to be effectively responsible for the non-relativistic classical motion of the spatial mean value of the wave-packet: $x_0\to x_0+p_0\Delta t/m-V_0\Delta t^2/2m$. We may furthermore notice that the corresponding shift in the momentum representation, which is responsible for the classical motion of the linear momentum mean value, is directly given by the phase containing the linear potential, $\left(-\frac{i}{\hbar}V_0{\hat{\textrm x}}(t-t_i)\right)$. We may therefore consider the term $\propto\frac{V_0{\hat{\textrm p}}}{2m}$ as a linear potential in the momentum space. In other words, the term $\propto\frac{V_0{\hat{\textrm p}}}{2m}$ may be considered the dual of the potential $V_0{\hat{\textrm x}}(t-t_i)$. The former is generated by the presence of the latter because the latter does not commute with the free Hamiltonian. The presence of both potentials gives symmetry to the evolution of the state in spatial and momentum representations and ensures that in both representations the mean value is evolved following the non-relativistic classical motion.
On the other hand, the phase $-\frac{1}{\hbar}\frac{V_0^2}{6m}(t-t_i)^3$ does not play any role on determining the classical evolution of the state. Such a phase originates from the second (and last) expansion term of the Zassenhaus formula, and it is therefore a higher correction with respect to other terms. Indeed, this phase would be the only one missing if we replaced, in the free plane wave $e^{\frac{i}{\hbar}\left(px-\frac{p^2}{2m}(t-t_i)\right)}$, the classical transformations $x\to x-\frac{V_0}{2m}(t-t_i)^2$, $p\to p-\frac{V_0}{m}(t-t_i)$, as one would do as a first attempt to guess the wave-function of a state subjected to a linear potential. An experiment aimed at ascertaining the existence of this last phase would thus probably be a useful test for quantum mechanics. To this aim, here we sketch a simple experimental scenario which permits such a measurement.
![Sketch of a simple experimental scenario which allows to generate a phase shift using the evolution operator in Eqs. , . []{data-label="fig:fig2"}](fig2.eps)
Let us consider two electron beams along the $z$ direction, where the electrons are in phase one with another [@Ton2985]. One of the two beams is accelerated and subsequently decelerated along an axis orthogonal to the beam direction, for instance $x$. In order to apply Eqs. -, the acceleration and deceleration must be due to a linear potential. As showed in Fig. \[fig:fig2\], this could be realized, for instance, by a series of three capacitors of lengths $L=v\Delta t$, $2L$ and $L$, where $\Delta t$ is the time the electron spends in the first capacitor and $v$ is the beam velocity along $z$ ($L$ must here be much larger than the spatial width of the electron state). For this example, we must consider the three dimensional generalization of Eq. , where ${\hat{\textrm p}}^2$ is replaced by ${\hat{\mathbf{p}}}^2={\hat{\textrm p}}_x^2+{\hat{\textrm p}}_y^2+{\hat{\textrm p}}_z^2$. Since position and momentum operators along different directions commute, such a replacement can be safely made. By applying such generalized evolution operator to the initial electron state ${\mathinner{|{{\bm}p_i}\rangle}}\simeq{\mathinner{|{p_x=0, p_y=0, p_z=p_0}\rangle}}$, the electron state after the electrostatic deflection (which lasts for a time $4\Delta t$) can be easily calculated to be $e^{-\frac{i}{\hbar}\frac{2V_0^2 L^3}{3m\,v^3}}{\hat{\textrm U}}_0(4\Delta t){\mathinner{|{{\bm}p_i}\rangle}}\equiv e^{-\frac{i}{\hbar}\frac{2V_0^2 L^3}{3m\,v^3}} {\mathinner{|{{\bm}p_i}\rangle}}$. We have here redefined ${\hat{\textrm U}}_0(4\Delta t){\mathinner{|{{\bm}p_i}\rangle}}\equiv {\mathinner{|{{\bm}p_i}\rangle}}$ since the phase given by the free evolution operator is shared by both beams and therefore not measurable. Thus, upon passing the capacitors, the beam acquires a phase-shift equal to $-\frac{1}{\hbar}\frac{2V_0^2L^3}{3m v^3}$ with respect to the other (non-deflected) beam. Such phase shift can be measured when the beams are recombined. In what follows, we shall denote the experimental apparatus sketched in Fig. \[fig:fig2\] as Phase Shift Generator (PSG).
From the above considerations we see that, when different beams are subjected to different accelerations, a phase difference proportional to the potential-difference squared may appear, if the potentials responsible for the accelerations can be approximated to linear. Therefore, Eqs. (\[eq:Utot\]), (\[eq:Utot2\]) and Fig. \[fig:fig2\] might be also useful for estimating the loss of coherence in dealing with charged particles.
Phase shifts of quantum states have been very useful in physics and are object of current research and debate (e.g., the gravitational phase shift [@COW; @Lit] and the Gouy phase [@G1; @G2]). Along the same lines, ascertaining the existence of the phase shift generated by the PSG would be interesting for testing quantum mechanics and might also have several applications. We shall see in Sec. \[sec:SpinE\] how such phase could be for example used in quantum information and spintronics.
Application to Stern&Gerlach experiment {#sec:SG}
=======================================
The SG experiment [@SG1922] is rightfully considered of fundamental and pedagogical importance for understanding quantum mechanics. The particles injected in the SG apparatus are subjected to a linear potential. The SG apparatus is therefore probably the best example for a clear application of equations -. The following brief analysis is also motivated by the fact that in textbooks the SG apparatus is normally explained with intuitive, semi-classical arguments (e.g., see in Ref. [@Sak1994]), while in literature it is more rigorously explained with an involved quantum mechanical formalism [@Scu1987].
In the SG experiment, Silver atoms are injected in the apparatus [@SG1922]. Out of the 47 electrons of the Silver atom, only the outermost electron contributes to the atomic spin, if we neglect the nuclear contribution (which is irrelevant to our discussion). Therefore it is common to consider the spin state of such electron as characterizing the spin state of the whole atomic system. Since no atomic excitations are to be considered, we may disregard any atomic internal structure. The potential for a SG whose magnetic field is along $x$ is ${\hat{\textrm V}}_{SG}\simeq-\left(\frac{{\mathrm{e}}\hbar}{2m_e}B_0\right){\hat{\textrm x}}\,\hat \sigma_x$, where ${\mathrm{e}}$ and $m_e$ are the electric charge and mass respectively, $\hat \sigma_x$ is the Pauli spin operator along the $x$ direction, while $B_0$ is the strength of the magnetic field. The state of the atoms before entering the SG is completely mixed. Thus, it may not be described by a ket state, but rather it can be described by the following density operator [@Sak1994; @Fratini2011]: $$\begin{array}{lcl}
\hat \rho(t_i)&=&{\displaystyle}{\mathinner{|{{\bm}p}\rangle}}{\mathinner{\langle{{\bm}p}|}}\,\otimes\,
\frac{1}{2}\Big({\mathinner{|{S_x,+}\rangle}}{\mathinner{\langle{S_x,+}|}} + {\mathinner{|{S_x,-}\rangle}}{\mathinner{\langle{S_x,-}|}}\Big)~,
\end{array}$$ where ${\bm}p=(0,0,p_0)$ and ${\mathinner{|{S_x, \pm}\rangle}}$ are spin-1/2 states along the $x$ direction. Setting $V_0\to {\hat{\textrm V}}_0 \equiv -\left(\frac{{\mathrm{e}}\hbar}{2m_e}B_0\right)\,\hat \sigma_x$ in Eq. (which is allowed, since $\hat \sigma_x$ commutes with any of the operators ${\hat{\textrm p}}_x$, ${\hat{\textrm p}}_y$, ${\hat{\textrm p}}_z$, ${\hat{\textrm x}}$, ${\hat{\textrm y}}$, ${\hat{\textrm z}}$), the evolution of the density operator can be easily computed [@Balashov]: $$\begin{array}{lcl}
\hat \rho(t)&=&{\hat{\textrm U}}(t, t_i)\hat \rho(t_i){\hat{\textrm U}}^\dag(t, t_i)\\[0.4cm]
&=&{\displaystyle}\frac{1}{2}\Big(
{\mathinner{|{{\bm}p_+}\rangle}}{\mathinner{\langle{{\bm}p_+}|}}\,\otimes\,{\mathinner{|{S_x,+}\rangle}}{\mathinner{\langle{S_x,+}|}} \;+\;{\mathinner{|{{\bm}p_-}\rangle}}{\mathinner{\langle{{\bm}p_-}|}}\,\otimes\,{\mathinner{|{S_x,-}\rangle}}{\mathinner{\langle{S_x,-}|}}\Big)~,
\end{array}$$ where ${\bm}p_+=\big(+\Delta p, 0, p_0\big)$ and ${\bm}p_-=\big(-\Delta p, 0, p_0\big)$, while $\Delta p=\frac{{\mathrm{e}}\hbar}{2m_e}B_0(t-t_i)$. The probability density of measuring the generic state ${\mathinner{|{{\bm}p'}\rangle}}{\mathinner{\langle{{\bm}p'}|}}$ after an interaction time $(t-t_i)$ is therefore $$\Tr\Big[{\mathinner{|{{\bm}p'}\rangle}}{\mathinner{\langle{{\bm}p'}|}}\hat \rho(t)\Big]=\frac{1}{2}\,\delta({\bm}p'-{\bm}p_\pm)~,$$ which is the well-known SG outcome.
Manipulating spin by employing electric fields {#sec:SpinE}
==============================================
Manipulating spin by means of electric fields is currently subject of applied research [@Hon2013; @Tsc2006; @Jan2010; @Rov2010; @Han2008]. The advantage of controlling spin by using electric rather than magnetic fields is that the former are easier to generate. Moreover, they allow for controlling spins independently one from another, which is a requirement for building quantum computers [@Duc2006; @Now2007]. Our goal here is to show how to manipulate spin of charged particles by employing constant electric fields. This will be achieved by combining a spin-beam splitter with a PSG apparatus. For our purposes and for simplicity, we will consider electrons and we will use the SG apparatus as spin-beam splitter. A brief discussion on the feasibility and on possible extensions is laid out at the end of the present section.
![a) A spin flipper built by employing SG and PSG apparatuses. b) By removing the PSG, the spin is not flipped. []{data-label="fig:fig3"}](fig3.eps)
We prepare an electron with, for example, momentum and spin along $z$. Its state is therefore described by ${\mathinner{|{{\bm}p}\rangle}}\otimes{\mathinner{|{S_z,\pm}\rangle}}$, with ${\bm}p=(0, 0, p_0)$. We let such electron sequentially pass through a SG, then through a PSG, and finally through a second SG. While the first SG is set along $x$ direction, the second SG is set along $-x$ direction. This gedanken experiment is sketched in Fig. \[fig:fig3\] (panel a). During the several steps, by using Eq. we find that the electron state is given by (up to an overall phase): $$\begin{array}{l}
{\mathinner{|{{\bm}p}\rangle}}\otimes{\mathinner{|{S_z,\pm}\rangle}}={\displaystyle}\frac{1}{\sqrt{2}}{\mathinner{|{{\bm}p}\rangle}}\otimes\Big(
{\mathinner{|{S_x,+}\rangle}}
\pm
{\mathinner{|{S_x,-}\rangle}}
\Big)\\[0.4cm]
\;\xrightarrow{SG\uparrow}{\displaystyle}\frac{1}{\sqrt{2}}\Big(
{\mathinner{|{{\bm}p_+}\rangle}}\otimes{\mathinner{|{S_x,+}\rangle}}
\pm
{\mathinner{|{{\bm}p_-}\rangle}}\otimes{\mathinner{|{S_x,-}\rangle}}
\Big)\\[0.4cm]
\;\xrightarrow{PSG}{\displaystyle}\frac{1}{\sqrt{2}}\Big(
{\mathinner{|{{\bm}p_+}\rangle}}\otimes{\mathinner{|{S_x,+}\rangle}}
\pm
e^{-\frac{i}{\hbar}\frac{2V_0^2L^3}{3m v^3}}{\mathinner{|{{\bm}p_-}\rangle}}\otimes{\mathinner{|{S_x,-}\rangle}}
\Big)\\[0.4cm]
\;\xrightarrow{SG\downarrow}{\displaystyle}\frac{1}{\sqrt{2}}{\mathinner{|{{\bm}p}\rangle}}\otimes\Big(
{\mathinner{|{S_x,+}\rangle}}
\pm
e^{-\frac{i}{\hbar}\frac{2V_0^2L^3}{3m v^3}}{\mathinner{|{S_x,-}\rangle}}
\Big)~,
\end{array}$$ where ${\bm}p_{\pm}$ has been defined previously and contains the linear momentum shift given by the Lorentz force exerted on the electron in the intermediate steps. By setting appropriate values for the PSG parameters, the wished final electron spin state is obtained. In particular, by setting $\frac{2V_0^2L^3}{3\hbar m v^3}=\pi$, the initial spin state is reversed. By removing the PSG, the spin is not flipped (Fig. \[fig:fig3\], panel b). This entails that it is the electric field in PSG that is responsible for the spin flip, while the magnetic field in the SG apparatuses is just used to feed the PSG.
The employment of a SG apparatus to split electron beams of different spins has been widely discussed in the past. Since the beginning ot the last century, it has been several times argued that the SG apparatus cannot successfully split electron beams. Conversely, more recently this viewpoint has been confuted and an effective spin-beam splitter for electrons using SG has been proposed (see Ref. [@Bate1997] and references therein). Here, we used the SG apparatus for simplicity, but any spin-beam splitter would work and would produce the shown results. Spin-beam splitters of nano- and meso-scopic dimensions have been in fact realized for electrons (e.g., [@Ch2008; @Pe2006; @X2006; @Fre2003]). Our scenario for spin manipulation is therefore feasible with the current state-of-the-art technology. A microscopic realization of PSG jointly with a spin-beam splitter may be applied, for instance, in quantum information and spintronics (including atomtronics, if charged atoms are employed) [@Wol2001; @Zu2004; @Pep2009; @Ru2004]. In fact, as showed in this section, such a combination works as a gate ${\mathinner{|{0}\rangle}}+{\mathinner{|{1}\rangle}}\to{\mathinner{|{0}\rangle}}+e^{i\mu}{\mathinner{|{1}\rangle}}$, where $\mu$ is any wished phase.
Summary {#sec:SumC}
=======
In summary, we used the Zassenhaus formula to re-derive the quantum mechanical evolution operator in the presence of a scalar linear potential. We discussed the form of the wave-function of the evolved state paying special attention to quantum tunneling. We then analyzed the phases that the evolved state acquires. We proposed an experimental scenario for measuring one particular phase given by the non-commutativity of momentum and position operators. We applied the evolution equations to rigorously re-analyze the Stern&Gerlach experiment and to show how to manipulate spin by using constant electric fields.
F.F. acknowledges Fundação de Amparo à Pesquisa do estado de Minas Gerais (FAPEMIG) and Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq). L.S. and F.F. acknowledge support by the Research Council for Natural Sciences and Engineering of the Academy of Finland. F.F. is thankful to Prof. S. A. Werner for suggestions and for kindly providing the slides of his talk given at NIST (op. cit.). F.F. is thankful to Hakob Avetisyan for useful discussions. The authors are thankful to an anonymous referee for his/her valuable comments and for having suggested important references to include.
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[^1]: This is a less stringent approximation than the assumption for the validity of WKB approach. Within WKB approach, which is widely used in quantum tunneling problems, the DeBroglie wave-length of the packet is considered to be small compared to the characteristic distance over which the potential varies appreciably [@Sak1994].
|
---
abstract: 'Exploiting the matrix-product-state based density-matrix renormalization group (DMRG) technique we study the one-dimensional extended ($U$-$V$) Hubbard model with explicit bond dimerization in the half-filled band sector. In particular we investigate the nature of the quantum phase transition, taking place with growing ratio $V/U$ between the symmetry-protected-topological and charge-density-wave insulating states. The (weak-coupling) critical line of continuous Ising transitions with central charge $c=1/2$ terminates at a tricritical point belonging to the universality class of the dilute Ising model with $c=7/10$. We demonstrate that our DMRG data perfectly match with (tricritical) Ising exponents, e.g., for the order parameter $\beta=1/8$ (1/24) and correlation length $\nu=1$ (5/9). Beyond the tricritical Ising point, in the strong-coupling regime, the quantum phase transition becomes first order.'
address:
- ' Institut für Physik, Ernst-Moritz-Arndt-Universität Greifswald, 17487 Greifswald, Germany '
- ' Computational Condensed Matter Physics Laboratory, RIKEN, Wako, Saitama 351-0198, Japan '
- ' The Rudolf Peierls Centre for Theoretical Physics, Oxford University, Oxford OX1 3NP, United Kingdom '
author:
- Satoshi Ejima
- Florian Lange
- 'Fabian H. L. Essler'
- Holger Fehske
title: Critical behavior of the extended Hubbard model with bond dimerization
---
Extended Hubbard model, (tricritical) Ising universality class
Introduction
============
Half a century has passed since it was proposed, yet the Hubbard model [@Hubbard1963] is still a key Hamiltonian for the investigation of strongly correlated electron systems. Originally designed to describe the ferromagnetism of transition metals, in successive studies the Hubbard model has also been used for heavy fermions and high-temperature superconductors. The physics of the model is governed by the competition between the itinerancy of the charge carriers and their local Coulomb interaction. In one dimension (1D), seen from a theoretical point of view, the Hubbard model is a good starting point to explore, for example, Tomonaga-Luttinger liquid behavior (including spin-charge separation).
While the 1D Hubbard model is exactly solvable by Bethe Ansatz [@EFGKK05], most of its extensions are no longer integrable. This is even true if only the Coulomb interaction between electrons on nearest-neighbor lattice sites is added. The ground-state phase diagram of this so-called extended Hubbard model (EHM) is still a hotly debated issue. At half filling, this relates in particular to the recently discovered bond-order-wave (BOW) state located in between spin-density-wave (SDW) and charge-density-wave (CDW) phases [@Na99; @Na2000]. To characterize the BOW state and determine its phase boundaries considerable efforts were undertaken in the last few years, using both analytical [@TF02; @TF04] and numerical [@Je02; @SBC04; @EN07] methods.
At present, quantum phase transitions between topologically trivial and nontrivial states arouse great interest [@GW09; @PTBO10; @PBTO12]. In this context, extensions of the half-filled EHM also attracted attention, mainly with regard to the formation of symmetry-protected-topological (SPT) states [@PTBO10]. Including an alternating ferromagnetic spin interaction [@LEF15] or an explicit dimerization [@EELF16] in the EHM, the SDW and BOW phases are completely replaced by an SPT insulator, whereby a quantum phase transition occurs between the SPT and the CDW, the area of which shrinks. Most interestingly, the SPT-CDW continuous Ising transition with central charge $c=1/2$ ends at a tricritical point, belonging to the universality class of the tricritical Ising model, a second minimal model with $c=7/10$ [@FQS84; @FQS85]. Above this point, the quantum phase transition becomes first order. In Ref. [@EELF16] it has been demonstrated that the transition region of the EHM with bond dimerization can be described by the triple sine-Gordon model by extending the former bosonization analysis [@BEG06]. The predictions of field theory regarding power-law (exponential) decay of the density-density (spin-spin) and bond-order correlation functions are shown to be in excellent accordance with the numerical data obtained by a matrix-product-states (MPS) based density-matrix renormalization group (DMRG) technique [@Wh92; @Sch11].
The Ising criticality of the EHM with explicit dimerization was established in early work [@BEG06] that also specifies the critical exponents. The critical exponents at the tricritical point should differ from those at the ordinary Ising transition because the tricritical Ising quantum phase transition belongs to a different universality class.
Simulating the neutral gap and the CDW order parameter by DMRG, in this paper we will determine the critical exponents at both Ising and tricritical Ising transitions. The paper is structured as follows. Section \[sec:model\] introduces the model Hamiltonians under consideration and discusses their ground-state properties. The critical exponents will be derived in Sect. \[sec:crit-expos\]. Section \[sec:summary\] summarizes our main results.
Model {#sec:model}
=====
Extended Hubbard model
----------------------
![DMRG ground-state phase diagram of the 1D EHM (\[H\_EHM\]) at half filling [@EN07]. The red dotted line gives the continuous SDW-BOW transition. The bold (thin) blue dashed line marks the continuous (first-order) BOW-CDW transition and the green dashed-dotted line denotes the first-order SDW-CDW transition. []{data-label="PD-EHM"}](fig1.pdf){width="\columnwidth"}
The Hamiltonian of the EHM is defined as $$\begin{aligned}
\hat{H}_{{\scalebox{0.6}{\rm EHM}}} &=&
-t \sum_{j\sigma}
(\hat{c}^\dagger_{j\sigma}\hat{c}_{j+1\sigma}^{\phantom{\dagger}}
+ {\rm H.c.})
\nonumber\\
&&+U \sum_{j}\left(\hat{n}_{j\uparrow}-\frac{1}{2}\right)
\left(\hat{n}_{j\downarrow}-\frac{1}{2}\right)
\nonumber\\
&&+ V \sum_{j} (\hat{n}_{j}-1)
(\hat{n}_{j+1}-1) \,,
\label{H_EHM}\end{aligned}$$ where $\hat{c}^\dagger_{j\sigma}$ ($\hat{c}^{\phantom{}}_{j\sigma}$) creates (annihilates) an electron with spin projection $\sigma=\uparrow,\downarrow$ at Wannier site $j$, $\hat{n}_{j\sigma}=\hat{c}^\dagger_{j\sigma}\hat{c}^{\phantom{}}_{j\sigma}$, and $\hat{n}_{j}=\hat{n}_{j\uparrow}+\hat{n}_{j\downarrow}$. In the Hubbard model limit ($V=0$), at half-filling, no long-range order exists. Instead the system shows fluctuating SDW order. The spin (charge) excitations are gapless (gapped) $\forall U>0$ [@EFGKK05]. At finite $V$, for $V/U\lesssim1/2$, the ground state is still a SDW. When $V/U$ becomes larger than 1/2 a 2$k_{\scalebox{0.6}{\rm F}}$-CDW is formed. As pointed out first by Nakamura [@Na99; @Na2000] and confirmed later by various analytical and numerical studies [@SBC04; @EN07; @SSC02; @TTC06], the SDW and CDW phases are separated by a narrow BOW phase below the critical end point, $(U_{\rm ce}^{{\scalebox{0.6}{\rm EHM}}}$,$V_{\rm ce}^{{\scalebox{0.6}{\rm EHM}}})\approx (9.25t,4.76t)$. In the BOW phase translational symmetry is spontaneously broken, which implies that the spin gap opens passing the SDW-BOW phase boundary at fixed $U<U_{\rm ce}^{{\scalebox{0.6}{\rm EHM}}}$. Increasing $V$ further, the system enters the CDW phase with finite spin and charge gaps. The BOW-CDW Gaussian transition line with central charge $c=1$ terminates at the tricritical point, $(U_{\rm tr}^{{\scalebox{0.6}{\rm EHM}}}, V_{\rm tr}^{{\scalebox{0.6}{\rm EHM}}})\approx (5.89t, 3.10t)$ [@EN07]. For $U_{\rm tr}^{{\scalebox{0.6}{\rm EHM}}}<U<U_{\rm ce}^{{\scalebox{0.6}{\rm EHM}}}$, the BOW-CDW transition becomes first order, characterized by a jump in the spin gap (see, Fig. 3 in Ref. [@EN07]). Figure \[PD-EHM\] summarizes the rich physics of the half-filled EHM.
![(a): Correlation length $\xi_\chi$ of the EHM as a function of $V/t$ for $U/t=4$ obtained from iDMRG. The dashed line indicates the BOW-CDW transition point. (b): von Neumann entropy $S_\chi$ as a function of logarithm of $\xi_\chi$ at $V\approx V_{\rm c}$ for $U/t=4$. The iDMRG data for $\ln\xi_\chi>6$ ($\chi\geq 1800$) provide us the numerically obtained central charge $c^\ast\simeq 0.996$ by fitting to Eq. (\[eq-vN\]). []{data-label="xi-ehm"}](fig2.pdf){width="\columnwidth"}
The criticality at the continuous BOW-CDW transition line can be verified numerically by extracting, e.g., the central charge from the the correlation length ($\xi_\chi$) and von Neumann entropy ($S_\chi$), where $\xi_\chi$ can be obtained from the second largest eigenvalue of the transfer matrix for some bond dimension $\chi$ used in a infinite DMRG (iDMRG) simulation [@Sch11; @Mc08]. Conformal field theory tells us that the von Neumann entropy for a system between two semi-infinite chains is [@CC04] $$\begin{aligned}
S_\chi=\frac{c}{6}\ln\xi_\chi +s_0
\label{eq-vN}\end{aligned}$$ with a non-universal constant $s_0$.
Figure \[xi-ehm\](a) shows iDMRG results of $\xi_\chi$ as a function of $V/t$ for fixed $U/t=4$. Since the system is critical in the SDW phase and at the BOW-CDW transition point, we find a rapid increase of $\xi_\chi$ in the SDW phase and a distinct peak at the BOW-CDW critical point ($V_{\rm c}/t \approx 2.160$) when we increase $\chi$ from 200 to 400. This indicates the divergence of the correlation length $\xi_\chi\to\infty$ as $\chi\to\infty$. Now, plotting the von Neumann entropy $S_\chi$ as a function of $\ln\xi_\chi$ and fitting the graph to Eq. (\[eq-vN\]), the criticality at $V=V_{\rm c}$ can be proved, as demonstrated by Fig. \[xi-ehm\](b). The obtained $c^\ast\simeq 0.996$ for iDMRG data with $\chi\geq1800$ corroborates the Gaussian transition resulting from a bosonization analysis [@TF02; @TF04]. Note that for the confirmation of the SDW-BOW transition much larger bond dimensions $\chi$ are required in order to make clear the convergence of $\xi_\chi$ in the BOW phase of Fig. \[xi-ehm\].
EHM with explicit bond dimerization
-----------------------------------
![Ground-state phase diagram of the 1D EHM with bond dimerization in the half-filled band sector [@EELF16]. The red solid line marks the PI-CDW phase boundaries for $\delta/t=0.2$. The tricritical Ising point \[$U_{\rm tr}$, $V_{\rm tr}$\] separates continuous Ising and first-order phase transitions. For comparisons, the phase boundaries of the pure EHM ($\delta=0$) were included. []{data-label="PD-EPHM"}](fig3.pdf){width="\columnwidth"}
Let us now add a staggered bond dimerization to the EHM, $\hat{H} = \hat{H}_{{\scalebox{0.6}{\rm EHM}}}+\hat{H}_{\delta}$, where $$\hat{H}_{\delta}= -t\sum_{j\sigma}\delta(-1)^j
(\hat{c}^\dagger_{j\sigma}\hat{c}_{j+1\sigma}^{\phantom{\dagger}}
+ {\rm H.c.})\, .
\label{H_EPHM}$$ Previous studies of this model have shown that the low lying excitations in the large-$U$ limit are chargeless spin-triplet and spin-singlet excitations [@GNT99; @Gi03; @GBSTK97; @NF81; @Ts92; @US96; @ETD97], whereby the dynamics is described by an effective spin-Peierls Hamiltonian. Moreover, at finite $U$, the Tomonaga-Luttinger parameters have been explored at and near commensurate fillings by DMRG [@EGN06]. Particularly for half filling, it has been proven by perturbative [@GGR05; @DM05] and renormalization group [@TF04; @SS02; @TO02] approaches that the system realizes Peierls insulator (PI) and CDW phases in the weak-coupling regime. According to weak-coupling renormalization-group results [@TF04], any finite bond dimerization $\delta$ will change the universality class of the continuous BOW-CDW transition (realized in the pure EHM) from Gaussian to Ising type. Thereby the PI-CDW transition in the weak-to-intermediate coupling regime belongs to the universality class of the two-dimensional (2D) Ising model [@TF04; @BEG06].
Even more interesting physics appears analyzing the intermediate-to-strong-coupling regime [@EELF16] by analogy with an effective spin-1 (EHM) system with alternating ferromagnetic spin interaction [@LEF15]: Here the continuous PI-CDW Ising transition line with central charge $c=1/2$ terminates at a tricritical point that belongs to the universality class of the 2D dilute Ising model with $c=7/10$. Above the tricritical Ising point the quantum phase transition becomes first order. Displaying the ground-state phase diagram, Fig. \[PD-EPHM\] summarizes these results. A field theoretical description of the tricritical transition region has been performed in terms of a triple sine-Gordon model [@EELF16], based on the bosonization analysis in Ref. [@BEG06], providing results for the decay of various correlation functions, such as the density-density, bond-order or spin-spin two-points functions. The predictions of field theory are in excellent agreement with iDMRG data.
Critical exponents {#sec:crit-expos}
==================
In the following, we give further evidence for the Ising respectively the tricritical Ising universality classes of the quantum phase transitions in the EHM with bond dimerization by calculating the critical exponents of various physical quantities. When approaching a continuous phase transition by varying a parameter (e.g., a coupling strength) $g$ of the Hamiltonian, the correlation length diverges as $$\begin{aligned}
\xi \propto \left|g-g_{\rm c}\right|^{-\nu}\, .\end{aligned}$$ Here, $g_{\rm c}$ denotes the (critical) value of $g$ at the transition point and $\nu$ is the corresponding critical exponent. Other quantities such as the order parameters or energy gaps also show power-law behavior. In this way the system is characterized by a set of universal exponents near the continuous phase transitions. The exact values of the most common exponents for the 2D Ising and tricritical Ising universality classes are listed in Table \[crit\_expo\].
---------------------------- ---------------------------- ------------------------- -------------
tricritical
[\[-1.5ex\][quantity]{}]{} [\[-1.5ex\][exponent]{}]{} [\[-1.5ex\][Ising]{}]{} Ising
magnetization $\beta$ 1/8 1/24
correlation length $\nu$ 1 5/9
pair correlation $\eta$ 1/4 3/20
---------------------------- ---------------------------- ------------------------- -------------
: Critical exponents belonging to the Ising and tricritical Ising universality classes in 2D [@LMC91; @DiFrancesco1997; @Mussardo2009]. The critical exponent $\eta$ for the pair correlation function has been confirmed in Ref. [@EELF16]. []{data-label="crit_expo"}
The exponents satisfy the following scaling relation $$\begin{aligned}
\frac{\nu}{2}(\eta+{\rm d}-2)=\beta\, , \end{aligned}$$ where d is the spatial dimension (in our case ${\rm d}=2$).
For the EHM with bond dimerization, $\beta$ and $\nu$ can be extracted from the CDW order parameter and the neutral gap, respectively. The CDW order parameter is defined as $$\begin{aligned}
m_{{\scalebox{0.6}{\rm CDW}}}=\frac{1}{L}\sum_j (-1)^j (\hat{n}_j-1)\,.\end{aligned}$$ The neutral gap is obtained from $$\begin{aligned}
\Delta_{\rm n}(L)=E_1(N)-E_0(N)\, ,\end{aligned}$$ where $E_{0}(N)$ \[$E_{1}(N)$\] denotes the energy of the ground state \[first excited state\] of a system with $L$ sites, $N$ electrons, and vanishing total spin $z$ component.
Ising transition
----------------
![Absolute value of the CDW order parameter in the vicinity of the Ising transition at fixed $U/t=4$. Symbols are iDMRG data; the dashed line displays the fitting function $|\langle m_{\rm CDW}\rangle|\propto(V-V_{\rm tr})^{\beta}$ with critical exponent $\beta=1/8$ (Ising universality class). Inset: Log-log plot of the order parameter for $V>V_{\rm tr}$ demonstrating the power-law decay with exponent $\beta$.[]{data-label="Ising-beta"}](fig4.pdf){width="\columnwidth"}
We now show that the critical exponents $\beta=1/8$ and $\nu=1$ follow from (i)DMRG simulations by varying $V$ at fixed $U$ and $\delta$, just as the corresponding phase transition line was obtained in Fig. \[PD-EPHM\]. Note that $\beta=1/8$ and $\nu=1$ were extracted in Ref. [@BEG06] by means of the DMRG method, varying $\delta$ for fixed $U$ and $V$.
Figure \[Ising-beta\] gives the CDW order parameter as a function of $V/t$, fixing $U/t=4$ and $\delta/t=0.2$, calculated by iDMRG technique with bond dimensions $\chi=800$. Obviously, in the CDW (PI) realized for $V>V_{\rm c}$ $(V<V_{\rm c})$, $|m_{{\scalebox{0.6}{\rm CDW}}}|$ is finite (zero). Using $V_{\rm c}/t \approx 2.5035$, the iDMRG data are well fitted by $(V-V_{\rm c})^{\,\beta}$ near the transition, where the critical exponent $\beta=1/8$ can be easily read off from a log-log plot; see inset of Fig. \[Ising-beta\].
![(a): Neutral gap $\Delta_{\rm n}$ near the Ising transition at fixed $U/t=4$ (symbols are DMRG data taken from Ref. [@EELF16]). (b): Log-log plots of $\Delta_{\rm n}$ as a function of $|V-V_{\rm c}|$, fitted by $|V-V_{\rm c}|^{\nu}$ with $\nu=1$ (Ising universality class). []{data-label="Ising-nu"}](fig5.pdf){width="\columnwidth"}
Extrapolating the values of the neutral gap $\Delta_{\rm n}$ to the thermodynamic limit, the critical exponent $\nu=1$ is verified, as demonstrated by Fig. \[Ising-nu\]. Increasing $V$ at fixed $U/t=4$, the neutral gap decreases linearly and closes at the Ising transition point. If $V$ grows further, $\Delta_{\rm n}$ opens again with linear slope. This is clearly visible in the log-log plots representation, both for $V>V_{\rm c}$ and $V<V_{\rm c}$; see Fig. \[Ising-nu\](b).
Perturbed tricritical Ising model
---------------------------------
![Absolute value of the CDW order parameter in the vicinity of the tricritical Ising point at fixed $U/t=10.56$. Symbols are iDMRG data; the dashed line displays the fitting function $|\langle m_{\rm CDW}\rangle|\propto(V-V_{\rm tr})^{\beta}$ with critical exponent $\beta=1/24$ (tricritical Ising universality class). Inset: Log-log plot of the order parameter for $V>V_{\rm tr}$ demonstrating the power-law decay with exponent $\beta$. []{data-label="TIM-beta"}](fig6.pdf){width="\columnwidth"}
As quoted above and demonstrated in Ref. [@EELF16], the tricritical point in the EHM with bond dimerization belongs to the universality class of the 2D tricritical Ising model with the critical exponents given in Table \[crit\_expo\]. Let us emphasize that it is exceptionally challenging to verify the critical exponents at the tricritical Ising point numerically, not least because one first has to determine the tricritical point itself, with high precision, varying $U$ and $V$ simultaneously [@EELF16].
The exponent $\eta$ characterizes the power-law decay of the CDW order-parameter two-point function at the critical point. As shown in Ref. [@EELF16] one has $$\langle (-1)^\ell(\hat{n}_{j+\ell}-1)(\hat{n}_{j}-1)\rangle\propto
\ell^{-3/20}\ ,\quad \ell\gg 1\ .$$ This establishes that $\eta=3/20$. In order to determine the exponents $\beta$ and $\nu$ one needs to consider the off-critical regime. We therefore consider the perturbation of the tricritical Ising conformal field theory by the “energy operator” $\epsilon(x)$, which has conformal dimensions $\left(\Delta_\epsilon,\bar{\Delta}_\epsilon\right)=
\left(\frac{1}{10},\frac{1}{10}\right)$ [@LMC91; @DiFrancesco1997; @Mussardo2009] $$H=H_{\rm CFT}+h\int dx\ \epsilon(x)\ .
\label{PCFT}$$ The perturbing operator has scaling dimension $d=1/5$ and is therefore relevant in the renormalization group (RG) sense. It generates a spectral gap $M$ that scales as $$M\sim Ch^{1/(2-d)}=Ch^{5/9},$$ where $C$ is a constant. This identifies the critical exponent $\nu=5/9$. The magnetization operator $\sigma(x)$ in the tricritical Ising model has scaling dimension $\left(\Delta_\sigma,\bar{\Delta}_\sigma\right)=\left(\frac{3}{80},\frac{3}{80}\right)$. In the perturbed theory (\[PCFT\]) it acquires a non-zero expectation value that scales as $$\langle\sigma(x)\rangle\sim D
h^{\Delta_\sigma/(1-\Delta_\epsilon)}
=D h^{1/24}\ ,$$ where $D$ is a constant. This identifies the critical exponent $\beta=1/24$.
![(a): DMRG data for the neutral gap $\Delta_{\rm n}$ in the vicinity of the tricritical Ising point where $U/t=10.56$. (b): Log-log plots of $\Delta_{\rm n}$ as a function of $|V-V_{\rm tr}|$, fitted by $|V-V_{\rm tr}|^{\nu}$ with $\nu=5/9$ (tricritical Ising universality class). []{data-label="TIM-nu"}](fig7.pdf){width="\columnwidth"}
The predictions of perturbed conformal field theory for $\beta$ and $\nu$ can be checked against numerical computations as follows. Fixing $U=10.56t$ ($\simeq U_{\rm tr}$), we first give the iDMRG results for the CDW order parameter $|\langle m_{{\scalebox{0.6}{\rm CDW}}}\rangle|$ as a function of $V$, cf. Fig. \[TIM-beta\]. Just as in the case of the Ising universality class, $|\langle m_{{\scalebox{0.6}{\rm CDW}}}\rangle|$ is finite (zero) for $V>V_{\rm tr}$ ($V<V_{\rm tr}$). The order parameter $|\langle
m_{{\scalebox{0.6}{\rm CDW}}}\rangle|$ now vanishes much more abruptly approaching the quantum phase transition point from above. Fitting the iDMRG data for $V>V_{\rm tr}$ to $(V-V_{\rm tr})^{\,\beta}$ with $V_{\rm tr}/t\approx
5.497$ and $\beta=1/24$ works perfectly, see the log-log representation.
In order to verify the field theory prediction for $\nu$ we examine the $L\to\infty$ extrapolated values of the neutral gap $\Delta_{\rm
n}$. Increasing $V$($<V_{\rm tr}$) at fixed $U/t=10.56$, $\Delta_{\rm n}$ is reduced but not linearly as in the Ising case (cf. Fig. \[Ising-beta\]), and closes at $V\approx V_{\rm tr}$ before it becomes finite again for $V>V_{\rm tr}$. Again the log-log representation can be used to extract the critical exponent for $|V-V_{\rm tr}|^\nu$, $\nu=5/9$, for both $V<V_{\rm tr}$ and $V>V_{\rm tr}$, in conformity with the tricritical Ising universality class.
Summary {#sec:summary}
=======
To conclude, we have investigated the criticality of the 1D half-filled extended Hubbard model (EHM) with explicit dimerization $\delta$. The BOW-CDW Gaussian transition with central charge $c=1$ of the pure EHM gives way to an Ising transition with $c=1/2$ at any finite $\delta$. The Ising transition line terminates at a tricritical point, which belongs to the universality class of the tricritical Ising model in two dimensions. The change of the universality class is verified numerically by (i)DMRG (see also [@EELF16]). Furthermore, we demonstrate that not only the Ising but also the tricritical Ising critical exponents $\beta$ and $\nu$ can be obtained with high accuracy by simulating the CDW order parameter and the neutral gap.
We thank M. Tsuchiizu for fruitful discussions. The DMRG simulations were performed using the ITensor library [@ITensor]. This work was supported by Deutsche Forschungsgemeinschaft (Germany), SFB 652, project B5 (SE and HF), and by the EPSRC under grant EP/N01930X/1 (FHLE). FL thanks RIKEN for the hospitality sponsored by the IPA program.
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abstract: 'Confidence intervals provide a way to determine plausible values for a population parameter. They are omnipresent in research articles involving statistical analyses. Appropriately, a key statistical literacy learning objective is the ability to interpret and understand confidence intervals in a wide range of settings. As instructors, we devote a considerable amount of time and effort to ensure that students master this topic in introductory courses and beyond. Yet, studies continue to find that confidence intervals are commonly misinterpreted and that even experts have trouble calibrating their individual confidence levels. In this article, we present a ten-minute trivia game-based activity that addresses these misconceptions by exposing students to confidence intervals from a personal perspective. We describe how the activity can be integrated into a statistics course as a one-time activity or with repetition at intervals throughout a course, discuss results of using the activity in class, and present possible extensions.'
author:
- |
Xiaofei Wang\
Department of Statistics, Yale University\
\
Nicholas G. Reich\
Department of Biostatistics and Epidemiology,\
University of Massachusetts at Amherst\
\
Nicholas J. Horton\
Department of Mathematics and Statistics, Amherst College
bibliography:
- 'refs.bib'
title: '**Enriching Students’ Conceptual Understanding of Confidence Intervals: An Interactive Trivia-based Classroom Activity**'
---
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[**Enriching Students’ Conceptual Understanding of Confidence Intervals: An Interactive Trivia-based Classroom Activity**]{}
[*Keywords:*]{} uncertainty, calibrating confidence, subjective probability
Introduction {#sec:introduction}
============
Confidence intervals are one of the most commonly used statistical methods to summarize uncertainty in parameter estimates from data analyses. However, both the formal concept of and intuition behind confidence intervals remain elusive to many students and data analysts. In the case of a 95% confidence interval for a proportion for a given sample size, one textbook definition is “95% of samples of this size will produce confidence intervals that capture the true proportion” [@book:1321656]. Another textbook definition reads “A plausible range of values for the population parameter is called a confidence interval” and later specifies for 95% confidence “Suppose we took many samples and built a confidence interval \[for the population mean\] from each sample...then about 95% of those intervals would contain the actual mean” [@diez2012openintro]. Studies have found that many students struggle with understanding definitions like these even after completing coursework at various levels [@Fidler:2005ur; @Kalinowski:2010tq; @Kaplan:2010vd].
In addition to a lack of intuition about the definition of a confidence interval, researchers have documented overconfidence in quantitative assessments of uncertainty in students and experts alike [@Alpert:A8vJTwl8; @soll2004; @jorgensen2004; @cesarini2006]. In this context, overconfidence has been defined as individuals having “excessive precision” in their beliefs about particular facts [@moore2008]. One study showed that when subjects were asked to provide 98% confidence intervals relating to numeric facts, only 57.4% of their intervals captured the true value [@Alpert:A8vJTwl8]. This display of overconfidence is not limited to students and novices, but is also exhibited by field experts [@hynes1976reliability; @christensen1981physicians].
The textbook definition of a confidence interval can be challenging to understand, and various pedagogical tools have been proposed to assist with comprehension. Some teachers have documented success by teaching confidence intervals with bootstrapping techniques [@Maurer:2015vq] or with visualizations of simulated samples [@Cumming:2007cl; @Hagtvedt:2008jv]. [@behar2013twenty] presented one analogy useful in understanding 95% confidence intervals: “It is like a person who tells the truth 95% of the time, but we do not know whether a particular statement is true or not.” Providing students with an opportunity to calibrate their intuitive understanding of what it means to be, say, 90% confident about a fact through short repetitive activities may enhance their ability to understand the strength of conclusions from a data analysis. In this manuscript, we describe an activity that aims to solidify an intuitive understanding of what it means to be “90% confident”.
Motivated by [@Alpert:A8vJTwl8], our short interactive classroom activity ties together formal concepts of confidence intervals to tangible information and questions about facts that students may or may not have some familiarity with and interest in. In brief, the instructor reads off ten trivia questions with specific quantitative answers. Students are asked to provide their answer to each question in the form of a 90% confidence interval (rather than just providing a point estimate). After all the trivia questions are read, the correct answers are revealed and students calculate the number of intervals that capture the correct answers within their intervals.
By providing immediate feedback and fostering a gently competitive spirit in the classroom, this activity incentivizes students to engage with the central conceptual challenges of confidence intervals as described in the GAISE College Report [@GAISEcollege].
The questions should be appropriate for the audience, and can be about topics that are timely and relate to cultural memes (for example, celebrities, sports teams, and TV shows). However, it is important to emphasize that this is not a test of knowledge, but rather of how well one knows and is able to quantify the limits of their knowledge. Specifically, if students are not familiar with a particular topic, they can (and should) respond with a wide interval. Assessing personal confidence interval coverage rates provides immediate feedback about how well-calibrated a student’s confidence level is.
The activity is simple, requires little preparation, and is appropriate for students at all different levels and backgrounds, including undergraduate and graduate students. The activity requires only a list of 10 questions with quantitative answers and can be used as early as in an introductory statistics course when the students are first exposed to confidence intervals. In an upper level course, the activity can be used early on for review. Given the simplicity of the exercise, this activity can be utilized in a classroom of twenty students or two hundred students with little modification.
In Section \[sec:activity\], we describe the activity procedure and necessary preparation. Section \[sec:results\] presents anecdotal and numerical evidence relating to the instructional value of the activity. Finally, in Section \[sec:discussion\], we present a discussion relating to the effectiveness of the activity.
Activity {#sec:activity}
========
Overview
--------
Before class, the instructor needs to prepare a series of ten or more trivia questions that each have specific quantitative answers. There are many sources for ideas. Trivia questions about the college or university may be popular (e.g. how many official student organizations are listed on the school website?). We have used the board game Wits and Wagers[^1] as one source of questions. The questions should ideally cover a broad range of topics so that most students will not know the answers exactly, but that educated guesses are possible. For example, a question such as “In what year was the Declaration of Independence signed?” does have a quantitative answer but would be a poor choice because most students would be able to answer 1776 without hesitation. Table \[tbl:example\] gives a series of example questions that might be used for this activity. These questions were obtained by one of the authors via the Wits & Wagers: Trivia Party iOS app[^2]. Appendix A contains a list of thirty additional questions that can be used for this activity.
We now outline the steps of the activity assuming that ten questions have already been selected for use.
1. Provide students with a blank sheet of paper and ask the students to make a numbered list from 1 to 10, leaving room for ten answers.
2. Tell students that they will hear a series of ten questions, each one having a numeric answer. Instead of writing down a specific value to answer each question, students should provide their answer in the form of a 90% confidence interval, i.e. (lower bound, upper bound). Optionally, provide an example question and a corresponding interval (not scored) so that students understand the format of the exercise.
3. Read each of the ten trivia questions, pausing for sufficient time (typically 20-30 seconds) in between questions to give students a chance to jot down their answer.
4. After all questions are read, explain the scoring process; as you read the answers to each question, the students should give themselves a point for every interval that captures the correct numeric response. (Alternatively, you might arrange for students to switch answers with a neighbor to have them do scoring for one other.)
5. Review the answers with the students and ask students to tally up their score, out of ten. (This process is often high-energy with lots of reaction from the students.)
6. Obtain and display a distribution of student scores. If there are a small number of students, a show of hands suffices for data collection. Otherwise an online survey or a clicker poll might be useful for collecting the information. The instructor might summarize results in a sketched stem-and-leaf plot or histogram.
7. Discuss the results as a group. A proposed starting point could be “what might you expect the results to look like?” or “what do the observed results indicate about the over- or under-confidence of the class as a whole?”
In our experience, students tend to score low the first time that they do this exercise (see Section \[sec:results\]); they tend to be overconfident in their answers. The last step of the exercise gives students an opportunity to reflect on both what it means to be overconfident and what the source of that miscalibration might be. As part of the discussion, one might ask what the expected score would be for a student who indeed provided 90% confidence intervals (nine out of ten). More advanced classes could compare the distribution to repeated draws from a Binomial distribution with $N=10$ and various probabilities of success. Some students may have scored poorly due to an overestimation of how well they know the subject areas in question. Other students may have been overeager to play a competitive trivia game and may have overlooked the specification of the 90% confidence level that should be considered as part of their answers. For all students, this activity serves to bridge the gap between a textbook definition of confidence intervals to a more intuitive understanding of confidence and uncertainty.
**Question** **Answer**
---- ---------------------------------------------------------------------------------------------------------- ------------
1 How old was actress Drew Barrymore when her first feature film was released? 5
2 How many cards are in an original Uno deck? 108
3 The first item ever sold on eBay was a broken laser pointer. In dollars, how much did it sell for? 14
4 According to the 1994 film Forrest Gump, what was Gump’s IQ? 75
5 In years, how long did it take Michelangelo to paint the ceiling of the Sistine Chapel? 4
6 In months, how long did it take Apple to sell 100 million iPods? 66
7 How many episodes aired of the TV sitcom Friends, not including specials? 236
8 In what year was the first modern Summer Olympics held? 1896
9 In miles, how long is the main wall of the Great Wall of China, excluding all of its secondary branches? 4163
10 How many states were part of the United States in 1860? 33
: Example Trivia Questions[]{data-label="tbl:example"}
For the convenience of the instructor, we have included a one-sheet summary of the activity with example questions, scoring, and discussion questions in the Supplementary Materials. This could be printed out to use in class.
Variations and Evaluation
-------------------------
We note that when exactly ten questions are utilized, students can ‘cheat’ the activity by giving nine extremely wide intervals and one obviously wrong interval. In doing so, they can almost definitely guarantee a score of nine out of ten. One way to fix this loophole is to modify the number of questions used. For instance, students might be asked 15 questions but only a random selection of 10 are used for scoring. Additionally, the activity could be extended by asking students to think about and propose a metric that could identify this kind of cheating; for example, a mean and standard deviation could be calculated from all intervals for a question, and students having multiple interval widths more than some $z$-scores above the mean could be flagged as potential cheaters.
The proposed activity can take many forms and be used for different purposes. As a group, the authors have used this activity with both graduate and undergraduate students, at one or more times during the semester, and with slightly different learning goals in mind. Two of the authors first use the activity in an undergraduate introductory statistics course during the class period immediately following the introduction of confidence intervals with a follow-up towards the end of the semester. Another has used it repeatedly throughout the semester (using different questions each time) in a graduate-level course on regression that is a required course for statistics Masters students. In this course, the goal of repeating the activity is to illustrate the challenges of calibrating uncertainty, i.e. avoiding “excessive precision”, and understanding what it means to be 90% confident about a fact. In each of the courses described above, the instructors also employ small simulation-based exercises to illustrate the properties of confidence intervals in a more formal statistical sense.
Apart from using this activity for understanding confidence intervals, the act of producing intervals and then validating whether the answers fall within those intervals serves as a way of calibrating one’s level of confidence. We believe students stand to benefit from participating in this exercise several times over the course of a semester, of course, on different trivia questions. In our experience, students score better with each additional iteration (see Section \[sec:results\]). Such incremental improvements suggest that students learn to better quantify their levels of uncertainty with practice.
In the graduate-level course where the activity was repeatedly used, the Comprehensive Assessment of Outcomes in a First Statistics course, or the CAOS test, was used to evaluate overall conceptual understanding of statistics [@delmas2007]. Specifically, the CAOS pre- and post-tests were administered at the beginning and end of the semester. We include a comparison of pre- and post-test scores from confidence interval-related questions in the Supplementary Materials.
Results {#sec:results}
=======
Over the course of one semester, the three authors used this activity two, two, and five times in their respective classes. We now describe the results with data consisting of student scores from these three classes. The data collection, management, and analysis were approved by the University of Massachusetts at Amherst Institutional Review Board (\#2014-2051) and Amherst College Institutional Review Board (\#16-008).
The authors that used the activity twice in a semester (Horton and Wang) did so in undergraduate introductory statistics courses with 24 and 27 students, respectively. The first time that the activity was used (Iteration 1) was at the beginning of the lesson on confidence intervals, during Week 7 of a 13-week semester. Students were asked to pre-read the chapter on confidence intervals for a single proportion prior to coming to class; class began with the activity without discussion of the reading material. The second time that the activity was used (Iteration 2) was during Week 11 after students had seen confidence intervals used in two-sample settings and in linear regression. Table \[tbl:data\_summary\] shows that the students performed poorly on their first attempt, averaging 4.5 and 3.3 in the two classes, and improved significantly on the second attempt, averaging 5.7 and 5.8. Identifying information was not tracked during either of the iterations, so it is not possible to track individual-level improvements.
The third author (Reich) used this activity in a graduate-level multivariate regression course five times spread approximately every two weeks throughout the semester and recorded results from each round of the activity. Students entered their data into a shared spreadsheet, with a unique student identifier across all rounds. Out of 14 students total, 11 participated in all five rounds of the activity, while two participated in all but the first round and one other participated in the third to fifth rounds. (One student declined to have their scores used in the subsequent data analysis.)
The longitudinal data from this class are plotted in Figure \[fig:analysis\], along with the overall mean score per iteration and model estimates of per-iteration success rates. The average number of intervals that covered the true values in the first and fifth iterations were 4 and 6.5, respectively. While there is substantial within-student variation (as shown by the student-specific lines), the overall trend of the median number of intervals that cover the truth increased with repetition of the activity. Over the five iterations, only in one instance was there an individual with all ten intervals covering the truth.
To enable inference on per-iteration improvement while accounting for within-subject correlations, we fit a logistic mixed-effects model with a random intercept for each individual student and fixed effect dummy variables for iteration. We used the [rstanarm]{} package [@rstanarm] in R version 3.3.2 (2016-10-31) [@rcoreteam] to fit the model with a Bayesian Markov Chain Monte Carlo algorithm, using standard weakly informative priors for the coefficients and variance parameters. While accuracy increased with each iteration, the biggest incremental score gain occurred between the first and second iterations, with an improvement of 1.5 to 2 questions answered correctly (Table \[tbl:data\_summary\]). Accuracy increased more slowly between iterations 2 through 5, although an average improvement of almost a single correct question was observed across these four iterations. Comparing the 90% posterior credible intervals (CI) for the number of questions answered correctly per iteration, only rounds 4 (90% CI: (4.7, 7)) and 5 (90% CI: (4.9, 7.2)) showed significant increases in accuracy over iteration 1 (90% CI: (2.4, 4.4)); the credible intervals for iterations 2 and 3 still overlapped with the credible interval for iteration 1.
------------------------------------------------ ----- ----- ----- ----- --------- --------- --------- ------- ---------
(lr)[2-3]{} (lr)[4-5]{} (lr)[6-10]{} Iteration 1 2 1 2 1 2 3 4 5
n 21 21 26 19 11 13 14 14 14
mode 5 3 1 4 6 6 8 3 7
median 5.0 6.0 3.0 6.0 4.0 6.0 6.5 6.0 6.5
mean 4.5 5.7 3.3 5.8 3.6 5.2 5.6 5.8 6.0
sd 1.8 2.1 2.3 2.2 2.5 2.8 3.0 2.8 1.8
estimated mean 3.3 5.3 5.6 5.9 6.1
estimated 90% CI 2.2-4.5 4.1-6.5 4.4-6.8 4.7-7 4.9-7.2
------------------------------------------------ ----- ----- ----- ----- --------- --------- --------- ------- ---------
: Summary Statistics of Activity By Class and Iteration. The last two rows represent model-based estimates.[]{data-label="tbl:data_summary"}
![Student scores from five iterations of the activity ($n=14$, shown in lighter colors, color-coded by student). The solid black line indicates the mean score across all students. The dashed blue line indicates the model-estimated posterior median number of questions answered correctly, with corresponding 90% confidence intervals shown in vertical blue lines.[]{data-label="fig:analysis"}](analysis-1){width="70.00000%"}
Discussion {#sec:discussion}
==========
Confidence intervals play a key role in inferential thinking. We have described an interactive activity to help solidify students’ understanding of confidence intervals. The activity is attractive because it requires no technology, is interactive and applicable to a wide variety of students, and fun, in the spirit of friendly competition between students. The activity can help reinforce statistical thinking and help students recognize overconfidence. However, it should be noted as a limitation of the activity that the intervals requested of the students are somewhat removed from the typical notion of a confidence interval; students are not computing standard errors and point estimates but are rather relying on their own judgement. Nonetheless, because the activity does not involve any calculations, which tends to be the typical entry point for teaching confidence intervals, it can be particularly helpful for promoting an intuitive understanding. Students with very little mathematical background, such as graduate students from non-quantitative fields, may find this activity a helpful way to understand confidence intervals without the distraction of the mathematical formulae.
It is interesting to note that while scores improve over repeated iterations of the activity, perfect calibration of confidence was not attained within the course of a single semester. Even with five iterations of the activity, student scores plateau between 5 to 6 correct answers out of 10, well below the desired level of 9. Moreover, students in the classes that only used the activity twice in a semester seemed to do about as well in the second iteration as those who had a third, fourth, or fifth try. This observation suggests that overconfidence is challenging to overcome. Examining whether the same achievement gap holds if students were asked to produce confidence intervals at different levels (i.e. 50% or 70%) could provide information on whether this is a problem inherent to individual’s judgment uncertainty, or whether 90% is a particularly tricky level to achieve.
We relied on the CAOS test (details provided in the Supplementary Materials) to provide a quantitative evaluation of student’s understanding of confidence intervals. The data suggested improvement in students’ overall understanding of the interpretation of confidence intervals, although the sample size was small (15 students took both tests) and improvements in understanding could also be attributed to other content in the course.
With the continued increase of written content by “data journalists” in the mass media (often including some technical measures of uncertainty), the value of accurate intuition about uncertainty will grow for not just students and practitioners of statistics but for the general public as well. The results shown in this manuscript serve as proof-of-concept that asking students to create confidence intervals for quantities that are of potential interest can foster a more intuitive understanding of confidence and uncertainty. Our experience is that students are consistently surprised by the results of this exercise, leading to “teachable moments” when students grasp how and why their interval coverage was too low. Therefore, activities such as the one presented here and others may serve an important role in creating informed consumers of modern data-driven content, whether journalistic or academic, by arming them with tools to interpret quantitative results.
Acknowledgements {#sec:acknowledgements}
================
First we would like to thank the Associate Editor and two anonymous reviewers who provided very helpful feedback on the first iteration of this manuscript. We also thank the students who participated in these activities. Finally, we thank Eric W. Bright, CFA, who introduced NGR to a version of this activity.
Appendix A: 30 Questions {#sec:appendix_a_30_questions .unnumbered}
========================
In this section, we provide thirty more questions in addition to the ones presented in Table \[tbl:example\] for convenience. The first ten questions are adapted from [@confidentdecision] (reprinted in [@supercrunchers], among other places), the next nine are taken from the Wits and Wagers board game, and the rest are written up by the authors.
**Question** **Answer**
---- -------------------------------------------------------------------------------------------------------------------- ------------
1 At what age did Martin Luther King die? 39
2 In miles, how long is the Nile River? 4,187
3 In 2016, how many countries are part of OPEC? 12
4 How many books were part of the Old Testament? 39
5 What is the diameter of the moon in miles? 2,160
6 In pounds, what is the weight of an empty Boeing 747? 390,000
7 In what year was Wolfgang Amadeus Mozart born? 1756
8 What is the gestation period, in days, of an Asian elephant? 645
9 In miles, what is the air distance from London to Tokyo? 5,959
10 How deep, in feet, is the deepest known point in the oceans? 36,198
11 In what year did an actress first earn \$1 million for a movie role? 1963
12 In feet, how tall is The Statue of Liberty including the pedestal? 305
13 On average, how many quarts of ice cream did an American eat in the year 2012? 20
14 In pounds, what was the weight of the heaviest domesticated cat ever recorded? 46.81
15 How many points did Michael Jordan average per game during his NBA career? 30.12
16 In years, how long does the US \$1 bill stay in circulation? 4.8
17 How many times could Rhode Island fit into the land area of Alaska? 423
18 How many inventions did Thomas Edison patent during his lifetime? 1,093
19 In what year was a woman first appointed to the Supreme Court? 1981
20 In 2012, how many customers frequented Disneyland per day on average? 44,000
21 As of the end of year 2015, the New York City subway system was constituted of how many subway stations? 469
22 How many James Bond films are there total as of 2016? 26
23 What is the total height of Yosemite Falls, the highest water fall in Yosemite National Park? 2,425
24 How many moons does Saturn have? 62
25 As of the end of the 2015-2016 season, how many different basketball teams have won at least one NBA Championship? 17
26 In degrees Fahrenheit, what is the average temperature at the North Pole in summer? 32
27 If Earth was hollow, how many moons could fit inside it? 50
28 How many times did Ron’s name appear in all of the Harry Potter books? 5,784
29 In millions, how many copies of Adele’s album ’25’ sold in the U.S. during its debut week? 3.38
30 How many goals were scored during the 2016 soccer tournament Copa America? 91
Sources for questions 20 to 30 {#sources-for-questions-20-to-30 .unnumbered}
------------------------------
20. <http://www.latimes.com/business/la-fi-disneyland-crowds-20150519-story.html>
21. <http://web.mta.info/nyct/facts/ridership/>
22. <https://en.wikipedia.org/wiki/List_of_James_Bond_films>
23. <https://en.wikipedia.org/wiki/Yosemite_Falls>
24. <https://en.wikipedia.org/wiki/Moons_of_Saturn>
25. <http://www.landofbasketball.com/championships/summary_of_winners.htm>
26. <http://www.landofbasketball.com/championships/summary_of_winners.htm>
27. <http://www.astronomy.org/programs/moon/moon.html>
28. <http://www.moviefone.com/2011/07/13/harry-potter-numbers-trivia/>
29. <http://www.billboard.com/articles/columns/chart-beat/6777959/adele-25-historic-chart-numbers>
30. <https://en.wikipedia.org/wiki/Copa_Am%C3%A9rica_Centenario>
Supplementary Materials
Activity Printout {#activity-printout .unnumbered}
=================
The first two pages serve as a convenient one-sheet (front and back) summary of the activity that can be printed out for use in class.
Activity Outline {#activity-outline .unnumbered}
----------------
Activity lasts approximately 10-15 minutes. Steps follow:
1. Each student starts out with a sheet of paper with numbered list from 1 to 10, leaving room for 10 answers.
2. You will read aloud 10 trivia questions each with a numeric answer. Instruct students to write down a 90% confidence interval in response to each question.
3. Read each of the trivia questions.
4. Review answers to the trivia questions. Students should score themselves: 1 point for every interval that captures the correct numeric answer.
5. Ask students to tally up their score out of 10.
6. Obtain and visualize a distribution of student scores.
7. Discuss the results as a group.
Possible Discussion Questions {#possible-discussion-questions .unnumbered}
-----------------------------
1. If a student provided 90% confidence intervals in all ten cases, how many points would we expect him/her to score?
2. If *every* student provided 90% confidence intervals in all ten cases, what would a histogram of scores look like for the class?
3. Examining the histogram (or stem-and-leaf plot) of scores from our class, do you think we were overconfident or underconfident?
4. How might we, as a class, do better at this exercise?
Sample Questions, Answers, and Scoring {#sub:sample_questions_and_answers .unnumbered}
--------------------------------------
In the table below, we reiterate the questions/answers presented in Table 1 and present sample responses for each. Responses that are **bolded** earn one point.
**Question** **Answer** **Sample Response**
---- ---------------------------------------------------------------------------------------------------------- ------------ ---------------------
1 How old was actress Drew Barrymore when her first feature film was released? 5 $\bm{(1, 7)}$
2 How many cards are in an original Uno deck? 108 $(52, 104)$
3 The first item ever sold on eBay was a broken laser pointer. In dollars, how much did it sell for? 14 $\bm{(3,200)}$
4 According to the 1994 film Forrest Gump, what was Gump’s IQ? 75 $\bm{(50, 110)}$
5 In years, how long did it take Michelangelo to paint the ceiling of the Sistine Chapel? 4 $(1, 3)$
6 In months, how long did it take Apple to sell 100 million iPods? 66 $(80,120)$
7 How many episodes aired of the TV sitcom Friends, not including specials? 236 $(250, 300)$
8 In what year was the first modern Summer Olympics held? 1896 $\bm{(1896, 1900)}$
9 In miles, how long is the main wall of the Great Wall of China, excluding all of its secondary branches? 4163 $\bm{(1000, 8000)}$
10 How many states were part of the United States in 1860? 33 $\bm{(0, 50)}$
Total score: 6 out of 10
Sample Visualization {#sub:sample_visualizations .unnumbered}
--------------------
{width="50.00000%"}
CAOS Test Results {#caos-test-results .unnumbered}
=================
In the graduate-level course where the activity was repeatedly used, the Comprehensive Assessment of Outcomes in a First Statistics course (CAOS) test, was used to evaluate overall conceptual understanding of statistics. The CAOS pre- and post-tests were administered at the beginning and end of the semester, respectively. We specifically evaluated the student responses to four questions (questions 28 through 31) on the CAOS pre- and post-tests that evaluate understanding about confidence intervals. The percentage of correct responses on all 4 questions increased from 61.0% on the pre-test to 76.7% on the post-test. Students showed larger improvements on Questions 28 and 29 than on Questions 30 and 31.
---------- ---------- ----------- --
question pre-test post-test
28 37.5 93.3
29 68.8 80
30 43.8 46.7
31 93.8 86.7
average 60.975 76.675
---------- ---------- ----------- --
: Summary statistics from the CAOS pre-test ($N$=16) and post-test ($N$=15) questions 28 through 31, which directly pertain to confidence intervals.[]{data-label="tbl:caos"}
[^1]: Crapuchettes, D. and N. Heasley (2005). . Bethesda, MD: North Star Games.
[^2]: <https://itunes.apple.com/us/app/wits-wagers-trivia-party/id637929057?mt=8>
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